Research Article Hybrid Wavelet-Postfix-GP Model for Rainfall Prediction of Anand Region of India

12 Pages·2015·1.86 MB·English

Hindawi Publishing Corporation
Advances in Artificial Intelligence
Volume 2014, Article ID 717803, 11 pages
http://dx.doi.org/10.1155/2014/717803
Research Article
Hybrid Wavelet-Postfix-GP Model for Rainfall Prediction of
Anand Region of India
VipulK.Dabhi1andSanjayChaudhary2
1InformationTechnologyDepartment,DharmsinhDesaiUniversity,Nadiad387001,India
2IICT,AhmedabadUniversity,Ahmedabad380009,India
CorrespondenceshouldbeaddressedtoVipulK.Dabhi;[email protected]
Received11January2014;Accepted15May2014;Published2June2014
AcademicEditor:DjamelBouchaffra
Copyright©2014V.K.DabhiandS.Chaudhary. This is an open access article distributed under the Creative Commons
AttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkis
properlycited.
Anaccuratepredictionofrainfalliscrucialfornationaleconomyandmanagementofwaterresources.Thevariabilityofrainfall
inbothtimeandspacemakestherainfallpredictionachallengingtask.Thepresentworkinvestigatestheapplicabilityofahybrid
wavelet-postfix-GPmodelfordailyrainfallpredictionofAnandregionusingmeteorologicalvariables.Thewaveletanalysisisused
asadatapreprocessingtechniquetoremovethestochastic(noise)componentfromtheoriginaltimeseriesofeachmeteorological
variable.ThePostfix-GP,aGPvariant,andANNarethenemployedtodevelopmodelsforrainfallusingnewlygeneratedsubseries
ofmeteorologicalvariables.Thedevelopedmodelsarethenusedforrainfallprediction.Theout-of-samplepredictionperformance
ofPostfix-GPandANNmodelsiscomparedusingstatisticalmeasures.TheresultsarecomparableandsuggestthatPostfix-GP
couldbeexploredasanalternativetoolforrainfallprediction.
1.Introduction thegiventimeseriesandtheindependenceoftheresiduals.
Moreover,theseapproacheslacktheabilitytoidentifynon-
An accurate prediction of rainfall is crucial for agriculture linearpatternsandirregularityinthetimeseries.
based Indian economy. Moreover, it also helps in the pre- Hence, in recent years, use of different machine learn-
vention of flood, the management of water resources, and ing techniques for modeling and prediction of rainfall has
generatingrecommendationsrelatedtocropforfarmers[1]. receivedmuchattentionofpractitioners[7].Hungetal.[8]
The variability of rainfall in both time and space makes developed a neural network for 1 to 3 hours ahead forecast
the rainfall prediction a challenging task. Moreover, the ofrainfallforBangkok.Theyusedmeteorologicalparameters
meteorologicalparametersneededfortherainfallprediction (air pressure, relative humidity, wet bulb temperature, and
are complex and nonlinear in nature. Practitioners have cloudiness)andtherainfallregisteredattheneighborstations
applied numerical [2, 3] and statistical [4, 5] models for asinputintheneuralnetwork.Theyusedthehydrometeoro-
the rainfall prediction. The Numerical Weather Prediction logicaldatathatcoversbothrainyandnonrainyperiodsfor
(NWP) models are deterministic models and approximate trainingtheneuralnetwork.Moustrisetal.[9]usedaneural
complexphysicalprocessesforweatherprediction.However, network approach for forecasting the monthly minimum,
themodelsarenotusefulforpredictionatsmallerscaledue maximum, mean, and cumulative precipitation for the four
to inherent limitations of these models to initial conditions meteorologicalstationsofGreece.TheynotedthatANNwas
and model parameterization. Practitioners have also used not able to predict the peaks in all cases. They suggested
autoregressive moving average (ARMA) and autoregressive increasing the size of training dataset to overcome this
integratedmovingaverage(ARIMA)techniquesfordevelop- problem [9]. Lin and Chen [10] used a neural network for
ing a model for the rainfall [6]. However, these approaches forecastingtyphoonrainfall.ANNisdevelopedwhichtakes
were developed based on the assumption of stationarity of typhoon characteristics (direction of typhoon movement,
2 AdvancesinArtificialIntelligence
latitudeandlongitudeofthetyphooncentre,themaximum [24] to develop a model that can explain the cause and
windspeed,andatmosphericpressureofthecentre)andthe effectrelationshipbetweenrainfallandrunoffprocessesata
spatialrainfallinformationofnearbyraingaugeasinputsand catchmentinSingapore.BabovicandKeijzer[25]employed
gives1houraheadforecastoftyphoonrainfall.Practitioners GPfordevelopingrainfall-runoffmodelsusingthehydrom-
applied ANN [11, 12] and approach based on chaos theory eteorologicaldataandtheavailabledomainknowledge.Khu
[13]fornumericalmodelerrorpredictioninhydrology.ANN etal.[26]appliedGPforreal-timerunoffforecastingatthe
andconceptsofchaostheoryareusedtoadjustthevalueof Orgeval catchment in France. The GP is used as an error
outputsproducedbythenumericalmodel. updatingstrategytoaccompanyarainfall-runoffmodel.The
The hydrometeorological time series can be considered GPproducedresultsarecomparedwiththoseobtainedusing
as a composition of stochastic (noise or fluctuations) and autoregressionandKalmanfilter.TheresultssuggestthatGP
structuredcomponents.Thestochasticcomponentobscures outperforms other strategies for real-time flow forecasting
themodelingoftimeseries.Thestructuredcomponentcan [26].
be extracted by removing the stochastic component from a The objective of the present work is to explore the
timeseries.Then,adeterministicmodelcanbedevelopedfor applicability of a hybrid wavelet-postfix-GP model for pre-
thestructuredcomponentofatimeseries.Practitionershave diction of daily rainfall of Anand station of Gujarat, India,
used following data preprocessing techniques for cleaning using meteorological variables. The wavelet analysis is used
hydrological time series: wavelet analysis (WA), principal to decompose the original series of each meteorological
component analysis (PCA), and singular spectrum analysis variable into various discrete wavelet (DW) subseries. The
(SSA). However, in recent years, the wavelet analysis has decompositionisusefultoidentifytheDWsubserieswhich
become an effective tool for analyzing nonstationary time have high correlation with the original rainfall series. The
series. effective DW subseries for each meteorological variable are
Partal and Cigizoglu [14] proposed a wavelet-ANN added to generate a final subseries. The objective behind
approachforpredictingthedailyprecipitationof12meteo- addition of DW subseries is to increase the correlation
rological stations of Turkey. They used meteorological data between the final subseries and the original rainfall series.
for precipitation prediction. Nasseri et al. [15] applied a ThePostfix-GP,aGPvariant,isthenusedtodevelopamodel
combination of back propagation algorithm and genetic thatcanexplaintherelationshipbetweenthefinalsubseriesof
algorithm (GA) for rainfall forecasting in western suburbs meteorologicalvariablesandtheoriginalrainfallseries.The
ofSydney.TheyusedGAtotrainandoptimizefeed-forward Postfix-GP uses linear individual representation and stack
neural network. They concluded that a combined approach based evaluation. This helps Postfix-GP to minimize both
outperformedanapproachthatusesaneuralnetworkalone. the memory requirement and evaluation time compared to
A modular artificial neural network (MANN) [16] is com- conventionaltreebasedrepresentation.TheevolvedPostfix-
bined with three data preprocessing techniques: moving GPmodelsarethenusedforout-of-samplepredictions.The
average(MA),PCA,andSSAforpredictionoftwomonthly performance of the evolved Postfix-GP model is measured
andtwodailyprecipitationseries. using mean absolute error (MAE), mean squared error
The symbolic regression technique can be used for (MSE),andcorrelationcoefficient(CC).
developing a model (mathematical model in a symbolic The rest of the paper is organized as follows. The next
form)thatcanexplaintherelationshipbetweenrainfalland sectionpresentshomogeneityanalysisofcollectedmeteoro-
meteorological parameters. The advantage of the symbolic logicaldata.Section3presentstheproposedhybridwavelet-
regression technique over traditional regression techniques Postfix-GP model for rainfallprediction. Section4 presents
isthatitsearchesforboththestructureandtheappropriate experimental settings. The evolved Postfix-GP model that
numeric coefficients of the model. Symbolic regression can representsrainfallasfunctionofmeteorologicalsubseriesis
be performed by means of genetic programming (GP) [17]. presentedinSection5.Thesectionalsopresentscomparison
Moreover, GP approach is preferred over other approaches of the out-of-sample predictive performance of wavelet-
for symbolic regression because the approach produces an Postfix-GPmodelandwavelet-ANNmodel.Conclusionsare
explicitmathematicalexpressionasasolution(model)[18]. presentedinSection6.
Theproducedmodelcanprovideaninsightintotheprocess
which gives rise to the data [19]. Moreover, the interpreta-
tion of the produced model allows us to combine evolved 2.DataandHomogeneityAnalysis
knowledge with already existing knowledge [20, 21]. An
exhaustive survey on open issues and approaches used by MeteorologicaltimeseriesofAnandregionfor12years(from
practitioners to deal with these issues in field of symbolic 1991to2002)werecollectedfromAnandAgricultureUniver-
regressionthroughGPispresentedin[22].KisiandShiri[23] sity,Anand,Gujarat,India.Wehaveused10yearsofdatafor
suggestedthatuseofGPispreferredinfollowingsituations: trainingand2yearsofdataforpredictions.Wehavecollected
(i)therelationshipbetweentherelevantvariablesarepoorly timeseriesdataforthefollowingmeteorologicalvariables:(i)
understood,(ii)determinationofoptimalsolutionisdifficult, minimumtemperature—𝑇Min,(ii)maximumtemperature—
(iii)anapproximatesolutionisacceptable,and(iv)thereisa 𝑇Max,(iii)meantemperature—𝑇Mean,(iv)relativehumidity—
largeamountoftrainingdatatobemodeled. RH, (v) evaporation—EP, (vi) 1-day previous rainfall—RF1,
Considering the mentioned advantages, many practi- and (vii) 2-day previous rainfall—RF2. The objective is to
tioners applied GP for rainfall prediction. GP is used in evolve a model (as a function of mentioned meteorological
AdvancesinArtificialIntelligence 3
Table1:Resultofdifferenthomogeneitytests.
Test 𝑇Min 𝑇Max 𝑇Mean RH EP RF
Pettitt 0.204 0.241 0.238 0.354 0.481 0.093
SNHT 0.372 0.466 0.500 0.528 0.225 0.151
BR 0.287 0.438 0.395 0.508 0.455 0.115
VNR 0.495 0.146 0.595 0.004 0.304 0.127
variables)fordailyrainfallusingahybridwavelet-Postfix-GP (v) normalize the values of newly generated subseries in
approach. the range (0, 1), (vi) divide the normalized dataset into
Homogeneitytestsareappliedtodetectthevariabilityof (a) training and (b) test dataset, (vii) evolve the model for
themeteorologicaldata.Severalfactorscanaffectthequality training data using Postfix-GP, and (viii) apply the evolved
ofmeteorologicaldata.Themainsourcesofinhomogeneity modelforout-of-samplerainfallprediction.Theobjectiveis
are station relocation, changes in measurement techniques toevolveamodel(asafunctionofmentionedmeteorological
and observational procedures, and changes in instruments. variables) for rainfall using a hybrid wavelet-Postfix-GP
Homogeneity test are useful to detect the break (shift in approach.Inthefollowingsubsections,wediscussthewavelet
themean)inthegiventimeseries.Asmanymeteorological transformandPostfix-GPinbrief.
parameters are highly variable in time and space, most of
the homogeneity tests are designed for monthly or yearly
3.1. Wavelet Transform. The properties of irregularity in
data and not for daily data. We applied the following four
shapeandcompactnessmakewaveletsanidealtoolforanal-
homogeneity tests to meteorological time series [27]: stan-
ysis of nonstationary signals. Fourier analysis decomposes
dardnormalhomogeneitytest(SNHT),Buishandrange(BR)
a signal into sine and cosine waves of various frequencies
test,Pettitttest,andvonNeumannratio(VNR)test.Thenull
whereaswaveletanalysisdecomposesasignalintoshiftedand
hypothesisforallthementionedtestsistheannualvalues𝑋𝑖 scaledversionsofthemotherwavelet.Theshifting(delaying)
ofthetestvariable𝑋whichareindependentandidentically
ofthemotherwaveletprovideslocalinformationofthesignal
distributed and series can be viewed as homogeneous [27].
intimedomainwhereasscaling(stretchingorcompressing)
ThealternativehypothesisforBRtest,SNHT,andPettittest
of the mother wavelet provides local information of the
presumesthatthereisabreakinthemeanoftheseriesand
signal in frequency domain [28]. The scaling and shifting
theseriescanbeviewedasinhomogeneous.Thereasonfor
operations applied to mother wavelet are used to calculate
applying more than one test to check homogeneity is that
wavelet coefficients that provide correlation between the
thesetestshavedifferentsensitivityindetectingabreak.For
waveletandlocalportionofthesignal.Fromthecalculated
example,theSNHTtestissensitiveindetectingabreaknear
wavelet coefficients, we can extract two types of compo-
the starting and the end of the series whereas Pettit and
nents: approximate coefficients and detail coefficients. The
Buishandtestsaresensitiveindetectingbreakinthemiddle
approximatecoefficientsrepresenthighscale,lowfrequency
oftheseries.
componentoftheoriginalsignalwhereasdetailcoefficients
The significance level is set to 1% for all tests. If the
representlowscale,highfrequencycomponent.
obtained 𝑃 value is lower than the selected significance
Continuouswavelettransform(CWT)operatesatevery
level,thenwerejectthenullhypothesis.Table1presentsthe
scalefromthatoftheoriginalsignaluptosomemaximum
results of homogeneity tests for the annual mean values of
scale. This distinguishes CWT from DWT (which operates
meteorologicalparameters.Asallthetestsgivevaluesgreater
at dyadic scales only). CWT is also continuous in terms
than0.01,thenullhypothesiscannotberejected(timeseries
of shifting: during computation, the analyzing wavelet is
is homogeneous). For the relative humidity, VNR test gives
shifted smoothly over the full domain of signal. The results
a 𝑃 value of 0.004, less than 0.01. However, as the 𝑃 values
of the CWT are wavelet coefficients, which are a function
associatedtotheothertestsaregreaterthan0.01,weaccept
of scale and position. Multiplying each coefficient by the
thenullhypothesis.
appropriatelyscaledandshiftedwaveletgivestheconstituent
waveletsoftheoriginalsignal.
The computation of wavelet coefficients at every scale
3.AHybridWaveletPostfix-GPModel
requires large computational time. To reduce the time, it
is preferred to calculate wavelet coefficients for selected
The architecture that integrates discrete wavelet transform
subset of scales and positions. If the scales and positions
(DWT) and Postfix-GP for modeling and prediction of
rainfalltimeseriesispresentedinFigure1.Thearchitecture are selected based on power of two (dyadic scales and
comprises the following main steps: (i) apply DWT on positions), then the analysis will be efficient and just as
meteorologicaldataandgenerateDWsubseriesateverylevel, accurate,nameddiscretewavelettransform(DWT)[29].The
(ii)calculatethecorrelationcoefficientbetweenthegenerated process of decomposition can be iterated, with successive
DWsubseriesandtheoriginalrainfallseries,(iii)identifythe approximationsbeingdecomposedinturn(discardingdetail
significantDWsubseries,(iv)addsignificantDWsubseries coefficients), so that original signal is broken down into
togenerateanewsubseriesforeachmeteorologicalvariable, manylower-resolutioncomponents.Thisprocessisreferred
4 AdvancesinArtificialIntelligence
Raw data
Check homogeneity of annual data Start
(time series)
Selection of mother wavelet and
decomposition level Generate semantically diverse
initial population
Repeat
Evaluate fitness of individuals of
for every Apply DWT on raw data (time series) to
population on training data
meteorological generate DW subseries at every level
variable
Calculate CC between every DW subseries
and original rainfall series Termination
criterion
satisfied
Identify significant DW subseries
(components)
Apply genetic operators to produce
Add significant components and calculate
semantically diverse offspring
CC between summed series and
original rainfall series
Return to the best solution
Training
data
Divide data set into: (i) training, (ii) test
End
data set
Test Best solution
data
Perform out-of-sample predictions
Figure1:HybridWavelet-Postfix-GParchitectureforrainfallmodelingandprediction.
as multiresolution analysis. We have selected Db4 (length- 3.2. Postfix Genetic Programming. Postfix-GP [30], a GP
4 Daubechies) [28] wavelet as mother wavelet because variant,adoptspostfixnotationforindividualrepresentation.
this is one of the commonly used wavelets for separating Individuals represented in form of postfix strings can be
fluctuations from the given time series. The smoothness easily evaluated using stack. Moreover, the individual rep-
of different wavelets depends on the number of vanishing resentationwithoutpointersassistsPostfix-GPtominimize
moments [29]. Db4 wavelet has four vanishing moments, a thefitnessevaluationtimeandrequiredmemorytostorethe
smallestlengthwaveletwithsmoothnessproperty.Wehave individual.EachPostfix-GPindividualcontainsthefollowing
setmaximumresolutionleveltovalue10fordecomposition three attributes: MinLength, MaxLength, and ValidLength.
ofeverymeteorologicaltimeseries. The MinLength and MaxLength attributes define the range
The forward discrete wavelet transform is employed ofsyntacticallyvalidPostfix-GPindividuals.TheValidLength
to decompose original time series of every meteorological attributereferstotheindexofthelastelementofanindividual
variable at different scale (maximum level 𝑛 = 10). The formingavalidpostfixexpression.AnindividualwithVali-
wavelettransformproduceshigh-pass(detail)coefficientsat dLengthgreaterthanMinLengthandlessthanMaxLengthis
everylevelandonelow-pass(approximation)coefficient.The consideredasvalid[30].
inversediscretewavelettransformisappliedontheproduced Postfix-GP employs idea of Stack count, introduced by
coefficients at every level to generate DW subseries, which Keith and Martin [31], to find out the ValidLength of an
areofthesamelengthastheoriginalseries.Thus,weobtain individual.TheStackcountofanelementiscalculatedasthe
𝑛 + 1 (𝑛 detail and one approximate) DW subseries for the numberofargumentspushedonthestackminusthenumber
original time series of every meteorological variable. The ofargumentspoppedoffthestackbytheelement.Forexam-
correlationcoefficientbetweenthegeneratedDWsubseries ple,thestackcountvalueforanoperandis1,whereasitis0
atdifferentlevelandtheoriginalrainfallseriesiscalculated. forunaryoperator.Furthermore,thetotalsumofstackcount
The number of DW subseries which have high correlation valuesmustbe1attheValidLengthpositionofanindividual.
withtheoriginalrainfallseriesisidentifiedandsummedup A Postfix-GP individual with its syntactically valid portion
to generate a new (final) subseries for that meteorological andcorrespondingtreerepresentationisdepictedinFigure2.
variable. The objective behind addition of DW subseries It should be noted that the transformation of syntactically
havinghighcorrelationwiththeoriginalrainfallseriesisto valid portion to a tree representation is not required for
reducethenumberofvariables(dimensionsorinputs)andto fitness calculation; it is shown for better comprehension of
increase the correlation between newly generated subseries thereader.
and the original rainfall series. This process is repeated for Table2 presents difference between an individual rep-
everymeteorologicaltimeseries. resentation scheme of Postfix-GP and other evolutionary
AdvancesinArtificialIntelligence 5
Terminal set={a,b}Function set ={+, −,∗, /, Q,S}
MinLength=5,MaxLength=12,ValidLength=8
Element position 1 2 3 4 5 6 7 8 9 10 11 12
Elements of individual b b ∗ b a S + ∗ a b b Q
Sum of stack count 1 2 1 2 3 3 2 1 2 3 4 4
∗
Identifying valid portion of an individual
∗ +
bb∗baS+∗
b b b S
Valid portion of an individual
a
Tree representation of valid portion of an individual
Figure2:Postfix-GPindividualrepresentation.
Table2:DifferencebetweenindividualrepresentationschemesofPostfix-GPandotherevolutionaryapproaches.
Evolutionaryalgorithms Genotype Phenotype Mapping
GA Linearstringoffixedlength Decodinggenotype Problemdependent
GP Parsetree Parsetree Traversingtree
GEP Linearstringoffixedlength Parsetree Karvanotation
Postfix-GP Linearstringoffixedlength Linearstring Postfixnotation
approaches, GA [32], GP [17], and gene expression pro- selectsbehaviorallydifferentparentsforgeneratingoffspring,
gramming(GEP)[33].ThestandardGArepresentsanindi- whichisusefultominimizethe“nochangetofitness”events
vidual using a fixed length binary string. The individual [39].Crossoveroftwodissimilarparentsislikelytoproduce
representation scheme of GA does not permit the model achangeinoffspring(solution)quality.
structuretovaryduringevolutionaryprocess.StandardGP Postfix-GP extracts all subtrees of an individual having
employs a variable length, tree structure for an individual ValidLengthgreaterthanMinLengthandtreattheextracted
representation. The representation scheme of GP is more subtrees as separate solutions during the evolutionary pro-
generalandflexiblethanGAsinceitallowsmodelstructure cess.Postfix-GPemploysone-pointmutationoperator,where
tovaryduringevolution.GEPisaGPvariantthatrepresents the chosen element of an individual is interchanged with a
an individual using multiple genes, where each gene repre- differentelementofthesamearity.Anarchiveisusedtostore
sents a small subexpression. Each GEP gene is composed “best-so-far”foundsolutions,usefultoexploitgoodsolutions
of two different domains—a head and a tail. GEP applies over a number of generations [40]. Postfix-GP uses MAE
Karva notation to transform a linear representation to tree as a standardized fitness measure. However, MAE measure
representation. Grammatical evolution (GE) [34] employs is not range bound. Therefore, we have normalized the
a string of integers to represent an individual (genotype). fitnessvalueofanindividualbetween0and1using(1).The
The string of integers is used to determine the sequence of normalizedfitnessisreferredtoasanadjustedfitnessofan
productionrulesincontext-freegrammar.Byfollowingthese individualandisusefultodifferentiatebetweenindividuals
sequences,anindividualcanbeconvertedintoanexpression havingclosestandardizedfitnessvalues,whichmayhappen
tree (phenotype). Adaptive logic programming (ALP) [35] in generations near the end of Postfix-GP run. Postfix-GP
also uses a string of integers for individual representation. is developed using.NET [41] framework on Windows XP
However,thestringofintegersisusedtoselectclausesina operating system. Zedgraph [42], an open source graph
logic program instead of production rules of a context-free library,isusedforplottingcharts.
grammar.
1
Postfix-GP produces initial population of individuals at AdjustedFitness= . (1)
random.TheValidLengthoftheseindividualswillhavevalue (1+Standardized Fitness)
inbetweenMinLengthandMaxLength.Postfix-GPemploys
semanticawaresubtreecrossover[36]toimprovepopulation 4.ExperimentalSettings
diversity among individuals. The operator checks semantic
equivalence [37, 38] of two subtrees, to be swapped, while We set a range of solution search space by specifying a
performing crossover operation. Moreover, the operator minimum and maximum number of elements (nodes) that
6 AdvancesinArtificialIntelligence
Table3:Correlationcoefficientbetweeneachsubseriesandtheoriginalrainfallseries.
Subseries 𝑇Min 𝑇Max 𝑇Mean RH EP RF1 RF2
DW −0.079 −0.029 −0.076 0.033 −0.127 −0.304 −0.013
1
DW −0.068 −0.093 −0.103 0.110 −0.124 0.130 −0.249
2
DW −0.060 −0.098 −0.097 0.101 −0.156 0.355 0.187
3
DW −0.033 −0.098 −0.087 0.114 −0.177 0.302 0.258
4
DW −0.027 −0.115 −0.094 0.132 −0.165 0.273 0.259
5
DW −0.043 −0.053 −0.056 0.052 −0.065 0.138 0.135
6
DW −0.030 −0.145 −0.102 0.160 −0.153 0.211 0.210
7
DW 0.230 0.117 0.211 0.225 0.025 0.251 0.250
8
DW 0.013 −0.045 −0.013 0.072 −0.044 0.081 0.081
9
DW −0.016 −0.018 −0.017 0.019 −0.026 0.019 0.019
10
Approximate 0.018 0.004 0.014 0.034 0.008 0.044 0.044
Summed 0.230 −0.236 0.211 0.356 −0.347 0.613 0.519
a solution can have during an evolutionary run. The Min- 5.Results
LengthandMaxLengthparametersofPostfix-GPareusedto
attainthistask.TheMinLengthandMaxLengthparameters 5.1. Identification of Important Components. The discrete
of Postfix-GP are set to values 10 and 40. The function wavelet transform(DWT) is applied on daily time series of
set includes both arithmetic and trigonometric operators: every meteorological variable to decompose the series into
{+, −, /, ∗, S, C, E, L, K, Q}, where S, C, E, L, and Q several DW subseries [14]. Each time series is decomposed
represent sine, cosine, exponential, logarithmic, and square up to 10 resolution levels. We have selected the resolution
root functions. The terminal set includes final subseries for levelof10becausewerequiredaddressingbothannualand
𝑇Min, 𝑇Max, 𝑇Mean, RH, EP, RF1, RF2} and a list of constants seasonal factors. The levels DW2, DW3, DW4, DW5, DW6,
in range [−10,...,10]. The population size and the number DW7, DW8, DW9, and DW10 correspond to the temporal
of generations are set to values 500 and 100. The crossover scale of 4, 8, 16, 32, 64, 128, 256, 512, and 1024 days. The
and mutation rates are set to 0.9 and 0.1. The Postfix-GP subseries,DW8,correspondtoatemporalscaleof256days
uses archive based roulette wheel selection [40]. Crossover (approximatelyoneyear).Thecorrelationcoefficientbetween
operation is performed between parents chosen from an thegeneratedDWsubseriesatdifferentlevelsandtheoriginal
archiveandcurrentpopulation rainfall series is calculated. Table3 presents the values of
the correlation coefficient between each DW subseries and
theoriginalrainfallseries.Thesecorrelationvaluesareused
MAE= 𝑁1 ∑𝑁 󵄨󵄨󵄨󵄨(𝑦𝑖−𝑦̂𝑖)󵄨󵄨󵄨󵄨. troaindfeatlelrpmreindeicttihoen.effective DW components (subseries) for
𝑖=1
AccordingtoTable3,forminimumandmeantempera-
MSE= 𝑁1 ∑𝑁 (𝑦𝑖−𝑦̂𝑖)2, (2) truairne,fatlhlesecroirersehlaatsiohnigbhetvwaleueen.ThDWis8resvuebaslesrtiheseafancdttthheatoarnigninuaall
𝑖=1 component of mean and minimum temperature subseries
cov(𝑌,𝑌̂) influences the amount of rainfall. The DW7 subseries of
𝑟= , maximum temperature has high correlation compared to
𝜎𝑌𝜎𝑌̂ other DW subseries. Moreover, the DW2, DW3, DW4, and
DW5 subseries of maximum temperature have little high
correlation compared to other DW subseries. Similarly, the
We have shown the evolved Postfix-GP solutions with
DW1, DW2, DW3, DW4, DW5, and DW7 subseries of
theirmeanabsoluteerror(MAE),meansquarederror(MSE),
evaporation have high correlation compared to other DW
and correlation coefficient (CC). Equations in (2) are used
tocalculatethesestatisticalmeasures,where𝑦𝑖,𝑦̂𝑖,𝑌,𝑌̂,and soufbmsearxiiems.uThmistesmugpgeersattsutrheatanshdoervtearptoirmateiopnerhioadvecocmorpreolnaetinotns
cov(𝑌,𝑌̂)standforgiven𝑖thobservation,corresponding𝑖th with the original rainfall series. The high correlation value
observationcalculatedbytheevolvedmodel,actualobserved isnoticedforDW8subseriesofrelativehumidity.Moreover,
series,estimatedseriesbytheevolvedmodel,andcovariance thecorrelationbetweentheoriginalrainfallandDW2,DW3,
between the observed and the estimated series. Value of 𝑟 DW4, DW5, and DW7 subseries of relative humidity is
closetozeroindicatesnocorrelationbetweenpredictedand littlehighcomparedtocorrelationbetweentherainfalland
observed values. We have used MSE to measure closeness remaining DW subseries of relative humidity. This suggests
between the given set of points and those generated by the that both annual and shorter time period components of
evolvedsolution. relativehumidityhavecorrelationwiththerainfall.
AdvancesinArtificialIntelligence 7
Table4:SelectedDWcomponentsforgeneratingfinalsubseries.
𝑇Min 𝑇Max 𝑇Mean RH EP RF1 RF2
DW +DW +
DW +DW + DW +DW + DW +DW + 2 3 DW +DW +
2 3 2 3 1 2 DW +DW + 3 4
DW DW +DW + DW DW +DW + DW +DW + 4 5 DW +DW +
8 4 5 8 4 5 3 4 DW +DW + 5 6
DW DW +DW DW +DW 6 7 DW +DW
7 7 8 5 7 DW 7 8
8
Relative humidity components:
dity 50 DW2+DW3+DW4+DW5+DW7+DW8
mi
u
e h 0
v
ati
Rel−50
500 1000 1500 2000 2500 3000 3500
Days
(a)
Evaporation components:
DW +DW +DW +DW +DW +DW
10 1 2 3 4 5 7
n
o
ati 5
or 0
p
a
v −5
E
−10
500 1000 1500 2000 2500 3000 3500
Days
(b)
Mean temperature components:DW
10 8
ure 5
erat 0
p
m −5
Te −10
500 1000 1500 2000 2500 3000 3500
Days
(c)
Figure3:Summedsubseries:(a)relativehumidity,(b)evaporation,and(c)meantemperature.
The high correlation between both shorter and yearly Usage of each selected DW subseries as input increases
components of relative humidity and rainfall is observed the dimension of the solution search space. Moreover, it
because of direct relation between these two variables. The also increases the complexity of the final model. Therefore,
humidity and rainfall data of the examined region show the selected DW subseries, having high correlation with
similar meteorological characteristics. The DW3 subseries the original rainfall series, are added together to form a
of one-day previous rainfall show high correlation with new (final) subseries. The final subseries are generated for
the original rainfall series. The DW2, DW4, DW5, DW6, everymeteorologicalvariable.TheadditionofselectedDW
DW7, and DW8 subseries also show little high correlation components(subseries)ishelpfultoimprovethecorrelation
compared to other DW subseries. The number of DW between the final subseries and the original rainfall series.
subseries which have high correlation with the original For example, the DW8 subseries of relative humidity have
rainfallseriesisidentifiedandsummeduptogenerateanew correlation value of 0.225. However, for the added (DW2 +
(final)subseriesforthatmeteorologicalvariable.Theselected DW3 +DW4 +DW5 +DW7 +DW8)subseries,thecorrela-
DW components for different meteorological variables are tionvalueincreasesto0.355.
presented in Table4. However, the selection of number of Theaddedsubseriesoftherelativehumidity,evaporation,
DW components (subseries) depends on the user. Partal andmeantemperatureareshowninFigure3.Thesummed
and Cigizoglu [14] suggested that applying a threshold on seriesofthemaximumtemperature,minimumtemperature,
correlationvaluewouldbehelpfultodeterminethenumber and previous day rainfall are presented in Figure4. We
ofDWcomponents(subseries). normalized the data of added subseries in the range (0, 1).
8 AdvancesinArtificialIntelligence
Minimum temperature components:DW
10 8
e
ur 5
erat 0
p
m −5
Te
−10
500 1000 1500 2000 2500 3000 3500
Days
(a)
Maximum temperature components:
DW +DW +DW +DW +DW
20 2 3 4 5 7
e
ur 10
at
er 0
p
m
−10
Te
−20
500 1000 1500 2000 2500 3000 3500
Days
(b)
One day previous precipitation components:
DW +DW +DW +DW +DW +DW +DW
100 2 3 4 5 6 7 8
80
nfall 4600
ai 20
R 0
−20
500 1000 1500 2000 2500 3000 3500
Days
(c)
Figure4:Summedsubseries:(a)minimumtemperature,(b)maximumtemperature,and(c)1-daypreviousrainfall.
120 notoverwhelmsmallvaluevariables.
100
5.2. Evolved Postfix-GP Solutions. The evolved Postfix-GP
mm) 80 solution(forterminalset{𝑇Min,𝑇Max,𝑇Mean,EP,RH,RF1})is
all ( 60 shownin(3).Itisnotedthatthesolutioncontainssignificant
nf meteorological variables (evaporation, minimum tempera-
ai 40
R ture, relative humidity, and maximum temperature) which
20 are useful for rainfall prediction. We got MSE = 48.2860,
0 adjustedfitness=0.3926,and𝑟=0.7469forthetrainingdata
0 100 200 300 400 500 600 700
Time (day) 𝑒(cos(cos(−10.6EP𝑇𝑖))−(𝑋𝑖/(sin(RF1RH)∗RH))+3.7292), (3)
Observed points
Postfix-GP predicted points where𝑋𝑖 =EP∗𝑒Log10(4.3EP+0.4941)/(𝑇𝑥+RH).
TheobservedandPostfix-GPmodelpredictedvaluesfor
Figure5:DailyrainfallpredictionbyPostfix-GPModelforthetest
the testing period are presented in Figure5. The model has
period(2years).
predictedthegeneralbehavioroftheobservedrainfalldata.
It has accurately predicted the summer days and estimated
zero rainfall for these days. However, the model performs
satisfactoryforestimatingmaximumrainfallvalues.
5.3.ComparisonofResultObtainedbyWavelet-Postfix-GPand
Wavelet-ANN Models. We have compared the performance
ofwavelet-Postfix-GPandwavelet-ANNmodelsforthedaily
rainfallprediction.Theselectionofarchitecture(numberof
layers and number of nodes in each layer) and the training
Thedatanormalizationassuresthatlargevaluevariablesdo algorithmisimportantdesignparametersofANN.Thereis
AdvancesinArtificialIntelligence 9
Table5:Statisticalmeasuresforthebestwavelet-Postfix-GPandwavelet-ANNmodelsfordifferentinputcombinationsfortestperiod.
Wavelet-Postfix-GP Wavelet-ANN
Modelinputs
MSE Adj 𝑅 ANNstructure MSE Adj 𝑅
Fit Fit
𝑇Min,𝑇Max,𝑇Mean,EP,RH,RF1,RF2 49.2494 0.4002 0.6661 7,5,1 52.6614 0.2804 0.6919
𝑇Min,𝑇Max,𝑇Mean,EP,RH,RF1 47.5766 0.4017 0.6794 6,6,1 50.4010 0.3545 0.7641
120 suggestthatPostfix-GPproducescomparableresultsandcan
100 beanalternativetoANNapproach.
m) 80
m 60
nfall ( 40
Rai 20
0 6.Conclusion
−20
0 100 200 300 400 500 600 700 We applied four homogeneity tests (SNHT, BR, Pettit, and
Time (day) VNR) to detect the variability of the meteorological (mini-
mum temperature, mean temperature, maximum tempera-
Observed points
ture,evaporation,andrelativehumidity)variablesrecorded
WANN predicted points
at Anand station. The results of homogeneity tests suggest
Figure6:DailyrainfallpredictionbyANNmodelforthetestperiod that the time series of the meteorological variables are
(2years). homogeneous.Theseriesofeverymeteorologicalvariableis
decomposedupto10resolutionlevelusingdiscretewavelet
transform.Thecorrelationcoefficientbetweenthegenerated
noruleavailabletofindoutthenumberofhiddenlayersand DW subseries at different levels and the original rainfall
appropriatenumberofnodesforeachhiddenlayer.However, seriesiscalculated.ThenumberofDWsubserieswhichhave
practitioners[43]foundthatANNwithonehiddenlayeris correlation with the original rainfall series is identified and
complex enough to model the nonlinear properties of the summeduptogenerateanewsubseriesforeverymeteoro-
hydrologic processes. We investigated different architecture logicalvariable.It isobservedthatbothannualandshorter
withdifferentactivationfunctionsandnumberofneuronsfor timeperiodcomponentsofrelativehumidityhavecorrelation
thehiddenlayerandselectedtheonewhichgivesanoptimal withrainfall.Theannualcomponentsofminimumandmean
result.Amultilayerfeed-forwardANNthatcomprisesseven temperatureshowcorrelationwithrainfallseries.Thenewly
nodes in the input layer, one hidden layer with five nodes, generatedmeteorologicalsubseriesareregardedasinputsand
and one node in the output layer is selected for the rainfall therainfallseriesasoutput.
prediction. Moreover, the input and hidden layers have ThePostfix-GPisthenemployedtodevelopamodelthat
an additional bias neuron. The ANN is trained with back canexplainrelationshipbetweentheinputsandtheoutput.
propagation (BP) learning algorithm. We set log-sigmoid Thedevelopedmodelisthenusedfortherainfallprediction.
transferfunctionbetweeninputandhiddenlayersandpure Thepredictionperformanceoftheevolvedmodelwasgood,
lineartransferfunctionbetweenhiddenandoutputlayers. giving low values for MAE and MSE and high value for
Figure6presentstheobservedandwavelet-ANNmodel thecorrelationcoefficient,suggestingthatPostfix-GPisable
predictedrainfallvalues.Themodelhaspredictedthegeneral to evolve an accurate and reliable model. The advantage of
behavior of the observed rainfall data. It is observed that Postfix-GPoverANNisthatitgivesanexplicitmathematical
wavelet-ANN model has produced negative prediction for nonlinear equation that describes the relationship between
some days having zero rainfall, which are not practically inputs and output. The predictive performance of Postfix-
possible.PartalandCigizoglu[14]foundthesimilarbehavior GPandANNmodelsiscompared.Theresultsshowthatthe
whileapplyingANNforpredictingdailyprecipitation.They Postfix-GPisafaircompetitorofANNapproach.Moreover,
noted that the negative prediction problem occurred due thepredictiveperformanceoftheevolvedPostfix-GPmodel
to extrapolation ability of feed-forward back-propagation for the nonrainy (zero rainfall) periods and rainy (highest
mechanism.Moreover,similartoPostfix-GP,theANNfailed rainfall)periodsissatisfactoryascomparedtoANNmodel,
inaccuratelypredictingthepeakvaluesofrainfallduringthe which produces negative prediction for some days of sum-
testperiod. merseason.Weconcludethatwavelet-Postfix-GPapproach
Table5 presents the statistical results for both wavelet- obtained good quality solutions for the tested rainfall pre-
Postfix-GP and wavelet-ANN hybrid models for different diction series and could be explored as an alternative tool
inputcombinationsforthetestperiod.ThePostfix-GPmodel forpredictingthehydrometeorologicalvariables.Ourfuture
hasproducedslightlybetterresultsthanANNfromtheMSE planistoapplyPostfix-GPfordevelopingmoreaccurateand
andfitnessviewpoint.TheANNmodelslightlyoutperforms reliablemodelsusingdifferentcombinationoffunctionand
Postfix-GP in terms of correlation coefficient. The results terminalsets.
10 AdvancesinArtificialIntelligence
ConflictofInterests artificialneuralnetwork,”ExpertSystemswithApplications,vol.
35,no.3,pp.1415–1421,2008.
The authors declare that there is no conflict of interests
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