1st Partial All

FORECASTING STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa Olascoaga

DEFINITION OF FORECASTINGIs predicting what will happen in the future.Is an estimate of the future level of some variable

Why do we forecast?

DEFINITION OF FORECASTINGMost companies forecast in order to help the firm in strategic planning activities such as: inventory purchasing, capacity planning, labor planning, etc.

NAIVE FORECASTING TECHNIQUES

Naive forecasts: a folk forecasting techniqueIn every day life situations we forecast using very simple techniqueThis technique is close to linear trend model

NAIVE FORECASTS

The value tomorow will be the same as today. Example: Number of visitors today was 120. Forecast NF1 for tomorow: 120.The value tomorow will be less (greater) by 10%. Example: Average temperature this month is 20 degrees. Forecast for the next month: Temperature will be 25 degrees (increase of 25%).

CURRENT SITUATION IN FORECASTINGThere exist a well defined set of forecasting methods There exists computer software that may be quite simply applied in forecastingExcel program allows to solve simple forecasting tasks

LAWS OF FORECASTINGForecasts are Always Wrong: No forecasting approach or model can predict the exact level of the future variable. deal with the risksForecasts for the near-term tend to be more accurate: Predicting tomorrows gas price will likely be more accurate than predicting gas price 6 weeks from now.Forecasts for groups of products or services tend to be more accurateForecasts are no substitute for calculated values

BASIC STEPS OF A FORECASTING TASKDefining the problemChoosing time-series dataAnalysing visually data paterns Choosing a modelCalculating a forecastEvaluating the forecasting accuracy, caculating errors

FORECASTING METHODS STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa Olascoaga

QualitativeRequires advice from experts. It is used when there is no available or applicable information. Used for intermediate or long term decisions.Used for immediate or short term decisions and:Past information from forecasted variable is availableInformation can be quantifiedIs reasonable to believe that past behaviour will continueQuantitativeFORECASTING

FORECASTING

FORECASTINGPast SalesPast Sales + Marketing Budget

Past GradesPast Grades + Practice TimeFuture SalesFuture Grades

PATTERNS IN A TIME SERIESA time series is a sequence of observations of a variable that are measured during a regular period of time (every hour, day, month or year)

They must be mesasured with equal intervals.

EXAMPLE OF TIME-SERIES DATANumber of visitors

Year199819992000200120022003Number420450440460470465

EXAMPLE OF TIME-SERIES DATA

PATTERNS IN A TIME SERIESThe pattern of the data helps us to understand the behaviour (trend) of the information.

A graph shows the relation between time and the value of the variable$time

PATTERNS IN A TIME SERIESThe pattern of the data helps us to undersdtand the behaviour of the information.

A graph shows the relation between time and the value of the variable

PATTERNS IN A TIME SERIESWhen trying to analyze a time series (be it sales, promotions etc) it's good to remember that there are four basic patterns:Horizontal (random, irregular variation)Trend (linear)Periodical (cyclical, seasonal)Complex (a combination of part or all listed above

HORIZONTAL PATTERNWhen the values of the time series fluctuate around a constant mean. Stationery time series are very easy to forecast.Retail or Stock market?

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Horizontal pattern

Patterns of the time-series data

1. Horizontal (random, irregular variation)

Example: Number of visitors in a small librarry

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3. Periodinis atsitiktinis pavidalas (savaitinis, nuo pirmadienio iki sekmadienio, pavaizduotos 3 savaits)

DienaLankytoj skaiius

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Horizontal pattern

Horizontal (irregular variations)

Trend

Periodinis atsitiktinis pavidalas

Periodical seasonal

Sudtingas atsitiktinis, kryptinis, periodinis pavidalas

Complex


2. Trend (close to the linear growth)

Example: Number of library visitors during the 1-6 week of the Winter semester

WeekNumber of library visitors

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Complex

Trend (close to the linear growth)

Linear



3. Periodical random (seasonal variations)



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4. Complex data pattern including random, trend and periodical variations


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Power

Trend (linear)

Linear

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Trend (logarithmic)

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Trend (power)

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Smoothing illustration

Trend (exponential)

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Smoothing the data variation

Number of visitors in a smal library

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Smoothing the data variation: a graphical presentation

TREND PATTERNConsists of a long-term increase or decrease of the values of the time series (close to the linear growth). Trend patterns are easy to forecast and are very profitable when found by stock traders!

Chart1

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Horizontal pattern

Irregular variation of the number of visitors

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D. ilyg






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Duomen svyravim ilyginimas

Klient skaiiaus kitimas firmoje 1-24 savait

SavaitKlient skaiius

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Ilyginti

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Duomen svyravimo ilyginimo grafin iliustracija

PERIODICAL PATTERNThe values of these time series are influenced by seasonal factors. A time series with seasonal patterns are more difficult to forecast but not too difficult.

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Horizontal pattern


Periodical seasonal


Trend


D. ilyg


3 Periodical random (seasonal variations)



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11063

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Klient skaiiaus kitimas firmoje 1-24 savait

SavaitKlient skaiius

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Duomenys

Ilyginti

Labiau ilyginti

Duomen svyravimo ilyginimo grafin iliustracija

COMPLEX PATTERNAre a mixture of the others. Which patterns can you find?.

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Horizontal pattern


Trend


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Number of visitors in a smaal library


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FORECASTING METHODSmoothing MethodHorizontal PatternTrend PatternSeasonal PatternMoving AveragesExponential SmoothingExponential Smoothing with Adaptive ResponseCenteredNon-centeredLinear Moving AveragesLinear Exponential SmoothingExponential Smoothing squared (Brown Method)1 Param: Brown M.2 Param: Holt M.Winters MethodMultiplicative Seasonality

What is the accuracy of a particular forecast?

How to measure the suitability of a particular forecasting method for a given data set?FORECASTING ACCURACY

MEASURES TO ESTIMATE ERRORSKey concepts:

Error (e)Mean (average) error (ME)Mean absolute error (MAE)Mean squared error (MSE)Percentage error (PE)Mean absolute percentage error (MAPE)

To preliminary evaluate a forecast and suitability of a method, various statistical measures may be used.

Error (e) of a forecast is measured as a difference between the actual (A) and forecasted values (F), that is, e = A F (+) undervalued: forecast under real (-) overvalued: forecast over real

The error can be determined only when actual (future) data are available.FORECAST ERROR

Mean (average) error (ME)

Not a very useful value as it tends to compensate. (+) : low forecasts (-) : high forecastsMEASURES TO ESTIMATE ERRORS

Mean absolute error (MAE)

Avoids the issue of negatives and positives.Also known as mean absolute deviationMEASURES TO ESTIMATE ERRORS

Mean squared error (MSE)

Also avoids the issue of negatives and positives.However this two measures are affected by the scale of measurementMEASURES TO ESTIMATE ERRORS

Percentage error (PE)

Is calculated for each period of timeMEASURES TO ESTIMATE ERRORS

Mean absolute percentage error (MAPE)

Is a relative or percentual measure of the error that allows to make comparisons when scales are differentMEASURES TO ESTIMATE ERRORS

MEASURES TO ESTIMATE ERRORS

STATISTICAL MEASURES OF GOODNESS OF FITThe Correlation Coefficient (R):Measures the strength and direction of linear relationships between two variables. It has a value between 1 and +1. A correlation near zero indicates little linear relationship, and a correlation near one indicates a strong linear relationship between the two variables

The Determination Coefficient(R2) measures the percentage of variation in the dependent variable that is explained by the regression or trend line. It has a value between zero and one, with a high value indicating a good fit.

In trend analysis the following measures will be used:

MOVING AVERAGE FORECASTING METHOD STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa Olascoaga

MAIN IDEA OF THE METHOD

The moving average uses the average of a given number of the most recent periods' value to forecast the value for the next period.

Moving average smoothes down the fluctuations in the data

SMOOTHING

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Linear

Trend (linear)

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Linear

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Linear trend

Logarithmic

Trend (logarithmic)

Logarithmic

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Logarithmic

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Logarithmic trend

Power

Trend (power)

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Power

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Trend (power)

Exponential

Trend (exponential)

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Exponential

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Trend (exponential)

Polynomial

Trend (polynomial)

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Polynomial

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PREREQUISITE DATA PATTERN

Moving average method is commonly used when: the pattern in the data does not have periodical (seasonal or cyclic) characteristics is neither growing nor declining rapidly.

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Complex





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Linear






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Trend (poly)


Power

Trend (linear)

Linear

1.31

2

3.6

4.62

5.33

6.67

7.82

8.61

10.22

10.85

Power

1.31

2

3.6

4.62

5.33

6.67

7.82

8.61

Linear trend

Exponential

Trend (logarithmic)

Logarithmic

1

4.67

5.95

7.32

8.53

9.6

10.14

10.8

11.45

11.75

Exponential

1

4.67

5.95

7.32

8.53

9.6

10.14

10.8

Logarithmic trend

Polynomial

Trend (power)

0.55

1.2

2.21

3.79

5.6

7.72

9.52

5.66

9

10.11

Polynomial

0.55

1.2

2.21

3.79

5.6

7.72

9.52

Trend (power)


Trend (exponential)

0.09

0.64

1.2

2.33

7.4

24.3

51.7

14.5

21.3

10


0.09

0.64

1.2

2.33

7.4

24.3

51.7

Trend (exponential)

Trend (polynomial)

3.36

1.29

1.43

2.66

4.47

6

6.48

5.95

3.36

1.29

1.43

2.66

4.47

6

6.48

5.95

Trend (polynomial)




140

261

343

435

558

652

737

850

960

1067

1141

1233

1350

1461

1544

1646

1760

1835

1957

2038

2163

2247

2337

2450

Data

Smoothed data

Moore smoothed data


FORMULAIf forecast for t period is denoted by Ft, and the actual value of the time-series was Yt-1 during period t-1, Yt-2 during period t-2, etc., then k period simple moving average is expressed as:

CHOOSING THE AVERAGING PERIODThe averaging period (value of n) must be determined by the decision-maker. It is important to try to select the best period to use for the moving average. As a general rule, the number of periods used should relate to the amount of random variability in the data. Specifically, the bigger the moving average period, the greater the random elements are smoothed.

EXAMPLE

MonthNumber of clientsMoving average (2)January50February30March2040April3025May (forecast)25

CALCULATION OF ERRORS: EXAMPLE

MonthNumber of clientsMoving average (2)ErrorPercentage error (MAPE)January50February30March2040-20100April3025517May (forecast)25Mean error7.558.5

EVALUATION OF MAAdvantage: Very simple method.Shortcomings: Not applicable when trend exists,No strict rule of choosing its parameter,The new and the old data are treated in the same way (while, in fact, the old data should be treated as being less signifficant).

EXAMPLECertain barbecue restaurant offers an entre of lamb on Friday. This rare dish has been ordered by 75, 64, 68 and 70 clients during the past four weeks. Prepare a demand forecast for the next week using a moving average of four periods.

EXAMPLEThe following chart shows the price (in U.S. Dlls.) per ounce of silver in the first trading day of each month. With a running average of 10 terms, predict the price of silver for the next month..

EXPONENTIAL SMOOTHING FORECASTING METHOD STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa Olascoaga


The exponential smoothing method main idea is to smooth down variations in the data. Forecast error for the previous period is taken into account. Time series should have irregular variations:

Chart1

40

61

43

35

58

52

37

50

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Horizontal pattern





140

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435

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652

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850

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1067



539

645

746

822

934

1016

118

1241.4

1344.3

1441.1

1525

1635

179

1813

1934

2041

2143

2229

2336

248

256



102

111

122

1316

146

1518

1631

1726

181

1929

206

215

2228

2321

2419

2514

2624

276

285

2926

3027

3136

139

224

37

45

539

645

746

822

934

1016

118

1269

1335

1433

1525

1635

179

1813

1934

2041

2143

2229

2336

248

256

2660

2751

2862

2956

3049

122

223

343

428

526

6122

728

818

923

1030

1137

1234

1336

1427

157

163

1720

184

Horizontal pattern


Trend


Periodical seasonal


Complex





11063

22369

33159

43964

55001

Complex


Linear






539

645

746

822

934

1016

118

1241

1344

1441

1525

1635

179

1813

1934

2041

2143

2229

2336

248

256

Linear

39

45

46

22

34

16

8

41.4

44.3

41.1

25

35

9

13

34

41

43

29

36

8

6

Periodical seasonal

Logarithmic




102

111

122

1316

146

1518

1631

1726

181

1929

206

215

2228

2321

2419

2514

2624

276

285

2926

3027

3136

139

224

37

45

539

645

746

822

934

1016

118

1269

1335

1433

1525

1635

179

1813

1934

2041

2143

2229

2336

248

256

2660

2751

2862

2956

3049

122

223

343

428

526

6122

728

818

923

1030

1137

1234

1336

1427

157

163

1720

184

Logarithmic

2

1

2

16

6

18

31

26

1

29

6

5

28

21

19

14

24

6

5

26

27

36

39

24

7

5

39

45

46

22

34

16

8

69

35

33

25

35

9

13

34

41

43

29

36

8

6

60

51

62

56

49

22

23

43

28

26

122

28

18

23

30

37

34

36

27

7

3

20

4

Trend (poly)


Power

Trend (linear)

Linear

1.31

2

3.6

4.62

5.33

6.67

7.82

8.61

10.22

10.85

Power

1.31

2

3.6

4.62

5.33

6.67

7.82

8.61

Linear trend

Exponential

Trend (logarithmic)

Logarithmic

1

4.67

5.95

7.32

8.53

9.6

10.14

10.8

11.45

11.75

Exponential

1

4.67

5.95

7.32

8.53

9.6

10.14

10.8

Logarithmic trend

Polynomial

Trend (power)

0.55

1.2

2.21

3.79

5.6

7.72

9.52

5.66

9

10.11

Polynomial

0.55

1.2

2.21

3.79

5.6

7.72

9.52

Trend (power)


Trend (exponential)

0.09

0.64

1.2

2.33

7.4

24.3

51.7

14.5

21.3

10


0.09

0.64

1.2

2.33

7.4

24.3

51.7

Trend (exponential)

Trend (polynomial)

3.36

1.29

1.43

2.66

4.47

6

6.48

5.95

3.36

1.29

1.43

2.66

4.47

6

6.48

5.95

Trend (polynomial)




140

261

343

435

558

652

737

850

960

1067

1141

1233

1350

1461

1544

1646

1760

1835

1957

2038

2163

2247

2337

2450

Data

Smoothed data

Moore smoothed data


EVALUATION OF ESAdvantage: Rather simple method; More recent data are considered as being more influential. Requires only two data values to smooth the future: the most recent observation, and the most recent estimate, also requires a default value for alpha ().Short term - inventoriesShortcomings: No definite rule of choosing its parametersThe value of should be determined by trial and error. (Test different values of to find the lowest MSE).

EVALUATION OF ESRecomendations: To predict long-term or smooth a series by removal of cyclical and irregular variations, select a small value for = 0

FORMULAFt+1= forecast for the period t + 1 Yt = real value of time series in period tFt = forecast for period t of time series = smoothing constant (0 1)

EXAMPLE

MonthNumber of clientsForecast(Exponential Smoothing)=.02January40February37March43April45May

Calculation of errors: Example

MonthNumber of clientsExponential Smoothing Forecast=ErrorSquared errorJanuary40February37 March43April45May

EXERCISESThe sales of cars in a Ford Agency are calculated using simple exponential smoothing. The agency believes that the sales have no seasonal variation or trend, thus this method works fine. However, each month of march the agency observes that the forecast for March is higher than that of february by 200 units. Supose that at the end of february of 2002, Yt = 600. During march of 2002, 900 cars were sold. With = 0.6, find the expected sales of cars for April 2002.

EXERCISESApril sales expected to be for 860 units

Real SalesForecast1- February6000.60.4March900800860

EXERCISESA distributor of construction materials is using exponential smoothing to forecast the monthly demand (in tons) of a material. For June the forecast demand was of 130 ton, however the sales were of 135 ton. For July, the forecast was for 133.75, and 132 ton were sold. Find the smoothing constant and forecast the demand for August.

EXERCISESThe smoothing constant equals .75 and the forecast the demand for August equals 132.4375 tons.

Real SalesForecastJune135130ft+1=a(yt)+ft(1-aJuly132133.75ft+1=ayt+ft-aftAugust132.4375ft+1 - ft =a(yt-ft)a= ft+1 - ft/(yt-ft)0.751- 0.25

EXERCISESThe weekly demand for car loans in the Washington Federal Credit Union in the last six weeks is shown in the table below. Find:

Find using exponential smoothing the forecast for week seven (try constants =0.1, 0.3 and 0.5)Using the MSE determine which of the three options provides the best forecast. Using the MAPE criteria determine which of the three options provides the best forecast.Calculate and interpret the corresponding values of the MAE. Calculate and interpret the corresponding values of the MPE.

WeekDemand of loans120218322424528623

EXERCISESThe weekly demand for car loans in the Washington Federal Credit Union in the last six weeks is shown in the table below. Find:

WeekDemand of loansForecastErrorAEESPEAPE1202183224245286237MEAN

ADAPTIVE RESPONSE RATE EXPONENTIAL SMOOTHING STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa Olascoaga


Adaptive exponential smoothing methods allow a smoothing parameter to change over time, in order to adapt to changes in the characteristics of the time series.

EVALUATION OF ARRESAdvantage: automatically adjusts the smoothing parameters based on the forecast error. The smoothed term is replaced by t, and the later is adaptive over timeShortcomings: The underlying rationale for time varying is less clear. as a random coefficient or autoregressive form, depending on the nature of the process.Another drawback for this model is that there is no explicit way to handle seasonality.A major flaw with smoothing models is their inability to predict cyclical reversals in the data, since forecasts depend solely on the past.

FORMULAe t = Yt Ft Et = et + (1)Et 1 = smoothed error Mt =letl+(1)Mt1 = abs. val. Of smoothed error =

= 0.2, a choice variable (chosen by the decision maker)

CHOOSING THE SMOOTHING CONSTANT The forecasting procedure for this model can be proceeded recursively:

Given the values of Yt and Ft, we can estimate et from (1). The et then is plugged into (2) and (3).Given the estimated value of , we can obtain t by using (4). Finally, we use t, Yt and Ft to predict Ft+1 by employing the first equation.

The smoothing constant () must be determined by the decision-maker. However, in this case = 0.2 is the general choice.

Example

EXERCISESA

STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa OlascoagaLINEARLY WEIGHTED MOVING AVERAGE


It calculates a second moving average from the original moving average.

EVALUATION OF ARRESAdvantage: This method is used to forecast series with a linear trend line since this method handles it better that the simpler one.Designed to handle trending data. One set of moving averages is calculated and then a second set is calculated as a moving average of the first set.Weighted Moving Average - place more weight on recent observations. Sum of the weights needs to equal 1

FORMULAFirst moving averageDifference between both moving averagesSecond moving averageAnother adjustment constantForecast value p periods to the future Note: n is periods to calculate the moving average and p is periods to forecast

ExampleCalculate de DMA for the following rental information. Consider a p=3.

TimeRents16542658366546725673667176938694970110703117021271013712147111572816

Example

Chart1

65411

65822

6656593

6726654

673670664.6666666667

671672669

693679673.6666666667

694686679

701696687

703699.3333333333693.7777777778

702702699.1111111111

710705702.1111111111

712708705

711711708

728717712

Rents

SMA

DMA

Time

Rents

Weekly Rents 3 Period Double Moving Average

Sheet1

TimeRentsSMADMAabForecastError

1654

2658

3665659

4672665

56736706656755

66716726696753681-10

7693679674684567815

869468667969376904

970169668770597001

107036996947056714-11

117027026997053710-8

1271070570270837082

1371270870571137111

147117117087143714-3

15728717712722571711

16727

Sheet2

Sheet3

EXERCISESLet 198.75 be the first linear moving average in the period 8, and 199.93 the value of the second moving average in that same period. Calculate the value of the forecast for period 9, 15 and 18. Consider n = 4.

EXERCISESFind the equation of the line and the forecast for 1998, using DMA for four periods (n=3)

STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa OlascoagaBROWN METHOD OF FORECASTING


Also known as Double Exponential Smoothing. Used to forecast a series with a linear trend.

EVALUATION OF BROWNAdvantage: Similar to the double moving average model.If the trend as well as the mean is varying slowly over time, a higher-order smoothing model is needed to track the varying trend.Uses two different smoothed series that are centered at different points in time. The forecasting formula is based on an extrapolation of a line through the two centers.

FORMULAEquationsUsed for t: 2, 3.

FORMULAEquationsUses a single coefficient, alpha, for both smoothing operations. Calculates the difference between single and double smoothed values as a measure of trend (at). It then adds this value to the single smoothed value together with an adjustment for the current trend (bt).

EXERCISESGiven b4 = 1.0696 and A4 = 96.064 calculate the value of the forecast for period five. Use =.2

Given the first and second observation whose values are 94 and 90, respectively determine the forecast for period 3. Use =.25

ExampleFind the linear equation with this method and the forecast for period 8,10 & 15. Use =.2

PeriodYtAtAtatbt15.425.335.345.656.967.277.281015

STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa OlascoagaHOLT METHOD


Is an extension of Browns double exponential smoothing but it uses two coefficients one for smoothng trend and other the slope .

=is the smoothing constant for the level = is the trend smoothing constant - used to remove random error


How do we find the best combination of smoothing constant?

Low values of alpha and beta should be used when there are frequent random fluctuations in the data.High values of alpha and beta should be used when there is a pattern such as trend in the data.

EVALUATION OF HOLTAdvantage: Reduces the effect of randomness (using the difference between the averages calculated in two successive periods). Updates the forecasting trend.Avoids a forecast with a reaction delayed growth.

Equations:

Where:

FORMULA

Start with:

Define the smoothing constantsFind At and Tt for all the periods

Find the quadratic equation

FORMULA

EXERCISESThe following table shows annual tax on income (millions of U.S. Dlls.) paid to the Government by residents of a city. Use the method of Holt with = 0.95 and = 0.5, to obtain the forecasting equation and the forecast for 1998.

Example

YearPeriodYtAtTtFt1995155.41996261.51997368.71998487.21999590.42000686.22001794.720028103.220039119200410122.4200511131.6200612157.6200713181200814217.8200915244.12012

ExampleUse the Holt method to predict the monthly sales of DVD's in Highland Appliance. It is known that in October 2010, At = 200 and Tt = 10. During November 2010, 230 were sold DVD's. Forecasts sales of DVDs for December 2010 using = = 0.5.

ExampleThe method of Holt, exponential smoothing for time trend with no seasonal variation series, is used to predict weekly sales of an agency Ford. The previous to last week sales were 50 cars per week, with a trend of 6 cars per week. 30 Cars were sold last week. After observing the sales of the previous to last week, predict the number of cars to be sold during this and the next week. Use = = 0.7. Discuss the result.

ExampleThe following table shows the sales of saws for the Acme tool Company.These are quarterly sales From 2008 through 2010.

YearQuartertsales200811500223503325044400200915450263503720048300201019350210200311150412400

ExampleFor Holt Method use: = .3 and =.1 to predict sales for 2008 Q3 and 2010 Q2Use Brown to forecast Q3 of 2011Use DMA: to forecast with a moving period of 4 Q2 2010

YearQuartertsales200811500223503325044400200915450263503720048300201019350210200311150412400

STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa OlascoagaQES METHOD


This method is used when there is a non-linear trend in the time series

EVALUATION OF QESAdvantage: This technique achieves good results when forecasting this type of series as it uses three smoothings to perform the calculation of forecast.

FORMULA

FORMULAConsider:

Calculate the first, second and third smoothing for t= 2,3,4,

Find the quadratic equation

EXERCISESCalculate using = 0.1 the quadratic equation for this model and the value for the forecast for period 3 & 7. Having the observed values for periods one to four: 105, 108, 111 and 103.

Example

EXERCISESFor the following exercises consider that the trend equation is quadratic with a smoothing coefficient : 0.3.

Find the equation knowing that: Y4= 9.25, A3= 8.22, A3=8.31 & A4= 8.4

There are only two observations for a new product are: 8.48 & 8.06. Find the equation and the forecast for time 4.

STATISTICS 2ITESM Campus GuadalajaraProf: Ing. Maria Luisa OlascoagaWINTERS METHOD


Another method for forecasting that can adequately handle the presence of seasonality in a time series is the Winters Method. (multiplicative seasonality)

Equations:

Where:

FORMULA

Start with:

Use formulas of the method

FORMULA

EXERCISEForecast for next year (Period 7, 8 & 9) using = 0.822 ; = 0.055 y = 0.001

PeriodSales (Yt)Level (At)Trend (T)Seasonal Index (SI)Forecast (Ft)136223853432434153826409749883879473

EXERCISE

PeriodSales (Yt)Level (At)Trend (T)Seasonal Index (SI)Forecast (Ft)1362 0.953 2385 1.013 3432 1.137 43413809.750 0.897 5382398.99310.258 0.953 371.29 6409404.67910.007 1.013 414.64 7 471.43 8 381.11 9 414.11

Outdoor Furniture Swings. Usually the customers buy more swings in the hot months than in the cold ones, so sales change with the seasons. Suppose that the Outdoor Furniture swings are very good and verbal advertising increases the number of people who buy them. Their data that reflects seasonality and trend are given in the following table. Calculate sales for next year using = 0.15 ; = 0.1 y = 0.2

EXERCISE

Outdoor Furniture Swings.

EXERCISE

Outdoor Furniture Swings. (L=4)

EXERCISE

YearQuarter1231606984223426631031631882124505964

Use the method of Winters for a time series with seasonality (quarters). The values registered for the four quarters of the first year are: 57.26 66.81, 68.39 and 77.99 respectively; Likewise known records of the four quarters of the second year are: 65.18, 74.02, 74.12 and 82.31. Use the forecasting method to find period 9. (= 0.3, = 0.2 and = 0.2)

EXERCISE

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