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7/31/2019 Meteorological Forecasting Using Type 2 Fuzzy Logic System
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Prepared By Selver Arslan
May 2012
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Time series represents a powerful mathemetical tool to model
the temporal evolution of phenomenon in physical , economic
and biological environment.In this light of this meteorological
forecasting can be rendered as a complex time series
prediction problem. To overcome the problem of the residual non-stationary of the
random components of data and to consider other sources of
uncertainity , such as the model uncertainity related to tuning
of the predictor parameters in this work we extend theapproach on very recent class of fuzzy logic systems.
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Most time series patterns can be described in terms of 2
basic components: Trends and seasonality.Trends represents a
general systematic linear or nonlinear component that changes
overtime.Seasonality may have a formally analogous nature.
But it repeats itself in systematic intervals over time.
We consider the measurements of temperature and relative
humidity collected 2004 in about one year in Torino Italy at a
sample rate 900 second .
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Missing data is a frequent occurring phenomenon in many
practical contexts.For example , the measurements can not be
acquired directly. Or the measured values are unreliable or
missing or interrupted data collection occurs.All these casesfrequently happen.
In these cases a few missed data can be extrapolated by a data
fitting prosedure.Unfortunately ,in our application many data
are missed as shown below.Thus , we only consider the valuesof T and RH collected during the first 7 months of the year.
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There is no automatic technique that can identify trend
components in the time series data; However as long as the
trend is monotonous (increasing or decreasing).The analysis is
typically not very difficult.When random components arepresent , the first step is smoothing.
Lastly a trend can be exracted by performing a data
fitting.Many monotonous time series data can be adequatelyapproximated by a linear function.
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Seasonal dependency is another general component of the time
series pattern.It is formally defined as correlational dependency
of the order k between each ith element of the series and the
(i-k)th element is measured by autocorrelation ; k is usually
called the lag.Seasonality can be visually identified in the seriesas a pattern that repeats every k elements
The seasonal component may be extracted by filtering or by a
time frequency analysis.In our application the maximum
frequency of the seasonal component is so slow that very highorder filters should be used.
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Once seasonality and trend have been identified,and a random
component has been reduced, then a residual component still
represents the core of the data.This is called the chaoticcomponent which depends on some initial state that does not
repeat over time and that can not be reduced or eliminated by
averaging.
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All the steps described above aim to exract a stationary time
series from the collected data.This issue is crucial in many
forecasting methods.Meteorological time series are quite
complex as shown.And many different components contribute
to the temporal evolution of the data.The non-stationary of theresidual component is clear and it can be also shown by
performing a local variance estimation of the time series, by
considering 100 samples at a time .In doing so we obtain the
plot shown below.
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STEP 1:First in a Type-1 FLS we expect a crisp value for the
input value , while in a type 2 FLS we can consider a
membership function for the input value x.This issue is crucial
when data are affected by random contributios or non-stationarynormal random component with a time dependent variance 2xsuch that x is between |1,2| We can easily build the MF as
shown below.
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Analogously , if the dominant uncertainity contribution relates
to the central value of the MF as it occurs in the fuzzy sets
pattitioning the input domain.Then an MF with uncertain
central value and fixed width is built as shown below.
Once the input domain of each input (data of the time series in
our context) has been partitioned by means of type 2 MFs such
as depicted in figure below , consider the generic jth fuzzy ruleof the defined predictor as :
If (Xn-p+1 is A1) and (Xn-p+2 is A2) and (Xn is Ap+1) then (Xn+1 is Ap+1)
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STEP 2 : At this point , the evaluation of each of eachantecedent is performed by implementing the intersection of
the MF of the input variable and the considered MF of the
input domain.The intersection can be accomplished by several
operators.Here , we implement the minimum operator.Each
antecedent has an interval of membership degrees associated
figure below and denoted as [lx1,ux1] and [lx2,ux2] where l and u
stand for lower and upper.
STEP 3 :To implement the and between two antecedents , weapply again the minimum operator to the 2 intervals achieved
above obtaining the interval [l,u] as shown figure below.This
interval represents the degree of the truth of antecedents in a
certain rule.
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STEP 4: Implication operators are needed to apply the degreeof truth of each rule to the consequent .In our context, with theoutput and the input domains the same, the MF of the output
variable considered in a rule are weighted by the interval [l,u]resulting in a clipping of the MFs of the output variable.Thesituation is depicted in figure below in the case of 2 firedconsequents.
STEP 5:This step involves fuzzy rules aggregation.In particular,
an orbetween different rules is implemented by the maximumoperator among the lower and upper degrees of each firedconsequent respectively.The result are two MFs a lower andupper one as shown in figure 12 where ,for instance weaggregate the two rules that have been activated in the previous
steps considering the uncertain region bounded by the lowerand upper MFs.This is footprint of uncertainity.
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STEP 6:
This step shows the important difference in using type-2 FLS with
respect to type 1 FLS. A type reducer has to be applied before
defuzzifying the output.Type reduction extracts an interval for the
output value from the uncertain region .Among different methods
proposed in the literature , here we apply the centroid type reduction.
The procedure works as follows.Define 3 vectors.Y vector represents
sampling points.C vector represents centers.S vector represents
spreads.The output value is between ylow and yhigh which is achived atmost Ns iteration.Center of the centroid is calculated according to
standart centroid center formula.
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This paper presents a novel approach to forecast Meteorological
phenomena. It is based on the signal decomposition and on an
architecture that uses a Type-2 FLS in order to predict the
chaotic components.Signal decomposition permits to focus the
prediction only on chaotic components. This procedure allows toobtain longterm prediction bounded errors. Using Type-2 FLS is
also veryimportant because this class of systems has shown
small sensitivity to measurements uncertainty and noise. The
whole system has been designed and simulated on historical
meteorological data previously acquired. First results show
promising performance with respect to other methodologies