CYCLEANALYTICS

FORTRADERS

CYCLEANALYTICSFORTRADERS

AdvancedTechnical

TradingConcepts

JohnF.Ehlers

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LibraryofCongressCataloging-in-PublicationData:Ehlers,JohnF.,1933-Cycleanalyticsfortraders:advancedtechnicaltradingconcepts/JohnF.Ehlers.pagescmISBN978-1-118-72851-2(cloth)—ISBN978-1-118-72841-3(ebk)—ISBN978-1-118-72860-4(ebk)1.Technicalanalysis

(Investmentanalysis)2.Investmentanalysis.I.Title.HG4529.E3882014332.63′2042—dc23

2013034306

CONTENTS

Cover

HalfTitle

TitlePage

CopyrightPage

Preface

AbouttheAuthor

Chapter1:UnifiedFilterTheory

TransferResponseNonrecursiveFilters

RecursiveFiltersGeneralizedFiltersProgrammingtheFiltersWaveAmplitude,Power,andDecibels(dB)

KeyPointstoRemember

Chapter2:SMAs,EMAs,orOther?

SimpleMovingAverages(SMAs)Exponential

MovingAverages(EMAs)WeightedMovingAverages(WMAs)MedianFilterKeyPointstoRemember

Chapter3:SmoothingFiltersonSteroids

NonrecursiveFiltersModifiedSimpleMovingAveragesModifiedLeast-Squares

QuadraticsSuperSmootherSuperSmootherFilterApplicationsKeyPointstoRemember

Chapter4:DecyclersDecycler

ConstructionDecyclerApplicationDecyclerOscillatorKeyPointstoRemember

Chapter5:Band-PassFilters

Band-PassFilterBand-PassFilterQAutomaticGainControl(AGC)SpectralDilationRemoval

Band-PassFilterMeasuringtheCyclePeriodKeyPointstoRemember

Chapter6:MarketStructureandtheHurstCoefficient

FractalDimensionComputingtheHurstCoefficientTheHurstCoefficientinActionDrunkard'sWalk

HypothesisforMarketStructureKeyPointstoRemember

Chapter7:SpectralDilation

FrequencyContentof

IndicatorOutputsRoofingFilterasanIndicatorImpactofSpectralDilationonConventionalIndicatorsKeyPointsto

Remember

Chapter8:Autocorrelation

BackgroundAutocorrelationAutocorrelationPeriodogramAutocorrelationReversals

KeyPointstoRemember

Chapter9:FourierTransforms

SpectralDilationDiscreteFourierTransform

(DFT)KeyPointstoRemember

Chapter10:CombFilterSpectralEstimates

SpectralDilationComputinga

CombFilterSpectralEstimateKeyPointstoRemember

Chapter11:AdaptiveFilters

AdaptiveRelative

StrengthIndex(RSI)AdaptiveStochasticIndicatorAdaptiveCCI(CommodityChannelIndex)AdaptiveBand-Pass

FilterAdaptiveIndicatorComparisonKeyPointstoRemember

Chapter12:TheEvenBetterSinewaveIndicator

EvenBetterSinewaveApproachEvenBetterSinewaveDescriptionUsingtheEvenBetterSinewaveIndicator

KeyPointstoRemember

Chapter13:Convolution

TheoreticalFoundationHeatMapDisplayComputing

ConvolutionKeyPointstoRemember

Chapter14:TheHilbertTransformer

AnalyticSignalsHilbertTransformer

MathematicsComputingtheHilbertTransformerTheHilbertTransformerIndicatorUsingtheHilbertTransformerto

ComputetheDominantCycleDualDifferentiatorPhaseAccumulationHomodyneKeyPointstoRemember

Chapter15:IndicatorTransforms

FisherTransformInverseFisherTransformCubeTransformKeyPointstoRemember

Chapter16:SwamiCharts

SwamiChartsOverviewSwamiChartsRSISwamiChartsStochasticRollYourOwnSwamiCharts

KeyPointstoRemember

Chapter17:Swing-TradingStrategies

ConventionalWisdomAnticipatingtheTurningPoint

SineWaveUniquenessSafetyValveExitingaTradeStopLossEvaluatingaTradingStrategyMonteCarloEvaluation

StockSpotter.comKeyPointstoRemember

AbouttheWebsite

Index

BookCard

PREFACE

It has been over 10 yearssince Rocket Science forTraders was published. Inthosedays, technicalanalysiswasprimarilytheprovinceoffutures traders, whileportfolio theory andfundamental analysis

comprised the conventionalwisdom for equity traders.However, there have beenprofound changes in themarketplace since that time.Futures trading has lostpopularity because ofscandalsinvolvingsegregatedcustomer accounts, the stockmarket's relentless trendupwardhasbeenbrokenwiththeresultthatbuy-and-holdisno longer a valid investmentstrategy, and new trading

vehicles such as exchange-traded funds (ETFs) haveevolved. In addition,commission rates havedecreased, and the Internethas made electronic tradingavailabletoeveryone.Allthishas caused investors to bemore involved in the tradingprocessandinterestedinself-directed trading. Majorbrokerage houses haveresponded by includingtechnical analysis tools in

theirtradingplatforms.Thisisatechnicalresource

bookwritten for self-directedtraders who want tounderstand the scientificunderpinnings of the filtersand indicators they use intheir trading decisions ratherthan to use the trading toolsonblindfaith.Thereisplentyof theory and years ofresearch behind the uniquesolutions provided in this

book, but the emphasis is onsimplicity rather thanmathematical purity. Inparticular, thesolutionsuseapragmatic approach to attaineffective trading results. Theconcepts are presented sothey can be understood withonlyabackgroundinalgebra.Thewritingstyle in thebookis intentionally terse so thereader doesn't need to wadethroughamountainofwordsto find the ideas being

presented. EasyLanguagecomputer code is used tocalculate and display theindicators. From myviewpoint, EasyLanguage isjust a dialect of Pascal withkey words for trading.Therefore,thecodeshouldbenearlyasreadableasEnglish.Cycles are unique because

they are one of the fewcharacteristicsofmarketdatathat can be scientifically

measured. However, cyclemeasurement is extremelycomplex. In themostgeneralsense,thereisatripleinfinityof parameters–period, phase,and amplitude–that must beidentified simultaneously tocompletely describe thecycles. Additionally, marketcycles are ephemeral and areoftenburiedinpurenoise.Sothe compromises begin. Oneof the first realizations that atrader must make is that

cycles cannot be the basis oftradesallthetime.Sometimesthe cycle swings areswamped by trends, and it isfolly to try to fight the trend.However, the cyclic swingscanbehelpful toknowwhento buy on a dip in thedirection of the trend.Traditional indicatorssuchasStochastics, relative strengthindex (RSI),moving averageconvergence/divergence(MACD), and commodity

channel index (CCI) aresubject to the sameconstraints,andthereforethisbook will lead to a greaterunderstandingofalltechnicalindicators.Most important, Cycle

Analytics for Traders willallowtraderstothinkoftheirindicators and tradingstrategies in the frequencydomain as well as theirmotions in the time domain.

This new viewpoint willenablethemtoselectthemostefficient filter lengths for thejob at hand.Thedescriptionsarewritten for understandingat several different levels.Traders with littlemathematical backgroundwillbeable toassessgeneralmarket conditions to theiradvantage. More technicallyadvanced traderswillbeableto create indicators andstrategies that automatically

adapt to measured marketconditions by usingcombinations of computercodethataredescribed.So what should a trader

take away from this book?These are a few of the newconceptsthatIhaverankedinpriority:

An awareness ofSpectral Dilation, andhowtoeliminateitor to

useittoyouradvantage.How to use automaticgain control (AGC) tonormalize indicatoramplitudeswings.Thinkingofpricesinthefrequency domain aswell as in the timedomain.An awareness that allindicators are statisticalrather than absolute, asimplied by their single-linedisplays.

Several advancedcookbook filters. Theseinclude theSuperSmoother, roofingfilter, even bettersinewave, decycler, andHilbert TransformIndicator.Several differentmethods of estimatingmarket spectra andsifting out the dominantcycle, with theautocorrelation

periodogram being thepreferredmethod.How to use transformsto improve the displayand interpretation ofindicators.

The concepts I havedeveloped and derived fromscientific principles are newand useful aids to short-termtrading. Ultimately, tradingcomes down to buying andselling decisions. These

decisionsarenevereasy, andinthefinalchapterIunitetheconceptswith a few tips andtricks that I have acquired inmy years of trading. Aboveall, trading should beapproached as a statisticalprocess. Even with a goodperformance of 60 percentwinning trades, 60 percent isalotcloserto50–50thanitisto 100 percent regarding asingle event. Therefore, theperformancejudgedbyafew

tradesis invalid,andIwouldencourage readers to stickwith a trading strategy theyhave developed with aprofitable history, albeithypothetical, and let thestatistics be the light tosuccessinthelongrun.Asevidenceofmywarped

senseofhumor, eachchapterstarts with a “Tom Swifty”pun that encapsulates theentire content of the chapter

and I hope serves as ananchor for the reader'smemory.Ithinkthecomputercode is often the mostsuccinctandefficientmethodof describing a concept.Accordingly,mystyleistobebrief, with plenty of poeticlicense with mathematicalnotation in an effort toconvey the concepts to mosttraders. Each chapterconcludeswiththesignificantpointstorememberfromthat

chapter.Iwishyouallgoodtrading.

JohnF.EhlersAugust2013

ABOUTTHEAUTHOR

John Ehlers is an electricalengineer, receivinghisBSEEand MSEE from theUniversity of Missouri. Hedid his doctoral work at TheGeorge Washington

University, specializing inFields & Waves andInformation Theory. He hasretiredasaseniorengineeringfellowfromRaytheon.Hehasbeen a private trader since1976.John is a pioneer in

introducing the MESAcycles-measuring algorithmand the use of digital signalprocessing in technicalanalysis. He developed

maximum entropy spectrumanalysis (MESA) over threedecades ago. The programhas evolved with theincreasedcapacityofmoderncomputers.John has written

extensivelyaboutquantitativealgorithmic trading usingadvancedDSP (digital signalprocessing) and has spokeninternationallyonthesubject.HisbooksincludeMESAand

Trading Market Cycles,Rocket Science for Traders,and Cybernetic Analysis forStocks and Futures. Hisapproach is unique. Anytechniquemust firstwork ontheoretical waveforms beforetestingagainstreal-worlddataisattempted.John is a cofounder of

StockSpotter.com.

CHAPTER1

UnifiedFilterTheory

“It is toocomplex,”saidTomsimply.

Simplicity is at the heart oftheconceptoflinearsystems.Inputdataaresuppliedtothesystem, and the systemprovides the resultant as anoutput. There is only oneinput and only one output.However,thesystembetweentheinputandoutputcanbeascomplex as desired. Theoutputdividedbytheinputisthe transfer response of thesystem. It is this transfer

response that describes theactionofthesystem.In this chapter you will

find the difference betweennonrecursive filters andrecursive filters, andcombinations of the two,enablingyoutoselectthebestfilter for each application. Inaddition, you will find thatthe responses in the timedomain and in the frequencydomain are intimately

connected. When designingfilters for trading, it isbeneficial to consider theresponse in both of thesedomains. It is important toremember that no filter ispredictive—filter responsesare computedon thebasisofhistoricaldatasamples.Bythinkingintermsofthe

transfer responses, you willeasily make the transitionbetween filter theory and

programming the filters inyourtradingplatform.

■ TransferResponseConsider a four-bar simplemoving average. The inputdataarethelastfoursamplesof price. The filter output isone-fourthof themost recentprice data plus one-fourth ofthe data sample delayed byonebarplusone-fourthofthedata sample delayed by two

bars plus one-fourth of thedata sampledelayedby threebars. Ifwe allow the symbolZ −1 to represent one unit ofdelay, then we can write anequation for the transferresponse of a simplemovingaverage(SMA)as:

(1-1)

Thevaluesof¼arecalledthe coefficients of the filter.In general, the filter

coefficients sum to 1, so theratiooftheinputtotheoutputis1if theinputisaconstant.Ifwechoosetogeneralizethefilter to be other than anSMA, the values of thecoefficients canbearbitrarilyassigned. Further, we canextend the filter to have anyarbitrary length. In this case,the filter transfer responsecanbewrittenas:

(1-2)

The interesting thingaboutthis equation is thatwe havenow written the transferresponse as a generalizedalgebraic polynomial. Thepolynomial can have as highanorderasdesired.Thefiltergeneralitycanbe

extended by writing thetransfer response as the ratiooftwopolynomialsas:

(1-3)

This equation completelydescribes the transferresponse of any filter. Theonly thing that differentiatesone filter fromanother is theselection of the coefficientsof the polynomials. It isimmediatelyapparentthatthemore fancy and complex thefilterbecomes,themoreinputdataisrequired.Thisisreally

badforfiltersusedintradingbecause using more datameans the filter necessarilyhasmorelag.Minimizinglagin trading filters is almostmore important than thesmoothing that is realized byusing the filter. Therefore,filters used for trading bestusearelativelysmallamountof input data and should benotbecomplex.Although mathematicians

will cringe at the notation,filter computations canperhaps be better understoodby simplifying Equation 1-3as:

Clearing fractionsbycrossmultiplying, we get anequation useful forprogramming:

(1-4)

Equation 1-4 says that thefilter output is comprised oftwo parts. The first part, thenumerator term, uses onlyinputdatavalues.Ifthatistheonly term used in the filter,the filter is said to benonrecursive. The secondpart, the denominator term,consists of previously

computed values of theoutput. Filters using anypreviously computed valuesof the output are said to berecursive. The distinction isimportant because it isdifficult to create recursivefilters in some computerlanguages used for trading.Parenthetically, thecoefficienta0 isusuallyunitytokeepthingssimple.

■NonrecursiveFiltersA nonrecursive filter is onewhere the output responsedepends only on the inputdata and does not use aprevious calculation of theoutput to partially determinethe current value of the

output. Nonrecursive filtershave wide applications andthereforehaveacquiredmanydifferent names. Among thealiasesare:

MovingaveragefiltersFinite impulse response(FIR)filtersTappeddelaylinefiltersTransversalfilters

SMA filters are a specialcaseofmovingaveragefilters

where all the filtercoefficients have the samevalue.One of themost important

filter characteristics to atrader is how much lag thefilter introducesat theoutputrelative to the input. Anonrecursive filter whosecoefficients are symmetricalabout the center of the filteralwayshasa lagequal to thedegreeofthefilterdividedby

two. For example, anonrecursive filter of degreesix will have a three-bardelay. This delay is constantforallfrequencycomponents.Since lag is very important,and since lag is directlyrelated to filterdegree, filtersused for trading mostgenerally are simple and areoflowdegree.If the a0 coefficient equals

one and all the other “a”

coefficientsarezero,themostgeneral transfer response isjust thesimplepolynomial inthe numerator. From thefundamental theorem ofalgebra, the polynomial canbe factored into as manycomplex roots as it hasdegrees. In other words, thetransfer response can bewrittenas:

(1-5)

The coefficients may becomplex numbers rather thanrealnumbers.Inthiscase,theroots of the polynomial arecalledthezerosofthetransferresponse. For example, thefour-bar SMA transferresponse is a polynomial ofdegreethreeandthereforehasthreerootsfactoredas:

(1-6)

This transfer response has

one real root and twocomplementary imaginaryroots. If we substitute anexponential as exp(−j2π f) =Z −1, in the real root portionofEquation1-5,wegetusingDeMoivre'stheorem:

(1-7)

This equation can be trueonly when the frequency ishalf the sampling frequency.Half the sampling frequencyis the highest frequency thatis allowable in sampled datasystemswithoutaliasing,andis called the Nyquistfrequency. In our case, thesamplingisdoneuniformlyatonce per bar, so the highestpossible frequency we canfilter is0.5cyclesperbar,ora period of two bars.

Equation 1-7 shows that thezero in the transfer responseoccursexactlyat theNyquistfrequency. We havesucceeded in completelycanceling out the highestpossible frequency in thefour-barSMA.We can see the frequency

characteristic of the transferresponse by starting with afive-element SMA and thengeneralizing.

Multiplying both sides ofthis equation by Z −1 andsubtracting that multiplicandfrom both sides of theequation,weobtain

We get the frequencyresponse of this five-elementSMA by making thesubstitutionZ−1=exp(−j2πf), where f is the samplingfrequency.Then,

The equality of theexponential expressions andthe sine equivalent will berecognized by readersfamiliar with complexvariables as DeMoivre'stheorem. For readerswithoutthismath background, pleaseacceptitonfaith.

Generalizing this result foran N-length SMA, we havethe transfer response of anSMA in the frequencydomainexpressedas:

But since the Nyquistfrequency is half thesampling frequency, thetransfer response in thefrequencydomainis

(1-8)where f = frequency relativetothesamplingfrequencyThe important conclusion

from this discussion is thatwe can think of the transferresponse with equal validityin the time domain or in thefrequencydomain.Whenweplottheresponse

ofthefour-elementSMAasafunction of frequency in

Figure1.1,weseethatwenotonly have a zero at theNyquistfrequency,butalsoatafrequencyof0.25.

Figure1.1FrequencyResponseofaFour-BarSimpleMovingAverage

The horizontal axis isplotted in termsof frequencyrather than the cycle periodthat is most familiar totraders.Frequencyandperiodhaveareciprocalrelationship,soafrequencyof0.25cyclesperbarcorrespondstoafour-bar period. The vertical axisistheamplitudeoftheoutputrelative to the amplitude ofthe input data in decibels. Adecibel (dB) is a logarithmic

measure of the power in theoutput.Figure1.1shows thatthere are zeros in the filtertransfer response in thefrequency domain as well asinthetimedomain.Theconceptof thinkingof

how a filter works in thefrequency domain as well ashow it works in the timedomain is central to theunderstanding of theindicators that will be

developed. Low frequenciesnear zero are passed frominput to output with little orno attenuation. Since higherfrequencies are blocked frombeing passed to the output,the SMA is a type of low-pass filter—passing lowfrequencies and blockinghigher frequencies. Low-passfiltersaredatasmoothersthatremove the higher-frequencyjitter in the input data thatoftenmakes the data hard to

interpret. The penalty traderspay for this smoothing is thelag introduced in the transferresponse.Low-passfiltersarenotthe

only filters that can begenerated with thegeneralized transfer responseofEquation1-3.Supposewearrange to have thecoefficientstobeas:

All other coefficients areequaltozero.Then the frequency

responseofthefilterisshowninFigure1.2.

Figure1.2FrequencyResponseofaTwo-BarDifferenceFilter

In this case, the higherfrequencies are passed, andthe lower frequencies areseverely attenuated by thefilter.Thisisanexampleofahigh-pass filter. Since trendscanbeviewed as pieces of avery long cycle, a high-passfilter is basically a detrenderbecause the low-trendfrequenciesarerejectedinitstransferresponse.Since the coefficients of a

simple high-pass filter areequivalent to just taking thedifferenceoftwoconsecutivesamples of input data, thedifference operation can beviewed as analogous to aderivative function in thecalculus. This conceptenables a high-pass filter tobe used in several differentways in trading to attempt tocreateapredictivewaveform.If the inputdataareassumedto be in a trend, then the

difference between any twodata samples is constant. Inthis case, adding thedifference to the current bardatapredicts thevalueof theinput data for the nextsample. Alternatively, if theinputdataareassumedtobeaquiescent sinewave, a tradercanuse the relationship fromcalculusas:

(1-9)

Ifthefrequencyofthesinewaveisknown,thehigh-passfilter not only provides awaveformthatleadstheinputdatawaveformby90degrees,but also provides the meansto normalize the outputamplitudetotheamplitudeoftheinputdata.Returning to Equation 1-2

for a generalizednonrecursive filter, and

factoring out a Z −N/2 term,weobtain:

(1-10)

SinceZ−N/2isapuredelayterm, and since exp(−j2π f )can be substituted for Z −1,Equation 1-10 is proof thatnonrecursive filters havingcoefficients symmetricalabout the center of the filterwill have a constant delay at

all frequencies. Further, thatdelaywillbeexactlyhalf thedegree of the transferresponsepolynomial.

■ RecursiveFiltersA recursive filter is onewhere the output responsedependsnotonlyontheinputdata but also on previousvalues of the output. Strictlyrecursive filters arecharacterizedbyusingonlyaconstantinthenumeratorandmultiple terms in the

denominatorofEquation1-3.Recursive filters also havewide applications andthereforehaveacquiredmanydifferent names. Among thealiasesare:

Exponential movingaveragefiltersInfinite impulseresponse(IIR)filtersLadderfiltersAutoregressivefilters

If the b0 coefficient is aconstantandalltheother“b”coefficientsarezero,themostgeneral transfer response isjust thesimplepolynomial inthe denominator. Thispolynomial can be factoredinto as many complex rootsas it has degrees. In otherwords, the transfer responsecanbewrittenas:

(1-11)

The coefficients may becomplex, rather than real,numbers. In this case, theroots of the polynomial arecalledthepolesofthetransferresponsebecauseazerointhedenominator of the transferresponse causes the transferresponse to go to infinity atthat point. One can visualizethe transfer response as thecanvasof a circus tent in thecontext of complex numbers,and the poles in the transfer

responseareanalogoustothetentpoles.Whileitispossibleto choose coefficients thatcause the transfer function to“blow up,” frequencies areconstrained to be realnumbers, and therefore it isrelatively easy to avoid thecomplexpolelocations.Consider the special case

ofarecursivefilterwhere

Then, Equation 1-3becomes

Then, using theconventional notation thatOutput[1] equals the output

onebarago:(1-12)

Equation 1-12 is exactlythe equation for anexponential moving average(EMA).Note that the sumofall of the coefficients on theright-handsideofEquation1-11 sum to1 so that the filterhasnonoisegain.Figure 1.3 shows the

frequency response of theEMAwhen alpha= 0.2.The

EMA is a type of low-passfilter, passing the lower-frequency components of theinput data and attenuating itshigher-frequencycomponents.Ifalphaismadeto be smaller, fewer of thelower-frequency componentsare allowed to pass, and thehigh-frequency componentsare attenuated to a greaterdegree. Conversely, if alphaismade to be larger, there islesssmoothing,and therefore

more higher-frequencycomponents of the input areallowedtopasstotheoutput.Therearenozeros (orpoles)inthetransferresponse.

Figure1.3ExponentialMovingAverageFrequencyResponseforα=0.2

ThelagofEMAfilterswillbederivedinChapter2.

■ GeneralizedFiltersAgeneralizedfilterusesboththe numerator anddenominator of Equation 1-3to achieve a wider range ofresponsesotherthanlow-passfilteringandhigh-passfilters.Some familiar aliases forthesegeneralizedfiltersare:

Autoregressive movingaverage(ARMA)filtersAutoregressiveintegrated movingaverage(ARIMA)filters

A band-pass filter can becreated by connecting a low-pass filter in tandem with ahigh-pass filter so that boththe low-frequency and high-frequency components areattenuated and componentsnear a selected frequency are

passed to the output.However, it is much moreefficient to create the band-passfilterresponsesimplybyselecting the propercoefficientsinEquation1-3.

■ProgrammingtheFiltersIt is most convenient toconsiderfiltersas“stonewall”filters that have only a passbandandastopbandwiththeboundary between themlocated at a critical cycleperiod. For example, a

smoothing low-pass filter onstock data might have acritical cycle period of 20bars because many stockshave monthly cycle periodsofapproximately20barsandthe smoothing wouldattenuate only those cycliccomponents shorter than 20bars.Band-pass filterswouldpass only cycle componentscentered at the critical cycleperiod. Band-stop filterswould reject only cycle

components also centered atthecriticalcycleperiod.With the exception of

nonrecursive filters, onlyfiltersofdegreetwoareuseddue to lag considerations. Inthis case, Equation 1-3reducestoEquation1-13:

(1-13)

UsingthenotationthatZ−1* Input = Input[1], etc., and

algebraically rearrangingEquation1-13,theexpressionthat can be used forprogramming becomesEquation1-14:

(1-14)

Beforeweassignthevalueofthe coefficients, it is mostconvenient to compute someconstants in terms of thecritical period. In thefollowing equations, the

arguments of thetrigonometricfunctionsareindegrees.1

(1-15)

WhereK = 1 for single-polefilters

K = 0.707 for two-polehigh-passfiltersK = 1.414 for two-polelow-passfilters

(1-16)Whereγ=Cos(360*/Period)

and δ = bandwith as afractionofPeriod(1-17)

Using these unifiedcalculations, the filtercoefficients for the varioustypes of filters can be foundinTable1.1.Table1.1FilterCoefficientsforVariousTypesofFilters

Simple two-pole filters arejust two single-pole filtersthat are serially connected.Filteringapreviously filteredoutputmeans thecoefficientsmustbeadjustedtoobtainthe

same−3-dBattenuationatthedesired critical period.Equation 1-15 accomplishesthisbycomputingalphatermat a period where theattenuation is approximately−1.5 dB for each of the twocomponent filters. Thus, thetotal attenuation of theserially connect set is −3decibelsatthecriticalperiod.A trader may also serially

connect any of the conical

filter types in Table 1.1 toobtain a more complexsystem to achieve an overalldesiredresult.Forexample,aband-pass filter output couldbetheinputtoanEMAfilterwithadifferentcriticalperiodtofurthersmooththedata.Asanother example, a low-passfiltercanbecombinedwithahigh-pass filter to create avery wide bandwidth band-passfilter.

All the filters inTable 1.1have application to tradingwith the exception of theband-stop filter. Using theparameters in Table 1.1 theband-stop filter would moreappropriately be called anotch filter. That is, itattenuates a very narrowrangeoffrequencies.Itcouldbe used to remove 60-hertzpower line frequency fromaudio equipment, forexample. Market cycles tend

to be ephemeris and havevery fuzzy bandwidths,making them difficult torejectbyfiltering.Evenmoresophisticatedband-stopfilterswith relatively wide stopbands do not seem to beeffective in removing cycliccomponents from the data.This is indeed unfortunatebecause removing cycliccomponents to isolate thetrends is a philosophicallyinteresting digital signal

processingapproach.

■ WaveAmplitude,Power, andDecibels(dB)The transfer response of afilter is expressed as theoutputrelativetotheinput.Inthe time domain, we canexamine the two waveforms

tomakeacomparison.Waveshave three components:frequency, amplitude, andphase. It is generally moreconvenient for trading todescribe only the relativeamplitudes and delays. Wecan eliminate phase byexamining power rather thanwave amplitude. Thedifferencebetweenpowerandwave amplitude is analogousto describing a lightbulb asusing 60 watts rather than

using 115 volts. Power isproportional to the square ofthe wave amplitude, and tobetter examine a wide rangeof power levels, relativepower isusuallymeasured ina logarithmic ratio calleddecibels.2Using V to represent

relative wave amplitude, thetransferresponseamplitudeisexpressedmathematicallyas

Some relative amplitudesusefultorememberare:

■ Key PointstoRemember

1.Allthecommonfiltersuseful for tradershaveatransfer response thatcanbewrittenasa ratiooftwopolynomials.2.Lag isvery importantto traders. Morecomplex filters can becreatedusingmoreinput

data,butmoreinputdataincreases lag.Sophisticated filters arenot very useful fortrading because theyincurtoomuchlag.3. Filter transferresponse can be viewedin the time domain andthe frequency domainwithequalvalidity.4. Nonrecursive filterscan have zeros in the

transfer response,enabling the completecancellation of someselected frequencycomponents.5. Nonrecursive filtershaving coefficientssymmetrical about thecenter of the filter willhave a delay of half thedegree of the transferresponse polynomial atallfrequencies.

6. Low-pass filters aresmoothers because theyattenuate the high-frequency componentsoftheinputdata.7. High-pass filters aredetrenders because theyattenuate the low-frequency componentsoftrends.8. Band-pass filters areboth detrenders andsmoothers because they

attenuate all but thedesired frequencycomponents.9. Filters provide anoutputonlythroughtheirtransfer response. Thetransfer response isstrictly a mathematicalfunction, andinterpretations such asoverbought, oversold,convergence,divergence, and so on

are not implied. Thevalidity of suchinterpretations must bemade on the basis ofstatistics apart from thefilter.

Notes1. Sanjit K. Mitra,Digital SignalProcessing, 2nd ed.(New York: McGraw-Hill, 2000), Section4.5.2.

2.Adecibelisone-tenthof a bel, named afterAlexanderGrahamBell.Abeloriginalmeanttheapproximate power lost

in one mile of standardtelephone cable, and isabout the least amountof change in audiodetectablebyahuman.

CHAPTER2

SMAs,EMAs,orOther?

“Everyone's children areabove average,” said

Tommeanly.

Moving averages arediscussed in this chaptersimply because they areubiquitous in technicalanalysis. There are manybetter filters for variousapplications that aredescribed in the followingchapters. Moving averageshave the advantage that theyare dirt-simple to compute.Theirdisadvantageisthatthe

produceonlya small amountof smoothing for the cost oflagintheircomputation.One little-known fact is

thatasimplemovingaverageconstitutesabestfittodataintheleast-squaressense.

■ SimpleMovingAverages(SMAs)SMAs are a special case ofnonrecursive filterswhere allthe coefficients are equal tounity divided by the numberof elements in the filter. At

any given point in the datastream,theoutputofthefilteris just the average of dataover the span of the filter.When time is advanced byonebar inonealgorithm, thefractional part of the oldestdata sample is discarded andthe fractional newest datasampleisadded.Thus, it isa“moving” average. It justdoesn't getmore simple thanthat,andsoitisdeemedtobeasimplemovingaverage.

Viewed from anotherperspective, it canbeviewedas a “window” that is passedacross a fixed data stream.Only those data sampleswithin the window areaveraged and provided as anoutput of the filter. Sincemodern computers are veryfast, I recommend this latterapproach because if thelengthoftheaverageisvariedacross the data set, thecomputations will fail. We

maywant tomake the lengthof the average vary whenadaptingthefiltertovolatilityorameasuredcycleperiod.Animpulseisatheoretical

data object that has zerowidthandinfiniteheight.Itisused to assess the filtertransfer response. Forexample, you can think of itas a clapper striking abell—the bell provides the transferresponse, and the ringing is

theoutput.Inotherwords,theringing is the impulseresponseofabell.Insampleddata systems, we canapproximate an impulse as adata stream that has a valueonly at one point and haszeros at all other datasamples.If we slide the SMA

window across the impulsedata stream, the filter willhaveanoutputonlywhenthe

impulse falls within thewindow of the filter. Thus,the impulse response of anSMA is finite, and thereforeanSMAisaspecialcaseofafinite impulse response (FIR)filter.Suppose we start a filter

design by trying to make itstransfer response be the bestfit to a straight line. If wehave more than two datapoints, this would be

impossible in the generalcase.However,wecandefinea best fit, where the sum ofthesquaresofthedifferencesbetween the data and ourstraight line is minimum. Inthe case of five data points,designatedasux,wecanwritethedesiredfunctionas

We can minimize thisfunction by taking the

derivatives of this functionwith respect to m and b andsetting those derivatives tozero. When we do this, wefind that the correct value ofb is just the average of thefive data points. This is trueregardless ofwhich five datapoints we select, so that anSMA is synonymous withfinding a best-fitting straightlinetothedata,atleastwhenthe lag of the filter isdisregarded.Ifindthistobea

fascinating little-known factaboutSMAs.In Chapter 1, we showed

that thelagofanonrecursivefilter having coefficientssymmetrical about its centerpoint and of length N isexactly (N−1) / 2.SinceanSMA is a special case of anonrecursive filter, it is alsotruethatanSMAlagisalso

Equation 1-8 is repeated

here as the amplitude of thetransferresponseofanSMA.

(2-1)At zero frequency,

Equation2-1evaluatesto1inthe limit.This starts the passbandoftheSMA.Thecriticalperiodofafilteroutputisthefrequencyatwhichtheoutputpowerof the filter ishalf thepower of the input wave atthat frequency. Since power

is proportional to the squareofthewaveamplitude,weareinterested in finding thefrequencyatwhichtheoutputwave has an amplitude of0.707(thesquarerootof0.5).Knowing this, we can selectan SMA that gives us theapproximate critical periodweseek.Figure2.1showstherelationship between thecriticalperiodandthenumberofelementsintheSMA.

Figure2.1SMACritical

PeriodasaFunctionofFilterLengthIt is a reasonable

approximation and easy toremember that the criticalperiodofanSMAistwiceitslength.The first zero in the

transfer response occurs thewhen the argument in thenumerator has a value of π.This occurswhen f = 1 / N.Subsequent zeros in the

transfer response occurwhenthe argument in thenumerator are integermultiples of π. The firstmaximum in the stop bandoccurswhen the argument inthenumeratorcausesthesinefunctiontobeatamaximum,or f = 3π / 2. At thisfrequency the value of thetransfer response isapproximately 2 / (3π), or−13.3 decibels (dB). For thesame reason, the next

maximum in the stop bandoccurs at f = 5π / 2, and theattenuation is 2 / (5π), or−17.8 dB. This amount ofattenuation is not substantial,and you will see methods tomodify an SMA to improveitsfilteringcapability.

■ ExponentialMovingAverages(EMAs)An EMA is computed bytaking a fraction of thecurrent price added to thecomplementary fraction oftheEMAcalculation one bar

ago. The first fraction isusually called alpha, and canhave a value between zeroand one. The two fractionsmust sum to unity, so thesecondfractionmusthavethevalue of one minus alpha.The equation to compute anEMAis:

(2-2)

The EMA becomes amoving average by movingfrom bar to bar, from left to

right,acrossthepricedata.Imagine a case where the

input data set has only onesampleatanamplitudeof1/α.When theEMA is appliedto these data, the first outputfrom the filter is unitybecause there was noprevious value for the EMA.On subsequent calculations,thepricevalueiszero,andsothe sequence of calculationsis:

Since the quantity (1 − α)must be less than one, theamplitude decays as theexponent of each succeedingcalculation from an impulseinput, hence the name

exponential moving average.Inprinciple,apartofanydatainput remains in subsequentcalculations, although thecontribution becomesvanishingly small. ThisattributemakesanEMApartof a general class of filterscalled infinite impulseresponse(IIR)filters.It is instructive toexamine

the EMA response to a stepfunction.A step functionhas

a series of constant valuesand then jumps to anotherseries of constant values.Assume the price has beenzerofora longtimeandthensuddenlyjumpsuptoavalueof1andmaintains thatvaluethereafter. On the first bar,theEMAwillhaveavalueofα. On the second bar, thevaluewillbeα+α*(1−α).On the third bar, the valuewillbeα+α*(1−α)+α*

(1−α)2,andsoon.TheEMAwill gradually approach thevalue of 1. A common errorinprogramming is to insertavalue for α, such as .2, andinserting another number for(1 − α), such as .9. The twoterms must sum to unity orthe recursive algorithm willleadtoerraticresultsormighteven cause your computer tocrash. You should alwayscheckyourcomputercode to

ensurethatthetwotermssumtounity. I amsocautiousonthis point that I assign thevalue α as a global variableand write out the EMAequation in terms of α. Byletting the computer do thework, I know the two termsmustsumcorrectly.Wecanassertthattheprice

onday“d”isd.Ifweassumethe lag of the EMA is “L,”then the current value of the

EMAis (d−L).Further, theprevious EMAwould have avalue of (d − L − 1), sinceprice is rising one unit perbar.Putting thesevalues intotheequationfortheEMA,weobtain

This equation shows that

we can select an acceptablelag and, from that lag,computethealphatermoftheEMA.Forexample,ifwecanaccept a three-bar lagresulting from the EMA, wewoulduseα=0.25.WecanalsorelateanEMA

to an SMA on the basis oftheir equivalent static lags.Recalling that the lag of anSMAis(N−1)/2foranN-bar SMA, we can substitute

this value of lag into thealphacalculationoftheEMAas

(2-3)

This is the relationshipbetween an N-bar SMA andthe alpha of an EMA that isquoted in most technicalanalysis books. Thefrequencyresponsesofafive-

bar SMA and its EMAequivalent are compared inFigure 2.2. Figure 2.3 showsthat the lagsof theSMAandEMA are equal only at zerofrequency.

Figure2.2FrequencyResponseComparisonofaFive-BarSMAandSimilarEMA

Figure2.3LagResponseofEMAComparedtoaConstantTwo-BarLagofaFive-BarSMA

Although the lags at staticfrequencies align, note thatthecriticalperiodoftheEMAisnotalignedwiththatoftheSMA.Wecanobtainabettercalculation for the fraction αby examining the transferresponseoftheEMA.

■ WeightedMovingAverages(WMAs)ItappearsthattheWMAwasinventedby a traderwhodidnothaveafirmgraspoffiltertheory in hopes of reducinglag.

A WMA is an FIR filterhaving coefficientsasymmetrically weightedabout the center point of thefilter. The most recent datasample is weighted mostheavily, the second mostrecent data sample weightedsomewhat less, and so on.The weighting does notnecessarily have to be linearwiththedatasample,butthatis the most common usage.For our graphical example, I

willuseafive-elementWMAwhosecoefficientsare [5432 1] / 15. Figure 2.4 showsthefrequencyresponseofthisfilter, and Figure 2.5 showsitslagresponse.

Figure2.4AWMAHasPoorRejectioninItsTransferResponse

Figure2.5LagofaFive-ElementWMAHasNoOutstandingCharacteristics

Figure 2.4 shows that aWMAhas poorer attenuationthan the equivalent SMAshown inFigure2.2 and thataWMA hasmore lag in thepassbandthantheequivalentEMAshowninFigure2.3.Inshort, a WMA has little torecommend itself comparedto theothercommonmovingaverages.

■ MedianFilterThere are times when amedian filter is the mostappropriate filter to use. It isbest applied when the datacontains impulsive noise orwhen there are widevariationsinthedata,suchasvolumedata.Itspurposeistoattempttodiscardoutlierdata

samplestoavoidhavingthemexertaninordinateimpactontheaverage.A median filter simply

picks the middle value of asample set of data after thesamples have been ranked.Thisselectioniseasiestwhenan odd number of datasamples is used. Todemonstrate the effect of amedian filter, consideradataset consisting of four

samples, eachhavingavalueof1andonesamplethathasavalue of 6.The data samplescanappearinanyorder.Afterranking, the center value isjust 1. Therefore the medianvalue is 1,which happens tobe the value of most of thedata samples. However, theaveragevalueis(4+6)/5=2. That one outlier datasample has severely skewedthe average, and the medianfilter best recovers the

“average” value of the dataset.

■ Key PointstoRemember

1. The critical perioddefiningthepassbandofan SMA isapproximately twice thelengthofthefilter.2. An N-length SMAwill have a lag of (N −1) / 2 bars at allfrequencies.

3. An EMA shouldalways be written asEMA=α*Input+(1−α) * EMA[1] to avoidcomputational errors inevaluating the fractionalcoefficient.4. The fractionalcoefficient of an EMAequivalent to an N-length SMA iscalculated asα=2 / (N−1).

5. AWMA has little ornoredeemingvirtue.6.Amedianfilterisbestused when the datacontain impulsive noiseor when there are wildvariations in the data.Smoothing volume dataisoneexampleofagoodapplication for amedianfilter.

CHAPTER3

SmoothingFiltersonSteroids

“Steroids are for guys

who want to cheat theiropponents,” said Tomcombatively.

The objective of smoothingfilters in trading is to get thehighest degree of smoothingpossiblewithin the constraintof inducing the least amountof lag. The purpose of thischapter is to develop thosekinds of filters for traders.Then, it isup to the trader tointerpret the results of the

filtering.Noneofthesefiltersare predictive.None of thesefiltersexplainmarketactivity.

■NonrecursiveFiltersAny nonrecursive filterhaving coefficientssymmetrical about the centerpointof the filter andhavinga polynomial of odd degreewillalwayshavearootofthepolynomial at the Nyquist

frequency. This is importantbecause completely rejectingthe highest possiblefrequency component in thefilteroutputgoes a longwaytowardnoisereduction.Noisereduction is the principalpurposeofasmoothingfilter.Since the degree of the

polynomial is one less thanthenumberofelementsinthefilter, nonrecursive filtershaving the number of

elementsas2,4,6,8,10,andso on will have thischaracteristic.Sincethelagofnonrecursivefiltersishalfthedegree of the filter, thesefilters will have lags of 0.5,1.5, 2.5, 3.5, and 4.5 bars,respectively.Filterdesignnotonly depends on the amountofsmoothingdesired,butalsoontheamountoflagthatcanbetolerated.As a shorthand notation,

[b0 b1 b2 b3 . . . bN] / S isintroduced to describe thecoefficients of nonrecursivefilters.Forexample,[1111]/ 4 describes the coefficientsof a four-element simplemoving average (SMA). Infact,afour-elementSMAfitsthecriterionofhavingazeroat the Nyquist frequency. Itsfrequency response is shownin Figure 3.1. This figureshows that a four-bar cycle

period (frequency = 0.25) isalso rejected. The criticalfrequency is the−3 dBpointin the response. The −3 dBpoint along the frequencyaxis is where the output isattenuatedtobehalftheinputpower.Inthiscasethecriticalfrequency is approximately0.1 cycles per bar,corresponding to a 10-barcycle period.The attenuationcharacteristic between a two-bar cycle period and a four-

barcycleperiodrisestoabout−12 decibels (dB), whichmeans thewaveamplitudeattheoutputisabout25percentoftheamplitudeattheinput.

Figure3.1FrequencyResponseofa[1111]/4NonrecursiveFilter

The expectation ofobtainingmoresmoothingbyincreasing the length of theSMA leads us to examine asix-element nonrecursivefilterwhoseelementsare[111 1 1 1] / 6. The frequencyresponse of this filter isshown in Figure 3.2.Increasing the degree of thepolynomial introducedanother zero in the transferresponse polynomial so that

two-, three-, and five-barcycle periods are completelyeliminated at the output.However, attenuationbetween the rejection pointsisnotchangedwithrespecttothe lobe number, and is onlyslightly increased from lobetolobe.Thecriticalperiodofthis filter is increasedcomparedtothefour-elementSMAtobeata13.7-barcycleperiod.

Figure3.2FrequencyResponseofa[111111]/6NonrecursiveFilter

Nonrecursivefiltersarenotrestricted to be SMAs. Thecoefficients can be weightedsymmetrically about thecenter of the filter. Amongthemost simpleof these is afilter whose coefficients are[1 2 2 1] / 6. The frequencyresponse of this filter isshown in Figure 3.3.Comparing this nonrecursivefiltertotheSMAoflikeorder(in Figure 3.1), we see that

not only have the cycleperiods that are completelyrejected are two- and three-bar cycle periods as opposedto two- and four-cycleperiods. In addition, theattenuation between therejection frequencies rises toa minimum of −25 dB,meaningtheoutputwaveformamplitude is less than 5.6percentoftheinputamplitudein this rejection band. Theincreased attenuation comes

at the expense of a decreasein the critical cycleperiod toa 7.4-bar cycle. The filterdelayremainsthesameat2.5bars.

Figure3.3FrequencyResponseofa[1221]/6NonrecursiveFilter

From our experience withSMAs,we can improve boththe smoothing and therejection band attenuation byincreasing the degree of thefilter (at the expense ofincreasedlag).Thefrequencyresponseof a [123321] /12 nonrecursive filter isshown in Figure 3.4. In thiscase, two-, three-, and four-bar cycle periods arecompletely eliminated from

theoutputofthefilterandtherejection in the stop bands isstill greater than −25 dB.Filteringisimprovedbecausethe critical period has beenincreased to be at a 10.9-barcycleperiod.All this is doneat the expense of a delay of3.5bars.Whilethisisaprettynicefilter,wecandobetter.

Figure3.4FrequencyResponseofa[123321]/12NonrecursiveFilter

■ ModifiedSimpleMovingAveragesAn interesting fact is that ifthevaluesofthefirstandlastelements of an SMA havinganevendegreearecutinhalf,then one is guaranteed adouble zero in the transferresponse at the Nyquist

frequency. For example, afive-element (four-degree)SMAwouldhavecoefficientsas [0.5 1 1 1 0.5] / 4. Thetransfer response of a five-element modified SMA isshown in Figure 3.5 forcomparisontotheresponseofa four-element SMA showninFigure3.1.Notonlydowehave the double zero at theNyquist frequency and thesamezeroata four-barcycleperiod, but we also have

increased the minimumattenuation between the twozeros to be −20 dB, whichmeanstheoutputamplitudeisonly 10 percent of the inputdata amplitude. Thisimproved filteringperformance comes at theexpenseofonlyahalfbarofdelaybecausewewentfromafour-element filter to a five-elementfilter.

Figure3.5Frequency

ResponseofaFive-ElementModifiedSimpleMovingAverage

Thefrequencyresponsesofthree-,five-,seven-,andnine-element modified SMAs areshowninFigure3.6.

Figure3.6ModifiedSimpleMovingAverageFrequencyResponses

■ ModifiedLeast-SquaresQuadraticsInstead of transfer response'shavingaleast-squaresbestfit,aswiththeSMA,wecanalsoarrange to obtain a least-squaresbestfittoaquadraticequation.Thismeanswe canaccommodate some bowing

in the data while getting ourleast-squares best fit. Then,wecanhalvethefirstandlastcoefficients to realize thedouble zero in the transferresponse at the Nyquistfrequency. Hamming1 givesthecoefficientsofsuchfiltersas:

Thefrequencyresponsesof

these filters are shown inFigure 3.7. The frequencyresponses inFigure3.7 showthattheseareclearlysuperiorsmoothing filters for trading.Thesefiltersincurtwo,three,four, and five bars of lag,respectively.

Figure3.7ModifiedLeast-SquaresQuadraticsFilterFrequencyResponses

■SuperSmootherOne of the advantages ofrecursive filters is that thetransition between the passbandandthestopbandcanbemuch sharper compared tothat of the nonrecursivefilters. As a matter ofcomputational simplicity, thecriticalperiodofthefiltercan

be established independentlyfrom the degree of the filter.On the downside, recursivefilters don't have the samedelay at all frequencies. Thisvariable delay across thespectrum leads to dispersiondistortion of the outputwaveform. Most of thedispersion occurs at thosefrequency components thathave been attenuated, so theeffect of the distortion is arelativelyminorone.

A Butterworth filter is afavorite analog filter becauseits frequency response ismaximally flat, near zerofrequency. Years ago, ItranslatedanalogButterworthfilters to their digitalapproximations having likedegrees in thenumeratoranddenominator of the transferresponse. The transferresponseischaracterizedbyasingle variable: the criticalperiod. The critical period is

that where the output isattenuatedby3dBrelativetotheinputdata.The digital Butterworth

filter approximationsincluded numerators whosecoefficientswere [121] and[1 3 3 1] for two- and three-pole configurations,respectively. However, forapplications to trading, Inoted that the primarycontributionof thenumerator

terms was to increase thefilter lag. So I createdmodified versions simply bydeleting all but the constantterm in the numerator of theButterworth transferresponse.Theequations for the two-

pole modified Butterworthfilterare:

(3-1)

The equations for thethree-pole modifiedButterworthfilterare:

(3-2)

The frequency responsesfor two- and three-polemodified Butterworth filters

whose critical periodwas setata10-barcycleareshowninFigure 3.8. The three-poleversion gives about 6 dBmore attenuation at theNyquist frequency than thetwo-poleversion.Thatmeanstheoutputwaveamplitudeofthe three-pole version wouldbehalfthewaveamplitudeatthe Nyquist frequency. Insome applications, thisdifference in filteredamplitudecouldbeimportant.

Figure3.8FrequencyResponseofTwo-andThree-PoleModifiedButterworthFiltersWherePeriod=10

The delay of the two- andthree-pole modifiedButterworth filters whosecriticalperiodwassetata10-bar cycle is shown in Figure3.9. The price to be paid forthe additional rejection bandattenuation is an increase ofalmost twobarsofadditionaldelaynear thecriticalperiod.In addition, there is greaterpercentage dispersiondistortion across the pass

band.

Figure3.9LagsofTwo-andThree-PoleModifiedButterworthFiltersWherePeriod=10

Having obtained goodfilteringwithminimumdelayand minimum delaydistortion,itisrelativelyeasytoobtainazerointhetransferresponse at the Nyquistfrequency by adding a two-element moving average inthenumerator.Thedelaycostis only half a bar, and thatcostseemsworthit.Bydoingthis, a filter I call theSuperSmoother is created.

The equations for theSuperSmootherfilterare:

(3-3)

The frequency response ofthe SuperSmoother filter

wherethecriticalperiodissetto10isshowninFigure3.10.

Figure3.10FrequencyResponseofaSuperSmootherFilterWherePeriod=10

■SuperSmootherFilterApplicationsA SuperSmoother filter isused anytime a movingaverage of any type wouldotherwise be used, with theresultthattheSuperSmoother

filter output would havesubstantially less lag for anequivalent amount ofsmoothing produced by themoving average. Forexample,afive-barSMAhasa cutoff period ofapproximately 10 bars andhas two bars of lag. ASuperSmoother filter with acutoffperiodof10barshasalag a half bar larger than thetwo-pole modifiedButterworth filter shown in

Figure3.9.Therefore, such aSuperSmoother filter has amaximum lag ofapproximately 1.5 bars andeven less lag into theattenuationbandof the filter.The differential in lagbetweenmoving average andSuperSmoother filter outputsbecomes even larger whenthecutoffperiodsarelarger.Market data contain noise,

and removal of noise is the

reason for using smoothingfilters. In fact, market datacontain several kinds ofnoise. I'll group one kind ofnoise as systemic, caused bythe random events of tradesbeing exercised. A secondkind of noise is aliasingnoise, caused by the use ofsampled data. Aliasing noiseis the dominant term in thedataforshortercycleperiods.Itiseasytothinkofmarket

data as being a continuouswaveform,butitisnot.Usingthe closing price asrepresentative for that barconstitutes one sample point.It doesn't matter if you areusing an average of the highand low instead of the close,you are still getting onesample per bar. Sincesampled data is being used,there are some DSP aspectsthat must be considered. Forexample,theshortestanalysis

period that is possible(without aliasing)2 is a two-bar cycle. This is called theNyquistfrequency,0.5cyclespersample.Aperfecttwo-barsine wave cycle sampled atthe peaks becomes a squarewave due to sampling.However, sampling at thecycle peaks cannot beguaranteed, and theinterference between thesampling frequency and the

data frequency creates thealiasing noise. The noise isreduced as the data period islonger. For example, a four-barcyclemeanstherearefoursamples per cycle. Becausethere are more samples, thesampled data are a betterreplica of the sine wavecomponent. The replica isbetter yet for an eight-bardata component. Theimproved fidelity of thesampled data means the

aliasing noise is reduced atlonger and longer cycleperiods.Therateofreductionis 6 dB per octave. Myexperience is that thesystemicnoise rarely ismorethan10dBbelowthelevelofcyclicinformation,sothatwecreate two conditions foreffective smoothing ofaliasingnoise:

1. It is difficult to usecycleperiodsshorterthat

two octaves below theNyquist frequency. Thatis, an eight-bar cyclecomponent has aquantization noise level12 dB below the noiselevel at the Nyquistfrequency. Longer cyclecomponents thereforehave a systemic noiselevel that exceeds thealiasingnoiselevel.2. A smoothing filter

should have sufficientselectivity to reducealiasing noise below thesystemic noise level.Since aliasing noiseincreasesattherateof6dB per octave above aselected filter cutofffrequency and since theSuperSmootherattenuationrateis12dBper octave, theSuperSmoother filter isan effective tool to

virtually eliminatealiasing noise in theoutputsignal.

I urge readers to universallyadapt the SuperSmootherfilterset toacutoffperiodof10 bars or so on all data toattenuate aliasing noise. Theoutput of the SuperSmootherfilter can be used directly asanindicatororasthesampleddata fed to any otherindicator.

■ Key PointstoRemember

1. The lag of allnonrecursive filtershaving coefficientssymmetrical about thecenter point of the filteris half the degree of thetransfer responsepolynomial. In otherwords, for a

nonrecursive filterhaving N elements, thelag of that filter will be(N−1)/2.2. Nonrecursive filterswith coefficientssymmetrical about thecenter point of the filterand having an evennumberof elementswillalwayshaveazerointhetransfer response at theNyquistfrequency.

3. An SMA having anodd number of elementsandwhose first and lastelements havecoefficients cut in halfareguaranteedofhavinga double zero in thetransfer response at theNyquistfrequency.4.Whereas SMAs are aleast-squaresbestfittoastraight line,nonrecursive filters can

also have coefficientsthat are a least-squaresbest fit to a quartic.Halving their endcoefficients alsoguaranteesadoublezerointheirtransferresponseat the Nyquistfrequency.5. Probably the bestfilterformostsmoothingapplications is a two-poleSuperSmoother.

6. A SuperSmootherfilter is recommendedfor universal use toremovealiasingnoise.

Notes1. R. W. Hamming,Digital Filters, 3rd ed.(Upper Saddle River,NJ:PrenticeHall,1997),49.

2. Aliasing is the falsesignals that result fromundersampling. Forexample, wagon wheelsoften appeared to turnbackwards in the early

cowboy motion picturesbecauseof the relativelyslowmovieframerate.

CHAPTER4

Decyclers“Thewiggles have beencanceled,” said Tomflatly.

If technical analysistechniques include adetrender,thencertainlythereis room for its corollary, adecycler.Adecyclerremovesthe cycle components andretains only the trendcomponents.

■ DecyclerConstructionThe concept of a decycler isreally pretty simple. Thecyclic components areremoved by the process ofcancellation. Figure 4.1showstheamplituderesponseofahigh-passfilter.Notethatthe amplitude of the filteroutput is almost the same as

thefilterinputamplitude(i.e.,0 dB).Thus, if the high-passfilter output is subtractedfrom the input data, theresidual only contains thelow-frequencycomponents.

Figure4.1FrequencyResponseofaHigh-PassFilterHavinga30-BarCutoffPeriod

The transfer response of adecycler is written as thedifference between input andtheoutputofahigh-passfilteras

Byputtingtheright-handsideof this equation over acommon denominator, weobtain

This transfer responseequation shows the decyclertobeaone-polefilterbecausethe denominator containsonlyafirst-orderpolynomial.Byexamination, thedecyclermustcloselyfollowthe inputdata because there is nodifference term in thenumerator of the transferresponse. The EasyLanguage

codetocomputethedecycleris written almost directlyfrom the transfer responseequationinCodeListing4-1.

CodeListing4-1.EasyLanguage

CodetoComputeaDecycler

{Decycler©2013JohnF.Ehlers}Inputs:Cutoff(60);

Vars:alpha1(0),HP(0),Decycle(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan“cutoff”barsalpha1=(Cosine(360/Cutoff)+Sine(360/Cutoff)-1)/Cosine(360/Cutoff);Decycle = (alpha1 / 2)*(Close+Close[1])+(1-alpha1)*Decycle[1];Plot1(Decycle);

■ DecyclerApplicationThe decycler removes theshorter cycle energy bycancellation, leaving thedecycler output to bebasicallyaone-pole low-passfilter.Adecyclerhavinga30-barcutoffperiodisusedasanexample. The amplituderesponse is shown in Figure

4.2, and the lag is shown inFigure 4.3. The amplituderesponse confirms the 6-dB-per-octaveattenuationroll-offrateofasingle-polefilter.Forexample, the response at afrequency of 0.1 bars percycle is approximately −10dB, and the response oneoctave higher at a frequencyof 0.2 bars per cycle isapproximately−16dB.

Figure4.2Amplitude

ResponseofaDecyclerHavinga30-BarCutoffPeriod

Figure4.3LagofaDecyclerHavinga30-BarCutoffPeriod

The really importantcharacteristic of the decycleris its exceptionally low lag.The very longest cyclecomponents are delayed lessthan five bars, and at afrequency of 0.05 cycles perbar (a 20-bar cycle period),the lag ison theorderof1.5bars. Higher-frequencycomponentsaredelayedevenless. The end result is thehigher-frequencywigglesthat

make it through the filterattenuation are roughlycoincidentwiththewigglesinthe price itself. This featuremakes the decycler an ideal“instantaneous trend line”that accurately portrays thetrendofthedata.A similar smoothed filter

outputcanbeproducedusinga SuperSmoother filter.However,whentheresultsofthe decycler are compared to

those of the SuperSmoother,thedecyclerwillalwayshavelesslag.However,adecycleris only a one-pole filter andthereforehasinferiorfilteringcapabilities. Therefore, adecycler should not be usedas a smoothing filter toremove aliasing noise.Rather, its role should berelegated to producing aninstantaneous trend linewhere the selected cutoffperiodisrelativelylarge.The

large cutoff period enablesthe decycler to attenuate thealiasing noise because it ismany octaves away from theNyquistfrequency.It is certainly possible to

create a decycler bysubtracting a two-pole high-pass filter output from theinput data. In fact, such adecyclerhassome interestingproperties because the longcycle components have

substantial lag, while theshort cycle componentscontinuetohaveminimallag.However, interpretation oftheindicatorismoredifficultbecause the larger spread inlag across the spectrum.Therefore, only the single-pole decycler isrecommended for use asinstantaneoustrendlines.

■ DecyclerOscillatorA decycler oscillator iscreated by subtracting theoutput of a high-pass filterhavingashortercutoffperiodfrom the output of anotherhigh-pass filter having alonger cutoff period. Thisway, both elements have azero in their transfer

responses at zero frequency.Thus, the very, very longcycle components (and thestatic term) are removed.There is a finite differencebetweentheoutputofthetwofilters in the frequency rangebetween their cutoff periods,but shorter cycle componentsare still removed bycancellation. As a result, thetrend line is displayed as anoscillator.TheEasyLanguagecodetocomputethedecycler

oscillator is shown in CodeListing4-2.

CodeListing4-2.EasyLanguage

CodeforDecyclerOscillator

{DecyclerOscillator©2013JohnF.Ehlers}

Inputs:HPPeriod1(30),HPPeriod2(60);

Vars:alpha1(0),

alpha2(0),HP1(0),HP2(0),Decycle(0);

alpha1=(Cosine(.707*360/HPPeriod1)+Sine(.707*360/HPPeriod1)-1)/Cosine(.707*360/HPPeriod1);

alpha2=(Cosine(.707*360/HPPeriod2)+Sine(.707*360/HPPeriod2)-1)/Cosine(.707*360/HPPeriod2);

HP1 = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP1[1]-(1-alpha1)*(1-alpha1)*HP1[2];

HP2 = (1 - alpha2 / 2)*(1 - alpha2 / 2)*(Close-2*Close[1]+Close[2]) + 2*(1-alpha2)*HP2[1]-(1-alpha2)*(1-alpha2)*HP2[2];Decycle=HP2-HP1;

Plot1(Decycle);

Plot2(0);

Thedecycleroscillatorcan

beuseful fordetermining thetransition between uptrendsand downtrends by thecrossing of the zero line.Alternatively, the changes ofslope of the decycleroscillator are easier toidentify than the changes inslopeoftheoriginaldecycler.Optimum cutoff periods caneasily be found byexperimentation.The decycler and decycler

oscillatorexamplesareshownin Figure 4.4, where thecutoffperiodis30bars.Inthecase of the decycleroscillator, the longer cutoffperiodissettobe60bars.

Figure4.4BothDecyclersShowtheInstantaneousTrendwithMinimumLag

■ Key PointstoRemember

1. A decycler filterfunctions the same as alow-passfilter.2. A decycler filter iscreated by subtractingtheoutputofahigh-passfilter from the input,thereby removing thehigh-frequency

components bycancellation.3. A decycler filter hasverylowlag.4. A decycler oscillatoris createdby subtractingtheoutputofahigh-passfilter having a shortercutoff period from theoutput of another high-pass filter having alongercutoffperiod.5. A decycler oscillator

shows transitionsbetween uptrends anddowntrends at the zerocrossings.

CHAPTER5

Band-PassFilters

“A little of the datanarrowly passed,” said

Tombroadly.

Perhapstheleastappreciatedand most underutilized filterin technical analysis is theband-pass filter. The band-pass filter simultaneouslydiminishes the amplitude atlowfrequencies,qualifying itasadetrender,anddiminishesthe amplitude at highfrequencies,qualifyingitasadata smoother. It passes onlythose frequency components

frominputtooutputinwhichthe trader is interested. Thefilteringproducedbyaband-passfilterissuperiorbecausetherejectioninthestopbandsis related to its bandwidth.The degree of rejection ofundesired frequencycomponents is calledselectivity.The band-stop filter is the

dualoftheband-passfilter.Itrejects a band of frequency

componentsasanotchat theoutput and passes all otherfrequency componentsvirtually unattenuated. Sincethe bandwidth of the deeprejection in the notch isrelatively narrow and sincethespectrumofmarketcyclesis relatively broad due tosystemicnoise, theband-stopfilter has little application intrading.

■ Band-PassFilterA band-pass filter can begenerated by cascading alow-pass filter and a high-pass filter. However, thisapproach results in reallypoor filtering unless a largenumber of data samples areusedtoincreasethedegreeofthe polynomials. However,

digital signal processing(DSP) enables us tomap theentire0 to0.5cyclesperbarrange of a simple high-passfilter into the lower half-bandwidth of the pass bandandtoalsomaptheentire0to0.5cyclesperbarrangeofanexponential moving average(EMA) low-pass filter intothe upper half-bandwidth ofthe pass band. When we dothis,thetransferresponseofasimple pass-band filter is

described by Equation 5-1.Theband-pass filterwasalsodescribedinTable1.1.

(5-1)Whereγ=Cos(360*/Period)

and δ = bandwith as afractionofPeriodλ=Cos(360/Period)

Since the degree of the

polynomial in thedenominator of the transferresponseistwo,thisisatwo-pole band-pass filter. Inprogramming language,where the delay ofN bars isdenoted by “[N]”, theequationforatwo-poleband-passfilterbecomes

(5-2)

The frequency response ofa two-pole band-pass filter

centered at a 10-bar cycleperiod, and having a passbandthatis30percentofthecenter frequency at the halfpower points (−3 dB) isshown in Figure 5.1. Notethatthisfilterhaszerosinthetransferresponsebothatzerofrequency and at theNyquistfrequency. The relativelysharp roll-off in the rejectbands occurs because thecomputational frequencies ofthe filter havebeen scaled to

the half-bandwidth of thefilter.

Figure5.1FrequencyResponseofaTwo-PoleBand-PassFilterTunedtoa10-BarCyclePeriodandHaving30PercentBandwith

Another example of thefrequencyresponseofaband-pass filter isshowninFigure5.2,where the inputdata isachirped frequency sine wavewhose period is continuouslyswept from a 10-bar cycleperiod to a 60-bar cycleperiod. The theoreticalclosing prices swing +/− 5fromacentralvalueof60.Inthis case, theband-pass filteris tuned to a 20-bar cycle

period. In Figure 5.2, it isclear that signals whoseperiodsarebothlessthanandgreater than a 20-bar cycleareattenuated.

Figure5.2FrequencyResponseofaBand-PassFilterTunedtoa20-BarCyclePeriod

Readers can easily make

their own chirped frequencysine wave data source inExcel.Make the first columnbeacounterfrom1to1,000.Theequationforaphaserateof change per sample so thecycleperiodvariesfrom10to60 as a counter varies from100to1,000isapproximatelyPhase = 0.1 − 0.000075 *Counter.Therefore,incellB1type“=60+5*Sin(6.28318*(0.1−0.000075*a1))”andthen copy cell B1 into cells

B2throughB1000.ColumnBcanbetheopeningprice.Youcanmakethedatalookalittlemore realistic by makingcolumn C the high price asone greater than price incolumnB,makingcolumnDasthelowpricebeonelowerthan the price in column B,and letting column E be theclosing price and have thesame value of the price incolumn B. For thoseplatformssensitivetodatesin

price data, export a real datafilehavingaboutfiveyearsofdatainCSVformattocapturethe dates. Then, copycolumnsB,C,D, andE intotheclipboardandthenEdit... Paste Special . . . Valuesinto the Open, High, Low,and Close columns,overwriting the values inthose columns with thechirped sine wave values.Finally, save the data file soyou can import it into your

tradingplatform.One of the amazing

characteristicsof aband-passfilter is that if the centerperiodofthefilteristunedtoa static sine wave whoseperiod is the same as thecenter period of the filter,thenthereisabsolutelynolagin the output. Figure 5.3shows the output of a 20-barband-pass filter having a 30percent bandwidth compared

to the 20-bar input data.However, if the band-passfilter is tuned to its previoushalf-power period of 17 barsper cycle, the 20-bar datacycleislongerthanthetunedfrequency of the filter. As aresult,theoutputresponsehasa 65-degree lag, as shown inFigure 5.4. However, if theband-passfilteristunedtoitsother previous half-powerperiod of 23 bars per cycle,the 20-bar data cycle is

shorter than the tunedfrequency of the filter. As aresult,theoutputresponsehasa65-degreelead,asshowninFigure 5.5. The phase shiftacross the bandwidth of theband-pass filter isunavoidable. The phase shiftis doubled each time thenumberofpolesisdoubledinthe transfer response.Therefore, it is important touse the simplest filterpossible in trading to

minimize phase shiftdistortion.

Figure5.3ABand-PassFilterPreciselyTunedHasNoLag

Figure5.465-DegreeLag

ResultsfromtheBand-PassFilter'sBeingTuned15PercentTooLong

Figure5.565-DegreeLeadResultsfromtheBand-PassFilter'sBeingTuned15PercentTooShort

■ Band-PassFilterQThequality factor,orQ,ofaband-pass filter is one of itsimportant characteristics.There are several interrelateddefinitions ofQ. Perhaps thesimplest is based onselectivity of the filter as theratioofitscenterfrequencytoits percentage bandwidth at

the half-power points. In ourexample of a 30 percentbandwidthfilter,theQwouldbe 3.33. Stated another way,Q=20/(23−17)=20/6=3.33.However, Q can also be

defined in terms of energyquantities. From therelationships in resonantelectronic circuits thedefinitionbecomes

(5-3)

For example, a band-passfilter having a Q = 3.33wouldhavetheenergystoredin the filter but only abouthalftheenergydissipatedpercycle. Such a filter wouldhave its ringing quicklyquenched.Suchafilterwouldalsoquicklyadapttonewanddifferentcyclecomponentsatits input. However, a high-Qfilter would ring out a longtime in response to animpulse input. A bell is an

example of such a high-Qfilter. High-Q filters havehigh selectivity in thefrequency domain, but theirringing obscures the cyclicinformation in thedata input.Therefore,theQofband-passfilters used in trading shouldbeaslowaspossibleandstillobtainthedesiredfiltering.Figure5.6 shows the input

dataas threecompletecyclesof a sine wave having a 20-

bar cycle period. Thesubgraph shows the outputresponseofaband-passfiltertuned to a 20-bar period andhavingaQ=3.33.Theoutputswingexceedsthehalf-poweramplitude within one fullcycleperiodafterthedataareapplied to the input. Further,the output swing is dampedbelow the half-poweramplitudewithinahalfcycleafter the data input isremoved. However, a

relativelyhighQfilterwillbeslow to rise in response tonewdataandwillcontinuetoringafterthedataisremoved,asshowninFigure5.7.Band-pass filters can be

more responsive in the timedomain by having a lowerQ(wider percentagebandwidth). For example,Figure5.8showstheresponseofaband-passfilterhavingaQ = 2. This filter quickly

respondstonewdataandjustasquicklyhas theoutput falloff after the input data areremoved.

Figure5.6TransientResponseofaBand-PassFilterHavingaQ=3.33

Figure5.7Transient

ResponseofaBand-PassFilterHavingaQ=10

Figure5.8TransientResponseofaBand-PassFilterHavingaQ=2

Using a band-pass filter

havinga30percentpassbandis a relatively goodcompromise betweenselectivity and transientresponsiveness for mosttradingapplications.

■ AutomaticGain Control(AGC)A band-pass filter describedbyEquations5-1and5-2willaccurately display theamplitude of the cyclicswings in the input data.Consequently, the amplitudeof these swings will vary

among stock ticker signalsbecausethepricesrangefrompenny stocks to well over$100 per share. A pennystock just cannot have aslarge a swing as a blue-chipstock because the basis ismuch smaller. Traders areused to indicators beingnormalized to a range like 0to 100 so that the indicatorscales the same regardless ofthe price amplitude swings.ThepurposeoftheAGCisto

provideaconsistent indicatorappearance independentlyfrom the range of the priceswings. The process of theAGCdividesthecurrentpriceby the absolute value of therecent maximum swing sothatthenormalizedwaveformhas a maximum of 1 or aminimumof−1.Ihaveneverseen the AGC conceptapplied to technicalindicators, which is a shamebecause it can be applied

universally.The specific kind of AGC

is called a fast attack−slowdecayAGC.Thenormalizingfactor is allowed to decay asmall amount before beingcompared to the next sampleof the signal so thenormalizing is not done onthe largest swing throughouthistory. If thenext sampleofthe signal is larger than thenormalizing factor, then the

normalizing factor isimmediately assigned thevalue of the next sample.However, if the next sampleof the signal is smaller thanthe normalizing factor, thenthe normalizing factor isallowed to decay anothersmall amount and then iscomparedtothethirdsample,and so on. The amount ofdecay is exponential if thenormalizing factor is notreset.IfthedecayfactorisK

for thefirstsample, thenit isK * K = K2 for the nextsample, and isK *K *K =K3 for the third sample, andsoon.Figure5.9depictstheAGC

action, where the absolutevalueof thesinewavesignalis shown by the solid line.The normalizing factor,shown by the dashed line,decays exponentially witheachsample.Whenthesignal

exceeds the normalizingfactor, thenormalizing factoris assigned the new value ofthe signal so that when theinputsignal isdividedby thenormalizing factor the AGCalgorithm output normalizedthe input value to be 1. Theprocess is repeated for eachnewsampleofdata.

Figure5.9TheAGCNormalizingFactorDecaysExponentiallyandIsRapidly

ResettotheAbsoluteValueoftheCyclicSwing

Whenchangesaremadein

the time domain, there arealways implications made inthe frequency domain.Considering theoretical sinewave signals, the effectivegainoftheAGCisthedecayfactor to the exponent of thehalf period of the sine waveas

About the shortest cycleperiod useful for trading is a10-bar cycle, and cycle

periods longer than 48 barscan often be consideredtrends. Therefore, we areconcernedwiththegainslopeover the range froma10-barperiodtoa48-barperiod.Thegainratioover thisrangecanbeexpressedas

Sincewemustacceptsomegain slope across the cycleperiods of interest, an

arbitrary but reasonableselection is 1.5 dB. Thenumericalvalueoftheratioiscomputedas:

Knowing the desired gainratioacross theband,wecancomputethedecayfactoras:

Iwillusethisvalueforthe

AGCdecayfactorthroughoutthe remainder of the bookunless the AGC is used foramplitude compensation as afunctionofthecycleperiod.

■ SpectralDilationRemovalThe effect of SpectralDilation will be fullydescribed in Chapter 7. In anutshell, theamplitudeof themarket data spectrum is notflat, and traders must besensitive to that fact when

employingfilters.The band-pass filter

attenuationincreases6dBperoctaveof thehalfbandwidth,starting at the upper andlowercutofffrequencies.Thisrelatively low rate ofattenuation is insufficient toremovetheeffectsofSpectralDilation on the long cycleperiod side of the filter.Therefore, a high-pass filtershouldbeconnectedinseries

with the band-pass filter toensure that the SpectralDilation distortions are notallowedtoreachtheoutput.The critical frequency of

thehigh-passfilterissettobeone octave below the loweredgeoftheband-passfiltertominimize the interaction ofthetwofilterswithinthepassband.Sincetheloweredgeoftheband-passfilterisat0.5*Bandwidth relative to the

center frequencyof thefilter,the critical frequency of thehigh-pass filter is set to be0.25 * Bandwidth relative tothe center frequency of theband-pass filter. That makesthe high-pass criticalfrequencyafulloctaveofthehalf-bandwidth below thecenterfrequency.

■ Band-PassFilterThe EasyLanguage code toimplement the band-passfilterisgiveninCodeListing5-1.Afterdeclaringvariables,the band-pass filtercalculation is preceded by ahigh-pass filter whose cutofffrequency is one half-bandwidth octave below the

lower-frequency criticalfrequency of the band-passfilter to avoid interferencewith the action of the band-pass filter while stillremoving the effects ofSpectralDilation.

CodeListing5-1.EasyLanguage

CodefortheBand-PassIndicator

{BandPassFilter©2013JohnF.Ehlers}

Inputs:Period(20),Bandwidth(.3);

Vars:alpha2(0),HP(0),gamma1(0),alpha1(0),beta1(0),BP(0),Peak(0),Signal(0),Trigger(0);

alpha2=(Cosine(.25*Bandwidth*360/Period)+Sine(.25*Bandwidth*360/Period)-1)/Cosine(.25*Bandwidth*360/Period);HP = (1 + alpha2 / 2)*(Close-Close[1])+(1-alpha2)*HP[1];

beta1=Cosine(360/Period);

gamma1=1/Cosine(360*Bandwidth/Period);

alpha1=gamma1-SquareRoot(gamma1*gamma1-1);BP = .5*(1 - alpha1)*(HP - HP[2]) + beta1*(1+alpha1)*BP[1]-alpha1*BP[2];

IfCurrentbar=1orCurrentBar=2ThenBP=0;

Peak=.991*Peak[1];

IfAbsValue(BP)>PeakThenPeak=AbsValue(BP);IfPeak<>0ThenSignal=BP/Peak;

Plot1(Signal);Plot2(0);alpha2=(Cosine(1.5*Bandwidth*360/Period)+Sine(1.5*Bandwidth*360/Period)-1)/Cosine(1.5*Bandwidth*360/Period);Trigger = (1 + alpha2 / 2)*(Signal-Signal[1])+(1-alpha2)*Trigger[1];

Plot6(Trigger);

An example of theindicatorplottedoverroughlyayearondailydataofDollarGeneral (symbol DG) isshown in Figure 5.10. Notethe band-pass filter correctly

identifies the peaks andvalleys in thedata.However,the band-pass filter is deadwrong when the prices gointoatrendasinMarch2012andJuneandJuly2012.Asadiscretionary indicator, onecansaytheband-passfilterisworking well when the filteroutput lookssimilar toasinewave. However, watch outwhen the filter outputcontainserraticsignals.

Figure5.10Band-PassIndicatorforDGPinpointsPeaksandValleys

An interesting addition to

the band-pass filter is thesecond high-pass filterserially connected to theoutputoftheband-passfilter.In this case, the high-passfilter is tuned to the high-frequency side criticalfrequency of the band-passfilter. Doing this creates aleadingwaveformresemblingthe band-pass filter output,but still has a phase leadcharacteristic. If a leadingwaveform is attempted by

taking the rate change of theband-pass filter output, theresultingwaveformwouldbetoo erratic to be of use as atrading indicator. As seen inFigure 5.11, the leadingtrigger crosses the band-passfilteroutputattheexactpeaksand valleys of the waveformexcept when the data aretrending.

Figure5.11TheLeadingTriggerWaveformClearly

FlagsPeaksandValleysoftheBand-PassIndicatorinRealTime

■ Measuringthe CyclePeriodFigure 5.10 shows the cyclecontent of the data swingingpretty much as a variableamplitude sine wave withvariable periodicity. Theband-pass filter can be usedas a relatively simple

measurementofthedominantcycle. A cycle is completewhen the waveform crosseszero two times from the lastzerocrossing.Therefore,eachsuccessive zero crossing ofthe indicator marks a halfcycle period. We canestablish the dominant cycleperiod as twice the spacingbetween successive zerocrossings.When we measure the

dominant cycle period thisway, it is best to widen thepass band of the band-passfilter to avoid distorting themeasurement simply due tothe selectivity of the filter.Using an input bandwidth of0.7 produces an octave-widepassband.Forexample,ifthecenter period of the filter is20andtherelativebandwidthis 0.7, the bandwidth is 14.Thatmeans the pass band ofthefilterextendsfrom13-bar

periods to 27-bar periods.That is, roughly an octaveexists because the longestperiod is twice the shortestperiod of the pass band. It isimperative that a high-passfilter is tuned one octavebelow the half-bandwidthedgeoftheband-passfiltertoensure a nominal zero meanofthefilteredoutput.Withouta zero mean, the zerocrossings can have asubstantialerror.Thecode to

measure the dominant cycleusing the band-pass filter isgiven in Code Listing 5-2.Since the measurement canvary dramatically from zerocrossing tozerocrossing, thecode limits the changebetween measurements to benomore than 25 percent.Anexample of the dominantcycle period measurement isshowninFigure5.12.

Figure5.12Band-PassFilter

MeasurementoftheDominantCyclePeriod

While measuring the

changing dominant cycleperiod via zero crossings ofthe band-pass waveform iseasy, it is not necessarily themost accurate method. Moreaccurate techniquesexamining the entire spectralcontent of the data arepresented in later chapters ofthisbook.

CodeListing5-2.DominantCycle

MeasuredbyZeroCrossingsoftheBand-PassFilter

{ZeroCrossingsPeriodMeasurer©2013JohnF.Ehlers}

Inputs:Period(20),Bandwidth(.70);

Vars:alpha2(0),HP(0),gamma1(0),alpha1(0),beta1(0),BP(0),Peak(0),Real(0),counter(0),DC(0);

alpha2=(Cosine(.25*Bandwidth*360/Period)+Sine(.25*Bandwidth*360/Period)-1)/Cosine(.25*Bandwidth*360/Period);HP = (1 + alpha2 / 2)*(Close-Close[1])+(1-alpha2)*HP[1];

beta1=Cosine(360/Period);

gamma1=1/Cosine(360*Bandwidth/Period);

alpha1=gamma1-SquareRoot(gamma1*gamma1-1);BP = .5*(1 - alpha1)*(HP - HP[2]) + beta1*(1+alpha1)*BP[1]-alpha1*BP[2];IfCurrentbar=1orCurrentBar=2ThenBP=0;

Peak=.991*Peak;

IfAbsValue(BP)>PeakThenPeak=AbsValue(BP);IfPeak<>0ThenReal=BP/Peak;

DC=DC[1];

IfDC<6ThenDC=6;

counter=counter+1;

IfRealCrossesOver0orRealCrossesUnder0ThenBeginDC=2*counter;If2*counter>1.25*DC[1]ThenDC=1.25*DC[1];If2*counter<.8*DC[1]ThenDC=.8*DC[1];counter=0;End;

Plot1(DC);

■ Key PointstoRemember

1. A band-pass filter isboth a detrender and asmoother combined intoonefilter.2.Attenuationofout-of-band frequencycomponents is superiorto that of high-passfilters and low-pass

filters because therejection isscaled to thebandwidthofthefilter.3. A band-pass filtertuned precisely to aconsistent dominantcyclehasnolag.4. A band-pass filtertunedtoaperiodshorterthan the dominant cycleproduces an output thatleads the swing in theinputdata.

5. A band-pass filtertuned to a period longerthan the dominant cycleproduces an output thatlags the swing in theinputdata.6.Theselectivityandthetransient response of aband-pass filter areinterrelated. A narrowpass-bandfilterisslowertoreacttochangesintheinputdata.Awiderpass-

band filter adapts morequicklytochangesintheinputdata.7. A serially connectedsingle-pole high-passfilter tuned one octavebelowthelow-frequencycritical frequency of theband-pass filter isnecessary to eliminatethe effects of SpectralDilationandtocreateanoutput having a

nominallyzeromean.8. A leading functioncan be created byserially connecting asingle-pole high-passfilter tuned to the high-frequency critical periodoftheband-passfilter.9. The dominant cycleperiodofthedatacanbeestimated by countingthe number of barsbetween zero crossings

of the band-pass filteroutput.

CHAPTER6

MarketStructureandtheHurst

Coefficient“The market has adefinite structure,” saidTomrandomly.

It is well understood thatwhitenoisehasnocorrelationin time, and random walk(Brownianmotion) noise hasno correlation betweenincrements. Brownian walkscan be generated from a

function where the spectraldensity isproportional to1/Fα,whereFisfrequency,αisapower law, and α = 1generally signifies “longmemory,” as I will describeinChapter8.Thus,1/Fnoiseis synonymous with long-rangedependence.Integrationof noise increases theexponentαby2,whereastheinverse operation ofdifferentiationdecreasesitby2. Therefore, 1/F α noise

cannot be obtained by thesimple procedure ofintegration or ofdifferentiation of suchconvenient signals. Thewidespread occurrence ofsignals exhibiting suchbehavior suggests that ageneric mathematicalexplanation might exist. Theubiquityof1/Fαnoiseisoneof the oldest puzzles ofcontemporary physics andscienceingeneral.

Figure 6.1 displays sometime series and, in the samecolors,theirassociatedpowerspectra.Suchtimeseriesarisein many natural systems.Power spectra are plotted inlog-log coordinates, as iscustomary,because

Figure6.1TimeSeriesandSpectralDensitiesforSeveralKindsofNoise

In other words, thelogarithmictransformrendersthe 1/F α power spectrum astraightlinewhoseslope,−α,can be easily estimated.Clearly, for such naturalsystems, observed byhumans, neither arbitrarilysmall nor arbitrarily largefrequencies can be recorded.These models are simply

relevant to physicallyplausiblesituations.Noise in economic data is

usually studied as long-rangedependence or longmemory.Ithasbeen shown repeatedlythat the autocorrelationfunctions of economic timeseries,suchasseriesofstockprices over days, weeks, ormonths, or thegross nationalproduct of various countriesover years, do not decay

exponentially as they wouldif the process generating theseries were a simpleautoregressive process.Instead, the autocorrelationfunctions of many economictime series reach a nonzeroasymptote and remain therefor the entire series, albeitoften at a low value,indicating that economicevents some distance in thepast continue to have aninfluence on current prices.

You only need to do anInternet search on “stockmarket spectral density” tofind a number of academicpapers, both measured andtheoretical, regarding thefrequency content of themarket.The Hurst coefficient is

one way to attempt to get ahandle on the slope of thepowerdensityofmarketdata.The Hurst coefficient varies

between 0 and 1, and isrelated to the α powercoefficient as H = 1 − α/2.TheHurstcoefficientismoreestimated than computed. Ifind the estimate using thefractal dimension is themostpractical for shorter-termmarket data. The Hurstcoefficient is related to thefractaldimensionasH= 2 −D. I would like to make itperfectly clear that the Hurstcoefficient or the fractal

dimension has no directpractical application totradingnotonlybecause it isan estimate, but alsobecauseit has no predictive value.These computations onlyreflectthegeneralstructureofthe market, and the answeryouget isdependentonyourassumptions. For example,theHurst coefficient changesdramatically with the lengthof data used in making theestimate.

■ FractalDimensionTo determine the fractaldimension of a generalizedpattern, we cover the patternwith a number “N” of smallobjects of several varioussizes“S.”Therelationshipofthenumberofobjects in twosetsofsizesis:

As an example, we canstart with a pattern that is aline segment 10meters long.We chose the two smalldimensions as S1 = 1 meterand S2 = 0.1 meter. Placingboxes along the line we canfit ten 1-meter boxes on thesegment, therefore N1 = 10.

Similarly, we can fit onehundred 0.1-meter boxes onthe same 10-meter linesegment. Therefore, N2 =100.Thefractaldimensionofthelinethencomputesto:

As a second example, wewill use the pattern as asquare that is10metersonasideinsteadofalinesegment.Retaining the same sizes of

our small boxes as 1 meterand 0.1 meter on a side,respectively,wegetN1=100and N2 = 10,000. When thesquare is our pattern, thefractal dimension thereforecomputestobe:

Aperfectsquarerepresentsan idealized geometry notfound in nature. Naturalfractals, such as that of a

seashore, lack the trueregularity of an algorithmicstructure but are self-similarinastatisticalsense.Thus,inorder todetermine thefractaldimension of natural shapes,we must average themeasured fractal dimensionmadeoverdifferentscales.We could measure the

fractaldimensionofpricesbycovering the curve with aseriesof smallboxes.This is

aburdensome task,but ifwetake into account that theprice samples are uniformlyspaced,we note that the boxcount is approximately theaverage slope of the curve.Therefore, we can estimatethe box count as the highestpriceduringanintervalminusthe lowest price during thatinterval,dividedbythelengthof the interval itself. Theequation for the number ofboxesisthen:

We compute the fractaldimension by computing Nover two equal intervals toget the averaging over eachinterval.Interval1coverstheperiod from0 toT bars ago.Interval 2 covers the periodfrom T to 2T bars ago.Therefore, N1 = (HighestPrice − Lowest Price) overthe interval from 0 to T,

dividedbyT.Similarly,N2=(Highest Price − LowestPrice) over the interval fromT to 2T, divided by T. WealsodefineanN3= (HighestPrice − Lowest Price) overthe entire interval from 0 to2T, divided by 2T. Sinceweare looking backwards intime, the slope computationofthefractaldimensionis:

The fractal dimensionvariesover the range fromD=1toD=2.SincetheHurstcoefficient is related to thefractaldimensionasH=2−D,sothatitvariesbetween0and 1. A value of 0.5indicates a true randomwalk(aBrowniantimeseries).Inarandom walk there is nocorrelation between anyelementandafutureelement.Acoefficientbetween0.5and1 indicates “persistent

behavior,”thatis,along-termdependence. A Hurstcoefficientbetween0and0.5exists with time series with“antipersistentbehavior”—inother words, “acountertrender”(atermthatIdetest).

■ Computingthe HurstCoefficientThe EasyLanguage code tocomputetheHurstcoefficientis given inCodeListing 6-1.The only user input is thelengthofdatatobeused.Thenumber can be arbitrarilylarge if you have sufficient

data.Theresultsarecriticallydependent on the input datalength selected. Afterdeclaring variables, thecoefficients of a 20-barSuperSmoother filter arecomputed. The computationsof N1, N2, and N3 are asdescribed in the previoussection. The fractaldimension is then convertedto the Hurst coefficient,which is subsequentlysmoothed in the

SuperSmootherfilter.

CodeListing6-1.EasyLanguage

CodetoComputetheHurstCoefficient

{HurstCoefficient©2013JohnF.Ehlers}

Inputs:Length(30);{Lengthmustbeanevennumber}

Vars:a1(0),

b1(0),c1(0),c2(0),c3(0),count(0),N1(0),N2(0),N3(0),HH(0),LL(0),Dimen(0),Hurst(0),SmoothHurst(0);

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/20);

b1=2*a1*Cosine(1.414*180/20);

c2=b1;

c3=-a1*a1;c1=1-c2-c3;

N3=(Highest(Close,Length)-Lowest(Close,Length))/Length;

HH=Close;

LL=Close;

Forcount=0toLength/2-1beginIfClose[count]>HHthenHH=Close[count];

IfClose[count]<LLthenLL=Close[count];

End;

N1=(HH-LL)/(Length/2);

HH=Close[Length/2];

LL=Close[Length/2];

Forcount=Length/2toLength-1beginIfClose[count]>HHthenHH=Close[count];IfClose[count]<LLthenLL=Close[count];

End;

N2=(HH-LL)/(Length/2);

IfN1>0andN2>0andN3>0thenDimen=.5*((Log(N1+N2)-Log(N3))/Log(2)+Dimen[1]);

Hurst=2-Dimen;SmoothHurst = c1*(Hurst+Hurst[1])/2+c2*SmoothHurst[1]+c3*SmoothHurst[2];

Plot1(SmoothHurst);

■ The HurstCoefficient inActionThe fractal dimensionindicator is applied toapproximately one year ofdata of Dollar General(symbol DG) in Figure 6.2using an input length of 30.With this setting, the Hurst

coefficient swings betweenshowing persistence andantipersistence. However,when the input length isincreasedto200,asshowninFigure 6.3, the uptrend isreflected as the Hurstcoefficient rising tobeabove0.5.

Figure6.2UsinganInputLengthof30,theHurstCoefficientSwingsbetweenShowingPersistenceand

Antipersistence

Figure6.3TheLonger-TermTrendisIndicatedbyaHurstCoefficientGreaterthan0.5WhenanInputLengthof200IsUsed

It is perhaps instructive to

describe action of the Hurstcoefficient from anothertheoretical perspective. InFigure 6.4, the indicator isapplied to a theoretical sinewave whose period iscontinuously increasing fromleft to right from a period of10 bars to 40 bars. A periodof20barsisapproximatelyinthe horizontal center of thechart. In this case, I havegiven the Hurst coefficientindicator an input of 20.The

periodicity in the persistentcase is due to the peaks andvalleysofthetimewaveformtending to “fill theboxes”ofthe fractal dimension. Fromthis perspective, the Hurstcoefficient can be used todefine trend modes (> 0.5)and cycle modes (< 0.5) inthe market relative to theselectedinputlength.

Figure6.4CyclePeriodsShorterthantheHurst

CoefficientInputParameterDisplayasAntipersistent;LongerCyclePeriodsDisplayasPersistent

■ Drunkard'sWalkHypothesis forMarketStructureThe efficient markets modelstatement that the price fullyreflects available informationhas been assumed to imply

that successive price changesareindependentofeachother.In addition, it has usuallybeenassumed thatsuccessivechanges are identicallydistributed. Together, thesetwohypothesesconstitute therandom walk model. Thismodel says that theconditional and marginalprobabilitydistributionsofanindependent random variableare identical. In addition, itsays that the probability

density function must be thesameforalltime.Thismodelisclearlyflawed.Ifthemeanreturn is constant over timethenthereturnisindependentof any information availableatagiventime.I assume that there are an

adequate number of tradersinvolved in making themarket that a statisticalanalysis involving a randomwalk is appropriate. There

mustbeseveralconstraintstosucharandomwalk.Thefirstconstraint is that the pricesare constrained to onedimension—theycanonlygoup or down. The secondconstraint is that time mustprogressmonotonically.I propose a philosophical

basis of market action fromextensive work usingconstrained random walks inthe physical sciences.1 The

expression of such a randomwalk is that of a drunkardmovingonaone-dimensionalarray of regularly spacedpoints. At regular intervalsthedrunkard flips a coin andmakesonesteptotherightorleft, depending on theoutcome of the coin toss. Attheendofn steps, he canbeatanyoneof2n+1sites,andthe probability that he is atanysitecanbecalculated.Letthe distance between the

points on the lattice be Δ L,and let the time betweensuccessivestepsbeΔT.IfΔLandΔTareallowed toshrinktozeroinsuchawaythat(ΔL)2 / ΔT remains constant tothediffusionconstantD,thenthe equation governing thedistribution of thedisplacement of the randomwalkerfromhisstartingpointis

This rather famous partialdifferential equation is calledthe diffusion equation. Thefunction P(x,t) can beinterpreted in two ways. Itcanbetakentoexpresseitherthe probability density or theconcentration of diffusingmatteratpositionxat time t.Following the latterinterpretation, it can, for

example, describe the wayheat flows up the stem of asilverspoonwhenplacedinahotcupofcoffee.To better understand the

theory of diffusion, imaginethe way a smoke plumeleaves a smokestack. Thinkabouthowthesmokerisesascomparable to how a trendcarries itself through themarket. A gentle breezedeterminestheangletowhich

the smoke, or trend, is bent.The widening of the smokeplume represents theprobability density of thesmokeparticlesasa functionof distance from thesmokestack.Thiswideningisanalogous to the decreasedaccuracy of the prediction offuture trend prices furtherintothefuture.The formulation of the

drunkard's walk has no

propertythatcanberegardedas the analog of momentum.Amorerealisticmodelofthemotion of a physical objectneeds to take into accountsome form of memory—weneed to know where theobject came from and thelikelihood that it willcontinuetomoveinthesamedirection. The simplestmodification of the randomwalkistoallowthecointossto determine the persistence

of motion. In other words,with probability p thedrunkardmakeshisnextstepin the same direction as thelastone,andwithprobability1−phemakesamoveintheopposite direction. Theordinary drunkard's walkoccurswhenp=1/2becauseeithermove is equally likely.The interesting featureof themodified drunkard's walk isthat as the distance betweenthe point and the time

between steps decreases, oneno longer obtains thediffusion equation, but ratherthefollowingequation:

in one dimension, where Tand c2 are constants. This isanother famous partialdifferential equation calledthe telegrapher's equation.This equation expresses theidea that diffusion occurs in

restricted regions, such thatx2<c2t2.Thatis,thepositionmustbelessthanthevelocityof propagation (c)multipliedby time (t). More important,the telegrapher's equationdescribes the harmonicmotion of P(x, t) just assurely as it describes theelectric wave traveling downapairofwires.Harmonic motion is

ubiquitous. It is the natural

response to a disturbance onany scale ranging from theatomic to the galactic. Youcan demonstrate the effect toyourself by holding a rulerover the edge of a table,bending the ruler down, andthen releasing it. Theresulting vibration isharmonic motion.Alternatively,youcanstretcha rubber band between yourfingers, pull the band to oneside, and then release it.The

oscillations of the rubberbandalsoconstituteharmonicmotion.Sincethereareplentyof opportunities for marketdisturbances, it is only asmall stretch to extend thesolution to the drunkard'swalk problem from physicalphenomena and use it todescribe the action of themarket.The drunkard's walk

solution can describe two

market conditions. The firstcondition, where theprobability is evenly dividedbetweenstepping to the rightortheleft,resultsinthetrendmode, described by thediffusion equation. Thesecond condition, where theprobability of motiondirectionisskewed,resultsinthecyclemode, describedbythe telegrapher's equation.The difference between thetwo conditions can be as

simpleasthequestionthatthemajority of traders areconstantlyaskingthemselves.Ifthequestionis“Iwonderifthe market will go up ordown?,” then the probabilityofmarketmovementisabout50−50, establishing theconditions for a trend mode.However, if the question isposed as “Will the trendcontinue?,” then theconditions are such that thetelegrapher'sequationapplies.

Asaresult,thecyclemodeofthemarketcanbeestablished.The telegrapher's equation

solution also describes themeandering of a river.Viewed as an aerialphotograph,everyriverintheworld meanders. Thismeandering is not due to alack of homogeneity in thesoil, but to the conservationof energy. (You canappreciate that soil

homogeneity is not a factorbecause other streams, suchas ocean currents, alsomeander in a nearlyhomogeneous medium.)Oceancurrentsarenotnearlyas visible as rivers and aretherefore not as familiar tomostofus.Everymeanderinariverisindependentofothermeanders, and thus all arecompletely random. If wewere to look at all themeanders as an ensemble,

overlaying one on top of theotherlikeamultiple-exposurephotograph, the meanderrandomness would alsobecome apparent. Thecomposite envelope of theriver paths would be aboutthe same as the cross-sectionof the smoke plume.However,ifweareinagivenmeander, we are virtuallycertainof thegeneralpathofthe river for a short distancedownstream.Theresultisthat

the river can be described ashaving a short-termcoherencybutisrandomoverthelongerspan.River meanders are the

kindofcycleswehaveinthemarket.We canmeasure andusetheseshort-termcyclestoour advantage if we realizethey can come and go in thelongerterm.Wecanextendouranalogy

to understand when short-

term cycles occur. Riversmeander in an attempt tomaintain a constant slope ontheirway to theocean. If theslope is too severe, themeanders have the sameeffect as a skierwhoweavesback and forth across theslopetoslowhisdescent.Theflow of a river physicallyadjusts itself for the purposeofenergyconservation.Ifthewaterspeedsup,thewidthofthe river decreases to yield a

constant flow volume. Thefaster flow contains morekinetic energy, and the riverattempts to slow it down bychanging direction. At thesametime,theriverdirectioncannot change abruptlybecauseof themomentumofthe flow of water.Meandering results. Thus,meanders cause the river totake the path of leastresistance in the sense ofenergy conservation. We

should think of markets thesame way. Time mustprogressassurelyastherivermust flow to the ocean.Overbought and oversoldconditions result from anattempt to conserve the“energy” of themarket. This“energy”arises from the fearandgreedoftraders.Again, itmay be useful to

test the principle ofconservation of energy for

yourself.Teara stripabout1inchwidealong thesideofastandardsheetofpaperabout11 inches long. Grasp eachend of this strip between thethumbandforefingerofeachhand.Nowmove your handstoward each other. Yourcompressionisputtingenergyinto this strip, and its naturalresponsecantakeoneoffourmodes. These modes aredetermined by the boundaryconditions thatyou forced. If

both hands are pointing up,the response is a singleupward arc, approximatingone alternation of a sinewave. If both hands arepointing down, the responseis a downward arc. If onehand is pointing up and theotherpointingdown,thestripresponse to the energy inputis approximately a full sinewave.Thefourlowestmodesare the natural responsesfollowing the principle of

conservation of energy. Youcan introduce additionalbendsinthestrip,butaminorjiggling will cause the paperto snap to one of the fourlowestmodes,with the exactmode depending on theboundary conditions that youimpose. The two full sinewave modes areapproximately the secondharmonic of the two single-alternationmodes.

The market only has asingledominantcyclemostofthe time, as shown by thespectrum measurements inChapters8through10.Whenmultiple cycles aresimultaneously present, theyare generally harmonicallyrelated.Thisisnottosaythatnonharmonic simultaneouscycles cannot exist, just thatthey are rare enough to bediscounted in simplifiedmodelsofmarketaction.The

general observation of asingle dominant cycle tendsto support thenotion that thenatural response to adisturbance is monotonicharmonicmotion.It is true that if you are a

hammer,therestoftheworldlooks like a nail. We musttakecaretorecognizethatallmarket action is not strictlydescribedbycyclesaloneandthatcycletoolsarenotalways

appropriate.Amorecompletemodel of the market can beachieved by recognizing thefactthattherearetimeswhenthe solution to thetelegrapher's equationprevails and times when thesolution to the diffusionequation applies. We cantherefore divide the marketaction into a cyclemodeandatrendmode.Byhavingonlytwo modes in our marketmodel we can switch our

trading strategy back andforthbetweenthem,usingthemore appropriate toolaccording to our situation.Since our digital signalprocessing tools analyzecycles,wecanestablishthatatrend mode is moreappropriateatanygiventimedue to the failure of a cyclemode.

■ Key PointstoRemember

1. The markets can becharacterized as noisehaving varying degreesoflong-termmemory.2. The Hurst coefficientis related to the powerspectral density of thenoise and to the fractaldimension. The use of

these terms isinterchangeable.3. The Hurst coefficientis more estimated thancomputed.4. The Hurst coefficienthas no predictive valueand therefore has nodirect usefulness intrading.5.TheappearanceoftheHurst coefficient iscritically dependent on

thelengthofdatausedtocomputeit.6. A Hurst coefficientless than 0.5 indicates acyclemode.7. A Hurst coefficientgreaterthan0.5indicatesatrendmode.8. A drunkard's walkmodel can also be usedto form a philosophicalbasisofmarketactivity.

Note1.G.H.WeissandR.J.Rubin,“RandomWalks:Theory and SelectedApplications,”Advancesin Chemical Physics 52(1982):363−505.

CHAPTER7

SpectralDilation

“The cycle swings aregetting bigger,” said

Tomexpansively.

Linear system theoryteaches that anyproblemcanbedividedintothree“boxes”:an input box, a transfer box,and an output box. Thetransfer box can be ascomplex as desired and canbe composed of many lesscomplex boxes wiredtogether. For this reason, Iassumed that if I applied afilter to inputdata theoutput

wouldcontainonlyfrequencycomponents defined by thefilter. I have certainly beenawareof thefractalnatureofmarketdataandthelogspiralused by Fibonaccians, but itjust did not occur tome thatthe concept would extend tothefrequencyrangeofswing-tradingfilters.I coined the term Spectral

Dilation to encapsulate thesignificance of the effect.

This effect has a profoundinfluence on all technicalindicators and is commonlyneglected because theseindicators are not viewedfrom the perspective of theirfrequency content. Theinterpretation of commonindicatorssuchastherelativestrength index (RSI) orStochastic are describedwiththe frequency distortions ofSpectralDilationimbeddedintheoutput.

■ FrequencyContent ofIndicatorOutputsA“roofingfilter”canbeusedtolimitthefrequencycontentofaninputbeforeproceedingtoconstructanindicator.Theroofing filter is composed of

a high-pass filter that passesonly frequency componentswhose periods are shorterthan 48 bars, for example. Italso is composed of aSuperSmoother filter thatpasses frequencycomponentswhoseperiodsarelongerthan10 bars. Thus, the roofingfilter is a wide bandwidthband-pass filter that passesonly frequency componentswhose periods fall between10barsand48bars.This,by

itself, is a simple indicator.The EasyLanguage code forthisindicatorisgiveninCodeListing7-1, and the indicatoroutput is displayed in Figure7.1.

Figure7.1RoofingFilterDisplayDoesNotHaveaZeroMean

While the roofing filter

indicator wiggles inproportion to the wiggles inthe price data as you wouldexpect, it is noticeable thattheindicatorisabovezeroforextended periods where themarket is in an uptrend. Inotherwords, the filter outputdoesnothaveazeromeanasyou would expect if theoutput consisted only ofcyclic components whoseperiods fall between 10 barsand48bars.

CodeListing7-1.HP-LPRoofing

FilterEasyLanguage

Code{HP-LPRoofingFilter©2013JohnF.Ehlers}

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),

Filt(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(360/48)+Sine(360/48)-1)/Cosine(360/48);

HP = (1 - alpha1 / 2)*(Close-Close[1])+(1-alpha1)*HP[1];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

Plot1(Filt);

Plot2(0);

The explanation for the

lackofazeromeanisthattheinput data has a spectralpowerdensityproportionalto1/Fα,andthereforeincreasesamplitudeat the rateof6dBperoctave,moreor less.Thehigh-passfilterhasarejectionslope that increasesattenuation directlyproportional to F 0.5, or rollsoff at the rate of 6 dB peroctave. This response issimilar to the frequency

responseshowninFigure1.2.Asaresult,theonlyeffectofthe high-pass filter is tobasically to equalize thefractal amplitude growth ofalllongerperiodsinthedata.Therefore, the filter outputstill contains all of thesefrequency components. Theonly way we can reduce theeffect of these lower-frequency components is tointroduce another high-passfilter, adding an additional 6

dBperoctaveroll-off,sothattheneteffectofthefilterisasoriginally surmised. I haveadded Filt2 to the code inCode Listing 7-2, and theeffect is shown as the solidline in Figure 7.2. Theoriginal roofing filterresponse is shown with thecrosshatched markers. Notethat in addition toestablishinganominally zeromean, removalof the longer-cycle components caused by

spectral dilation removes lagfromtheroofingfilteroutput.

Figure7.2ZeroMeanRoofingFilterOutputContainsOnlyDesiredFrequencyComponents

Since the roofing filter

consists of the serialconnection of a one-polehigh-pass filter, aSuperSmoother, and anotherone-polehigh-passfilter, it isclearthebetterformulationoftheroofingfilterwouldbetheserial connection of a two-pole high-pass filter and theSuperSmoother.

CodeListing7-2.EasyLanguage

CodeforaZeroMeanRoofing

Filter{ZeroMeanRoofingFilter©2013JohnF.Ehlers

}

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Filt2(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(360/48)+Sine(360/48)-1)/Cosine(360/48);

HP = (1 - alpha1 / 2)*(Close-Close[1])+(1-alpha1)*HP[1];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

Filt2= (1 - alpha1 / 2)*(Filt-Filt[1])+(1-alpha1)*Filt2[1];

Plot1(Filt);

Plot2(Filt2);

Plot6(0);

■ RoofingFilter as anIndicatorThe roofing filter does anexcellent job of using onlythe frequency componentsbetween its upper and lowercriticalperiods.Allthatneedsto be done to create anindicator from the roofing

filter is to add moregenerality by allowing thehigh-pass and low-passcriticalperiodsbesuppliedasindicator inputs. In the codeof Code Listing 7-3, the twohigh-pass filters have beenimplementedasasingletwo-polehigh-passfilter.The ideal time to buy is

whenthecycleisatatrough,and the ideal time to exit along position or to sell short

iswhenthecycleisatapeak.These conditions are flaggedby the filter crossing itselfdelayed by twobars, and areincluded as part of theindicator.

CodeListing7-3.EasyLanguage

CodeforaRoofingFilterIndicator

{

RoofingFilterIndicator©2013JohnF.Ehlers}

Inputs:LPPeriod(40),HPPeriod(80);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Filt2(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/HPPeriod)+Sine(.707*360/HPPeriod)-1)/Cosine(.707*360/HPPeriod);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPPeriod);

b1=2*a1*Cosine(1.414*180/LPPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

Plot1(Filt);

Plot2(Filt[2]);

Plot6(0);

Anexampleof the roofingfilter indicator is shown inFigure 7.3, where the LPperiod rather arbitrarily hasbeen set at 40 and the HP

period also has beenarbitrarily set at 80 bars. Acasual glance shows theroofing filter to be anexcellent indicator whosesignals are not marred bycomputational lag. The twoinputs can be set to get thebest response for a particularstock, exchange-traded fund,or future symbol. Thisindicator gives excellentguidance for discretionarytrading, but additional rules

wouldberequiredtocreateagood mechanical tradingsystemfromit.If a normalized amplitude

indicator is desired, theautomaticgaincontrol(AGC)code fragment described inChapter5 canbe addedaftertheFiltcalculation.

Figure7.3TheRoofingFilterIsanOutstandingTechnicalIndicator

■ Impact ofSpectralDilation onConventionalIndicatorsConventional indicators arenot immune to the effects ofspectral dilation. Forexample, a Stochastic

indicator remains near itsupperboundwhenthemarketisinanuptrendeventhougharelatively short lookbackperiod is used. This, ofcourse,isduetothepresenceofthelargerandlongercycliccomponents in the data. Idescribe the effect on theStochastic indicator withreference to Figure 7.4. AStochastic indicator basicallydisplays the current closingprice relative to the highest

and lowest values in thelookback range. If you stopandthinkaboutit,thecurrentprice will tend to be in thehigh end of the range whenthemarketisinanuptrend.Ina broad sense, the classicStochastic indicator acts as aone-pole filter because thereisjustasingledifferencetermin the numerator of itscalculation. The firstsubgraphinFigure7.4showsmycalculationofaStochastic

overa20-barlookbackrange.This calculation is the sameasthestandard,exceptIuseaSuperSmoother for thesmoothing instead ofmovingaverages. The bottomsubgraph shows an identicalcalculation except that theStochastic calculations areprecededbytheroofingfilterthat constrains the cycliccomponentstofallwithintherange from 10-bar to 48-barperiods. Clearly, the

Stochastic indicator has beendramatically altered by theinsertionoftheroofingfilter.

Figure7.4SpectralDilationRemovalHasaDramaticImpactontheStochasticIndicator

The interpretation and use

oftheStochasticindicator,orany other indicator, precededby the roofing filter will notbe addressed.Thosekindsofapplicationscanfillbooksallby themselves. Therefore, Iwill leave it to the reader toexplore how standardindicators can be improvedsimply by inserting theroofing filter before theindicator computations arebegun.

Asapointofreference,theEasyLanguage code used tocreatethebottomsubgraphinFigure 7.4 is given in CodeListing7-4.

CodeListing7-4.EasyLanguage

CodetoComputetheModifiedStochasticIndicator

{ModifiedStochasticIndicator©2013JohnF.Ehlers}

Inputs:Length(20);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),HighestC(0),

LowestC(0),count(0),Stoc(0),MyStochastic(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*

(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

HighestC=Filt;

LowestC=Filt;

Forcount=0toLength-1BeginIfFilt[count]>HighestCthenHighestC=Filt[count];IfFilt[count]<LowestCthenLowestC=Filt[count];

End;

Stoc=(Filt-LowestC)/(HighestC-LowestC);

MyStochastic = c1*(Stoc+Stoc[1])/2+c2*MyStochastic[1]+c3*MyStochastic[2];

Plot1(MyStochastic);

WellesWilder defined theRSIas

RSisshorthandforrelativestrength. That is, CU is thesum of the difference inclosing prices over theobservationperiodwherethatdifference is positive.Similarly, CD is the sum ofthe difference in closing

prices over the observationperiod where that differenceis negative, but the sum isexpressed as a positivenumber. When we substituteCU/CDforRSandsimplifytheRSIequation,weget

In other words, the RSI isthe percentage of the sum ofthedeltaclosesuptothesumofallthedeltaclosesovertheobservation period. The only

variable here is theobservation period. To havemaximum effectiveness theobservation period should behalf of the measured period.If the observation period ishalf thedominantcycle, thenfor a pure sine wave, thecloses up is exactly equal tothetotalclosesduringpartofthe cycle from the valley tothepeak.Inthiscase,theRSIwould have a value of 100.During another part of the

cycle—the next half cycle—therewould be no closes up.During this half cycle, theRSI would have a value ofzero.So,inprinciple,halfthemeasuredcycle is thecorrectchoice for the RSIobservation period. In Figure7.5 I have shown a standardRSI (except I used aSuperSmoother filter) in thefirst subgraph with amodifiedRSIthatisprecededby a roofing filter in the

second subgraph. Themodified RSI is far moreresponsive and has less lagbecause the low frequencycomponents have beenremoved by the high-passpartof the roofing filter.TheRSI is functionally the sameas a single-pole high-passfilterbecause theCUterminthe numerator is just asummation of one-bardifferences.

Figure7.5TheModifiedRSIIsMoreResponsivethantheStandardRSI

TheEasyLanguagecodeto

compute the modified RSIindicator is given in CodeListing7-5.Thecomputationstarts with the roofing filter,followed by the routinecalculations for an RSI. Ihave included an optionalSuper-Smoother to producethe MyRSI variable. Theindicatorlagcanbereducedalittle by not using asmoothing filter, but thedecreased lag comes at theexpense of the indicator's

beingmoreirregular.

CodeListing7-5.EasyLanguage

CodetoComputetheModifiedRSI

Indicator{ModifiedRSIIndicator©2013JohnF.Ehlers}

Inputs:Length(10);

Vars:alpha1(0),

HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),ClosesUp(0),ClosesDn(0),count(0),Denom(0),MyRSI(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

ClosesUp=0;

ClosesDn=0;

Forcount=0toLength-1BeginIfFilt[count]>Filt[count+1]ThenClosesUp=ClosesUp+(Filt[count]-Filt[count+1]);IfFilt[count]<Filt[count+1]ThenClosesDn=ClosesDn+(Filt[count+1]-Filt[count]);

End;

Denom=ClosesUp+ClosesDn;

IfDenom<>0andDenom[1]<>0ThenMyRSI=c1*(ClosesUp/Denom+ClosesUp[1]/Denom[1])/2+c2*MyRSI[1]+c3*MyRSI[2];

Plot1(MyRSI);

Plot2(.7);

Plot6(.3);

■ Key PointstoRemember

1. Spectral Dilation isinherentlypartofmarketdata, arising from thefractalnatureofthedata.That is, longer cycleperiods necessarily havecorrespondingly largeramplitudeswings.2. Spectral Dilation is

virtually ignored in allcommon technicalindicators.3. Ignoring SpectralDilation has led toerroneous, and oftenwacky, interpretation ofmarketactivity.4. Spectral Dilation canbecorrectedbyplacingaroofing filter before thecomputation ofindicators.

5. A roofing filterestablishes a near zeromean of detrended databy allowing only cycliccomponents within thefilter pass band to bepassed on for furtheranalysis.6. The roofing filter,itself, can be asuccessful technicalindicator.7. Insertion of the

roofing filter willdramatically change theuseand interpretationofstandard indicators,basically making themallnewindicators.

CHAPTER8

Autocorrelation“The correlations oftenrepeat,” said Tomperiodically.

Thisisanimportantchapterfor algorithmic tradersbecause the autocorrelationprocess yields a number ofunique and useful indicators.Autocorrelation is a simpleconcept.Themostrecentdatastream of a ticker symbol iscorrelatedwith a data streamof that same ticker symboldelayedbyaparametercalledlag. This computation isimportant to traders because

unconventional processingleads to ascertaining theperiodicity of the price datawithout the distortion ofSpectral Dilation.Autocorrelation also leads tosome particularly insightfulturningpointsignals.

■BackgroundPeterSwerling isbestknownfor the class of statistically“fluctuating target”scatteringmodels he developed in theearly 1950s to characterizethe performance of pulsedradar systems, referred to asSwerling targets. He notedthat the return radar echoeswere noisy because ofsemirandom reflections from

different parts of the aircraftbecause of the changingaspect of aircraft relative tothe radar transmitter. Therewere different kinds offluctuations due to targetshape and size, radarwavelength,andsoon.Somefluctuations would occurpulse to pulse, and otherswouldvarymoreslowly,suchas from scan to scan of theantenna. In fact, his earlywork led to the design of

modern stealthy aircraft. Thenoisy radar echoes weresuccessfully modeled as aconstant plus a randomnumber with memory. Interms recognized by traders,the echoes were modeled asan exponential movingaverage (EMA) passesnumbers. The time constantoftheEMAwasdifferentforthevariousmodels,andmorecomplex models includedseveral EMAs. The Swerling

model is entirely consistentwiththe1/Fαspectralmodelthat uses random inputswithlong-termmemory.Since there has been a

mountain of opinionregarding the randomness ofthemarket,itisreasonabletoapply a Swerling-like modeltoward the generation ofsynthetic data. Code Listing8-1 shows theEasyLanguagecode for generating such

syntheticdata,andFigure8.1shows a presentation in thesubgraph below real marketdata.Theresult issubjective,butappearstobeareasonableapproximation to realmarketmovement. There is norelationship between the realpricesinthetopsubgraphandthe synthetic prices in thelower subgraph. Sincerandomnumbersareused,thedisplay will change everytime the indicator is

computed. Therefore, Figure8.1 is simplyoneexampleofsynthetic prices. I probablycould have made thesimulation look a little morerealistic if I had taken morecare randomizing the dailypriceranges.

CodeListing8-1.EasyLanguageCodetoCreate

SyntheticPricesUsingRandomNumberswithMemory

{SyntheticPrices©2013JohnF.Ehlers}

Inputs:alpha1(.05);

Vars:Opn(0),Hgh(0),Lw(0),Cls(0);

Cls = alpha1*(45+Random(10))+(1-alpha1)*Cls[1];

Hgh=Cls+Random(.125);

Lw=Cls-Random(.125);

Opn=Cls+(Random(.25)-.125);

IfOpn>HghThenOpn=Hgh;

IfOpn<LwThenOpn=Lw;

Plot1(Opn,“O”,Blue,0,3);

Plot2(Hgh,“H”,Blue,0,2);

Plot3(Lw,“L”,Blue,0,2);

Plot4(Cls,“C”,Blue,0,3);

Figure8.1RandomNumberswithMemoryProduceReasonableSyntheticPrices

Since synthetic prices

createdbytakinganEMAofrandom numbers are areasonable approximation torealmarket prices, the pricescan be viewed as randomnumbers with memory. Alogical extension is that wecan gain insight into marketactivitybycorrelatingcurrentprices with prices in therecent history to takeadvantageofthememorypartof the model. At least thatwasmypremise.

■AutocorrelationIf we correlate a waveformcomposed of perfectlyrandomnumbersbyitself,thecorrelation will be perfect.However,ifwelagoneofthedata streams by just one bar,the correlation will bedramatically reduced. In along memory process with

normally distributed randomnumbers the autocorrelationfollowsthepowerlaw:

whereC is a constant and αhasanominalvalueofunity.One of the underlying

principles of technicalanalysisisthatmarketdatadonot follow this power law ofan efficient market, and wetherefore can extractinformation from the partial

correlation of theautocorrelationfunction.For example, assume the

data being examined is aperfect sine wave whoseperiod is 20 bars. Theautocorrelationwithzero lag,averagedoveronefullperiodof the sine wave, is unity.That is, the correlation isperfect. Introducing a lag ofonebarintheautocorrelationprocess causes the average

correlation to be decreasedslightly. Introducing anotherbar of lag further decreasesthe average correlation, andso on. That is, until a lag of10 bars is reached. In thiscase, the positive alternationofthesinewaveiscorrelatedwith the negative alternationof the lagged waveform andthenegativealternationofthesine wave is correlated withthepositivealternationofthelagged waveform, with the

result that perfectanticorrelation has beenreached. Continued lagincreases causes the averagecorrelation to increaseuntilalag of 20 bars is reached.When the lag is equal to theperiod of the sine wavewaveform, the correlation isagain perfect. In thistheoretical example, thecorrelation values as afunction of lag vary exactlyasasinewave.

Market data areconsiderably messier thanpurely random numbers orperfect sine waves butcontain features of both.However, the characteristicsthat are uncovered byautocorrelation offer uniquetrading perspectives. Sincethe process is relativelycomplicated, describing howthe computer code in CodeListing 8-2works is the bestpathtowardunderstanding.

CodeListing8-2.EasyLanguageCodeforthe

AutocorrelationIndicator

{AutocorrelationIndicator©2013JohnF.Ehlers}

Inputs:AvgLength(0);

Vars:alpha1(0), Filt2= (1 - alpha1 / 2)*(Filt-Filt[1])+(1-alpha1)*Filt2[1];HP(0),a1(0),

b1(0),c1(0),c2(0),c3(0),Filt(0),M(0),N(0),X(0),Y(0),

Lag(0),count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Color1(0),Color2(0),Color3(0);

Arrays:Corr[48](0),AuroraRaw[48](0),Aurora[48](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*

(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//Pearsoncorrelationforeachvalueoflag

ForLag=0to48Begin//SettheaveraginglengthasMM=AvgLength;IfAvgLength=0ThenM=Lag;//InitializecorrelationsumsSx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;

//AdvancesamplesofbothdatastreamsandsumPearsoncomponentsForcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End;//Computecorrelationforeachvalueoflag If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));//Scaleeachcorrelationtorangebetween0and1 Corr[Lag] = .5*(Corr[Lag]+1);

End;

//PlotasaHeatmap

ForLag=3to48BeginIfCorr[Lag]>.5ThenBegin//VariescolorfromgreenatCorr[Lag]=1toyellow Color1 = 255*(2-2*Corr[Lag]);Color2=255;EndElseBegin//VariescolorfromyellowtoredatCorr[Lag]=0Color1=255;

Color2=2*255*Corr[Lag];End;Color3=0;IfLag=3ThenPlot3(3,“S5”,RGB(Color1,Color2,Color3),0,4);IfLag=4ThenPlot4(4,“S4”,RGB(Color1,Color2,Color3),0,4);IfLag=5ThenPlot5(5,“S5”,RGB(Color1,Color2,Color3),0,4);IfLag=7ThenPlot7(7,“S7”,RGB(Color1,Color2,Color3),0,4);IfLag=7ThenPlot7(7,“S7”,RGB(Color1,Color2,Color3),0,4);IfLag=8ThenPlot8(8,“S8”,RGB(Color1,Color2,Color3),0,4);

IfLag=9ThenPlot9(9,“S9”,RGB(Color1,Color2,Color3),0,4);IfLag=10ThenPlot10(10,“S10”,RGB(Color1,Color2,Color3),0,4);IfLag=11ThenPlot11(11,“S11”,RGB(Color1,Color2,Color3),0,4);IfLag=12ThenPlot12(12,“S12”,RGB(Color1,Color2,Color3),0,4);IfLag=13ThenPlot13(13,“S13”,RGB(Color1,Color2,Color3),0,4);IfLag=14ThenPlot14(14,“S14”,RGB(Color1,Color2,Color3),0,4);IfLag=15ThenPlot15(15,“S15”,RGB(Color1,Color2,Color3),0,4);IfLag=16ThenPlot16(16,“S16”,RGB(Color1,Color2,Color3),0,4);IfLag=17ThenPlot17(17,“S17”,RGB(Color1,Color2,Color3),0,4);IfLag=18ThenPlot18(18,“S18”,RGB(Color1,Color2,Color3),0,4);IfLag=19ThenPlot19(19,“S19”,RGB(Color1,Color2,Color3),0,4);IfLag=20ThenPlot20(20,“S20”,RGB(Color1,Color2,Color3),0,4);IfLag=21ThenPlot21(21,“S21”,RGB(Color1,Color2,Color3),0,4);IfLag=22ThenPlot22(22,“S22”,RGB(Color1,Color2,Color3),0,4);IfLag=23ThenPlot23(23,“S23”,RGB(Color1,Color2,Color3),0,4);IfLag=24ThenPlot24(24,“S24”,RGB(Color1,Color2,Color3),0,4);IfLag=25ThenPlot25(25,“S25”,RGB(Color1,Color2,Color3),0,4);IfLag=26ThenPlot26(26,“S26”,RGB(Color1,Color2,Color3),0,4);

IfLag=27ThenPlot27(27,“S27”,RGB(Color1,Color2,Color3),0,4);IfLag=28ThenPlot28(28,“S28”,RGB(Color1,Color2,Color3),0,4);IfLag=29ThenPlot29(29,“S29”,RGB(Color1,Color2,Color3),0,4);IfLag=30ThenPlot30(30,“S30”,RGB(Color1,Color2,Color3),0,4);

IfLag=31ThenPlot31(31,“S31”,RGB(Color1,Color2,Color3),0,4);IfLag=32ThenPlot32(32,“S32”,RGB(Color1,Color2,Color3),0,4);IfLag=33ThenPlot33(33,“S33”,RGB(Color1,Color2,Color3),0,4);IfLag=34ThenPlot34(34,“S34”,RGB(Color1,Color2,Color3),0,4);IfLag=35ThenPlot35(35,“S35”,RGB(Color1,Color2,Color3),0,4);IfLag=36ThenPlot36(36,“S36”,RGB(Color1,Color2,Color3),0,4);IfLag=37ThenPlot37(37,“S37”,RGB(Color1,Color2,Color3),0,4);IfLag=38ThenPlot38(38,“S38”,RGB(Color1,Color2,Color3),0,4);IfLag=39ThenPlot39(39,“S39”,RGB(Color1,Color2,Color3),0,4);IfLag=40ThenPlot40(40,“S40”,RGB(Color1,Color2,Color3),0,4);IfLag=41ThenPlot41(41,“S41”,RGB(Color1,Color2,Color3),0,4);IfLag=42ThenPlot42(42,“S42”,RGB(Color1,Color2,Color3),0,4);IfLag=43ThenPlot43(43,“S43”,RGB(Color1,Color2,Color3),0,4);IfLag=44ThenPlot44(44,“S44”,RGB(Color1,Color2,Color3),0,4);IfLag=45ThenPlot45(45,“S45”,RGB(Color1,Color2,Color3),0,4);IfLag=46ThenPlot46(46,“S46”,RGB(Color1,Color2,Color3),0,4);IfLag=47ThenPlot47(47,“S47”,RGB(Color1,Color2,Color3),0,4);IfLag=48ThenPlot48(48,“S48”,RGB(Color1,Color2,Color3),0,4);

End;

The only input into theautocorrelation indicator isthe length over which theaveragingistobedone.Withthe default value of zero, theaveraging length is equal toeach lag. That way, theaveraging contains themaximum number of datasamples without overlappingthe original data stream andthe laggeddata stream.Afterdeclaring the variables andthe arrays, undesired long

wave cyclic components areremovedbya two-polehigh-pass filter tuned to 48 bars.This value was selectedbecause it is more than twomonths of daily data and isfour hours of five-minuteintraday bars. It is unlikelythatlongerwaveswouldhavea beneficial impact on cyclictrading.Thehigh-passfilterisfollowed by theSuperSmoother of Equation3-3, and whose transfer

response is displayed inFigure 3.10. The 10-barperiod for this filter wasselected for practical tradingreasons.Thinkofitthisway:The shortest period possibleto use (the Nyquistfrequency) has only twosamples per period. If youwere trading this period andgotahighsample,youcouldenter only on the next bar atthe low sample. This wouldbe an easy trade if the cycle

period were consistent, butmarket cycles are notconsistent. The situation isnot much better at cycliccomponentsanoctavesloweratfourbarsperperiod.Thereisjustnotenoughtimetogetthesignalandthenimplementthe trade. At another octaveslower at eight samples percycle, one can start toconsider rapid tradingbecausethereareseveralbarsof follow-through after the

signalisobtained.Inaperfectworld, you would be tradingasoftenasevery fourbars—onehalfoftheshortestusefulcyclic period. Thus, thecombination of the high-passfilter and the SuperSmootherformaroofingfilterthatpassonly the desired range offrequency components usefulfortrading.Textbook Pearson

correlation is accomplished

for each of the lag periodsfrom 1 to 48, and acorrelation value is assignedto each lag period. Thevariable Y is the base datastream, and thevariableX isthe data stream delayed bythe lag variable for eachcalculation loop. The valuesofX,Y,X*X,X*Y,andY* Y are summed over theaveraginglengthbeforebeingapplied to the correlationequation. The default

averaging length is equal tothe lag so that the fullduration of the lag iscorrelated and averaged. Thecorrelation resulting from thecomputations varies fromminus 1 to plus 1, andtherefore is rescaled to varybetween 0 and 1 for plottingpurposes.There is a considerable

amount of information to bedisplayed, and so aheatmap

isusedtomakesenseofitall.Thegeneralschemeisthatforeach horizontal axis timeposition the verticaldisplacement is equal to thelag from 3 to 48. Thecorrelationvalueforeachlagisconvertedtoacolor.Color1isredandColor2isgreen.When the correlation has avalue of 1, there is onlygreen. At a correlation valueof 0.5, both red and greenhave their maximum values,

creating yellow by theircombination. When thecorrelationvalueiszero,thenthere isonly red. In thiswaya “stoplight” display iscreated,where green denotesperfect correlation, reddenotes perfectanticorrelation, and yellowrepresents intermediatevalues.The autocorrelation

indicator applied to

approximatelyoneyear'sdataof Dollar General (symbolDG) is shown in Figure 8.2.Following along the bottomoftheindicatorsubgraph,theautocorrelation indicatorturns red at every pricereversal, indicatinganticorrelation. The indicatoris green between the pricereversals, indicating goodcorrelation of the databetween the price reversals.Rising vertically in the

subgraph, the indicatormorphsintoamorecloudlikedisplay because of theadditionalsmoothingoverthelonger lag periods. Theseclouds tend to give mixedmessages. For example, thedataareanticorrelatedforlagperiods around 20 barsbecauseof thered“cloud”inthat area. However, the dataare well correlated for lagperiods between 30 and 40barsoverthesameperiod.

Figure8.2AutocorrelationIndicatorShowsCorrelationtoItselfOveraRangeofLagPeriods

Since averaging tends to

obscure the meaning of thedisplay in the short term,where the display can beusefulfortrading,itwouldbeinteresting to see the displaywheretheaveraginglengthisonly three bars for all valuesof the lag period. Three barsis the minimum amount ofdata that can be used foraveraging. We do this bychanging the input from itsdefault value of 0 to a valueof 3.The resulting display is

showninFigure8.3.

Figure8.3AutocorrelationIndicatorwithanAveragingPeriodofThree

Aside from appearing

psychedelic, there are twodistinct characteristics of theautocorrelation indicatorusing minimum averaging.First,thereisasharpreversalfrom red to green and fromgreen to red at the timing ofprice reversals forallperiodsof lag. Second, there is avariation of the thickness ofthe bars and the number ofbarsovertheverticalrangeofthe indicator as a functionoftime.

The meaning of theindicator becomes clearerwhen applied to a theoreticalsinewavewhoseperiodis20bars in Figure 8.4. In thiscase,thesharpreversalsfromred to green and vice versaoccur exactly threebars afterthe peaks and valleys of thesine wave. Carefulexaminationoftheseparationofthelagperiodsbetweenthebottom of a red bar to thebottom of the next red bar

aboveit(orgreenbarsifyouprefer) shows that theseparation is exactly 20bars.Inotherwords,theseparationyields the periodicity of thewaveform. Knowing thischaracteristic, we can extractthe spectral information fromtheautocorrelationfunction.

Figure8.4AutocorrelationofaTheoretical20-BarSineWave

■AutocorrelationPeriodogramConstruction of theautocorrelation periodogramstartswiththeautocorrelationfunction using the minimumthree bars of averaging. Thecyclic information isextracted using a discrete

Fourier transform (DFT) ofthe autocorrelation results.This approach has at leastfour distinct advantages overother spectral estimationtechniques.Theseare:

1. Rapid response. Thespectralestimatesstarttoformwithin a half-cycleperiodoftheirinitiation.2.Relativecyclicpoweras a function of time isestimated. The

autocorrelation at allcycleperiodscanbelowif there are no cyclespresent, for example,during a trend. Previousworks treated themaximum cycleamplitude at each timebarequally.3.Theautocorrelation isconstrained to bebetween minus one andplus one regardless of

the period of themeasured cycle period.Thisobviatestheneedtocompensate for SpectralDilation of the cycleamplitude as a functionofthecycleperiod.4. The resolution of thecyclic measurement isinherently high and isindependent of anywindowing function ofthepricedata.

The autocorrelationperiodogram is describedwith reference to theEasyLanguage code in CodeListing 8-3. Theautocorrelation periodogramstarts with the same code asin Code Listing 8-2. Afterdeclaring variables,prefiltering is accomplishedby a 48-bar high-pass filterfollowed by a 10-barSuperSmoother described inEquation 3-3. The

autocorrelation isaccomplished for each valueof lag using a textbookPearsoncorrelationapproach.The DFT is accomplished

by correlating theautocorrelation at each valueof lag with the cosine andsine of each period ofinterest. The sum of thesquares of each of thesevalues represents the relativepowerateachperiodfromthe

familiar trigonometricequation:

There can be largevariations in the powermeasurementfrombartobar,and EMA is used to smooththe power measurement ateachperiod.Inthenextblockofcodea

fast attack−slow decayautomaticgaincontrol(AGC)is used to normalize the

spectral components and todevelop the variance of thespectralpowerovertime.TheAGCconceptwas introducedin Chapter 5. If the currentpower is greater than thevariable MaxPwr, then thevariable MaxPwr isimmediately set to the valueof the current power.However,ifthecurrentpowerislessthanMaxPwr,thentheMaxPwr is allowed to decayto0.991ofitspreviousvalue.

The dominant cycle isextracted from the spectralestimate in the next block ofcodeusingacenter-of-gravity(CG) algorithm. The CGalgorithm measures theaverage center of two-dimensional objects. It iscomputedby summing theYvalues and independentlysumming the X * Y values.Dividing the latter by theformer yields the averageposition along the X axis

wherealltheYvaluesreside.In the case of computing thedominant cycle, theYvaluesare power and the X valuesare the periods. Thus, thealgorithm computes theaverage period at which thepowers are centered. That isthe dominant cycle. Thedominantcycleisavaluethatvaries with time and can beused to automatically tuneother indicators such as theband-pass filter, commodity

channel index (CCI), relativestrength index (RSI),Stochastic,andsoon.The spectrum values vary

between 0 and 1 after beingnormalized.These values areconvertedtocolors.Whenthespectrum is greater than 0.5,the colors combine red andgreen, with yellow being theresultwhenspectrum=1andredbeing theresultwhen thespectrum = 0.5. When the

spectrum is less than0.5, thered saturation is decreased,with the result the color isblack when spectrum = 0.Since themaximum value ofthe spectrum is unity, I haveincludedanoptionalblockofcode (which has beencommented out by the curlybrackets) that providesadditional visual resolutionby raising the spectralcomponents to a higherpower. The selection of the

powertobeusedisarbitrary.

CodeListing8-3.AutocorrelationPeriodogramEasyLanguage

Code{AutocorrelationPeriodogram©2013JohnF.Ehlers}

Vars:AvgLength(3),M(0),N(0),

X(0),Y(0),alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lag(0),count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Period(0),Sp(0),Spx(0),MaxPwr(0),DominantCycle(0),Color1(0),Color2(0),Color3(0);

Arrays:Corr[48](0),CosinePart[48](0),

SinePart[48](0),SqSum[48](0),R[48,2](0),Pwr[48](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//PearsoncorrelationforeachvalueoflagForLag=0to48Begin//SettheaveraginglengthasMM=AvgLength;

IfAvgLength=0ThenM=Lag;Sx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;Forcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End; If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));End;

ForPeriod=10to48BeginCosinePart[Period]=0;SinePart[Period]=0;ForN=3to48BeginCosinePart[Period]=CosinePart[Period]+Corr[N]*Cosine(370*N/Period);SinePart[Period]=SinePart[Period]+Corr[N]*Sine(370*N/Period);End;SqSum[Period]=CosinePart[Period]*CosinePart[Period]+SinePart[Period]*SinePart[Period];

End;

ForPeriod=10to48BeginR[Period,2]=R[Period,1];R[Period,1]=.2*SqSum[Period]*SqSum[Period]+.8*R[Period,2];

End;

//FindMaximumPowerLevelforNormalization

MaxPwr=.995*MaxPwr;

ForPeriod=10to48BeginIfR[Period,1]>MaxPwrThenMaxPwr=R[Period,1];

End;

ForPeriod=3to48BeginPwr[Period]=R[Period,1]/MaxPwr;

End;

//ComputethedominantcycleusingtheCGofthespectrum

Spx=0;

Sp=0;

ForPeriod=10to48BeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;

End;

IfSp<>0ThenDominantCycle=Spx/Sp;

Plot2(DominantCycle,“DC”,RGB(0,0,255),0,2);

{

//IncreaseDisplayResolutionbyraisingtheNormPwrtoahighermathematicalpower(optional)

ForPeriod=10to48BeginPwr[Period]=Power(Pwr[Period],2);

End;

}

//PlotasaHeatmap

Color3=0;

ForPeriod=10to48BeginIfPwr[Period]>.5ThenBeginColor1=255; Color2 = 255*(2*Pwr[Period]-1);End

ElseBeginColor1=2*255*Pwr[Period];Color2=0;End;IfPeriod=3ThenPlot3[0](3,“S5”,RGB(Color1,Color2,Color3),0,4);IfPeriod=4ThenPlot4[0](4,“S4”,RGB(Color1,Color2,Color3),0,4);IfPeriod=5ThenPlot5[0](5,“S5”,RGB(Color1,Color2,Color3),0,4);

IfPeriod=6ThenPlot6[0](6,“S6”,RGB(Color1,Color2,Color3),0,4);IfPeriod=7ThenPlot7[0](7,“S7”,RGB(Color1,Color2,Color3),0,4);IfPeriod=8ThenPlot8[0](8,“S8”,RGB(Color1,Color2,Color3),0,4);IfPeriod=9ThenPlot9[0](9,“S9”,RGB(Color1,Color2,Color3),0,4);IfPeriod=10ThenPlot10[0](10,“S10”,RGB(Color1,Color2,Color3),0,4);IfPeriod=11ThenPlot11[0](11,“S11”,RGB(Color1,Color2,Color3),0,4);IfPeriod=12ThenPlot12[0](12,“S12”,RGB(Color1,Color2,Color3),0,4);IfPeriod=13ThenPlot13[0](13,“S13”,RGB(Color1,Color2,Color3),0,4);IfPeriod=14ThenPlot14[0](14,“S14”,RGB(Color1,Color2,Color3),0,4);IfPeriod=15ThenPlot15[0](15,“S15”,RGB(Color1,Color2,Color3),0,4);IfPeriod=16ThenPlot16[0](16,“S16”,RGB(Color1,Color2,Color3),0,4);IfPeriod=17ThenPlot17[0](17,“S17”,RGB(Color1,Color2,Color3),0,4);IfPeriod=18ThenPlot18[0](18,“S18”,RGB(Color1,Color2,Color3),0,4);IfPeriod=19ThenPlot19[0](19,“S19”,RGB(Color1,Color2,Color3),0,4);IfPeriod=20ThenPlot20[0]

(20,“S20”,RGB(Color1,Color2,Color3),0,4);IfPeriod=21ThenPlot21[0](21,“S21”,RGB(Color1,Color2,Color3),0,4);

IfPeriod=22ThenPlot22[0](22,“S22”,RGB(Color1,Color2,Color3),0,4);IfPeriod=23ThenPlot23[0](23,“S23”,RGB(Color1,Color2,Color3),0,4);IfPeriod=24ThenPlot24[0](24,“S24”,RGB(Color1,Color2,Color3),0,4);IfPeriod=25ThenPlot25[0](25,“S25”,RGB(Color1,Color2,Color3),0,4);IfPeriod=26ThenPlot26[0](26,“S26”,RGB(Color1,Color2,Color3),0,4);IfPeriod=27ThenPlot27[0](27,“S27”,RGB(Color1,Color2,Color3),0,4);IfPeriod=28ThenPlot28[0](28,“S28”,RGB(Color1,Color2,Color3),0,4);IfPeriod=29ThenPlot29[0](29,“S29”,RGB(Color1,Color2,Color3),0,4);IfPeriod=30ThenPlot30[0](30,“S30”,RGB(Color1,Color2,Color3),0,4);IfPeriod=31ThenPlot31[0](31,“S31”,RGB(Color1,Color2,Color3),0,4);IfPeriod=32ThenPlot32[0](32,“S32”,RGB(Color1,Color2,Color3),0,4);IfPeriod=33ThenPlot33[0](33,“S33”,RGB(Color1,Color2,Color3),0,4);IfPeriod=34ThenPlot34[0]

(34,“S34”,RGB(Color1,Color2,Color3),0,4);IfPeriod=35ThenPlot35[0](35,“S35”,RGB(Color1,Color2,Color3),0,4);IfPeriod=36ThenPlot36[0](36,“S36”,RGB(Color1,Color2,Color3),0,4);IfPeriod=37ThenPlot37[0](37,“S37”,RGB(Color1,Color2,Color3),0,4);IfPeriod=38ThenPlot38[0](38,“S38”,RGB(Color1,Color2,Color3),0,4);IfPeriod=39ThenPlot39[0](39,“S39”,RGB(Color1,Color2,Color3),0,4);IfPeriod=40ThenPlot40[0](40,“S40”,RGB(Color1,Color2,Color3),0,4);IfPeriod=41ThenPlot41[0](41,“S41”,RGB(Color1,Color2,Color3),0,4);IfPeriod=42ThenPlot42[0](42,“S42”,RGB(Color1,Color2,Color3),0,4);IfPeriod=43ThenPlot43[0](43,“S43”,RGB(Color1,Color2,Color3),0,4);

IfPeriod=44ThenPlot44[0](44,“S44”,RGB(Color1,Color2,Color3),0,4);IfPeriod=45ThenPlot45[0](45,“S45”,RGB(Color1,Color2,Color3),0,4);IfPeriod=46ThenPlot46[0](46,“S46”,RGB(Color1,Color2,Color3),0,4);IfPeriod=47ThenPlot47[0](47,“S47”,RGB(Color1,Color2,Color3),0,4);IfPeriod=48ThenPlot48[0]

(48,“S48”,RGB(Color1,Color2,Color3),0,4);End;

The autocorrelationperiodogram for DollarGeneral (symbol DG) overapproximately one year ofdata is shown in Figure 8.5.The spectrum display is intimesynchronizationwiththeprice bars above it. Thevertical scale is the cyclelength in termsofnumberofbars. Using this scale, the

spectrum display is equallyapplicable to intradaydataasit is to daily or weekly data.The cycle strength is coloredfromyellowtoblackasifthemeasurement is going fromwhite hot through red hot tocold. It is obvious from thespectrum display that marketcycles are evanescent—theycomeandgoandchangetheirperiodicity over time. Theblue line throughroughly thecenter of the strongest

components in the spectrumis the computed dominantcycle.

Figure8.5AutocorrelationPeriodogramforDG

Figure 8.5 is a time-

frequency representation(TFR)thatiseasilyusablebytraders by interpreting colorsasthestrengthofthespectralcomponents.Academia1 usescontour plots and waterfalldiagrams to display theresultsoftheircalculations.Awide range of techniques isemployed to measure thespectra of market data,including wavelets. Many ofthese techniques require

analytic data that are noteasily available to traders.Mostimportant,noneofthesetechniques havecompensation for SpectralDilation.

■AutocorrelationReversalsOne of the distinctivecharacteristics of Figures 8.3and 8.4 is that theautocorrelation shifts fromgreen to red or from red togreen at all values of lag atthe cyclic reversals of the

price. Therefore, all we needdo to determine thesereversalsistosumthebar-to-bar differences of theautocorrelation functionacrossallvaluesoflag.Whenthe sum is large a turningpointhasbeenidentified.TheEasyLanguage code tocompute these turning pointsisshowninCodeListing8-4.The indicated reversals arevery sensitive to thesmoothing of the price data.

Therefore, the LPLength ismade available as anindicator input todecreaseorincrease the number ofindicatedreversalsasdesired.The AvgLength parameter

is also made available as anindicator because thisaveraging also impacts thenumber of indicatedreversals. Care should betaken when increasing thevalue of this input because

the lag of the indicatorincreases in direct proportiontotheincreaseofthevalueoftheAvgLength.Typicaldelayof the indicatorwillbeaboutthree bars when theAvgLength parameter is settoavalueof3.TheHPLengthparameteris

also made available as aninput for complete flexibilityof the indicator. However,changing its value has a

relatively minor impact ontheindicatedreversals.

CodeListing8-4.EasyLanguage

CodetoCompute

AutocorrelationReversals

{AutocorrelationReversals©2013JohnF.Ehlers}

Inputs:HPLength(48),

LPLength(10),AvgLength(3);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),M(0),N(0),X(0),Y(0),Lag(0),count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),SumDeltas(0),Reversal(0);

Arrays:

Corr[48,2](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPLength);

b1=2*a1*Cosine(1.414*180/LPLength);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//Pearsoncorrelationforeachvalueoflag

ForLag=3to48BeginCorr[Lag,2]=Corr[Lag,1];//SettheaveraginglengthasMM=AvgLength;IfAvgLength=0ThenM=Lag;

//InitializecorrelationsumsSx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;//AdvancesamplesofbothdatastreamsandsumPearsoncomponentsForcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End;//Computecorrelationforeachvalueoflag If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag,1]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));//Scaleeachcorrelationtorangebetween0and1 Corr[Lag, 1] = .5*(Corr[Lag,1]+1);

End;

SumDeltas=0;ForLag=3to48BeginIf(Corr[Lag,1]>.5andCorr[Lag,2]<.5)Or(Corr[Lag,1]<.5andCorr[Lag,2]>.5)ThenSumDeltas=SumDeltas+1;

End;

Reversal=0;

IfSumDeltas>24ThenReversal=1;

Plot1(Reversal);

The autocorrelationreversals indicator is shownin Figure 8.6 using twodifferent values of theLPLength input. Thereversalsinthefirstsubgraphare computed using anLPLength of 20, and wouldbe suitable for trading

systems holding positions inthe range from10bars to 20bars. The reversals in thebottom subgraph arecomputedusinganLPLengthof 10, and would be moresuitable for more activetrading systems holdingpositionsfromafewdaystoaweek.

Figure8.6TheNumberofIndicatedReversalsIsDecreasedbyIncreasingthe

ValueoftheLPLengthParameter

■ Key PointstoRemember

1. The autocorrelationperiodogram has theleast latency of allmarket data spectralestimates.2. The autocorrelationperiodogramautomatically estimatesthe totalcyclicpower in

thedataasa functionoftime.3. The autocorrelationperiodogramprovidesanaccurate estimate of themarketspectrumwithoutthe need forcompensation forSpectralDilation.4. The autocorrelationperiodogram resolutionis independent of anywindowing function of

thepricedata.5. Price reversals areindicated by a dramaticswitch in correlations atall values of lag. Thesereversals can be used asareliableindicator.

Note1. Boualem Boushash,Time-Frequency SignalAnalysis (Kidlington,Oxford, UK: ElsevierLtd.,2003).

CHAPTER9

FourierTransforms

“The spectrum displaysare colorful,” said Tom

plainly.

In technical circles the termFFT (fast Fourier transform)is nearly synonymous withfrequency measurement, andtherefore the expectation isthatFouriertransformswouldbe the techniqueofchoice tomeasure market cycles.However,doingsomeanstheviolation of three basicrequirements of Fouriertransforms. These

requirementsare:1.Thedata sampleusedis a representative of aninfinitely long string ofdata.2. The data must bestationary (unchangingcharacteristics) over thesample.3. Only an integernumber of cycles overthesamplelengthcanbeanalyzed to avoid end

effectsintheanalysis.As we saw in Chapter 8,

marketcyclesareevanescent.They come and go andchange periodicity. This factalone is not consistent withrequirements 1 and 2 above.Supposewestartwith64barsof data. The smallest integernumber of cycles fitting inthis data is one (a 64-barcycle). The next shortestallowablecycleperiodis64/

2=32barcyclebecause2isnextlargestintegernumberofallowed cycles. The nextshortest allowed cycle periodis64/3=21.3bars,andthenext shortest is 64 / 4 = 16bar cycle, and so on. Theshortest allowed cycle periodis a two-bar cycle, becauseany shorter cycle periodviolatestheNyquistsamplingcriteria of requiring at leasttwo samples per cycle. Thepointisthatitisunrealisticto

expect a relatively short 12-bar market cycle to remainconsistent in amplitude andfrequency even over arelatively short data sampleof 64 bars. But even if thestationarity constraint ismet,the short data sample yieldsan extremely poormeasurement resolution. Forexample, the closest spectrallines between a 16-bar cycleand a 21.3-bar cycle yield aresolution of 5 bars, or 31

percent relative to the 16-barcycle. The only way toimprove the resolution is toincreasethelengthofthedatasample, which exacerbatestheproblemswithstationarityand averaging out short-termresults over the longer term.Further, increasing the datalength increases the latencyofthemeasuredresult,furthercontributingtolag.Nonetheless, I have been

able to bludgeon thetechnology by taking liberallicense with themathematicsto compute a Fouriertransform that produces aspectralestimatethatisusefulfortraders.Theapproachistocompute a moving discreteFourier transform (DFT).Each computation is a blockcalculationoverthehistoricaldata window, and the entireblockcomputationisrepeatedfor each succeeding bar of

data.Oneofthetricksusedistocompute theDFTin termsofcycleperiodrather than inthe more conventionalcomputation in terms offrequency.Traders can relateto a 20-bar cycle period, buttheequivalent0.05cyclesperbar frequency has noparticular meaning. Anotherkey feature to making theDFT usable is to display thespectral estimate as a heatmap.

■ SpectralDilationItisobvious,oncestated,thatannual cycle swings aregreater than monthly cycleswings, that monthly cycleswings are greater than dailycycle swings, and that dailycycle swingsaregreater thanhourly cycle swings. This isSpectral Dilation. Spectral

Dilation is approximatelylinearly proportional to theperiod of the cycle beingconsidered. If one is seekingthe dominant cycle over arelativelysmallrangeofcycleperiods,SpectralDilationcanbe ignored. However, if thedominant cycle is beingextracted from datacontaining several octaves ofcycle periods, then fractaldilationmustbecompensatedinthecomputationalprocess.

I discovered the necessityfor Spectral Dilationcompensation whencorrelatingthespectralresultsusing the autocorrelationperiodogram and the DFTdescribedinthischapter.Theautocorrelation periodogramdescribed in Chapter 8requires no compensationbecause the relativeamplitudes of the spectralcomponents are extractedfrom the correlation

waveforms that areconstrainedtoswingbetween−1and+1.However,spectralcomponents used in thecomputation of the DFT aresubject to Spectral Dilation.Therefore, Spectral Dilationcompensation is required fora correct spectral estimateusingtheDFTalgorithm.

■ DiscreteFourierTransform(DFT)TheDFT isaccomplishedbycorrelating the data with thecosineandsineofeachperiodof interest over the selectedwindow period. The sum of

the squares of each of thesecorrelated values representsthe relative power at eachperiod from the familiartrigonometricequation:

The code to compute theDFT is shown in CodeListing 9-1. The only userinput is whether to selectSpectral Dilationcompensation. The defaultselection is to use the

compensation. The defaultwindow length is 48 bars,selected to be one full cycleperiod of the longest cycleperiod to be computed. Thearrays are sized for thelongest cycle period's being48 bars. After declaring thevariables, the data areprefiltered in a high-passfilter having a 48-bar cutoffperiod to be consistent withour longest computed cycleperiod and then smoothed in

theSuperSmoother,describedinEquation3-3,andhavinga10-bar cutoff period to beconsistent with the shortestcycleperiodofinterest.EasyLanguageexecutesthe

entire code from top tobottom for each new bar ofdata from oldest to newest.Computation of the DFT foreach new data bar starts bymultiplying the data with asinewaveandacosinewave,

respectively, divided by theperiod being calculated andsumming the products overthe window length. In thiscase, thewindow lengthwasmade to be equal to theperiod of the longest cycleperiod to be computed. Thesine part and cosine part ofthe wave at each period aresquared and summed tocompute the power at eachperiod. The wave amplitudeof each period computation

was divided by the valuefractaldilationcompensation.If the spectral dilationcompensation input is set tofalse,thewaveamplitudesaresimplydividedby1,withtheeffect that there is nocompensation.Inthenextblockofcode,a

fast attack−slow decayautomaticgaincontrol(AGC)is used to normalize thespectral components and to

minimize the variance of thespectralpowerovertime.TheAGCconceptwas introducedin Chapter 5 of this book;however, in thiscase,wearescalingonthebasisofpowerrather than on waveamplitude. Therefore, thecorrect decay factor is thesquare root of 0.991, or0.995.If thecurrentpowerisgreater than the variableMaxPwr, then the variableMaxPwrisimmediatelysetto

the value of the currentpower. However, if thecurrent power is less thanMaxPwr,thentheMaxPwr isallowed to decay to 0.995 ofitspreviousvalue.The dominant cycle is

extracted from the spectralestimate in the next block ofcodeusingacenter-of-gravity(CG) algorithm. The CGalgorithm measures theaverage center of two-

dimensional objects. It iscomputedby summing theYvalues and independentlysumming the X * Y values.Dividing the latter by theformer yields the averageposition along the X axiswherealltheYvaluesreside.In the case of computing thedominantcycle, theYvaluesare power and the X valuesare the periods. Thus, thealgorithm computes theaverage period at which the

powers are centered. That isthe dominant cycle. Thedominantcycleisavaluethatvaries with time and can beused to automatically tuneother indicators such as theband-pass filter, commoditychannel index (CCI), relativestrength index (RSI),Stochastic,andsoon.The spectrum values vary

between 0 and 1 after beingnormalized.These values are

convertedtocolors.Whenthespectrum is greater than 0.5the colors combine red andgreen, with yellow being theresultwhenspectrum=1andredbeing theresultwhen thespectrum = 0.5. When thespectrum is less than0.5, thered saturation is decreased,with the result that the coloris black when spectrum = 0.Since themaximum value ofthe spectrum is unity, I haveincludedanoptionalblockof

code (which has beencommented out by the curlybrackets) that providesadditional visual resolutionby raising the spectralcomponents to a higherpower. The selection of thepowertobeusedisarbitrary.

CodeListing9-1.EasyLanguage

CodetoComputea

DFTSpectralEstimate

{DiscreteFourierTransform(DFT)©2013JohnF.Ehlers}

Inputs:SpectralDilationCompensation(true);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Period(0),N(0),M(0),MaxPwr(0),Comp(0),

Sp(0),Spx(0),DominantCycle(0),Color1(0),Color2(0);

//ArraysaresizedtohaveamaximumPeriodof48bars

Arrays:CosinePart[48](0),SinePart[48](0),Pwr[48](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilter

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*

(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//ThisistheDFT

ForPeriod=10to48BeginComp=Period;IfSpectralDilationCompensation=FalseThenComp=1;CosinePart[Period]=0;SinePart[Period]=0;//FindCosineandSinecorrelatedcomponents,compensatedforfractaldilationForN=0to47BeginCosinePart[Period]=CosinePart[Period]+Filt[N]*Cosine(360*N/Period)/Comp;SinePart[Period]=SinePart[Period]+Filt[N]*Sine(360*N/Period)/Comp;End;Pwr[Period]=CosinePart[Period]*CosinePart[Period]+SinePart[Period]*SinePart[Period];End;

//FindMaximumPowerLevelforNormalization

MaxPwr=.995*MaxPwr[1];

ForPeriod=8to48BeginIfPwr[Period]>MaxPwrThenMaxPwr=Pwr[Period];End;

//NormalizePowerLevelsandConverttoDecibels

ForPeriod=10to48BeginIFMaxPwr>0ThenPwr[Period]=Pwr[Period]/MaxPwr;End;

//ComputethedominantcycleusingtheCGofthespectrum

Spx=0;

Sp=0;

ForPeriod=10to48BeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;

End;

IfSp<>0ThenDominantCycle=Spx/Sp;Plot2(DominantCycle,“DC”,RGB(0,0,255),0,2);

{

//IncreaseDisplayResolutionbyraisingtheNormPwrtoahighermathematicalpower(optional)

ForPeriod=10to48BeginPwr[Period]=Power(Pwr[Period],3);

End;}

//PlottheSpectrumasaHeatmap

ForPeriod=10to48Begin//ConvertPowertoRGBColorforDisplayIfPwr[Period]>=.5ThenBeginColor1=255; Color2 = 255*(2*Pwr[Period]-1);End;IfPwr[Period]<.5ThenBeginColor1=255*2*Pwr[Period];

Color2=0;End;IfPeriod=10ThenPlot10(10,“S10”,RGB(Color1,Color2,0),0,4);IfPeriod=11ThenPlot11(11,“S11”,RGB(Color1,Color2,0),0,4);IfPeriod=12ThenPlot12(12,“S12”,RGB(Color1,Color2,0),0,4);IfPeriod=13ThenPlot13(13,“S13”,RGB(Color1,Color2,0),0,4);IfPeriod=14ThenPlot14(14,“S14”,RGB(Color1,Color2,0),0,4);IfPeriod=15ThenPlot15(15,“S15”,RGB(Color1,Color2,0),0,4);IfPeriod=16ThenPlot16(16,“S16”,RGB(Color1,Color2,0),0,4);IfPeriod=17ThenPlot17(17,“S17”,RGB(Color1,Color2,0),0,4);IfPeriod=18ThenPlot18(18,“S18”,RGB(Color1,Color2,0),0,4);IfPeriod=19ThenPlot19(19,“S19”,RGB(Color1,Color2,0),0,4);IfPeriod=20ThenPlot20(20,“S20”,RGB(Color1,Color2,0),0,4);IfPeriod=21ThenPlot21(21,“S21”,RGB(Color1,Color2,0),0,4);IfPeriod=22ThenPlot22(22,“S22”,RGB(Color1,Color2,0),0,4);IfPeriod=23ThenPlot23(23,“S23”,RGB(Color1,Color2,0),0,4);IfPeriod=24ThenPlot24(24,“S24”,RGB(Color1,Color2,0),0,4);IfPeriod=25ThenPlot25(25,“S25”,RGB(Color1,Color2,0),0,4);IfPeriod=26ThenPlot26(26,“S26”,RGB(Color1,Color2,0),0,4);IfPeriod=27ThenPlot27(27,“S27”,RGB(Color1,Color2,0),0,4);IfPeriod=28ThenPlot28(28,“S28”,RGB(Color1,Color2,0),0,4);IfPeriod=29ThenPlot29(29,“S29”,RGB(Color1,Color2,0),0,4);IfPeriod=30ThenPlot30(30,“S30”,RGB(Color1,Color2,0),0,4);IfPeriod=31ThenPlot31(31,“S31”,RGB(Color1,Color2,0),0,4);IfPeriod=32ThenPlot32(32,“S32”,RGB(Color1,Color2,0),0,4);IfPeriod=33ThenPlot33(33,“S33”,RGB(Color1,Color2,0),0,4);IfPeriod=34ThenPlot34(34,“S34”,RGB(Color1,Color2,0),0,4);IfPeriod=35ThenPlot35(35,“S35”,RGB(Color1,Color2,0),0,4);IfPeriod=36ThenPlot36(36,“S36”,RGB(Color1,Color2,0),0,4);

IfPeriod=37ThenPlot37(37,“S37”,RGB(Color1,Color2,0),0,4);IfPeriod=38ThenPlot38(38,“S38”,RGB(Color1,Color2,0),0,4);IfPeriod=39ThenPlot39(39,“S39”,RGB(Color1,Color2,0),0,4);IfPeriod=40ThenPlot40(40,“S40”,RGB(Color1,Color2,0),0,4);IfPeriod=41ThenPlot41(41,“S41”,RGB(Color1,Color2,0),0,4);IfPeriod=42ThenPlot42(42,“S42”,RGB(Color1,Color2,0),0,4);IfPeriod=43ThenPlot43(43,“S43”,RGB(Color1,Color2,0),0,4);IfPeriod=44ThenPlot44(44,“S44”,RGB(Color1,Color2,0),0,4);IfPeriod=45ThenPlot45(45,“S45”,RGB(Color1,Color2,0),0,4);IfPeriod=46ThenPlot46(46,“S46”,RGB(Color1,Color2,0),0,4);IfPeriod=47ThenPlot47(47,“S47”,RGB(Color1,Color2,0),0,4);IfPeriod=48ThenPlot48(48,“S48”,RGB(Color1,Color2,0),0,4);

End;

AsamplecomputationandspectraldisplayoftheDFTisshown in Figure 9.1.Harmonic components areclearly present (at periods of

half and one-third thestronger dominant cycleperiod). Nonetheless, thespectral estimate comparesfavorably with the preferredautocorrelation periodogram,as shown in Figure 9.2. Thisresult is remarkableconsidering all theconcessions made to puretheory.

Figure9.1AnExampleofaDFTSpectralEstimate

Figure9.2DFTSpectral

EstimateComparesFavorablywithanAutocorrelationPeriodogram

■ Key PointstoRemember

1. The practicalcomputation of a DFTfor market datadisregards theoreticalrequirements.2.TheDFTfortradersiscomputed in terms ofcycle period rather thanfrequency for ease of

interpretation.3. The DFT calculationshould includecompensation forSpectralDilation.4. The DFT results ascomputed in Figure 9.1compare favorably withthe results of theautocorrelationperiodogram.

CHAPTER10

CombFilterSpectralEstimates

“Youcantakethattothe

bank,” said Tomrepeatedly.

Band-pass filters weredescribed in Chapter 5. Theconcept of this chapter is toarrange a bank of thesefilters, each tuned to adifferent cycle period andoverlapping,andapplyingthesame data to the input ofthese filters. The band-passfilterscanbepicturedasteethin a comb, only the teeth

overlap. With this combstructure the relativeamplitudes of the signals attheoutputofthefilterswouldconstitute an estimate of thespectral content of the inputdata. The measurementresolution and,correspondingly,thetransientresponse of themeasurementisbycontrolofthebandwidthof each filter. It is just thateasy. This is the mostconceptuallypureandeasyto

understand of all the spectralestimateapproaches.

■ SpectralDilationBefore beginning, it isimperative to again addressthe issueofSpectralDilationof market data cycleamplitudes. This is acharacteristic intrinsic to thenature of market data.Compensation for SpectralDilation is mandatory when

making cycle periodmeasurements over arelatively large range and istherefore included in thisspectral estimation approach.Spectral Dilation of marketcycles was discussed morefullyinChapter7.

■ComputingaComb FilterSpectralEstimateSince the concept ofcomputingaspectralestimatebased measuring the outputpowers of a bank ofoverlapping band-pass filters

issostraightforward,thebestwayofdescribingtheprocessis with reference to theEasyLanguage code in CodeListing10-1.

CodeListing10-1.EasyLanguage

CodetoComputeaCombFilter

SpectralEstimate{

CombBandPassSpectrum©2013JohnF.Ehlers}

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Comp(0),beta1(0),gamma1(0),delta1(0),N(0),M(0),Period(0),Sp(0),Spx(0),MaxPwr(0),DominantCycle(0),Color1(0),Color2(0),Color3(0);

Arrays:

BP[48,48](0),Pwr[48](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

ForN=10to48Begin

ForM=48DownTo2BeginBP[N,M]=BP[N,M-1];End;End;

IfCurrentBar>12ThenBeginForN=10to48BeginComp=N;IfSpectralDilationCompensation=FalseThenComp=1;beta1=Cosine(360/N);gamma1=1/Cosine(360*Bandwidth/N);alpha1=gamma1-SquareRoot(gamma1*gamma1-1);BP[N,1]=.5*(1-alpha1)*(Filt-Filt[2])+beta1*(1+alpha1)*BP[N,2]-alpha1*BP[N,3];Pwr[N]=0;ForM=1toNBeginPwr[N]=Pwr[N]+(BP[N,M]/Comp)*(BP[N,M]/Comp);End;End;End;

//FindMaximumPowerLevelforNormalization

MaxPwr=.995*MaxPwr;

ForPeriod=ShortestPeriodto48BeginIfPwr[Period]>MaxPwrThenMaxPwr=Pwr[Period];

End;

//NormalizePowerLevelsandConverttoDecibels

ForPeriod=10to48BeginIfMaxPwr>0ThenPwr[Period]=Pwr[Period]/MaxPwr;End;

//ComputethedominantcycleusingtheCGofthespectrum

Spx=0;

Sp=0;

ForPeriod=ShortestPeriodtoLongestPeriodBeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;

End;

IfSp<>0ThenDominantCycle=Spx/Sp;Plot2(DominantCycle,“DC”,RGB(0,0,255),0,2);

{

//IncreaseDisplayResolutionbyraisingthePwrtoahighermathematicalpower

ForPeriod=10to48BeginPwr[Period]=Power(Pwr[Period],3);

End;}

ForN=10to48BeginColor3=0;IfPwr[N]>.5ThenBeginColor1=255; Color2 = 255*(2*Pwr[N]-1);EndElseBeginColor1=2*255*Pwr[N];Color2=0;End;IfN=10ThenPlot10(N,“S10”,RGB(Color1,Color2,Color3),0,5);IfN=11ThenPlot11(N,“S11”,RGB(Color1,Color2,Color3),0,5);IfN=12ThenPlot12(N,“S12”,RGB(Color1,Color2,Color3),0,5);IfN=13ThenPlot13(N,“S13”,RGB(Color1,Color2,Color3),0,5);IfN=14ThenPlot14(N,“S14”,RGB(Color1,Color2,Color3),0,5);IfN=15ThenPlot15(N,“S15”,RGB(Color1,Color2,Color3),0,5);IfN=16ThenPlot16(N,“S16”,RGB(Color1,Color2,Color3),0,5);IfN=17ThenPlot17(N,“S17”,RGB(Color1,Color2,Color3),0,5);IfN=18ThenPlot18(N,“S18”,RGB(Color1,Color2,Color3),0,5);IfN=19ThenPlot19(N,“S19”,RGB(Color1,Color2,Color3),0,5);IfN=20ThenPlot20(N,“S20”,RGB(Color1,Color2,Color3),0,5);IfN=21ThenPlot21(N,“S21”,RGB(Color1,Color2,Color3),0,5);

IfN=22ThenPlot22(N,“S22”,RGB(Color1,Color2,Color3),0,5);IfN=23ThenPlot23(N,“S23”,RGB(Color1,Color2,Color3),0,5);IfN=24ThenPlot24(N,“S24”,RGB(Color1,Color2,Color3),0,5);IfN=25ThenPlot25(N,“S25”,RGB(Color1,Color2,Color3),0,5);IfN=26ThenPlot26(N,“S26”,RGB(Color1,Color2,Color3),0,5);IfN=27ThenPlot27(N,“S27”,RGB(Color1,Color2,Color3),0,5);IfN=28ThenPlot28(N,“S28”,RGB(Color1,Color2,Color3),0,5);IfN=29ThenPlot29(N,“S29”,RGB(Color1,Color2,Color3),0,5);IfN=30ThenPlot30(N,“S30”,RGB(Color1,Color2,Color3),0,5);IfN=31ThenPlot31(N,“S31”,RGB(Color1,Color2,Color3),0,5);IfN=32ThenPlot32(N,“S32”,RGB(Color1,Color2,Color3),0,5);IfN=33ThenPlot33(N,“S33”,RGB(Color1,Color2,Color3),0,5);IfN=34ThenPlot34(N,“S34”,RGB(Color1,Color2,Color3),0,5);IfN=35ThenPlot35(N,“S35”,RGB(Color1,Color2,Color3),0,5);IfN=36ThenPlot36(N,“S36”,RGB(Color1,Color2,Color3),0,5);IfN=37ThenPlot37(N,“S37”,RGB(Color1,Color2,Color3),0,5);IfN=38ThenPlot38(N,“S38”,RGB(Color1,Color2,Color3),0,5);IfN=39ThenPlot39(N,“S39”,RGB(Color1,Color2,Color3),0,5);IfN=40ThenPlot40(N,“S40”,RGB(Color1,Color2,Color3),0,5);IfN=41ThenPlot41(N,“S41”,RGB(Color1,Color2,Color3),0,5);IfN=42ThenPlot42(N,“S42”,RGB(Color1,Color2,Color3),0,5);IfN=43ThenPlot43(N,“S43”,RGB(Color1,Color2,Color3),0,5);IfN=44ThenPlot44(N,“S44”,RGB(Color1,Color2,Color3),0,5);IfN=45ThenPlot45(N,“S45”,RGB(Color1,Color2,Color3),0,5);IfN=46ThenPlot46(N,“S46”,RGB(Color1,Color2,Color3),0,5);IfN=47ThenPlot47(N,“S47”,RGB(Color1,Color2,Color3),0,5);IfN=48ThenPlot48(N,“S48”,RGB(Color1,Color2,Color3),0,5);

End;

The arrays and plottingroutines have been sized toaccommodate the range ofband-pass filter periods from10 to 48 bars.After the datahave been preconditioned bythe combination of the two-pole high-pass andSuperSmoother filters toallow only the cycliccomponents of interest to beanalyzed, the band-passfilters for each of thechannels is filled with

historical values to allowcomputation and averaginglaterinthecode.Since the entire

EasyLanguage code isimplemented from top tobottomwith eachnewbarofdata, the next block of codeselects each band-passchannel and computes thecurrent filtered value at theoutput of each channel. Thewave amplitude of each

channel is divided by itscenter period to compensatefor the AGC frequency roll-off that comes later, squaredto compute the power outputof the channel, and averagedoverthelengthofthechannelperiod.Inthenextblockofcodea

fast attack−slow decayautomaticgaincontrol(AGC)is used to normalize thespectral components and to

minimize the variance of thespectralpowerovertime.TheAGCconceptwas introducedinChapter5;however,inthiscase, we are scaling on thebasisofpowerrather thanonwave amplitude. Therefore,thecorrectdecayfactoristhesquare root of 0.991, or0.995.If thecurrentpowerisgreater than the variableMaxPwr, then the variableMaxPwrisimmediatelysettothe value of the current

power. However, if thecurrent power is less thanMaxPwr,thentheMaxPwr isallowed to decay to 0.995 ofitspreviousvalue.Ofcourse,anyactioninthe

timedomainhas implicationsin the frequency domain. Inthiscase,thefastattack−slowdecayAGChas the effect ofan exponential decay in thetimedomain. Itseffect in thefrequencydomainisthesame

as an exponential movingaverage (EMA). Therefore,the frequency roll-off of thisde facto filter must becompensated for during thespectrum computation. TheAGC frequency responseslope is approximately 6 dBper octave across thefrequency band of interest,and therefore dividing eachchannel band-pass waveamplitude by the respectivechannel period provided a 6

dB per octave compensationfor the AGC action. Forreference, an octave is a 2:1frequency or period ratio. Afactor of 2 in the waveamplitude is a factor of 4 inthe power ratio, and a factorof4 inpower corresponds to6dB.The dominant cycle is

extracted from the spectralestimate in the next block ofcodeusingacenterofgravity

(CG) algorithm. The CGalgorithm measures theaverage center of two-dimensional objects. It iscomputedby summing theYvalues and independentlysumming the X * Y values.Dividing the latter by theformer yields the averageposition along the X axiswherealltheYvaluesreside.In the case of computing thedominant cycle, theYvaluesare power and the X values

are the periods. Thus, thealgorithm computes theaverage period at which thepowers are centered. That isthe dominant cycle. Thedominantcycleisavaluethatvaries with time and can beused to automatically tuneother indicators such as theband-pass filter, commoditychannel index (CCI), relativestrength index (RSI),Stochastic,andsoon.

The spectrum values varybetween 0 and 1 after beingnormalized.These values areconvertedtocolors.Whenthespectrum is greater than 0.5,the colors combine red andgreen, with yellow being theresultwhenspectrum=1,andredbeing theresultwhen thespectrum = 0.5. When thespectrum is less than0.5, thered saturation is decreased,with the result that the coloris black when spectrum = 0.

Since themaximum value ofthe spectrum is unity, I haveincludedanoptionalblockofcode (which has beencommented out by the curlybrackets) that providesadditional visual resolutionby raising the spectralcomponents to a higherpower. The selection of thepowertobeusedisarbitrary.Anexampleofthespectral

estimate computed by the

array of band-pass filtersusing a pass-band value of0.3 is shown in Figure 10.1.This estimate is comparedwith the autocorrelationperiodogram and DFTmethods in Figure 10.2. Thethree techniques provide aconsistent estimate of thespectral content of the data.In my opinion, theautocorrelation periodogramis the superior approachbecause themeasurementhas

less latency, has a widerrange of amplitude swings,does not require historicalaveraging, and does notrequire Spectrum Dilationcompensation. For thesereasons I will be using theautocorrelation periodogramapproach to compute thedominant cycle in theremainderofthisbook.

Figure10.1CombFilterSpectralEstimateforDG

Figure10.2Comparisonof

theCombFilter,DFT,andAutocorrelationPeriodogramSpectralEstimatesforDG

■ Key PointstoRemember

1. Market data spectracan be estimated byapplying the data to abank of band-passfilters, each tuned to adifferent center periodwith overlapping passbands. The relativeoutput power of the

filters provides anestimate of the cycliccontentofthedata.2. Compensation forSpectral Dilation shouldbe included in thecomputationsforacombfilter estimate of thespectrum.3. The autocorrelationperiodogram is thepreferred method ofobtaining spectral

estimatesofmarketdata.

CHAPTER11

AdaptiveFilters

“We should adjust theindicators,” said Tom

rigidly.

Adaptive filters can haveseveral different meanings.For example, PerryKaufman's adaptive movingaverage (KAMA)1 andTushar Chande's variableindex dynamic average(VIDYA)2 adapt to changesin volatility. By definition,these filters are reactive toprice changes, and therefore

theyclosethebarndoorafterthe horse is gone. Theadaptive filters discussed inthis chapter are the familiarStochastic, relative strengthindex (RSI), commoditychannel index (CCI), andband-pass filter. The keyparameter in eachcase is thelookback period used tocalculate the indicator. Thislookbackperiodiscommonlyafixedvalue.However,sincethe measured cycle period is

changing,aswehaveseeninprevious chapters, it makessense to adapt theseindicators to the measuredcycle period. When tradablemarket cycles are observed,theytendtopersistforashortwhile. Therefore, by tuningthe indicators to themeasurecycle period they areoptimized for currentconditionsandcanevenhavepredictivecharacteristics.

Thedominant cycleperiodis measured using theautocorrelation periodogramalgorithm. That dominantcycle dynamically sets thelookback period for theindicators. I employmy ownstreamlined computation forthe indicators that providesmoother and easier tointerpret outputs thantraditional methods. Further,theindicatorcodeshavebeenmodified to remove the

effects of spectral dilation.Thisbasicallycreatesawholenewsetofindicatorsforyourtradingarsenal.

■ AdaptiveRelativeStrengthIndex(RSI)Welles Wilder had justintroduced the RSI3 when Ibegan trading. I asked mybrokerwhy14dayswasusedin the calculation. His

response was something like“because Wilder said so.”You have no idea howunsatisfying that responsewas to an engineer, and thatstarted me on my decades-long quest to find theoptimum indicator settings. Ithink I can now describethoseoptimumsettings.Wilder's original RSI was

definedby

where

“ClosesUP” meant that iftoday'sclosewashigher thanyesterday's close, we wouldadd the difference in closingprices to the average.Similarly, “closes DOWN”meant that if today's closewas lower than yesterday's,we would add the positive

differenceinclosingpricestotheaverage.Wecansimplifythe RSI equation if wesubstitute RSI = CU / CD.Then,

Increasing generality, CUcan mean the sum of closesupoveranyinterval,andCD

can mean the sum of closesdown over the same period.Additionally, the factor of100 is recognized asproviding a simple scaling.So,RSIisjusttheratioofthesum of closes up to the sumof the absolute value of allthedifferentialclosingprices.

Thequestionthenbecomeswhat is thebest lengthtouse

fortheRSIsummation.Ifweuseanextremelylonglength,then the closes up will beapproximately half of theabsolute value of all theclosing price differentials. Inthis case, the RSI wouldhover near 0.5 andwould bebasicallyworthless.However,if the length used forsummationisveryshort,thenthe RSI would be stuck at 1for extended periods inuptrends andwould be stuck

at 0 for extended periods indowntrends. In a perfectworldwherepricesswingasasine wave having the periodof the dominant cycle, theRSI would just touch thevalueof1iftheupswinghalfofthedominantcycleisusedfor thesummation.Similarly,theRSIwould just touch thevalue of 0 if the downswinghalf of the dominant cycle isused for the summation.Therefore,togettheoptimum

swing of the RSI indicator,the best adaptive summationlength to use is half themeasureddominantcycle.The adaptive RSI starts

with the computation of thedominant cycle using theautocorrelation periodogramapproach. Rather thandescribing the computations,the reader is referred to thedescription in Chapter 8.Code Listing 11-1 contains

the EasyLanguage code forthe adaptive RSI, and theidentification of the RSIindicator itself following thedominant cycle calculation isnoted by the comment neartheend.Sincetheobjectiveisto use only those frequencycomponents passed by theroofingfilter,thevariableFiltis used as a data input ratherthan closing prices. Ratherthanindependentlytakingtheaverages of the numerator

and denominator, I chose toperform smoothing on theratio using theSuperSmoother filterdescribed in Chapter 3. Thecoefficients for theSuperSmoother filters havepreviously been computed inthe dominant cyclemeasurementpartofthecode.I have included referencelinesatthe30percentand70percent levels, but these caneasily be adjusted to 20

percent/80 percent orwhatever else serves thepurpose.

CodeListing11-1.EasyLanguageCodefortheAdaptiveRSI

{AdaptiveRSI(c)2013JohnF.Ehlers}Vars:AvgLength(3),M(0),N(0),

X(0),Y(0),alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lag(0),count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Period(0),Sp(0),Spx(0),MaxPwr(0),DominantCycle(0),Color1(0),Color2(0),Color3(0);Arrays:Corr[48](0),CosinePart[48](0),SinePart[48](0),

SqSum[48](0),R[48,2](0),Pwr[48](0);//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48barsalpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];//SmoothwithaSuperSmootherFilterfromequation3a1=expvalue(-1.414*3.14159/10);b1=2*a1*Cosine(1.414*180/10);c2=b1;c3=-a1*a1;c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];//PearsoncorrelationforeachvalueoflagForLag=0to48Begin//SettheaveraginglengthasMM=AvgLength;IfAvgLength=0ThenM=Lag;Sx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;

Forcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End; If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));End;ForPeriod=10to48BeginCosinePart[Period]=0;SinePart[Period]=0;ForN=3to48BeginCosinePart[Period]=CosinePart[Period]+Corr[N]*Cosine(360*N/Period);SinePart[Period]=SinePart[Period]+Corr[N]*Sine(360*N/Period);End;SqSum[Period]=CosinePart[Period]*CosinePart[Period]+SinePart[Period]*SinePart[Period];End;ForPeriod=10to48BeginR[Period,2]=R[Period,1];R[Period,1]=.2*SqSum[Period]*SqSum[Period]+.8*R[Period,2];End;//FindMaximumPowerLevelforNormalizationMaxPwr=.991*MaxPwr;ForPeriod=10to48Begin

IfR[Period,1]>MaxPwrThenMaxPwr=R[Period,1];End;ForPeriod=3to48BeginPwr[Period]=R[Period,1]/MaxPwr;End;//ComputethedominantcycleusingtheCGofthespectrumSpx=0;Sp=0;ForPeriod=10to48BeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;End;IfSp<>0ThenDominantCycle=Spx/Sp;IfDominantCycle<10ThenDominantCycle=10;IfDominantCycle>48ThenDominantCycle=48;//AdaptiveRSIstartshere,usinghalfthemeasureddominantcyclefortuningVars:ClosesUp(0),ClosesDn(0),Denom(0),MyRSI(0);ClosesUp=0;ClosesDn=0;Forcount=0toIntPortion(DominantCycle/2-1)BeginIfFilt[count]>Filt[count+1]ThenClosesUp=ClosesUp+(Filt[count]-Filt[count+1]);IfFilt[count]<Filt[count+1]ThenClosesDn=ClosesDn+(Filt[count+1]-Filt[count]);End;

Denom=ClosesUp+ClosesDn;IfDenom<>0andDenom[1]<>0ThenMyRSI=c1*(ClosesUp/Denom+ClosesUp[1]/Denom[1])/2+c2*MyRSI[1]+c3*MyRSI[2];Plot1(MyRSI);Plot2(.7);Plot6(.3);

TheadaptiveRSIhasbeenapplied to Dollar General(symbol DG) over roughlythe last year in Figure 11.1.Note that the indicatorreaches zero or one onlywhen the prices have had acyclic swing over half theperiodofthedominantcycle.

These events happenrelatively rarely, and showthat the indicator is properlytuned.

Figure11.1AdaptiveRSIIsProperlyTunedtoHalftheDominantCyclePeriod

■ AdaptiveStochasticIndicatorThe basic idea of thestochastic indicator is that itmeasures the current closingprice relative to the lowestclose over the observationrange, and normalize thatrelative price to the range

betweenthehighestcloseandthe lowest close over theobservation range. Sincefinding the highest close andlowest close over thedominant cycle period isassured only by searchingover the entire dominantcycle period, the entiredominantcycleperiodisusedfor the observation range inthecalculations.The adaptive stochastic

indicator starts with thecomputation of the dominantcycle using theautocorrelation periodogramapproach. Rather thandescribing the computations,the reader is referred to thedescription in Chapter 8.Code Listing 11-2 containsthe EasyLanguage code forthe adaptive stochasticindicator, and theidentification of the indicatoritself following the dominant

cycle calculation is noted bythe comment near the end.Since the objective is to useonly those frequencycomponents passed by theroofingfilter,thevariableFiltis used as a data input ratherthanclosingprices.

CodeListing11-2.EasyLanguageCodeforthe

AdaptiveStochasticIndicator

{AdaptiveStochastic(c)2013JohnF.Ehlers}Vars:AvgLength(3),M(0),N(0),X(0),Y(0),alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lag(0),

count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Period(0),Sp(0),Spx(0),MaxPwr(0),DominantCycle(0);Arrays:Corr[48](0),CosinePart[48](0),SinePart[48](0),SqSum[48](0),R[48,2](0),Pwr[48](0);//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48barsalpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];//SmoothwithaSuperSmootherFilterfromequation3a1=expvalue(-1.414*3.14159/10);b1=2*a1*Cosine(1.414*180/10);

c2=b1;c3=-a1*a1;c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];//PearsoncorrelationforeachvalueoflagForLag=0to48Begin//SettheaveraginglengthasMM=AvgLength;IfAvgLength=0ThenM=Lag;Sx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;Forcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End; If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));End;ForPeriod=10to48Begin

CosinePart[Period]=0;SinePart[Period]=0;ForN=3to48BeginCosinePart[Period]=CosinePart[Period]+Corr[N]*Cosine(360*N/Period);SinePart[Period]=SinePart[Period]+Corr[N]*Sine(360*N/Period);End;SqSum[Period]=CosinePart[Period]*CosinePart[Period]+SinePart[Period]*SinePart[Period];End;ForPeriod=10to48BeginR[Period,2]=R[Period,1];R[Period,1]=.2*SqSum[Period]*SqSum[Period]+.8*R[Period,2];End;//FindMaximumPowerLevelforNormalizationMaxPwr=.995*MaxPwr;ForPeriod=10to48BeginIfR[Period,1]>MaxPwrThenMaxPwr=R[Period,1];End;ForPeriod=3to48BeginPwr[Period]=R[Period,1]/MaxPwr;End;//ComputethedominantcycleusingtheCGofthespectrumSpx=0;Sp=0;ForPeriod=10to48BeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;End;

IfSp<>0ThenDominantCycle=Spx/Sp;IfDominantCycle<10ThenDominantCycle=10;IfDominantCycle>48ThenDominantCycle=48;//StochasticComputationstartshereVars:HighestC(0),LowestC(0),Stoc(0),SmoothNum(0),SmoothDenom(0),AdaptiveStochastic(0);HighestC=Filt;LowestC=Filt;Forcount=0toDominantCycle-1BeginIfFilt[count]>HighestCthenHighestC=Filt[count];IfFilt[count]<LowestCthenLowestC=Filt[count];End;Stoc=(Filt-LowestC)/(HighestC-LowestC);AdaptiveStochastic = c1*(Stoc+Stoc[1])/2+c2*AdaptiveStochastic[1]+c3*AdaptiveStochastic[2];Plot1(AdaptiveStochastic);Plot2(.7);Plot6(.3);

Thefirststep is to identify

the highest “close” and thelowest “close” over theperiod of the currentdominant cycle.The variableStoc iscomputedas the ratioofthecurrentvalueofFilt tothe price range and is theunfiltered value of theadaptive stochastic.TheStocvariable is filtered using theSuperSmoother filter firstdescribe inChapter3.Ratherthanindependentlytakingtheaverages of the numerator

and denominator, I chose toperform smoothing on theratio Stoc using theSuperSmoother filterdescribed in Chapter 3. Thecoefficients for theSuperSmoother filters havepreviously been computed inthe dominant cyclemeasurementpartofthecode.I have included referencelinesatthe30percentand70percent levels, but these caneasily be adjusted to 20

percent/80 percent orwhatever else serves thepurpose.The adaptive Stochastic

has been applied to DollarGeneral (symbol DG) overroughlythelastyearinFigure11.2. An eyeball scancomparing the prices and theadaptivestochasticshowsthatthis indicator accuratelyportrays the relativedetrended prices over the

length of the dominant cycleperiod, and has manycharacteristics in commonwiththeadaptiveRSI.

Figure11.2AdaptiveStochasticAccuratelyDisplaystheRelativeDetrendedPricesovertheChannelLengthoftheDominantCyclePeriod

■ AdaptiveCCI(CommodityChannelIndex)The CCI indicator was firstdescribed by DonaldLambert4 as a means of

statistically estimating wherethe current prices lie withinthe length of a channel. Hebasically compares thecurrent price relative to amoving average of pricesoverthelengthofthechannelto the standard deviation ofprices over the same length.A standard deviation is justthe square root of theaveraged squared differenceof the prices from theirmoving averages. As a

shorthandwayof thinkingofit,thebasicideaoftheCCIistaking the ratio of the“instantaneous deviation” tothe standard deviation.Lambertalsodividestheratioby 0.015, which is the sameas multiplying the ratio by66.7. This scaling factor sets+100 and −100 asapproximations to the plusand minus one standarddeviation if the data had aGaussian probability

distribution. That is not agood assumption, which canbe proven heuristically bymeasuring the dataprobability distributions overalotofdata.Thetimelengthtobeused

for the channel in thecalculations is widely variedin the literature. In all cases,thelengthisratherarbitrarilyestablishedtofittheindicatorto some preconceived event.

It seems tome that it wouldbe better to use one fullperiod of the dominant cycleas the length of data to beused.TheadaptiveCCIindicator

startswiththecomputationofthe dominant cycle using theautocorrelation periodogramapproach. Rather thandescribing the computations,the reader is referred to thedescription in Chapter 8.

Code Listing 11-3 containsthe EasyLanguage code forthe adaptive CCI indicator,and the identification of theindicatoritselfisnotedbythecomment near the endfollowing thedominant cyclecalculation. Since theobjective is touseonly thosefrequencycomponentspassedby the roofing filter, thevariableFilt isusedasadatainput rather than the averageofthehigh,low,andcloseas

wasdonebyLambert.

CodeListing11-3.EasyLanguageCodefortheAdaptiveCCI

{AdaptiveCCI(c)2013JohnF.Ehlers}Vars:AvgLength(3),M(0),N(0),X(0),Y(0),alpha1(0),HP(0),

a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lag(0),count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Period(0),Sp(0),Spx(0),MaxPwr(0),DominantCycle(0);Arrays:Corr[48](0),CosinePart[48](0),SinePart[48](0),SqSum[48](0),R[48,2](0),Pwr[48](0);//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48barsalpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*

(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];//SmoothwithaSuperSmootherFilterfromequation3a1=expvalue(-1.414*3.14159/10);b1=2*a1*Cosine(1.414*180/10);c2=b1;c3=-a1*a1;c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];//PearsoncorrelationforeachvalueoflagForLag=0to48Begin//SettheaveraginglengthasMM=AvgLength;IfAvgLength=0ThenM=Lag;Sx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;Forcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;

Syy=Syy+Y*Y;End; If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));End;ForPeriod=10to48BeginCosinePart[Period]=0;SinePart[Period]=0;ForN=3to48BeginCosinePart[Period]=CosinePart[Period]+Corr[N]*Cosine(360*N/Period);SinePart[Period]=SinePart[Period]+Corr[N]*Sine(360*N/Period);End;SqSum[Period]=CosinePart[Period]*CosinePart[Period]+SinePart[Period]*SinePart[Period];

End;

ForPeriod=10to48BeginR[Period,2]=R[Period,1];R[Period,1]=.2*SqSum[Period]*SqSum[Period]+.8*R[Period,2];

End;

//FindMaximumPowerLevelforNormalization

MaxPwr=.991*MaxPwr[1];

ForPeriod=10to48BeginIfR[Period,1]>MaxPwrThenMaxPwr=R[Period,1];

End;

ForPeriod=3to48BeginIfMaxPwr<>0ThenPwr[Period]=R[Period,1]/MaxPwr;

End;

//ComputethedominantcycleusingtheCGofthespectrum

Spx=0;

Sp=0;

ForPeriod=10to48BeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;

End;

IfSp<>0ThenDominantCycle=Spx/Sp;

IfDominantCycle<10ThenDominantCycle=10;

IfDominantCycle>48ThenDominantCycle=48;

Vars:

Price(0),AvePrice(0),RMS(0),Num(0),Denom(0),Ratio(0),MyCCI(0);

//AdaptiveCCIstartshere,usinghalfthemeasureddominantcyclefortuning

Price=Filt;

AvePrice=0;

Forcount=0toDominantCycle-1Begin;AvePrice=AvePrice+Price[count];

End;

AvePrice=AvePrice/DominantCycle;

RMS=0;

Forcount=0toDominantCycle-1Begin;RMS=RMS+(Price[count]-AvePrice[count])*(Price[count]-AvePrice[count]);

End;

RMS=SquareRoot(RMS/DominantCycle);

Num=Price-AvePrice;

Denom=.015*RMS;

Ratio=Num/Denom;

MyCCI = c1*(Ratio+Ratio[1])/2+c2*MyCCI[1]+c3*MyCCI[2];

Plot1(MyCCI);

Plot2(100);

Plot6(-100);

The first step in thecomputation is to find theaverage “price.” It isimportant to use this long

form to compute the averageprice rather than the fastalgorithmofdroppingoff1 /N of the oldest price andadding 1 / N of the currentprice to the average becauseN is varying from bar to bardue to the tuning using thedominant cycle period. Oncethe moving average iscomputed, it is used tocomputetherootmeansquare(RMS)of theprices.RMS issynonymous with the

standard deviation. Aftercreating the numerator andthe denominator, the ratio issmoothed using theSuperSmoother filterdescribed in Chapter 3. Thecoefficients for theSuperSmoother filters havepreviously been computed inthe dominant cyclemeasurementpartofthecode.I have included referencelines at the +100 and −100levels.

TheadaptiveCCIhasbeenapplied to Dollar General(symbol DG) over roughlythe past year in Figure 11.3.This indicator accuratelydisplays the level of thecurrent prices within achannel. The duration of thechannel length is determinedby the measured dominantcycle period. The adaptiveCCI exceeds the plus orminusone standarddeviationonlyoccasionally.

Figure11.3AdaptiveCCIShowsthePriceLocationwithintheChannelLengthDeterminedbytheMeasuredDominantCyclePeriod

■ AdaptiveBand-PassFilterIf a band-pass filter can bedesigned, it just makes sinceto tune that filter to themeasured dominant cycle toeliminate all the otherfrequency components thatare of no interest. The band-

pass filter was described inChapter5.Here, theadaptiveband-pass indicator startswith the computation of thedominant cycle using theautocorrelation periodogramapproach. Code Listing 11-4contains the EasyLanguagecode for the adaptive band-pass indicator, and theidentification of the indicatoritselfisnotedbythecommentnear the end following thecalculation of the dominant

cycle.

CodeListing11-4.EasyLanguage

CodetoComputetheAdaptive

Band-PassFilter{AdaptiveBandPassIndicator(c)2013JohnF.Ehlers

}

Inputs:Bandwidth(.3);

Vars:AvgLength(3),M(0),N(0),X(0),Y(0),alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lag(0),count(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Period(0),Sp(0),Spx(0),MaxPwr(0),DominantCycle(0);

Arrays:Corr[48](0),

CosinePart[48](0),SinePart[48](0),SqSum[48](0),R[48,2](0),Pwr[48](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//Pearsoncorrelationforeachvalueoflag

ForLag=0to48Begin//SettheaveraginglengthasMM=AvgLength;IfAvgLength=0ThenM=Lag;Sx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;Forcount=0toM-1BeginX=Filt[count];Y=Filt[Lag+count];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End; If (M*Sxx - Sx*Sx)*(M*Syy-Sy*Sy)>0ThenCorr[Lag]=(M*Sxy-Sx*Sy)/SquareRoot((M*Sxx-Sx*Sx)*(M*Syy-Sy*Sy));

End;

ForPeriod=10to48BeginCosinePart[Period]=0;SinePart[Period]=0;ForN=3to48BeginCosinePart[Period]=CosinePart[Period]+Corr[N]*Cosine(360*N/Period);SinePart[Period]=SinePart[Period]+Corr[N]*Sine(360*N/Period);End;SqSum[Period]=CosinePart[Period]*CosinePart[Period]+SinePart[Period]*SinePart[Period];

End;

ForPeriod=10to48BeginR[Period,2]=R[Period,1];R[Period,1]=.2*SqSum[Period]*SqSum[Period]+.8*R[Period,2];

End;

//FindMaximumPowerLevelforNormalization

MaxPwr=.995*MaxPwr;

ForPeriod=10to48BeginIfR[Period,1]>MaxPwrThenMaxPwr=R[Period,1];

End;

ForPeriod=3to48BeginPwr[Period]=R[Period,1]/MaxPwr;

End;

//ComputethedominantcycleusingtheCGofthespectrum

Spx=0;

Sp=0;

ForPeriod=10to48BeginIfPwr[Period]>=.5ThenBeginSpx=Spx+Period*Pwr[Period];Sp=Sp+Pwr[Period];End;

End;

IfSp<>0ThenDominantCycle=Spx/Sp;

IfDominantCycle<10ThenDominantCycle=10;

//AdaptiveBandPassindicatortunesaBandPassfilterto90%oftheperiodoftheDominantCycle

Vars:gamma1(0),alpha2(0),beta1(0),BP(0),Peak(0),Signal(0),

Lead(0),LeadPeak(0),LeadSignal(0);

beta1=Cosine(360/(.9*DominantCycle));

gamma1=1/Cosine(360*Bandwidth/(.9*DominantCycle));

alpha2=gamma1-SquareRoot(gamma1*gamma1-1);

BP = .5*(1 - alpha2)*(Filt - Filt[2]) + beta1*(1+alpha2)*BP[1]-alpha2*BP[2];

IfCurrentbar=1orCurrentBar=2ThenBP=0;

Peak=.991*Peak[1];

IfAbsValue(BP)>PeakThenPeak=AbsValue(BP);

IfPeak<>0ThenSignal=BP/Peak;

Lead = 1.3*(Signal+Signal[1]-Signal[2]-Signal[3])/4;

LeadPeak=.93*LeadPeak[1];

IfAbsValue(Lead)>LeadPeakThenLeadPeak=Absvalue(Lead);

IfLeadPeak<>0ThenLeadSignal=.7*Lead/LeadPeak;

Plot1(Signal);

Plot14(.9*Signal[1]);

Plot2(0);

Plot6(.707);

Plot10(-.707);

One way to make a band-pass filter have a leadingphasecapabilityistotunethefilter to aperiod shorter thanthe period of the cycle beingmeasured. In this case, thebandwidth of filter is set to

0.3.That is30percentof thetuned center period.Therefore,thehalfbandwidthis 15 percent. We tune thefilter tobe10percent towardthe shorter period from thedominant cycle period toprovide the phase lead whilestill having the data ofinterest be within the filterbandwidth. This provides aphase lead of the dominantcycle tobe somethingon theorder of 60 degrees, or one-

sixth of a cycle. If thedominantcyclewere18bars,for example, then thedetuning of the filter wouldproduce a 3-bar lead. Thisleading function is not huge,butitissignificant.Aconvenienttriggerlineis

included in the adaptiveband-pass filter to signal themore highly likely buy andsell points. The trigger iscomputeas90percentof the

amplitude of the adaptiveband-pass filter line and isdelayedbyonebar.Whiletheline crossings occur after thepeak of the band-pass filter,phase lead provides for thegenerationofa timelysignal.Significant trading signalsshould also include thecriteria that the line crossingoccuratgreaterthanthe+0.7and less than the −0.7referencelines.

The adaptive band-passfilter has been applied toDollar General (symbol DG)over roughly thepast year inFigure 11.4. This indicatoraccurately displays cyclicentry points when the twoindicator lines cross outsidethe plus or minus 0.7referencelines.

Figure11.4TheAdaptiveBand-PassFilterShowsBuyingandSelling

Opportunities

■ AdaptiveIndicatorComparisonIt is instructive to view theadaptive indicators on asingle chart so they can becompared on a head-to-headbasis. This comparison isavailable inFigure 11.5. Themost striking difference is

that the band-pass filter hasmore high-frequencycomponents. It is alsoapparent to me that the RSI,Stochastic,andCCIallprettymuchindicatethesamethingin the broad general sense.Without indicator transformsto enhance interpretation,which we cover in Chapter15, there isnooverwhelmingreason toselectone indicatoroveranother.

Figure11.5ComparisonoftheAdaptiveIndicators

■ Key PointstoRemember

1. When length is aninput parameter to anyindicator, that indicatorcan often be optimizedby tuning that length tothe dominant cycleperiod or a fractionthereof.2.Thepreferredmethod

of computing thedominant cycle is usingthe autocorrelationperiodogram.3. The RSI indicator isoptimized over half theperiod of the measureddominant cycle to reachfull-scaleswingswithoutoversaturation.4. The stochasticindicator is optimizedover the full period of

the measured dominantcycle to ensure thehighest close and thelowest close have beenincluded.5. The CCI indicator isoptimized over the fullperiod of the measureddominantcycletoensureacompletemeasurementof the standarddeviation.6.Theband-passfilteris

optimized by tuning thefilterto90percentofthemeasured dominantcycle period. Thisprovidesanapproximate60-degree phase lead intheindicator.

Notes1. P. J. Kaufman,Trading Systems andMethods, 3rd ed. (NewYork: John Wiley &Sons, 1998), pp. 436–438.

2. T. S. Chande and S.Kroll, The NewTechnical Trader (NewYork: John Wiley &Sons,1994).

3. J.WellesWilder, Jr.,New Concepts inTechnical TradingSystems (Winston-Salem, NC: Hunter,1978).

4. D. Lambert,Commodities Magazine,October1980.

CHAPTER12

TheEvenBetterSinewave

Indicator“This is heavy stuff,”saidTomlighty.

I first introduced theSinewave Indicator as earlyas 1996.1,2 The concept wasbased on the idea that notechnical indicator ispredictive.All indicators relyon historical data, and aretherefore their signals are

always lagging. However,cycles exist in the market,and it is often true that anobserved cycle will continuefor a short while into thefuture. Therefore, if we cantransfer the cyclic dataswingstoasinewave,wecanthen artificially advance thephase of the sine wave tocreateapredictiveindicator.Two conditions must be

satisfied to make the

Sinewave Indicator useful.First,themarketmustbeinacyclemode.Making a cyclicprediction when the data arein a trend is basically futile.Second, the period of thedominant cycle must beestimated with reasonableaccuracy. The downside oftheSinewaveIndicatoristhattheamplitudeswingsmustbefrom−1 to +1, and thereforethereisnodirectindicationofwhenthemarketisinatrend

mode.Since the roofing filter

givescontrolofthefrequencycontent of the data to beanalyzed, the basic idea is touse the frequency content toan advantage. This controlenables the Even BetterSinewave Indicator to notonly enter a trade as quicklyas possible, but also use thetrend component to give theconfidence to stay with an

existing trade without beingwhipsawed.

■ Even BetterSinewaveApproachThe original SinewaveIndicator was created byseeking the dominant cyclephase angle thathad thebestcorrelation between the pricedata and a theoreticaldominant cycle sine wave.

The Even Better SinewaveIndicator skips all the cyclemeasurements completelyand relies on a strongnormalization of thewaveform at the output of amodified roofing filter. Themodifiedroofingfilterusesasingle-polehigh-pass filter todeliberatelyretainthelonger-periodtrendcomponents.Thesingle-pole high-pass filterbasicallylevelstheamplitudeof all the cycle components

that would otherwise belarger with longerwavelengths due to SpectralDilation.Therefore,whenthewaveform is normalized tothe power in the waveformover a short period of time,the longer wavelengthcontributions tend to be anindication to stay in a tradewhenthemarketisinatrend.TheEvenBetter Sinewave

Indicator works

extraordinarilywellwhen themarket is in a trend mode.This means that thespectacular failures of mostswing wave indicators aremitigated when the expectedprice turning point does notoccur.Although I have not

studied it extensively, itappears that the Even BetterSinewave Indicator workswellonfutures intradaydata.

It takes a position in thecorrectdirectionand tends tostay with the good tradeswithout excessivewhipsawing.

■ Even BetterSinewaveDescriptionThe Even Better SinewaveIndicator is described withreference to theEasyLanguage code given inCode Listing 12-1. The onlyuser input for the indicator isthe expected maximum

duration of the trades whenthe market is in a trend.Making the duration inputparameter shorter willincreasethenumberoftradesover a given time span, andwill have a minor effect ofentering the trades a littlesooner. Conversely,increasing the duration inputparameter has the primaryeffect of holding andindicated trade positionlonger.

CodeListing12-1.EvenBetter

SinewaveIndicatorEasyLanguage

Code{EvenBetterSinewaveIndicator©2013JohnF.Ehlers

}

Inputs:Duration(40);

Vars:alpha1(0),HP(0),a1(0),

b1(0),c1(0),c2(0),c3(0),Filt(0),count(0),Wave(0),Pwr(0);

//HighPassfiltercycliccomponentswhoseperiodsareshorterthanDurationinput

alpha1=(1-Sine(360/Duration))/Cosine(360/Duration);

HP = .5*(1 + alpha1)*(Close-Close[1])+alpha1*HP[1];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//3BaraverageofWaveamplitudeandpower

Wave=(Filt+Filt[1]+Filt[2])/3;

Pwr=(Filt*Filt+Filt[1]*Filt[1]+Filt[2]*Filt[2])/3;

//NormalizetheAverageWavetoSquareRootoftheAveragePower

Wave=Wave/SquareRoot(Pwr);

Plot1(Wave);

After declaring variables,thedataisfilteredinasingle-pole high-pass filter. Asingle-pole high-pass filter isdeliberately selected to

equalize the data spectrumand allow trendingcomponentsthroughthefilterintotheindicator.Thedegreeof the spectrum allowedthrough the filter isestablished by the durationinput. The value of theduration input is set to beapproximatelythelengthofatradepositioninacontinuingtrend. The default value ofthe duration input is 40 bars,soamaximumtradeduration

of about two months can beexpected with this setting.Increasing the duration inputwill increase the maximumdurationofatradeinatrend.This has the implication thatyouwill be working throughsome additional drawdownsthroughout the tradewith thebenefit that you will not bewhipsawedoutofaprofitabletrendingtrade.The high-pass filtered data

is then filtered in aSuperSmoother filter whosecritical period is set to 10bars.After the data is passed

through themodified roofingfilter thethree-baraverageofthe wave amplitude and thepower are computed. Poweris proportional to the squareof the wave amplitude. TheEvenBetterSinewaveisthencomputedbynormalizing the

averaged wave amplitude tothe square root of theaveraged power, causing theindicator to swing between−1and+1.

■ Using theEven BetterSinewaveIndicatorThe Even Better SinewaveIndicator is shown in Figure12.1, using the defaultduration setting of 40 bars.This means that the

maximum length of a tradewhenthemarketisinatrendis about twomonths. This isbecause the high-pass filterstarts to attenuate the cycliccomponents that are longerthan40bars.

Figure12.1TheDefaultEvenBetterSinewaveIndicatorHoldsaPositionforaboutTwoMonthsinaTrend

Interpretation of the Even

Better Sinewave is simple.Holdalongpositionwhentheindicator value is near +1,and hold a short position (orgo flat if trading to the longsideonly)when the indicatorvalueisnear−1.The trade duration in a

trendcanbeextendedjustbyincreasing the duration inputvalue. For example, theindicator using a durationvalue of 40 is shown in

Figure12.2.Sureenough,theindicated longposition in theuptrend is extended. Doingthiscausessomeoftheshort-term reversals during thetrend to be eliminated. Theduration input parameter isadjusted to fit your tradingstyle.

Figure12.2TradePositionLengthisIncreasedbyIncreasingtheDurationInputParameter

Conversely, decreasing the

duration input parametershortens the maximum tradelengthwhenthemarketisinatrend,beingmoresensitivetotheshorterwavelengthsinthedata. For example, Figure12.3 shows the indicatorwhenthedurationinputissetto20bars.Somecare shouldbetakenwhenshorteningtheduration input because theresulting shorter trades aremore sensitive to entry andexittimingandcomputational

lagoftheindicator.

Figure12.3DecreasingtheDurationInputParameterMaketheEvenBetterSinewaveIndicatorMoreSensitivetoShort-TermVariationsintheData

■ Key PointstoRemember

1. The Even BetterSinewave Indicator is avariant of the roofingfilter,usingasingle-polehigh-passfilter.2. The unambiguoussignals of the EvenBetter SinewaveIndicator are generated

bynormalizingthewaveamplitude to the squarerootofthepower.3.TheonlyinputfortheEven Better SinewaveIndicator is the durationparameter.4. The durationparameter controls themaximum length of anindicated position bysettingthecriticalperiodofthehigh-passfilter.

Notes1. John F. Ehlers, “Stayin Phase,” Stocks &Commodities,November 1996, Vol.14,No.11,p.69.

2.JohnF.Ehlers,RocketScience for Traders:Digital SignalProcessing Applications(Hoboken, NJ: JohnWiley & Sons, 2001),

Chapter9.

CHAPTER13

Convolution“I need to know whenthe market reverses,”saidTompointedly.

Oneofthemajorobjectivesof technical analysis is todecisively identify a majorreversalsothatonecantradethe market primarily in thedirectionoftheensuingtrend.Convolution is just the tickettomeetthatobjective.In mathematics,

convolution is an operationon two functions thatproduces a third function.Convolution is similar to

cross-correlation between thetwo input functions—with atwist. A somewhat antiquename for convolution isfaltung,whichmeans foldingin German. It is this foldingconcept that makesconvolution useful fortrading.In this chapter, we will

spare you the details of themathematics and will jumpfrom the theoretical concept

tousefultradingexamples.

■ TheoreticalFoundationConsiderwhat happens at anidealizedmarketbottom.Theprices decrease linearly untilthe bottom is reached andthenincreaselinearlyafterthebottom has occurred. If wefold these idealized pricesabout themarket bottom, thetwo price segments are

perfectly correlated. That is,we have cross-correlated twomarket segments that havebeen folded at the horizontalpoint of the market bottom.This perfect correlationoccurs only at the idealizedmarket bottom that, in fact,establishes the need forprefiltering before thecorrelation is calculated sothat a relatively highcorrelation can be achievedusingrealdata.

Figure 13.1 shows twographic examples ofconvolution in an initialuptrend.Intheleft-handcase,the folding has been done inthe middle of a continuingtrend.Theresultantisthatthetwo price segments are notcorrelated after the foldingoperation. However, if thefolding isdoneatexactly themarketpeak,asshownintheright-handcase,thetwopricesegments are exactly

correlated after the foldingoperation.

Figure13.1HighCorrelationsOccurOnlyatMajorMarketTurningPoints

Since high correlationexists only at the marketturningpoint,theconvolutionindicator is dependent on thelookback period used in thecalculation. Assuming thetwo price segments have anequal timeduration, thepeakcorrelation occurs at half thelookback period of theindicator. For example, if a13-bar period is used, themarket peak would appear

witha7-bardelay.Thesamemarket peak would appearwith a 19-bar delay if a 39-bar lookback period wereused in the convolutioncomputation.The caseof themarketpeak'snotoccurringatthefoldingpointisillustratedin Figure 13.2, showing howthe correlation decreaseswhen the folding point doesnotoccuratthemarketpeak.

Figure13.2GoodCorrelation

OccursOnlyWhentheFoldingIsDoneattheMarketReversalPoint

Stillanothercase isshownin Figure 13.3, where themarket trend is brieflyinterruptedandthenresumes.As long as the lookbackperiodisequaltoorlessthantwice the distance betweenthetwoverticaldottedlines,ahigh-correlation event willoccur at half the lookbackperiod. However, when thelookback period becomeslonger, thenoncorrelated line

segment is included in theconvolution computation andthe overall correlation isreduced.

Figure13.3InterruptionsinMarketTrendsHaveHighCorrelationsforOnlyShortPeriods

When the correlation isplotted as a conventionalindicator,thehigh-correlationpointwillshowupataspikethat is delayed half thelookback period from thecurrent data bar. If thelookback period is short, thecorrelation spikes will berelatively current, but therewill be a lot of them. If thelookback period is long,convolution basically reports

ancienthistory,andtheshort-term interruptions will beeliminated.Consequently,theconvolution display isdependent on the lookbackperiod used in thecomputation.

■ Heat MapDisplayThe convolution indicatorcovers a range of lookbackperiods.Thevertical scale ofthe indicator is the lookbackperiod, and the indicatorvalue is converted to colors.An indicator is repeatedlycalculated for each lookbackperiod for each new bar of

data.Theresultisaheatmapdisplay that is in time syncwith the price data and thevalues of the indicator areshown for the full range ofusefullookbackperiods.When the convolution

computationsaredisplayedascolors over the range oflookback periods, theresultingchartshowsthehighcorrelation points as“plumes” that point back

preciselytothemajormarketreversals.Theinterruptionsinmarket trends show up asforeshorted and vestigialplumes.The convolution for

approximately the last yearon the Dollar General(symbol DG) is shown inFigure 13.4. By taking thedirection of the trend intoaccount, market tops aredisplayed as red plumes

(signaling a reversal to thedownside), and marketbottoms are displayed asgreen plumes (signaling areversal to the upside). Theforeshortened plumes inJanuary 2011 signal that theuptrendfromthefallof2010is still in play. For theremainder of 2011, the fiveredplumes and the sixgreenplumes absolutely nail themajor market reversals. Thereversal points are located at

thebottomoftheconvolutionsubgraph, and the longplumes identify each as amajor turning point. Since afinite amount of data isrequired tomake the shortestcalculation, convolution ismovedfourbarstothelefttobetter correlate the indicatorwith the actual turningpoint.This technique is similar tothat of a centered movingaverage.

Figure13.4ConvolutionNailedEachMarketReversaloverthePastYear

■ ComputingConvolutionFor those interested incomputing the convolutionindicator on their own, theprocess is relativelystraightforward. A filter tominimize the effects ofSpectralDilationprecedestheactual calculation, and theshortest and longest periods

ofthisfilterareprofitasuserinputs.Thedefaultvaluesare40 bars and 80 bars. Theshortest period can bereduced to reducecomputational lag, but at theexpense of additional displayartifacts near the bottom ofthe display. After declaringthe variables and the arrays,undesired long-wave cycliccomponentsareremovedbyatwo-pole high-pass filtertuned to the longest period.

The high-pass filter isfollowed by theSuperSmoother of Equation3-3, whose transfer responseis displayed in Figure 3.10.The 40-bar period for thisfilter was selected to reducethe displays of all but themajor turning points. Sincethe market data are fractal(thelongercycleperiodshavethelargerswings),wewanttoavoid spurious displays ofrelatively minor turning

points.Thus,thecombinationofthehigh-passfilterandtheSuperSmoother form a“roofing”filterthatpassesthedesired range of frequencycomponents useful forconvolution.Textbook Pearson

correlation is accomplishedfor each of the lookbackperiods from 1 to 48, and acorrelation value is assignedtoeach lookbackperiod.The

variable X is the base datastream, and thevariableY isthe data stream folded backforeachcalculationloop.ThevaluesofX,Y,X*X,X*Y,andY*Y are summed overthe lookback length beforebeing applied to thecorrelation equation. Thecorrelation resulting from thecomputations varies from −1to +1. These correlationvalues are applied to an

inverse Fisher transform2 tosharpen the display and thenrescaled to vary between 0and1forplottingpurposes.There is a considerable

amount of information to bedisplayed, and a heatmap isused to make sense of it all.Thegeneralschemeisthatforeach horizontal axis timeposition the verticaldisplacement is equal to thelag from 3 to 48. The

correlationvalueforeachlagisconvertedtoacolor.If theslope of the prices over thelookback period is positive,then the hue is red. Thesaturation of the red color isdetermined by the value ofthecorrelation.Iftheslopeofthe prices over the lookbackperiod is negative, then thehue is green. The saturationof the green color isdetermined by the value ofthecorrelation.

TheEasyLanguagecodetocompute the Convolutionindicator is given in CodeListing13-1.

CodeListing13-1.EasyLanguage

CodetoComputeandDisplayConvolution

{ConvolutionIndicator(c)2013JohnF.Ehlers

}

Inputs:ShortestPeriod(40),LongestPeriod(80);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),N(0),II(0),X(0),Y(0),Sx(0),Sy(0),Sxx(0),Syy(0),Sxy(0),Color1(0),Color2(0),Color3(0);

Arrays:

Corr[48](0),Slope[48](0),Convolution[48](0);

//HighPassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(1.414*360/LongestPeriod)+Sine(1.414*360/LongestPeriod)-1)/Cosine(1.414*360/LongestPeriod);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/ShortestPeriod);

b1=2*a1*Cosine(1.414*180/ShortestPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

ForN=1to48BeginSx=0;Sy=0;Sxx=0;Syy=0;Sxy=0;ForII=1toNBeginX=Filt[II-1];Y=Filt[N-II];Sx=Sx+X;Sy=Sy+Y;Sxx=Sxx+X*X;Sxy=Sxy+X*Y;Syy=Syy+Y*Y;End; If (N*Sxx - Sx*Sx)*(N*Syy-Sy*Sy)>0ThenCorr[N]=(N*Sxy-Sx*Sy)/SquareRoot((N*Sxx-Sx*Sx)*(N*Syy-Sy*Sy));Slope[N]=1;IfFilt[IntPortion(.5*N)]<FiltThenSlope[N]=-1;Convolution[N]=(1+(ExpValue(3*Corr[N])-1)/(ExpValue(3*Corr[N])+1))/2;

End;

//PlotasaHeatmap

ForN=3to48BeginIfSlope[N]>0ThenBegin

Color1=255*Convolution[N];Color2=0;End;IfSlope[N]<0ThenBeginColor1=0;Color2=255*Convolution[N];End;Color3=0; If N = 2 Then Plot2[4](2,“S2”,RGB(Color1,Color2,Color3),0,4); If N = 3 Then Plot3[4](3,“S3”,RGB(Color1,Color2,Color3),0,4); If N = 4 Then Plot4[4](4,“S4”,RGB(Color1,Color2,Color3),0,4); If N = 5 Then Plot5[4](5,“S5”,RGB(Color1,Color2,Color3),0,4); If N = 6 Then Plot6[4](6,“S6”,RGB(Color1,Color2,Color3),0,4); If N = 7 Then Plot7[4](7,“S7”,RGB(Color1,Color2,Color3),0,4); If N = 8 Then Plot8[4](8,“S8”,RGB(Color1,Color2,Color3),0,4); If N = 9 Then Plot9[4](9,“S9”,RGB(Color1,Color2,Color3),0,4); If N = 10 Then Plot10[4](10,“S10”,RGB(Color1,Color2,Color3),0,4); If N = 11 Then Plot11[4](11,“S11”,RGB(Color1,Color2,Color3),0,4); If N = 12 Then Plot12[4]

(12,“S12”,RGB(Color1,Color2,Color3),0,4); If N = 13 Then Plot13[4](13,“S13”,RGB(Color1,Color2,Color3),0,4); If N = 14 Then Plot14[4](14,“S14”,RGB(Color1,Color2,Color3),0,4); If N = 15 Then Plot15[4](15,“S15”,RGB(Color1,Color2,Color3),0,4); If N = 16 Then Plot16[4](16,“S16”,RGB(Color1,Color2,Color3),0,4); If N = 17 Then Plot17[4](17,“S17”,RGB(Color1,Color2,Color3),0,4); If N = 18 Then Plot18[4](18,“S18”,RGB(Color1,Color2,Color3),0,4); If N = 19 Then Plot19[4](19,“S19”,RGB(Color1,Color2,Color3),0,4); If N = 20 Then Plot20[4](20,“S20”,RGB(Color1,Color2,Color3),0,4); If N = 21 Then Plot21[4](21,“S21”,RGB(Color1,Color2,Color3),0,4); If N = 22 Then Plot22[4](22,“S22”,RGB(Color1,Color2,Color3),0,4); If N = 23 Then Plot23[4](23,“S23”,RGB(Color1,Color2,Color3),0,4); If N = 24 Then Plot24[4](24,“S24”,RGB(Color1,Color2,Color3),0,4); If N = 25 Then Plot25[4](25,“S25”,RGB(Color1,Color2,Color3),0,4); If N = 26 Then Plot26[4](26,“S26”,RGB(Color1,Color2,Color3),0,4);

If N = 27 Then Plot27[4](27,“S27”,RGB(Color1,Color2,Color3),0,4); If N = 28 Then Plot28[4](28,“S28”,RGB(Color1,Color2,Color3),0,4); If N = 29 Then Plot29[4](29,“S29”,RGB(Color1,Color2,Color3),0,4); If N = 30 Then Plot30[4](30,“S30”,RGB(Color1,Color2,Color3),0,4); If N = 31 Then Plot31[4](31,“S31”,RGB(Color1,Color2,Color3),0,4); If N = 32 Then Plot32[4](32,“S32”,RGB(Color1,Color2,Color3),0,4); If N = 33 Then Plot33[4](33,“S33”,RGB(Color1,Color2,Color3),0,4); If N = 34 Then Plot34[4](34,“S34”,RGB(Color1,Color2,Color3),0,4); If N = 35 Then Plot35[4](35,“S35”,RGB(Color1,Color2,Color3),0,4); If N = 36 Then Plot36[4](36,“S36”,RGB(Color1,Color2,Color3),0,4); If N = 37 Then Plot37[4](37,“S37”,RGB(Color1,Color2,Color3),0,4); If N = 38 Then Plot38[4](38,“S38”,RGB(Color1,Color2,Color3),0,4); If N = 39 Then Plot39[4](39,“S39”,RGB(Color1,Color2,Color3),0,4); If N = 40 Then Plot40[4](40,“S40”,RGB(Color1,Color2,Color3),0,4); If N = 41 Then Plot41[4]

(41,“S41”,RGB(Color1,Color2,Color3),0,4); If N = 42 Then Plot42[4](42,“S42”,RGB(Color1,Color2,Color3),0,4); If N = 43 Then Plot43[4](43,“S43”,RGB(Color1,Color2,Color3),0,4); If N = 44 Then Plot44[4](44,“S44”,RGB(Color1,Color2,Color3),0,4); If N = 45 Then Plot45[4](45,“S45”,RGB(Color1,Color2,Color3),0,4); If N = 46 Then Plot46[4](46,“S46”,RGB(Color1,Color2,Color3),0,4); If N = 47 Then Plot47[4](47,“S47”,RGB(Color1,Color2,Color3),0,4); If N = 48 Then Plot48[4](48,“S48”,RGB(Color1,Color2,Color3),0,4);

End;

■ Key PointstoRemember

1. Convolution issynonymous with“folding.” Theconvolution indicatorfinds price reversalpointsbyidentifyingthehighestcorrelationofthedata by folding it in thetime dimension about

thelookbackperiod.2. The major pricereversals are identifiedby plumes pointingbackward to the time ofthepricereversal.3.Topsandbottomscanbe identified by colorusing the price slope toassignthecolorvalues.

Notes1. John Ehlers and RicWay, “SwamiChartsConvolution,” Stocks &Commodities, October2012, Vol. 30, No. 10,pp.10–13.

2. John Ehlers, “TheInverse FisherTransform,” Stocks &Commodities, 2004,Vol. 22, No. 5, pp.

38−42.

CHAPTER14

TheHilbertTransformer

“The wave has animaginary component,”

saidTomrealistically.

An analytic signal allowsfor time-variable parametersand isageneralizationof thephasor concept, which isrestricted to time-invariantamplitude, phase, andfrequency. The analyticrepresentation of a real-valued function or signalfacilitatesmanymathematicalmanipulations of the signal.For example, computing the

phaseofasignalorthepowerin the wave is much simplerusinganalyticsignals.The Hilbert transformer is

the technique to create ananalytic signal from a realone.TheconventionalHilberttransformeristheoreticallyaninfinite-length FIR filter.Evenwhenthefilterlengthistruncatedtoausefulbutfinitelength, the induced lag is fartoo large to make the

transformer useful fortrading.I have therefore created a

modified Hilbert transformerthat has only a small lag butstill has performancecharacteristics superior tothose of a truncated FIRHilberttransformer.The modified Hilbert

transformercanbeusedasaneffective trading indicatorbecause the quadrature

component output leads thereal output in the samemanneracosinewaveleadsasine wave. In this sense, amodified Hilbert transformerisapredictiveindicator.In this chapter, I

demonstrate three differentways to measure the time-variantdominantcycleperiodusing analytic signals. Theseare all basically measuringtheinstantaneousphaseofthe

signal and, in essence,summing the phases toestimate the cycle period.Sincethesmoothinglengthisusually less than one fullcycle period, the results arefairlynoisyandare thereforeonly marginally useful fortrading.Iincludethecodeforthese dominant cyclemeasurements primarily forcompleteness and becausesomeone smarter than memay be able to use them

successfully.

■ AnalyticSignalsWithout getting allmathy onyou, I will try a simplifieddescription of the theory ofnumbers for you to reach anunderstanding of analyticsignals.Thenumbersthatyouknow and love extend fromminusinfinitytoplusinfinityalongastraightline.Thisline

includes integers, rationalnumbers, irrational numbers,and even zero. This is notexactly as trivial as it seemsbecausetheRomannumerals,for example, had no conceptof zero. It wasn't until 976A.D. that the Persianencyclopedist Muhammadibn Ahmad al-Khwarizmi(fromwhomwegetthewordalgorithm) used the notationand wasn't until 1202 A.D.thatFibonnaci introduced the

concept of zero to Europe atthe beginning of theRenaissance. Of course, thatwas beforemargin calls, andso the idea of negativenumbers didn't arise untilrelatively recently. With thatdigression, the main idea isthatall“real”numberscanbepicturedasbeingonastraightline.A straight line is not the

only constructwe can use to

visualize a numberingsystem. For example, if weplace numbers on a planesurface we can describenumbers by their location onthe plane. We do this byusing a “real” axis and an“imaginary” axis that is atright angles to the real axis.Thenamesarenotnecessarilysignificant except in ahistoricalcontext.Afterall,ifwe can have irrationalnumbers we can have an

imaginaryaxis.Figure 14.1 shows the

relevance of analytic signalstotradersusingindicators.Asa function of time the sinewave starts at zero andincreases toward itsmaximum value while thecosine wave starts at itsmaximum value anddecreases in amplitude. Notethat the cosine wave crosseszero just as the sine wave

reaches its maximum valueand has zero slope. Thecosine wave represents therate of change of the sinewave. These two waveformscan also be represented asvectors in the complex planehavingtherealandimaginaryaxes. The waves in the timedomain are represented byprojections on the imaginaryaxisas the twovectors rotatecounterclockwise. A fullcycle in the time domain is

completed when the vectorsrotate a full 360 degrees inthecomplexplane.

Figure14.1SineandCosineWavesasAnalyticSignals

Notethatthevectorsinthe

complex plane are at rightangles, and are said to beorthogonal. Because of theright angle property, I oftenrefertothesinecomponentasthe in-phase component, andthe cosine component is thequadrature component. Moregenerally, orthogonalitymeansthat theproductof thetwo vectors is uncorrelated.For example, the sum ofsin(x) * cos(x) over a fullcycle iszero.Knowingabout

analytic signals opens thedoor to making someinteresting calculations usingmarket data. For example,sin2(x) + cosine2(x) is equalto a constant.We also knowthis frombasic trigonometry.Failingtocomputethesumofthesquaresasaconstantisanindication of the failure ofcreatingananalyticsignal.Inthis case, that failure meansthe market is probably

transitioning between a cyclemodeandatrendmode.Analytic signals also open

thepossibilityofmakingveryrapid measurements of thedominant cycle period byexamining the ratechangeofphase angle of the vectors inthe complex plane. Forexample if the vector anglechanges 18 degrees per bar,the dominant cycle periodmustbe20barsbecause that

rateofchangegivesatotalof360 degree over the 20-barperiod.

■ HilbertTransformerMathematicsA Hilbert transformer is justanFIRfilterwhosepurposeisto create a quadraturecomponentfromtherealdatastream. The filter length istheoretically infinite, but canbe truncated as an

approximation. The filtercoefficients can be expressedasSin2(πn/2)/n,wherenisthe position from the centerof the filter. Coefficientsolder thanhalf thefilterhaveanegativesign.Forexample,a 23-element Hilberttransformer filter hascoefficientsas:

The frequency response of

this filter is shown in Figure14.2. This filter outputresponse has ripplethroughout its pass band androll-offatzerofrequencyandtheNyquist frequencydue totruncationof thecoefficients.Ideally, the Hilberttransformer is an all-passfilter.Wecanmanipulate thecoefficients to reduce theamount of in-band ripple;however, it is not worth theeffort. The problem is that

this FIR filter has a 11-barlag.More generally, the FIRfilterlagis(N−1)/2,wherethere are an odd number ofelements in the filter. In ourcase, a lag of 11 bars is justunacceptablefortrading.

Figure14.2FrequencyResponseofa23-ElementHilbertTransformer

The lag of a Hilberttransformer can be reducedby reducing the number ofelements in the filter. In thelimit, the shortest truncatedHilbert transformer has thecoefficientsas:

This shorter Hilberttransformerisnicefortradingbecause ithasonlyaone-barlag. However, the frequencyresponseof the three-element

Hilbert transformer has asubstantial amplitude slopeacross the pass band, asshown in Figure 14.3. Thepass band of interest extendsfrom approximately a 50-barperiod(frequency=0.02)toa10-bar period (frequency =0.1). The shorter Hilberttransformer is also notacceptable for tradingbecause of the hugeamplitude variation over thepassband.

Figure14.3FrequencyResponseofaThree-ElementHilbertTransformer

Figure14.4TheHilbertTransformerIndicatorHastheCorrectPhaseandAmplitudeasShownbytheResponsetoaChirpedSineWaveWhosePeriodVariesfrom10Barsto40Bars

I have included the

EasyLanguage code tocompute the Hilberttransformer as a classictruncated FIR filter in CodeListing14-1becauseitcanbeuseful for comparing resultsto the modified Hilberttransformer I will derive.After declaring variables, thecodebeginsbycomputingtheroofing filter as acombination of a two-polehigh-pass filter and aSuperSmoother filter. The

automaticgaincontrol(AGC)technique described inChapter5isusedtocomputethe normalized Real signal.ThevariableRealisbasicallythe same as the variable Filtexcept that it has anormalized amplitude. Theimaginary signal, calledImag, is computed from theFIR filtering of the variableReal.ThentheRealandImagsignals are plotted.Note thatthe plot for the imaginary

signalisartificiallymovedtothe left by 11 bars tocompensate for the ladinduced by the FIR filter. Ifthe Real and Imag signalswere to be used for trading,Real would have to bedelayed by 11 bars to havethe correct relationship toImag.

CodeListing14-1.EasyLanguage

CodetoComputeaClassicHilbertTransformer

{ClassicHilbertTransformer©2013JohnF.Ehlers}

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Real(0),Imag(0),IPeak(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

IPeak=.991*IPeak[1];

IfAbsvalue(Filt)>IPeakThenIPeak=AbsValue(Filt);Real=Filt/IPeak;

//TruncatedHilbertTransformFIRfilterasatestImag=(.091*Real+.111*Real[2]+.143*Real[4]+.2*Real[6]+.333*Real[8]+Real[10]-Real[12]-.333*Real[14]-.2*Real[16]-.143*Real[18]-.111*Real[20]-.091*Real[22])/1.865;

Plot1(Real);

Plot2(0);

Plot6[11](Imag);

■ Computingthe HilbertTransformerThe Hilbert TransformIndicator is described withreference to theEasyLanguagecodeshowninCode List 14-2. Thecomputations begin with theroofing filter as with the

classic Hilbert transformerexcept that thecriticalperiodof the SuperSmoother isprovided as a user-selectableinput.Thisgivestheindicatormore flexibility in providinggreaterorlessersmoothingasdesired.

CodeListing14-2.EasyLanguage

CodetoCompute

theHilbertTransformer

{HilbertTransformer©2013JohnF.Ehlers}

Inputs:LPPeriod(20);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),QFilt(0),Real(0),Imag(0),IPeak(0),

QPeak(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPPeriod);

b1=2*a1*Cosine(1.414*180/LPPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

IPeak=.991*IPeak[1];

IfAbsvalue(Filt)>IPeakThenIPeak=AbsValue(Filt);Real=Filt/IPeak;

QFilt=(Real-Real[1]);

QPeak=.991*QPeak[1];

IfAbsvalue(QFilt)>QPeakThenQPeak=AbsValue(QFilt);Imag=QFilt/QPeak;

Plot1(Real);

Plot2(0);

Plot6(Imag);

The Real component iscomputedexactlyasitwasinthe classic Hilberttransformer, by normalizingFilttoitsrecentpeakvalueas

determined by the AGCalgorithm.The imaginary component

calculation starts by takingthe one bar difference of theReal variable. This isanalogous to taking thederivative of a sine wave togenerate a cosine wave. Theone-bar differencecomputation provides thephasequadrature requiredforthe imaginary component.

The amplitude correction isprovided by a second AGCaction performed on theimaginary component alone.Having the phase quadratureandthenormalizedamplitudefulfills all the requirementsfortheimaginarycomponent.The lag of the Real and

Imag analytic signals is verysmall, and no compensationin the plotting routine isrequired. In other words, an

indicator has been createdthat has a predictivecapability.The outstanding

performance of the Hilberttransformer is bestdemonstrated by its responsetoachirpedsinewavewhoseperiodcontinuously increasesfrom 10 bars to 40 bars.Throughout the entire range,the indicator maintains thecorrect phase and amplitude

relationship between the realand imaginary components.This performance is vastlysuperior to thatof theclassicHilbert transformer using atruncatedFIRfilter.

■ The HilbertTransformerIndicatorFigure 14.5 shows the realandimaginarycomponentsoftheHilberttransformerbelowthepricechart.Basically, thereal component moves withthe general direction of theprices, and the imaginary

component is a predictiveindicator for the realcomponent in thesamesensethat a cosine wave is apredictor of a sine wave.Although the defaultLPPeriod input is set to 20bars in an attempt to smooththe indicator the imaginarycomponent is still too erratictobeuseful.

Figure14.5InitialApplicationoftheHilbert

TransformertoDG

Code Listing 14-3 showsthe EasyLanguage code forthe Hilbert transformerindicator.Thecodeisexactlythe same as for the Hilberttransformer except that aSuperSmoother filter hasbeen added to smooth theimaginarycomponent.

CodeListing14-3.EasyLanguage

CodetoCompute

theHilbertTransformerIndicator

{HilbertTransformerIndicator©2013JohnF.Ehlers}

Inputs:LPPeriod(20);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),QFilt(0),Real(0),

Quadrature(0),IPeak(0),QPeak(0),Imag(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPPeriod);

b1=2*a1*Cosine(1.414*180/LPPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

IPeak=.991*IPeak[1];

IfAbsvalue(Filt)>IPeakThenIPeak=AbsValue(Filt);Real=Filt/IPeak;

Quadrature=(Real-Real[1]);

QPeak=.991*QPeak[1];

IfAbsvalue(Quadrature)>QPeakThenQPeak=AbsValue(Quadrature);Quadrature=Quadrature/QPeak;

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Imag = c1*(Quadrature+Quadrature[1])/2+c2*Imag[1]+c3*Imag[2];

Plot1(Real);

Plot2(0);

Plot6(Imag);

The performance of theHilbert transformer indicatoris demonstrated in Figure14.6.Thesmoothingaffordedby the added SuperSmootherfilter destroys theorthogonality of theimaginary componentbecause it introduces severalbars of lag. However, theSuperSmoother made the

imaginary component usefulby smoothing out its erraticbehavior. Now the predictedturningpoint to theupside iswhen the imaginarycomponent crosses over thereal component. This givesyoutimetoactuallyplacethetradebefore the turningpointhas occurred. Similarly, thepredicted turningpoint to thedownside is when theimaginarycomponentcrossesunder the real component.

You have some control overthe indicator through theLPPeriodinputparameter.

Figure14.6TheImaginaryComponentoftheHilbertTransformerIndicatorPredictstheTurningPointsoftheRealComponent

The problem with

predictingtheturningpointisthat it may not be a swingturningpointatall.Rather, itcan be the onset of a newtrend.Inthiscase,theturningpoint predictionwill be deadwrong. One way to mitigatean incorrect turning pointprediction is to first take thetrade in the predicteddirection, but then be quickandagiletoexitthetradeifitisnotconfirmedby theEvenBetterSinewaveIndicator.

■ Using theHilbertTransformerto Computethe DominantCycleI want to emphasize that theonlyreasonforincludingthis

section is for completeness.Unless you are interested inresearch, I suggest you skipthis section entirely. Tofurther emphasize my point,do not use the code fortrading. A vastly superiorapproach to compute thedominant cycle in the pricedata is the autocorrelationperiodogram. The code isincluded because the readermay be able to capitalize onthealgorithmsinawaythatI

donotsee.Allthealgorithmsencapsulated in the codeoperate reasonably well ontheoretical waveforms thathavenonoisecomponent.Myconjectureat this time is thatthe sample-to-sample noisesimply swamps thecomputation of the ratechange of phase, andtherefore the resultingcalculations to find thedominant cycle are basicallyworthless. The imaginary

component of the Hilberttransformer cannot besmoothedaswasdone in theHilbert transformer indicatorbecause the smoothingdestroys the orthogonality oftheimaginarycomponent.

■ DualDifferentiatorThe first algorithm tocompute the dominant cycleis called the dualdifferentiator.Inthiscase,thephaseangleiscomputedfromthe analytic signal as thearctangent of the ratio of theimaginary component to thereal component. Further, the

angular frequency is definedas the rate change of phase.We can use these facts toderivethecycleperiod.Fromthe definition of thederivative of an arctangent,the mathematics of thisprocessare:

Simplifyingandsolvingforthe cycle period instead offrequency,weobtain:

Where: IP = In-phasecomponent(Real)

Q = Quadraturecomponent(Imag)IDot=RateofchangeofIPQDot = Rate of change

ofQThe code to compute the

dominantcycleusingthedualdifferentiatorisgiveninCodeListing14-4.

CodeListing14-4.EasyLanguage

CodetoComputetheDominantCycleUsingthe

DualDifferentiatorMethod

{MeasuringtheDominantCycleusingtheDualDifferentiator©2013JohnF.Ehlers}

Inputs:LPPeriod(20);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),QFilt(0),Real(0),Quad(0),Imag(0),

IPeak(0),QPeak(0),IDot(0),QDot(0),Period(0),DomCycle(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPPeriod);

b1=2*a1*Cosine(1.414*180/LPPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;

Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

IPeak=.991*IPeak[1];

IfAbsvalue(Filt)>IPeakThenIPeak=AbsValue(Filt);

Real=Filt/IPeak;

Quad=(Real-Real[1]);

QPeak=.991*QPeak[1];

IfAbsvalue(Quad)>QPeakThenQPeak=AbsValue(Quad);Imag=Quad/QPeak;

IDot=Real-Real[1];

QDot=Imag-Imag[1];

If(Real*QDot-Imag*IDot)<>0ThenPeriod=6.28318*(Real*Real + Imag*Imag) / (-Real*QDot+Imag*IDot);

IfPeriod<8ThenPeriod=8;

IfPeriod>48ThenPeriod=48;DomCycle = c1*(Period+Period[1])/2+c2*DomCycle[1]+c3*DomCycle[2];

Plot1(DomCycle);

■ PhaseAccumulationThe next algorithm tocompute the dominant cycleis the phase accumulationmethod. The phaseaccumulation method ofcomputing the dominantcycleisperhapstheeasiesttocomprehend. In thistechnique, we measure the

phase at each sample bytaking the arctangent of theratio of the quadraturecomponent to the in-phasecomponent. A delta phase isgenerated by taking thedifference of the phasebetween successive samples.At each sample we can thenlook backwards, adding upthe delta phases. When thesum of the delta phasesreaches360degrees,wemusthave passed through one full

cycle, on average. Theprocess is repeated for eachnewsample.The phase accumulation

methodofcyclemeasurementalways uses one full cycle'sworthof historical data.Thisis both an advantage and adisadvantage. The advantageis the lag in obtaining theanswer scales directly withthe cycle period. That is, themeasurementofashortcycle

period has less lag than themeasurement of a longercycle period. However, thenumber of samples used inmaking the measurementmeanstheaveragingperiodisvariable with cycle period.Longeraveragingreducesthenoise level compared to thesignal. Therefore, shortercycle periods necessarilyhave a higher output signal-to-noiseratio.

The code to compute thedominant cycle using thephaseaccumulationmethodisgiveninCodeListing14-5.

CodeListing14-5.EasyLanguage

CodetoComputetheDominantCycleUsingthe

Phase

AccumulationMethod

{MeasuringtheDominantCycleusingPhaseAccumulation©2013JohnF.Ehlers}

Inputs:LPPeriod(20);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),QFilt(0),Real(0),Quad(0),Imag(0),

IPeak(0),QPeak(0),Phase(0),DeltaPhase(0),InstPeriod(0),count(0),PhaseSum(0),DomCycle(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPPeriod);

b1=2*a1*Cosine(1.414*180/LPPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

IPeak=.991*IPeak[1];

IfAbsvalue(Filt)>IPeakThenIPeak=AbsValue(Filt);Real=Filt/IPeak;

Quad=(Real-Real[1]);

QPeak=.991*QPeak[1];

IfAbsvalue(Quad)>QPeakThenQPeak=AbsValue(Quad);Imag=Quad/QPeak;

//UseArcTangenttocomputethecurrentphaseIfAbsValue(Real)>0thenPhase=ArcTangent(AbsValue(Imag/Real));

//ResolvetheArcTangentambiguity

IfReal<0andImag>0thenPhase=180-Phase;

IfReal<0andImag<0thenPhase=180+Phase;

IfReal>0andImag<0thenPhase=360-Phase;

//Computeadifferentialphase,resolvephasewraparound,andlimitdeltaphaseerrors

DeltaPhase=Phase[1]-Phase;

IfPhase[1]<90andPhase>270thenDeltaPhase=360+Phase[1]-Phase;

//LimitDeltaPhasetobewithintheboundsof10barand48barcycles}

IfDeltaPhase<10thenDeltaPhase=10;

IfDeltaPhase>48thenDeltaphase=48;

//SumDeltaPhasestoreach360degrees.Thesumistheinstantaneousperiod.

InstPeriod=0;

PhaseSum=0;

Forcount=0to40beginPhaseSum=PhaseSum+DeltaPhase[count];IfPhaseSum>360andInstPeriod=0thenbeginInstPeriod=count;End;End;

//ResolveInstantaneousPerioderrorsandsmooth

IfInstPeriod=0thenInstPeriod=InstPeriod[1];DomCycle = c1*(InstPeriod+InstPeriod[1])/2+c2*DomCycle[1]+c3*DomCycle[2];

Plot1(DomCycle);

■HomodyneThe third algorithm forcomputing the dominantcycle is the homodyneapproach. Homodyne meansthe signal is multiplied byitself. More precisely, wewanttomultiplythesignalofthe current bar with thecomplex value of the signalone bar ago. The complexconjugate is, by definition, a

complex number whose signof the imaginary componenthasbeenreversed.Expressingthe signal in polarcoordinates,thearithmeticis:

The interesting result isthatwegetboththesquareofthe signal amplitude and theangular frequency (2π /Period) from the productbecausethedifferenceintimebetweensamples(tn−tn−1)is

justonebar.Inprinciple,thismeans that we can get theinstantaneous cycle period injust two successive samples.The added benefit is that wealso get the square of thesignal amplitude. Thecalculations are carried outusing the in-phase andquadraturecomponentsratherthanconvertingthemtopolarcoordinates. Either way, theresultswillbethesame.

Since thedifferentialangleis small in radian measure,the angle, the tangent of theangle and the arctangent ofthe angle are allapproximately the same.Therefore,thedominantcycleis computed by dividing 2πby the ratio of the imaginarycomponent to the realcomponent.Ideally,theanglecannot be negative becausetime cannot run backwards.Frankly, the purpose of the

smoothing filter is to covertheerraticcalculationscausedbynoise.The code to compute the

dominant cycle using thehomodynemethodisgiveninCodeListing14-6.

CodeListing14-6.EasyLanguage

CodetoComputetheDominant

CycleUsingtheHomodyneMethod

{MeasuringtheDominantCycleusingaHomoDyneDiscriminator©2013JohnF.Ehlers}

Inputs:LPPeriod(20);

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),QFilt(0),

Real(0),Quad(0),Imag(0),IPeak(0),QPeak(0),Re(0),Im(0),Period(0),DomCycle(0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/LPPeriod);

b1=2*a1*Cosine(1.414*180/LPPeriod);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

IPeak=.991*IPeak[1];

IfAbsvalue(Filt)>IPeakThenIPeak=AbsValue(Filt);Real=Filt/IPeak;

Quad=(Real-Real[1]);

QPeak=.991*QPeak[1];

IfAbsvalue(Quad)>QPeakThenQPeak=AbsValue(Quad);Imag=Quad/QPeak;

Re=Real*Real[1]+Imag*Imag[1];

Im=Real[1]*Imag-Real*Imag[1];IfIm<>0andRe<>0ThenPeriod=6.28318/AbsValue((Im/Re));

IfPeriod<10ThenPeriod=10;

IfPeriod>48ThenPeriod=48;DomCycle = c1*(Period+Period[1])/2+c2*DomCycle[1]+c3*DomCycle[2];

Plot1(DomCycle);

■ Key PointstoRemember

1. The classic Hilberttransformer uses atruncatedFIRfilter.2. The classic Hilberttransformer cannot beused for tradingbecausethe FIR filter introducesa large amount of lag,and shortening the FIR

filter introduces largeamplitude variationsacrossthepassband.3. A modified Hilberttransformer can becreated using a one-bardifference to establishphase quadrature andAGC to provideamplitudecompensation.4. The Hilberttransformer indicatorprovides predictive

turning points for swingtrading. Thesepredictionscanbewrongif a new trend isestablished rather than anewswing.5. Using the Hilberttransformer to computethedominantcycleisthepricedataisnotadvised.

CHAPTER15

IndicatorTransforms

“Now I see,” said Tomblindly.

An indicator transform is ameans to shape the indicatorto aid interpretation and useof the indicator itself. Otherthan shaping the display,these transforms have noimpact on the indicator. Forexample, they induce no lag.While relatively trivial toimplement, these transformshave a huge impact on theinterpretation and use of theindicators. There are

primarily three indicators ofinterest to traders: the Fishertransform, the inverse Fishertransform, and the cubetransform. Each has itspreferred application, whichwedescribeinthischapter.

■ FisherTransformAcommon fallacy in theuseof technical indicators is thattheir probability distributionfollows a normal, orGaussian, probabilitydistribution. That just is notso, as you can prove foryourself by analyzing areasonableamountofdata. If

thedatadonothaveanormalprobability distribution, thenthe calculation of a standarddeviation is based on a falsepremise. The purpose of theFisher transform is to takeany indicator having anominally zero mean andbounded between the limitsof −1 to +1 and convert theamplitude so that thetransformed indicator has anapproximate normalprobabilitydistribution.

Plottinganindicatorscaledto a normal probabilitydistribution has a hugeadvantage on indicatorinterpretation. If thetransformed indicator has avalueof−1, ithasavalueofnegative one standarddeviation and therefore thereis a32percent chancepriceswill go lower. If thetransformed indicator has avalueof−2, ithasavalueofnegative two standard

deviations, and thereforethere is only an 8 percentchance prices will go lower.This is a high-probabilitybuyingopportunity.Atalevelof−3,thenegative3standarddeviations means there isonlya2percentchanceoftheprices going lower. Thetransformed values aresymmetrical, so positivedeviations are high-probability indications toexita long position or to sell

short.The equation for a Fisher

transformis

Since an input value ofunitycancausetheratiotogoto infinity, the inputmust belimited to 0.999, whichcorresponds to an output onthe order of three standarddeviations. Similarly, theargument of the natural

logarithm cannot be allowedtogotozero,andsotheinputmust also be limited at−0.999.The transfer response of

theFishertransformisshownin Figure 15.1. The straightline shows the range ofinputs, and the curved lineshows the correspondingrange of outputs. When theinputs are less than anabsolute value of 0.5, the

outputs are almost the sameas the inputs.However, thereis a greater and greatermagnification of the outputsas the inputs approach theabsolute value of 1. It is thisnonlinear magnification thatproduces the “long tails” inthe output probabilitydistribution, thuscreating theresulting normal probabilitydistribution.

Figure15.1Transfer

ResponseoftheFisherTransformShowsGreaterMagnificationoftheOutputsastheInputsApproachanAbsoluteValueof1

To implement the Fisher

transform on the adaptiveRSI, replace the plotstatements of Code Listing11-1with the followingcodefragment:

CodeListing15-1.EasyLanguage

CodeFragmenttoAddtheFisherTransformtotheAdaptiveRSI

IndicatorVars:TranslatedRSI(0),AmplifiedRSI(0),Fish(0);

TranslatedRSI=2*(MyRSI-.5);

AmplifiedRSI=1.5*TranslatedRSI;

IfAmplifiedRSI>.999ThenAmplifiedRSI=.999;

IfAmplifiedRSI<-.999ThenAmplifiedRSI=-.999;

Fish=.5*Log((1+AmplifiedRSI)/(1-AmplifiedRSI));

Plot1(Fish);

Plot2(2);

Plot6(-2);

ThevariableMyRSIrangesbetween zero and one, andthereforetoaccommodatetheconditions for the Fishertransform, this variable mustbe translated and dilated torange between −1 and +1. IftheMyRSI variable does notrange fullybetweenzeroandone,youcanshorten theRSIlookback period to be lessthan half the measureddominant cycle, or you cansimply multiply it by a

magnificationfactorasIhavedone in the code fragment.The amplifying factor wasselectedtomaketheindicatorrarely exceed the two sigmapoints in the output. TheamplifiedRSI is then limitedto be within the range of−0.999 to +0.999 to avoid acomputer crash, and then isused to compute the Fishertransform. The plus andminus twostandarddeviationlevels are included in the

indicatordisplay.AnexampleoftheadaptiveRSIusingtheFisher transform is shown inFigure 15.2. The trend up isheralded by the Fisherizedindicator's falling below −2.Then, the trend down issignaled by the Fisherizedindicator rising above +2.These major turning pointsare relatively rare events andare perhaps best used inassociation with trendtrading.

Figure15.2AdaptiveRSIwithFisherTransformClearlyIdentifiesMajorTurningPoints

Since the Stochastic and

commodity channel index(CCI) indicators measurerelativepositioninachannel,theytendtobeneartheupperor lower limitsduring trends.That means that they do nothaveazeromeanintheshortrun and therefore are notgood candidates on which toapplytheFishertransform.Aband-pass filter would be amuchbettercandidate,butthemajor cyclic turning pointsare already easy to identify

and don't require a morecomplexstatisticalapproach.

■ InverseFisherTransformWhereastheFishertransformexpands the base indicatornear the extremes, thepurposeof the inverseFishertransform is to compress theindicatorintheseregions.Bycompressing values near the

extremes many extraneousand irrelevant wiggles areremoved from the indicator,making interpretation of therealmeaning of the indicatorbefarsimpler.Inasense,theinverse Fisher transform actsasasoftlimiter.The equation for the

inverseFishertransformis

The input may have as

largeavalueasdesired.Iftheinput values are very large,then theexponential terms inboth the numerator anddenominator are very largecompared to one, and so theoutput is approximately 1. Ifthe input values are verylarge, but negative, then theexponential terms inboth thenumerator and denominatorare very small compare toone, with the result that theoutput is approximately −1.

TheKintheexpressionisanoptional amplifying termbecause conventionalindicators already fallbetween the limits of−1 and+1,andsoamplifyingthemisnecessarytoreapthebenefitsof the soft limiting of theinverseFishertransform.The transfer response of

the inverse Fisher transformis shown inFigure15.3. Theinputvaluesareshownbythe

straight line, and the outputvalues are shown by thecurved line. This graphicalrepresentation shows howlarger and larger absolutevalues of the input arecompressed to essentially beunity.

Figure15.3TransferResponseoftheInverseFisherTransformShowstheCompressionofLargerValuesoftheInput

To implement the inverse

Fisher transform on anadaptive Stochastic indicator,replacetheplotstatementsofCode Listing 11-2 with thecode fragmentgiven inCodeListing15-2.Ihavechosentoamplify the adaptiveStochastic by a factor of 3when taking the inverseFishertransform.

CodeListing15-2.EasyLanguage

CodeFragmenttoImplementtheInverseFisher

TransformontheAdaptiveStochasticIndicator

Vars:IFish(0);

Value1 = 2*(AdaptiveStochastic-.5);IFish=(ExpValue(2*3*Value1)-1)/(ExpValue<(2*3*Value1)+1);

Plot1(IFish);

Plot4(.9*IFish[1]);

I have also included atrigger line,which is just theinverse Fisher transformdelayed by one bar andattenuated to90percent.Thecrossings of the inverseFisher transform line and thetrigger lineprovideclearandunequivocal indications forthe buy and sell points, as

demonstrated in the exampleofFigure15.4.

Figure15.4TheInverseFisherTransformoftheAdaptiveStochasticIndicatorGivesClearandUnambiguousIndicationsoftheProperBuyandSellPoints

The use of the inverse

Fisher adaptive Stochasticindicator involves thecrossings of the inverseFisher line and the triggerline.WhentheinverseFisherline crosses over the triggerline, then buy. When theinverse Fisher line crossesunder the trigger line, thensell short, or if you choose,exitthelongposition.The inverse Fisher

transform should be one of

the biggest weapons in yourarsenal of trading indicatorsandstrategies.Itsuseasasoftlimiter is universal and canhelp remove many of thedistracting and irrelevantsquiggles in your indicators.Theonlyconstraintshouldbethat the original input has anearlyazeromean.Youmayneed to amplify the input toattain the desiredcompression at the output. Ifyou want a simple “digital”

output, just amplify the inputbyalargeamount.

■ CubeTransformThe purpose of the cubetransform is to compress thesignals near zero of anindicatorthatswingsbetweenthe limitsof−1and+1. Thisishandyforanindicatorsuchas the adaptive band-passfilter where you areconcerned only with the

larger swings, and thesquiggles in the middle aremerely distractions. Thetransform consists of merelycubing the indicator values.Doingthis,thevaluesnear+1and−1arenearlyunchanged,but the smaller absolutevaluesareseverelyreducedinamplitude. Of course, thisonly works if the effects ofspectral dilation have beenremovedbyaroofingfilterorsome equivalent technique.

The cubing of the display isonly effective when theoriginal indicator has anominally zero mean. Thetransfer response of the cubetransform is shown in Figure15.5. In this figure, the inputvalues are shown by thestraight line and the outputvalues are shown by thecurvedline.

Figure15.5TheCubeTransformCompresses

SmallerValuesoftheInput

The effect of the cubetransform is shown in Figure15.6, where the adaptiveband-pass filter is shown inthe first subgraph, and theadaptiveband-passfilterwiththe cube transform is shownin the second subgraph. Thepurposeofthecubetransformistoclarifythemajorswingsby compressing thedistractions of the smallerswingvalues.

Figure15.6TheCubeTransformClarifiesSwingingSignalsbyCompressingtheSmallerAmplitudeSwings

■ Key PointstoRemember

1. The purpose of atransformer is shape theindicator to aidinterpretationanduseoftheindicatoritself.2. Transformers do notinducelag.3.AFisher transformofan indicator swinging

between−1and+1witha nominal zero meanplots that indicator interms of standarddeviations with a nearlynormal probabilitydistribution.4. An inverse Fishertransform acts as a softlimiter to removeextraneous wiggles inindicators have anominalzeromean.

5. A cube transformcompresses the smallervalues of an indicatorswinging between −1and+1.

CHAPTER16

SwamiCharts“Now that's a hotdisplay,” said Tomcoolly.

Technical indicators fortrading are long on precisionbut short on accuracy. Forexample, you know exactlywhenarelativestrengthindex(RSI) crosses the 80 percentthreshold.Inmyopinion,thisprecision property ofindicators gives traders afalse sense of security andobscures the fact thatindicators are basicallydisplays of a statistical

process.Further,mosttradersdon't have a clue regardingthe proper setting of theindicator lookback period.The end result is that theprecise indicator can showyouprettymuchanythingyouwant, dependingonhowyousettheparameters.Partofthesolution, as discussed inChapter 11, is to make theindicator adaptive to themeasuredcycleperiod.

In this chapter, I describeanother completely differentperspective, which is toaccept that indicators areinherently inaccurate and abetter approach is to look atthat indicatorovera rangeoflookback periods and viewthe short-term movement inthecontextofthelonger-termindictor results. In a sense,the indicator isusedas fuzzylogic. This is the purpose ofSwamiCharts.

Almost any oscillator-typeindicator can be plotted as aSwamiChart. If you do this,youwillquicklyisolatethoseindicators that have the bestperformance. You will alsofind, as I did, that there aresome indicators that indicatenothing at all. As a quickvisual example, Figure 16.1shows a conventional RSIandaconventionalStochasticonthesamechart.Theyseemto be giving different

information. Part of theanswer is that neither hasbeen compensated forSpectralDilation.The restofthe answer is that whenviewed in context, these twoindicatorsshowjustaboutthesamething.

Figure16.1RSIandStochasticIndicatorsSeemtoImpartDifferentInformation

■SwamiChartsOverviewSwamiCharts retain the corefunctionality of the technicalindicators with which you'realready familiar, whilepacking much moreinformation into an easy-to-interpretheatmapchart.With

SwamiCharts, you nowvisualize each indicator overa range of lookback periodsto revealabetterviewof theindicator's truer meaning incontext.SwamiCharts provide a

better solution because youcan now visualize contextover a wide range oflookbackperiods.Trendsandcycles emerge moresuccinctly,whilecontinuation

patternsandreversalsbecomeeasier to identify.SwamiCharts solves the datalookback problem byenabling you to viewmarketconditions at a glance, afeature that will dramaticallyfacilitateyourtrading.SwamiCharts condenses

muchmoreinformationabouteach indicator's truemeaninginto an easy-to-interpret heatmap chart. SwamiCharts

indicators are created bycomputing the technicalindicator multiple times overa range of lookback periods.The vertical scale of theSwamiChart indicator is thatrange of those lookbackperiods.Foreachdatabarandfor each lookback period thevalue of the indicator iscomputedintheconventionalway. After the indicator iscomputed, a color value isassigned to the matrix

location defined by the barlocation in the horizontalplaneandthelookbackperiodin the vertical plane. Thecomplete matrix looks like aheatmapwhen drawn belowthe price bars in a subgraph.Theresultingbasicdisplay iseasy to interpret. Greenmeans increasing values, redmean decreasing values, andyellow means intermediatevalues.That is, the heatmapis a so-called “stoplight”

chart.

■SwamiChartsRSIThe RSI indicator wasdescribed in Chapter 11,wherewemade the indicatoradaptive to the measureddominant cycle. In this case,we compute multiple RSIsover the range of lookback

periodsofinterestandplotallof them as a heat map. TheSwamiCharts RSI isdescribed with reference tothe EasyLanguage code inCodeListing16-1.

CodeListing16-1.EasyLanguage

CodetoComputetheSwamiCharts

RSI

{SwamiChartsRSI©2013JohnF.Ehlers

}Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lookback(0),ClosesUp(0),ClosesDn(0),count(0),Denom(0),Color1(0),Color2(0),Color3(0);

Arrays:Ratio[48,2](0),MyRSI[48,3](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

ForLookback=5to48Begin//rangeincludeshalftheshortestperiodofinterestRatio[Lookback,2]=Ratio[Lookback,1];MyRSI[Lookback,3]=MyRSI[Lookback,2];MyRSI[Lookback,2]=MyRSI[Lookback,1];ClosesUp=0;

ClosesDn=0;Forcount=0toLookback-1BeginIfFilt[count]>Filt[count+1]ThenClosesUp=ClosesUp+(Filt[count]-Filt[count+1]);IfFilt[count]<Filt[count+1]ThenClosesDn=ClosesDn+(Filt[count+1]-Filt[count]);End;Denom=ClosesUp+ClosesDn;IfDenom<>0ThenRatio[Lookback,1]=ClosesUp/Denom;//SmoothMyRSIwitha10barSuperSmoother MyRSI[Lookback, 1] = c1*(Ratio[Lookback,1]+Ratio[Lookback,2])/2+c2*MyRSI[Lookback,2]+c3*MyRSI[Lookback,3];//EnsureMyRSIdoesnotfalloutsidecolorlimitsIfMyRSI[Lookback,1]<0ThenMyRSI[Lookback,1]=0;IfMyRSI[Lookback,1]>1ThenMyRSI[Lookback,1]=1;End;

//PlotasaHeatmap

//Plottingdatamustrangebetween0and1

ForLookback=5to48Begin//StopLightColorsIfMyRSI[Lookback,1]>=.5ThenBegin Color1 = 255*(2-2*MyRSI[Lookback,1]);Color2=255;Color3=0;EndElseIfMyRSI[Lookback,1]<.5ThenBeginColor1=255;

Color2=255*2*MyRSI[Lookback,1];Color3=0;End;IfLookback=5ThenPlot5(5,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=6ThenPlot6(6,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=7ThenPlot7(7,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=8ThenPlot8(8,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=9ThenPlot9(9,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=10ThenPlot10(10,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=11ThenPlot11(11,“S11”,RGB(Color1,Color2,Color3),0,4);IfLookback=12ThenPlot12(12,“S12”,RGB(Color1,Color2,Color3),0,4);IfLookback=13ThenPlot13(13,“S13”,RGB(Color1,Color2,Color3),0,4);IfLookback=14ThenPlot14(14,“S14”,RGB(Color1,Color2,Color3),0,4);IfLookback=15ThenPlot15(15,“S15”,RGB(Color1,Color2,Color3),0,4);IfLookback=16ThenPlot16(16,“S16”,RGB(Color1,Color2,Color3),0,4);IfLookback=17ThenPlot17(17,“S17”,RGB(Color1,Color2,Color3),0,4);IfLookback=18ThenPlot18(18,“S18”,RGB(Color1,Color2,Color3),0,4);IfLookback=19ThenPlot19(19,“S19”,RGB(Color1,Color2,Color3),0,4);IfLookback=20ThenPlot20(20,“S20”,RGB(Color1,Color2,Color3),0,4);IfLookback=21ThenPlot21(21,“S21”,RGB(Color1,Color2,Color3),0,4);IfLookback=22ThenPlot22(22,“S22”,RGB(Color1,Color2,Color3),0,4);IfLookback=23ThenPlot23(23,“S23”,RGB(Color1,Color2,Color3),0,4);IfLookback=24ThenPlot24(24,“S24”,RGB(Color1,Color2,Color3),0,4);IfLookback=25ThenPlot25(25,“S25”,RGB(Color1,Color2,Color3),0,4);IfLookback=26ThenPlot26(26,“S26”,RGB(Color1,Color2,Color3),0,4);IfLookback=27ThenPlot27(27,“S27”,RGB(Color1,Color2,Color3),0,4);IfLookback=28ThenPlot28(28,“S28”,RGB(Color1,Color2,Color3),0,4);IfLookback=29ThenPlot29(29,“S29”,RGB(Color1,Color2,Color3),0,4);IfLookback=30ThenPlot30(30,“S30”,RGB(Color1,Color2,Color3),0,4);

IfLookback=31ThenPlot31(31,“S31”,RGB(Color1,Color2,Color3),0,4);IfLookback=32ThenPlot32(32,“S32”,RGB(Color1,Color2,Color3),0,4);IfLookback=33ThenPlot33(33,“S33”,RGB(Color1,Color2,Color3),0,4);IfLookback=34ThenPlot34(34,“S34”,RGB(Color1,Color2,Color3),0,4);IfLookback=35ThenPlot35(35,“S35”,RGB(Color1,Color2,Color3),0,4);IfLookback=36ThenPlot36(36,“S36”,RGB(Color1,Color2,Color3),0,4);IfLookback=37ThenPlot37(37,“S37”,RGB(Color1,Color2,Color3),0,4);IfLookback=38ThenPlot38(38,“S38”,RGB(Color1,Color2,Color3),0,4);IfLookback=39ThenPlot39(39,“S39”,RGB(Color1,Color2,Color3),0,4);IfLookback=40ThenPlot40(40,“S40”,RGB(Color1,Color2,Color3),0,4);IfLookback=41ThenPlot41(41,“S41”,RGB(Color1,Color2,Color3),0,4);IfLookback=42ThenPlot42(42,“S42”,RGB(Color1,Color2,Color3),0,4);IfLookback=43ThenPlot43(43,“S43”,RGB(Color1,Color2,Color3),0,4);IfLookback=44ThenPlot44(44,“S44”,RGB(Color1,Color2,Color3),0,4);IfLookback=45ThenPlot45(45,“S45”,RGB(Color1,Color2,Color3),0,4);IfLookback=46ThenPlot46(46,“S46”,RGB(Color1,Color2,Color3),0,4);IfLookback=47ThenPlot47(47,“S47”,RGB(Color1,Color2,Color3),0,4);IfLookback=48ThenPlot48(48,“S48”,RGB(Color1,Color2,Color3),0,4);

End;

Since the MyRSI iscomputed for all values over

the range of lookbackperiods,itmustbeanarrayinEasyLanguage rather than avariable. The RSI iscomputed relative to theoutput of the roofing filterrather than price closes toeliminate the effects ofSpectral Dilation. After thecomputation of the ratio, theindicator is smoothed in aSuperSmoother filter, so wemust create the three mostrecent instances ourselves in

the code. This is becauseEasyLanguage retainshistoricalvalues forvariablesbut not for arrays. In fact,ratiomustbeanarrayalsoforthe same reasons. Thecoefficients for the 10-barSuperSmoother filter arecomputed and are the samefor all lookback periods, andso can be outside thelookback loop to make thecalculations a little moreefficient.

Now, let's see how theSwamiCharts RSI indicatorslook in Figure 16.2. Wow!Now you can see the entiremarket activity at a glance.Theshort-termdownturnsareshown in the context ofcaution to upside trends.When themarket turnsdownthe short-term red areasexpand into the longer-termdown market. Then, at theright-hand edge of the chart,you can see the short-term

risingareaexpandingintothelonger-term cautionary areas.The RSI naturallyequivocates at the longerlookbackperiodsbecause thevalueoftheclosesupstarttoapproach the value of closesdown over the longeraverage.

Figure16.2SwamiChartsRSIContainsMuchMoreInformationthantheStandardRSI

■SwamiChartsStochasticThe Stochastic indicator wasalsodescribed inChapter11,wherewemade the indicatoradaptive to the measureddominant cycle. In this case,we compute multipleStochastics over the range of

lookback periods of interestandplotallofthemasaheatmap. The SwamiChartsStochastic is described withreference to theEasyLanguage code in CodeListing16-2.

CodeListing16-2.EasyLanguage

CodetoComputetheSwamiCharts

Stochastic{SwamiChartsStochastic©2013JohnF.Ehlers)

Vars:alpha1(0),HP(0),a1(0),b1(0),c1(0),c2(0),c3(0),Filt(0),Lookback(0),count(0),HighestC(0),LowestC(0),Color1(0),Color2(0),Color3(0);

Arrays:Ratio[48,2](0),

Stoc[48,3](0);

//Highpassfiltercycliccomponentswhoseperiodsareshorterthan48bars

alpha1=(Cosine(.707*360/48)+Sine(.707*360/48)-1)/Cosine(.707*360/48);

HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close-2*Close[1]+Close[2])+2*(1-alpha1)*HP[1]-(1-alpha1)*(1-alpha1)*HP[2];

//SmoothwithaSuperSmootherFilterfromequation3

a1=expvalue(-1.414*3.14159/10);

b1=2*a1*Cosine(1.414*180/10);

c2=b1;

c3=-a1*a1;

c1=1-c2-c3;Filt = c1*(HP+HP[1])/2+c2*Filt[1]+c3*Filt[2];

//ComputeStochasticforeachlookbackperiod

ForLookback=5to48BeginRatio[Lookback,2]=Ratio[Lookback,1];Stoc[Lookback,3]=Stoc[Lookback,2];Stoc[Lookback,2]=Stoc[Lookback,1];HighestC=Filt;LowestC=Filt;Forcount=0toLookback-1BeginIfFilt[count]>HighestCthenHighestC=Filt[count];IfFilt[count]<LowestCthenLowestC=Filt[count];End;Ratio[Lookback,1]=(Filt-LowestC)/(HighestC-LowestC);//Smoothwitha10barSuperSmoother.Coefficientsarecomputedabove Stoc[Lookback, 1] = c1*(Ratio[Lookback,1]+Ratio[Lookback,2])/2+c2*Stoc[Lookback,2]+c3*Stoc[Lookback,3];//EnsureStoc[Lookback,1]doesnotexceedthe0to1limitsIfStoc[Lookback,1]<0ThenStoc[Lookback,1]=0;IfStoc[Lookback,1]>1ThenStoc[Lookback,1]=1;End;

//PlotasaHeatmap

ForLookback=5to48Begin//StopLightColorsIfStoc[Lookback,1]>=.5ThenBegin Color1 = 255*(2-2*Stoc[Lookback,1]);Color2=255;Color3=0;End

ElseIfStoc[Lookback,1]<.5ThenBeginColor1=255;Color2=255*2*Stoc[Lookback,1];Color3=0;End;IfLookback=5ThenPlot5(5,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=6ThenPlot6(6,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=7ThenPlot7(7,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=8ThenPlot8(8,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=9ThenPlot9(9,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=10ThenPlot10(10,“S10”,RGB(Color1,Color2,Color3),0,4);IfLookback=11ThenPlot11(11,“S11”,RGB(Color1,Color2,Color3),0,4);IfLookback=12ThenPlot12(12,“S12”,RGB(Color1,Color2,Color3),0,4);IfLookback=13ThenPlot13(13,“S13”,RGB(Color1,Color2,Color3),0,4);IfLookback=14ThenPlot14(14,“S14”,RGB(Color1,Color2,Color3),0,4);IfLookback=15ThenPlot15(15,“S15”,RGB(Color1,Color2,Color3),0,4);IfLookback=16ThenPlot16(16,“S16”,RGB(Color1,Color2,Color3),0,4);IfLookback=17ThenPlot17(17,“S17”,RGB(Color1,Color2,Color3),0,4);IfLookback=18ThenPlot18(18,“S18”,RGB(Color1,Color2,Color3),0,4);IfLookback=19ThenPlot19(19,“S19”,RGB(Color1,Color2,Color3),0,4);IfLookback=20ThenPlot20(20,“S20”,RGB(Color1,Color2,Color3),0,4);IfLookback=21ThenPlot21(21,“S21”,RGB(Color1,Color2,Color3),0,4);IfLookback=22ThenPlot22(22,“S22”,RGB(Color1,Color2,Color3),0,4);IfLookback=23ThenPlot23(23,“S23”,RGB(Color1,Color2,Color3),0,4);IfLookback=24ThenPlot24(24,“S24”,RGB(Color1,Color2,Color3),0,4);IfLookback=25ThenPlot25(25,“S25”,RGB(Color1,Color2,Color3),0,4);IfLookback=26ThenPlot26(26,“S26”,RGB(Color1,Color2,Color3),0,4);IfLookback=27ThenPlot27(27,“S27”,RGB(Color1,Color2,Color3),0,4);IfLookback=28ThenPlot28(28,“S28”,RGB(Color1,Color2,Color3),0,4);

IfLookback=29ThenPlot29(29,“S29”,RGB(Color1,Color2,Color3),0,4);IfLookback=30ThenPlot30(30,“S30”,RGB(Color1,Color2,Color3),0,4);IfLookback=31ThenPlot31(31,“S31”,RGB(Color1,Color2,Color3),0,4);IfLookback=32ThenPlot32(32,“S32”,RGB(Color1,Color2,Color3),0,4);IfLookback=33ThenPlot33(33,“S33”,RGB(Color1,Color2,Color3),0,4);IfLookback=34ThenPlot34(34,“S34”,RGB(Color1,Color2,Color3),0,4);IfLookback=35ThenPlot35(35,“S35”,RGB(Color1,Color2,Color3),0,4);IfLookback=36ThenPlot36(36,“S36”,RGB(Color1,Color2,Color3),0,4);IfLookback=37ThenPlot37(37,“S37”,RGB(Color1,Color2,Color3),0,4);IfLookback=38ThenPlot38(38,“S38”,RGB(Color1,Color2,Color3),0,4);IfLookback=39ThenPlot39(39,“S39”,RGB(Color1,Color2,Color3),0,4);IfLookback=41ThenPlot41(41,“S41”,RGB(Color1,Color2,Color3),0,4);IfLookback=42ThenPlot42(42,“S42”,RGB(Color1,Color2,Color3),0,4);IfLookback=43ThenPlot43(43,“S43”,RGB(Color1,Color2,Color3),0,4);IfLookback=44ThenPlot44(44,“S44”,RGB(Color1,Color2,Color3),0,4);IfLookback=45ThenPlot45(45,“S45”,RGB(Color1,Color2,Color3),0,4);IfLookback=46ThenPlot46(46,“S46”,RGB(Color1,Color2,Color3),0,4);IfLookback=47ThenPlot47(47,“S47”,RGB(Color1,Color2,Color3),0,4);IfLookback=48ThenPlot48(48,“S48”,RGB(Color1,Color2,Color3),0,4);

End;

SincetheStociscomputedfor all values over the range

of lookback periods, it mustbeanarray inEasyLanguagerather than a variable. TheStochastic is computed usingtheoutputoftheroofingfilterrather than price closes toeliminate the effects ofSpectral Dilation. After thecomputation of the ratio, theindicator is smoothed in aSuperSmoother filter, and sowemustcreatethethreemostrecent instances ourselves inthe code. This is because

EasyLanguage retainshistoricalvalues forvariablesbut not for arrays. In fact,ratiomustbeanarrayalsoforthe same reasons. Thecoefficients for the 10-barSuperSmoother filter arecomputed and are the samefor all lookback periods, andso can be outside thelookback loop to make thecalculations a little moreefficient.

The SwamiChartsstochastic is shown inFigure16.3.TheprimarydifferencesfromtheSwamiChartsRSIisthat the display of the short-term downturns are morevivid, and there ismuch lessyellowatthelongerlookbackperiods. Thus, there is lessequivocation regarding themarketcalls.

Figure16.3SwamiChartsStochasticEquivocatesLess

thantheSwamiChartsRSIRegardingTrends

Reviewing Figure 16.4,

where the SwamiCharts RSIand the SwamiChartsStochasticarecompared, it isapparent that both indicatorsare telling you pretty muchthe same thing about themarket. This consistency ofviewpoint is hardly apparentin Figure 16.1, where thestandard versions of theindicatorsarecompared.

Figure16.4SwamiChartsRSIandSwamiChartsStochastic

BasicallyRevealtheSameInformationaboutMarketActivity

■ Roll YourOwnSwamiChartsYou can write your ownSwamiCharts customindicator if you create thearrays for all lookbackperiods, scale the indicatorvalues to fall between 0 and1, and use the heat map

display code without anyfurther changes except forinserting the names of thearrays in the color convertersectionofthecode.Notethatthe heatmap plotting routineis exactly the same for allthree examples in thischapter.If you wanted to make a

SwamiCharts commoditychannel index (CCI), forexample, you would only

needtoadd100totheresult.This would make theindicatorrangebetween0and200. Then, divide by 200,with the end result that theindicator ranges between 0and1.Even if you never really

use a SwamiCharts indicatorfor trading it will beworthwhile to program yourfavorite indicator in theSwamiCharts format. This

will enable you to comparethat indicator to others andassess its effectiveness.Having programmed anumber of indicators asSwamiCharts, I can tell youthattherearemorethanafewindicators that don't indicateanything at all. If you visitwww.StockSpotter.com orwww.SwamiCharts.com, youwill get a sneak preview ofsome of the advancedindicators I have created

using the SwamiChartsmodel.

■ Key PointstoRemember

1. SwamiCharts giveyou “the view from30,000 feet.” That is,yougetabird's-eyeviewofmarketactivityfromaSwamiChartindicator.2. Many indicatorsbasically tell you thesamethingaboutmarket

activity.3. A SwamiChartsrendition of an indicatorwill quickly reveal itsvalueorlackthereof.

CHAPTER17

Swing-TradingStrategies

“Writing tradingstrategies is simple,”

Tomsaideasily.

Well, I think Tom reallymeant that trading systemsshouldbesimple,butwritingeffective ones is not easy.Trading systems should besimple to avoid curve fittingto the data set on which thesystem is developed. Forexample, if you used oneyear's worth of daily data todevelop a system and yoursystem was written as a

polynomialoforder250,thenyou could have theoreticallymatched that polynomialexactlytothedata.However,applying that polynomial toanother year's data wouldmost likely have disastrousresults. Therefore, keeping atrading strategy simpleimproves the likelihood thatthe strategy will be robustacross multiple symbols andmultiple time frames. If youuse few parameters in your

system, it is less likely thestrategy is customized to thedataset.Given that the strategy is

simple, the next criterion isthat the entry and exit rulesbe predictive. Most rule-based strategies depend onindicators,setups,orpatterns.Sadly,allof thesedependonhistorical data and are betterat documenting what hashappened rather than

predicting what will happen.Additionally, setups andpatterns are often based onjust a few observations andare therefore anecdotal orheuristic.Thefact thatsetupsor patterns do not havesufficient instances to bestatistically significant isusually overlooked.Therefore, finding apredictive indicator isprobably the most difficultpart of developing a trading

strategy.The majority of this book

describesadvancedfiltersandindicators to help yourtrading. Ultimately, theindicators are used as an aidto making buy and selldecisions, whether thedecisions are based ondiscretion or on analgorithmic rule set.Indicators use parameters.For example, a moving

average uses length of theaverage as a parameter. Thepower of modern tradingplatformsenablesyoutovarythe parameters to maximizetrading performance over theselecteddataset.Theprocessis called optimization. I canassureyouthat theprocessisanything but optimum andcanlullyouintoafalsesenseof confidence in yourprospective strategy. Theprocess of optimization

should include a sensitivityanalysiswhereyouvaryeachparameter by itself and thenview the performance over afullrangeoftheparameter.Ifthe optimization does notprovide a gentle peak so theparameter is effective over arange of settings, then yourcandidate strategy isguaranteed not to be robust.When testing your strategy,you should produce enoughtrades so the results are

statistically significant. Onerule of thumb is that youshouldhaveatleast30tradesforeachparameterusedintheprospective strategy. Forexample, if you use fourparameters in your strategy,you should examine theresults over at least 120trades. Ipersonally think thiscountisonthelowside.Nobody wants to lose

money on a trade, and stop-

loss criteria are often builtinto the strategy tominimizelosses. Remember, the nameof theprocedure is stop loss,and half that name is loss.Therefore, stop-loss rulesshouldbeused sparinglyandnot be imbedded so that thestoplossisanintegralpartofthestrategy.Myprocedure isto develop the strategywithouttheuseofastop-lossrule. After the main part ofthe strategy is satisfactory, I

then examine the maximumadverse excursionsencountered,andtheninsertastop-loss rule that limitsonlythe maximum losses withoutinterfering much with thewinningtrades.Thefinalgeneralaspectof

strategy development thatcannotbeoverlookedis“out-of-sample” testing.Youhaveundoubtedly developed thestrategy on a given data set

that consists of one symbolover a selected period oftime. Out-of-sample testingmeans that you apply yourtrading strategy to the samesymbol over a different timerange or apply your tradingstrategy to multiple differentsymbols,allofwhichisdonewithoutchangingparameters.If you reoptimize, you aresimply cheating, and you areonly cheating yourselfbecause you are bound to be

disappointedwhenyouapplyyour strategy to real trading.However, if you getsatisfactoryresultsdoingthis,then your strategy is robustand you can trade it withgreaterconfidence.

■ConventionalWisdomBy analogy to gambling,trading strategy performanceis most easily characterizedbypercentwinningtradesandprofit factor. Profit factor istheratioofgrosswinningstogross losses, comparable to

the payout on a wager.Percent winning trades isself-explanatory. There is nounique combination of thesecharacteristics that lead to asuccessfultradingstrategy,sothetradingstyleisoptional.Trend following usually

means trying to identify anentry opportunity and thenexit the trade quickly if thetrend does not develop. Thismeansonecanmakea lotof

money on a few trades, butthere will be many losingtrades along the way.Therefore, trading the trendmeansbeingwillingtoacceptarelativelylowpercentageofwinning trades to achieve arelatively high profit factor.Alternatively, a trend tradermay elect to hold a positionthrough adversity to profit inthe long run. Holding tradesthrough adversity usuallymeans accepting large

drawdowns. In my view,trend trading works becausethe general market isunboundedandhasanupsidebias due to economic growthand inflation. I assert thatthere are no predictiveindicators for trends, andtherefore this styleof tradingdoes not fit my preferredprocess.Predictive indicators are

moreapplicabletoshort-term

processes such as swing ormomentum trading because,like the weather, predictingthe future is fragile andevanescent. If the indicatorsare indeed predictive, short-term trading is characterizedby a higher percentage ofwinning trades. The profitfactor of short-term trades isgenerally smaller becausespectral dilation limits theswings of the short-termmarket movements. The

general class of indicatorsthat support short-termtradingarecalledoscillators.Asdescribed inChapter6,

we are dealing with randomvariables when trying todescribe the market.Additionally, tradersgenerallyprefer to err on thesideofcautionbecauselossesare painful. Theseconsiderations lead to atrading philosophy thatwaits

for confirmation of theturning point of an indicatorbeforesignalingatradeentry.For example, theconventional Stochasticindicator rule could be “waituntil the oscillator crossesabove 20 percent beforemakinga longposition trade,and wait until the oscillatorcrosses below 80 percentbefore reversing to a shortposition trade.” This rule setwaits until the Stochastic

indicator moves in thedirection of the desired tradebefore making the tradedecision. As an example,assume the market has amonthly cycle. We wouldthen want to set the lengthparameter of a stochasticindicator to 10 because thereare roughly 10 days up and10 days down in a monthlycycle. When we invoke theconventional Stochasticindicator rule using the two-

pole Stochastic indicator ofChapter 7, we obtain theequity curve of Figure 17.1when applied to 10 years oftheS&PFuturesdata.

Figure17.1UsingaConventionalStochasticIndicatorRuleIsNotProfitable

These trading results areawful! Just a little analysiscan show what went wrong.

First, the roofing filter has alagofabout twobars,almostliketheSuperSmootherfilter.There is basically no lagcomputingtheStochastic,andthen there is another two-barlag of the SuperSmootherfilter that smooths theStochastic calculation. Sothereisatotaloffourbarsoflag just to compute theindicator. We don't get asignalforabout25percentofthe move in time, or about

three bars, after the turningpoint of the indicator.Further, we cannot make atradeentryuntil thedayafterthe signals are given.Therefore, our trade entriesandreversalsareabout8barslate on a 10-bar move. Thismeans we are trading almostexactly opposite of the waythe market is moving in amonthly cycle. This is thepenalty for waiting forconfirmation.

■ Anticipatingthe TurningPointI think the bestway to tradean oscillator is to anticipatethe turning point and rely onthe statistical return to themean as being predictive. Iwrote apaperon this subjectthat was awarded the 2008

Runner-Up Winner of theMarket TechnicianAssociation'sCharlesH.DowAward.1 In contrast to theconventional wisdom, theStochastic indicatoranticipation of the turningpoint trading rule would be“make a long position tradewhen the oscillator crossesbelow20percent,andreversetoashortpositiontradewhenthe oscillator crosses above

80percent.” The anticipationrule enables us to get thetradingsignalnominallythreebars before the turning pointoftheindicator.Westillhavethe four bars ofcomputationallagandtheonebarlagofentryfollowingthesignal.Therefore,ournet lagin making the trade entry isonly two bars relative to thereal turningpoint inthedata.In otherwords, out timing isjust about right. Invoking the

anticipated trading rule usingthe two-pole Stochasticindicator of Chapter 7, weobtain the equity curve ofFigure 17.2 when applied to10 years of the S&P Futuresdata.The anticipation rule is

equally applicable to mostoscillators. Exampleoscillatorsinthisbookwouldbe the band-pass filter inChapter 5 and the two-pole

RSI in Chapter 7. Theanticipation rule using thetwo-pole Stochastic indicatoris not intended to be acomplete trading system. Forexample, the 10-bar periodwas selected fordemonstration purposes. Thecomputation period of theindicatorcanbeoptimizedforagivensymboltoproducethebest timing for the tradeentries and exits. Further, nostop-loss values have been

used.

Figure17.2UsingtheAnticipatedStochasticIndicatorTurningPointRuleIsGenerallyProfitable

The purpose of describingthe conventional wisdom ofwaiting for confirmation as

contrasted with anticipatingthe turning points with noother changes of parametersor rules is to sensitizeyou tothe impacts of computationallag.

■ Sine WaveUniquenessAccording to Fourieranalysis, any complexwaveformcanbesynthesizedusing a combination of sinewave components. Thatmakesasinewaveaprimitivefrom which all patterns canbe formed. It isbest touseaprimitive in trading in the

interest of robustness.Further, it is difficult enoughjust tomeasureoneprimitiveinthemarketdata,letaloneacomplex set that createsanecdotal patterns. With asufficiently small bandwidth,theband-passfiltercreatesanoutput that can becharacterized as a sine wavehaving a slowly varyingamplitude and phase. Armedwiththisinformation,wecanexamine the use of sine

waves in providing tradingsignals.A sine wave can be

generatedbytheprojectionofa rotating phasor on itsvertical axis,2 as depicted inFigure 17.3. As the phasorrotates, the sine wave risesfrom zero to its maximum,backtozero,toitsminimum,andthenbacktozeroagaintocomplete one full cycleperiod.Notethatthe“tail”of

the phasor is pinned at theorigin.Ifthetailofthephasorwere offset from the origin,then the sinewavewould bedistortedandwouldnothavea zero mean. Since marketdatahaveSpectralDilation,itis crucial that the SpectralDilation effects be removed,as discussed in Chapter 5,before working with thefilteredsinewave.

Figure17.3ASineWaveCan

BeGeneratedbyaRotatingPhasorThatIsPinnedattheOrigin

The position of the phasorat any instant in time is thephaseangleofthesinewave.

Thereisaspecialrelationshipif the phase angle betweentwophasorsis90degrees,ora quarter of a cycle. Asshown in Figure 17.4, aphasor leading the sinewavephasor by 90 degrees createsacosinewave.Thefactthatacosine wave leads a sinewave by 90 degrees isimportantbecause itgivesusa way to artificially advancethe turning point of a band-passfilteroutputbyaquarter

of a cycle.By advancing thewaveformaquarterofacyclewe can negate or reduce thecomputationallagrequiredbythefilter.

Figure17.4ACosineWaveLeadsaSineWaveby90Degrees(aquarterofacycle)

Fromthecalculusweknowthat the rateofchangeof thesinewave isexactlyacosinewave whose amplitude ismodifiedby the argument.Aone-bardifferencecreatestherate of change of the sine

wave, so an amplitude-correctedcosinewavecanbecomputed in code as (usingEasyLanguagenotation):

Yes,itreallyisthatsimpleto create a leading tradingsignal using the output of aband-pass filter if you knowor estimate the period of thesinewave.The object of using the

cosinewaveistocreateabuy

signal when the cosine waveis at or near a trough and tocreate a sell signal when thecosine wave is at or near apeak. One way to do this isby noting that the sine wavecrosses below zero when thecosine wave is at its troughand the sine wave crossesover zero when the cosinewave is at its peak. In otherwords,wecancreatebuyandsell signals simply based onthezerocrossingsof thesine

wave.It is a little more

conservative to create atrigger signal that lags thecosinewavebyabaror two,as shown in Figure 17.5.Usingadelayedcosinewaveas a trigger means givingback a little of the lag wehavestruggletominimize.

Figure17.5BuyandSellSignalsCanBeCreatedbyCrossoverswithaCosine

WaveandtheCosineWaveDelayedbyaBarorTwo

Some whipsaw trades canbe eliminated by using thezero crossings of the sinewave as an entry signal onlywhen confirmed by thecrossover of the cosinewavewith itself delayed by onebar.

■SafetyValveSince effective swing tradesrequire anticipation of theprice turning, it is inevitablethat therewillbecaseswhenthepricekeepsongoingandbasically becomes a newtrend movement. In thesecases, your anticipation putsyouonexactlythewrongsideof the trade. You thereforeneed a rule that closes out a

position when losing tradeslikethisareexperienced.Youbasicallyhave twooptions toexitthetrade:basetheexitonpriceoron time in the trade.As a practicalmatter, I havenot found setting a stop lossto be an effective method ofexitingalosingtradeintheseconditions because the stopvalue has to be set so tightthatone iswhipsawedoutofa trade that ultimately couldturnouttobeprofitable.

If theexit rule fora losingtrade is based on price, onecan rely on the fact that theswing trade is expecting areversion to the mean, andthereforethepricearevaryingwithin a channel. If theanticipated reversal does notoccur,thenitislikelythattheprices will have a channelbreakout. A simple channelcan be established using aSuperSmoother filter on thehighs and lows of the price

bars. The channel can bewidened, if desired, byadding a fraction of theaverage bar range to thesmoothed high prices andsubtracting it from thesmoothed low prices. If youare in a long position tradeandthepricefallsoutsidethelowerchannel, thenit is timeto exit the trade at a loss,perhaps reversing to a shortposition to capture thedowntrend that you did not

anticipate.If the safety valve is

operatingon a time line, youmust be sensitive to theperiodicity of the cycle youare trading. For example, amonthly cycle generallyconsistsofa10-barmoveupfollowed by a 10-bar movedown. Therefore, yourexpected time to be in anygiven trade is about 10 bars.If you are not in a profitable

position by the time you areabout halfway into theexpected trade duration, it isprobably time to bail out ofthe trade at a loss. Being alittle more aggressive, youcan also establish a “quickout” rule that basically saysthat if you are not profitablein the first bar or two of atrade, exit the trade andwaitforabetteropportunity.I rely on technology for

trading and therefore do notrely on psychology much,even though psychologycertain has a place indiscretionarytrading.Thereisone unmistakablepsychologicalsignal toexitatrade. That is, if you eventhink about hoping a tradewill turnaroundandmoveinyourfavor,thenexitthetradeimmediately. Hope has anegativevalueintrading.

■ Exiting aTradeWhen trading futures, it iscommon to take both longand short positions. Using aswing-trading strategy in thiscase, the exit of a longpositioncanbecometheentryfor the next short position.Alternatively, when tradinglong-onlypositions, the short

entry signal can become theexitsignalforalongpositiontrade.Using a short entry signal

asanexitsignalwhentradingstocks is less than optimalbecause the stock prices donot move symmetrically.There is an upside bias tostock prices, and pricesgenerallymoveslower to theupside than to the downside.Therefore, anticipating a

downside move often willresultinanearlyexittowhatotherwise could have been averyprofitabletrade.Abetterexitstrategywhen

trading stocks follows thegeneralprincipleof“letyourprofits run.”A practicalwayof implementingsuchanexitwould be to smooth theclosing prices with aSuperSmoother filter, and ifthefilteroutputcrossesbelow

the filter output delayed byseveral bars, then exit thetrade. The smoothed priceswill always be above thedelayedsmoothpricesaslongas the prices are moving totheupside.Thus,youwillbekept in a profitable longposition longer. Of course,youwillgiveupalittleoftheaccrued profit when theprices turn down, but that isthe price you must pay toavoidthewhipsawexit.

■StopLossMy experience is that a stoploss will decimate therobustness of a tradingstrategy if it is built into thestrategy and becomes anintegral part of it. Rather, astoplossisbestleftonlyasaguardagainstextremelylargelosses.Using a stop loss thisway will maintain therobustness of the core

strategyyouhavebuilt.Therearealargenumberofwaystoimplement a stop loss rule.The simple rule that worksformeistoletthestopvaluejust be a percentage of theentryprice. InEasyLanguagenotation, the stop-loss rulesare:

The PctLoss is an inputthat can be establisheddifferently from tickersymbol to ticker symbol. Ifind that a value from2 to 5percent works for moststocks. Since leverage isinvolved with futures, thepercentage input probablyshouldbealittlesmaller.

■Evaluating aTradingStrategyIt is common to evaluate atradingstrategybylookingatits historical track record inthe form of an equity curve.This can lead to anunwarranted enthusiasm forthe trading strategy or, less

likely, abandonment of aperfectly good strategy. Thereason is that there isconsiderablevariabilityintheequity curves produced by astrategy having the givencharacteristicsofprofitfactorandpercentwinningtrades.By determining whether a

trade is a winner or a loserusingthepercentagewinsanda random number generator,applying the payout

probability toeach trade,andsumming the randomlyselected trades, you canprovide realistic expectationsfor the equity growthproducedbythesystem.Onlyin this sense canrandomization be introducedto establish performance.Simply winning or losing isnota randomoccurrence.AnExcel spreadsheet can showthe randomized cumulativetrading profits. Just press F9

and the spreadsheet willrecompute 500 randomizedtrades. You canindependently input differentvalues of profit factor andpercent winning trades tovisually assess the impact ofthesevariables.The variability in equity

curves, even for exampleshaving relatively high profitfactors and percent winningtrades, indicates the wisdom

of diversification in yourportfolio.Ifallthesymbolsinyourportfolioarestatisticallyindependent, then thedeviation in the equity curveistheoreticallyreducedbythesquare root of two for eachdoubling in the number ofelements. For example, youwould halve the deviation ofthe equity curve if you usedonly four symbols. Youwould halve the deviationagain if you used eight

symbols. Therefore, youwould quickly reach a pointof diminishing returns byadding symbols. Continuingtheexample,itwouldtake16simultaneously tradedsymbols to halve thedeviation again. Further,findingalargesetofsymbolshaving statisticallyindependent trades isunlikely.The following are the

directions to recreate thespreadsheet.Weneed to firstinsert the two importantstatistics.IncellA1,type“%Winners” without quotationmarks, and in cell B1, type55. In cell A2, type “ProfitFactor” without quotationmarks and in cell B2, type1.5.Thevaluesof55and1.5areonlyinitialvaluesandarerepresentative of a goodsystem.Theentries intocellsB1 and B2 are system

statistics thatyoucanchangeto visualize their impact onequitygrowth.In row 4, insert headings

for five columns as“random,” “trade profit,”“cum profit,” “trade #,” and“cum avg profit.” Thequotation marks for theheadersareunnecessary.In cell A5, input

“=RAND()” withoutquotationmarks.Thiscreates

a random number having auniformprobabilitydensityinthe range between 0 and 1.This random number iscompared to the probabilityof a win by inserting“=IF(A4<$B$1/100,$B$2,−1)”without the quotation marksintocellB5.Thisconditionalstatement says that if therandom number falls withinthewinning probability, thenassign the payout probability(theprofitfactor)tothetrade;

otherwise, assign a value of−1 to the trade. This is theoutcome of the trade. In cellC5, input “=B5” withoutquotation marks because thefirst trade profit is the sameas the cumulative profit forthe first trade. Insert 1 intocell D5 as the first tradenumber.IncellE5, insert theequation “=D5*(($B$2+1)*$B$1/100 −1)”withoutthequotationmarks.

Copyallofrow5intorow6.Thenchange cellC6 tobe

“=C4 + B5” without thequotation marks. This sumsthetradeprofitsincolumnC.Next, insert the equation“=D5+1”without quotationmarks in cell D6. Thisincrementsthetradenumber.Finally, copy all of row 6

and paste it into rows 7through 504. Now you have

the data for your analysisover 500 trades.Make a lineplot of columns C and E tographtheMonteCarloequitygrowth and cumulativeaveragetradeprofittogether.JustpressF9 to recompute

the spreadsheet. You willcreate a new randomizedequity growth curve becauseall the randomnumbershavechanged. Repeat as often asyou desire to get a feeling

thatyouknowwhattoexpect.You can change the data incellsB1andB2toassess thesensitivityofsuccesstothesebasicparameters.

■MonteCarloEvaluationMonte Carlo analysis is thebestwaytoevaluateatradingstrategy if you have theprogramming capability. Thecalculations are relativelystraightforward. Suppose youhave a sufficiently largenumber of trades that youhave confidence in the

strategy'sworkinginavarietyof conditions. If workingfutures, this could besomething like trading everytwo weeks on the averageover the past 10 years. Ifworkingstocks, thiscouldbeusing all the symbols in theS&P500 indexover thepastseveralyears.The computations start by

computing the profit per dayfor each of the trades in the

real trading history. Thenplacealltheseprofitsperdayin the proverbial hat. Next,drawaprofitperdayfromthehat, record its value, andreplace it in the hat. Repeatthe drawing 260 times tocreate a randomized year'sworth of trading. Note andrecordtherandomizedannualprofit in a “bin,” which is arelativesmall rangeofprofit.Then, repeat the annualizeddrawing 10,000 times and

place the annualized profitinto its correct bin. Doingthis, it is possible to havesomeyears thathavenothingbut losing trades and otheryears that have nothing butwinning trades. Of course,there is a wide range ofcombinations between thetwo extremes. The resultingnumberofcounts ineachbinwill have an approximatenormal(Gaussian)probabilitydistribution shape. An

example of a Monte Carloanalysis of trades can befound atwww.StockSpotter.com.3Since the Monte Carloanalysis results have theshapeofanormalprobabilitydistributionfunction,youcaneasily establish theexpectation (average profit)of the trading strategy.Additionally, you canestimate the standard

deviation in profitability, soyouwon't be surprisedwhenyour actual results don'tmatchyourexpectation.

■StockSpotter.comIn this chapter, I haveoutlined the tips andtechniques to write aneffectiveshort-termorswing-trading strategy. Rather thangivingaspecificcodelisting,Ihavepainted the techniquestobeusedwithabroadbrushbecause there are so many

variables that writing atrading strategy is analogousto writing music. For fulldisclosure, I am thecofounder ofStockSpotter.com.StockSpotter contains anumberoffreeindicatorsthatuse the principles I havedescribed in this book. Theyeliminate the effects ofspectral dilation. Theyminimize lag. They areuncommonly smooth due to

multipole filtering.SwamiCharts are alsoincluded.StockSpotter isprimarily a

site for short-term trading,andhasallthenecessarytoolsto support this role.Data forover 4,000 U.S. stocks andexchange-traded funds isincluded. Screeners scan thedata every day to identifyunique situations conduciveto short term trading. Since

StockSpotter is a cloudcomputing service, there arefour customized watch liststhat you can modify andmonitor for trading signals.You can import other lists,such as the IBD Nifty Fifty,into the watch list to get adouble filtering opportunityforyourtradingsignals.My cofounder, Ric Way,

has implementeddata-miningtechniques to supplement the

tradingrulesproducedbytheindicators. This data miningincludes difficult-to-pinpointcharacteristics such astrendiness, cycle amplitude,volatility, and so on. All thetrading signals, both entryand exit, are given after themarket close for exercise atthemarketontheopenofthenext tradingday.That is, thetrading signals are given inadvance. Then these signalsare transparently tracked as

hypotheticaltrades.Youhavecompleteaccesstoalltradingresults, including trade-by-tradelistingandMonteCarloanalysis.Ifyourinterestisprimarily

in short-term trading ratherthan coding indicators orwriting trading strategies, Irecommend that you tryStockSpotter.com as yourtoolofchoice.

■ Key PointstoRemember

1. The roofing filter orits equivalent must beused with the indicatorsused to generate swing-trading signals toeliminate the distortionsintroduced by SpectralDilation. That is, theindicators used to create

the signals must have azeromean.2. Effective swing-trading signals mustanticipate the priceturning points tomitigate the lagintroduced in computingthe indicators. Usingconventional oscillators,this is accomplished bycreating a long tradingsignalwhentheindicator

crosses under a lowerthreshold and creating ashort trading signalwhen the indictorcrosses over an upperthreshold.3. A sine wave is aunique primitive. It canbe created from pricedata using a relativelynarrow band-pass filterdescribedinChapter5.4.Acosinewavecanbe

created by taking theone-bar difference of asine wave. The cosinewave is an anticipatingsignal because it leadsthe sine wave by aquarterofacycle.Usinga cosine wave is asyntheticway to removelag.5. A leading long entrysignal can be generatedwhen a cosine wave

crosses over itselfdelayedbyabarortwo.A leading short entrysignal can be generatedwhen a cosine wavecrosses under itselfdelayedbyabarortwo.6. A less conservativemethodgenerates a longentry signal when thesinewave crosses underzero and generates ashort entry signal when

the sine wave crossesoverzero.7. Swing trades musthave a safety valve exitbecause the trades areenteredinanticipationofa price turning point.The safety valve can bebased on either time intrade or a price breakout.8. Swing-trade entrysignals indicate position

reversals for always-in-the-marketstrategies.9. Exit signals whentrading stocks to thelong side are bestaccomplished using aSuperSmoother filtercrossing under itselfdelayed by a few bars.This techniqueminimizeswhipsawsandunwarranted early tradeexits.

10. Stop-loss techniquesshould be used only tolimitmajorlosses.11.MonteCarloanalysisis the best and mostreliable way to assesstrading strategyperformance.

Notes1.www.stockspotter.com/In/TechnicalPapers.aspx

2.www.stockspotter.com/In/MesaIndicatorHelp.aspx

3.www.stockspotter.com/In/MonteCarloProfit.aspx

ABOUTTHEWEBSITE

This book includes acompanion website with theEasyLanguagecomputercodefound in the book. This willhelp you utilize the codewhile avoiding the

aggravation of keying it inyourself and debugging theerrors. All the code in thebook is fully functional andyou can either copy it fromthetextfilesandpaste it intoa new indicator you havecreated in TradeStation, oryou can import the *.ELDfilesdirectly.To access the site, go to

www.wiley.com/go/EhlersandenterthePINcodefound

at the back of this book. E-book readers will findadditional instructions forrequestingaPINcode.

INDEX

AAGCAliasingAlphaAnalyticSignalAnticipateAutocorrelationAutocorrelationPeriodogram

AutomaticGainControl

BBandPassfilterBandStopBandwidthButterworth

CCCICoefficientCompressConvolution

CorrelationCriticalperiodCube

DdBDecibelDecyclerDelayDFTDiffusionEquationDiscreteFourierTransformDispersion

DominantcycleDrunkard'swalkDualDifferentiatorDuration

EEMAEvenBetterSinewaveExponentialMovingAverage

FFilterFiniteImpulseResponse

FIRFisherFourierFractaldimensionFrequencyresponse

HHeatmapHighPassHilbertHomodyneHurstcoefficient

IIIRImaginaryInPhaseInfiniteImpulseResponseInputInstantaneoustrendlineIntradayInverseFisher

LLag

LowPass

MMedianFilterModified least-squaresquadraticsModified Simple MovingAverageMonteCarloMovingaverage

NNoise

NonrecursiveNormalize

OOrthogonalOscillatorOutput

PPearsonCorrelationPercentWinsPerformancePeriodmeasurement

PeriodogramPhaseAccumulationPhaseleadPhasorPowerPredictiveProbabilityDistributionProgramming

QQQuadratureQuantization

RRandomRandomwalkRealRecursiveReversalsRoofingfilterRSI

SSafetyValveSelectivity

SimpleMovingAverageSMASmoothingSpectralDensitySpectralDilationStochasticStopLossStrategySuperSmootherSwamiChartsSwerlingSyntheticprices

TTelegraphersEquationTradingsystemTransferresponseTransformTransformerTransientresponseTrendTurningpoint

WWave

WaveamplitudeWeightedMovingAverageWMAwww.StockSpotter.comwww.SwamiCharts.com

ZZerocrossingZeromean

EasyLanguageComputerCodefor

CycleAnalyticsforTraders

YourpurchaseofCycleAnalyticsforTradersincludesaccess

totheJohnEhlers'sEasyLanguagecomputer

code.Pleasevisit

www.wiley.com/go/ehlerstorequest

anaccesscodeandaccessthis

content.

JohnEhlersISBN:978-1-118-72841-3

(ePDF)978-1-118-72860-4(ePub)Fortechnicalsupport,please

visitwww.wiley.com.Fortelephonesupport,please

contactusat:1-800-762-2974(U.S.),1-

317-572-3994(International).

WILEY