research article forecasting spei and spi drought indices...

Research ArticleForecasting SPEI and SPI Drought Indices Usingthe Integrated Artificial Neural Networks

Petr Maca and Pavel Pech

Department of Water Resources and Environmental Modeling Faculty of Environmental SciencesCzech University of Life Sciences Prague Kamycka 1176 Suchdol 165 21 Prague 6 Czech Republic

Correspondence should be addressed to Petr Maca macafzpczucz

Received 3 June 2015 Revised 26 October 2015 Accepted 5 November 2015

Academic Editor Jose Alfredo Hernandez

Copyright copy 2016 P Maca and P PechThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The presented paper compares forecast of drought indices based on two different models of artificial neural networks The firstmodel is based on feedforward multilayer perceptron sANN and the second one is the integrated neural network model hANNThe analyzed drought indices are the standardized precipitation index (SPI) and the standardized precipitation evaporation index(SPEI) andwere derived for the period of 1948ndash2002 on twoUS catchmentsThemeteorological andhydrological datawere obtainedfromMOPEX experiment The training of both neural network models was made by the adaptive version of differential evolutionJADEThe comparison of models was based on six model performance measuresThe results of drought indices forecast explainedby the values of four model performance indices show that the integrated neural network model was superior to the feedforwardmultilayer perceptron with one hidden layer of neurons

1 Introduction

Droughts are natural disasters and extreme climate eventswith large impact in different areas of economy agriculturewater resources tourism and ecosystems The reviews ofsignificant drought events their impacts descriptionmitiga-tion and propagation in time are presented in detail in [1ndash3]

The drought indices are essential tools for explaining theseverity of drought events They are mainly represented ina form of time series and are used in drought modelingand forecasting [4]The intercomparison of different droughtindices connected with the development of forecasting toolswas studied in large number of research studies [5ndash9]

The recent development of artificial neural networks(ANN) has a significant impact on the application of thosetechniques for the forecasting of drought indices The ANNmodels are mostly represented by the nonlinear data drivenblack box modeling techniques Empirical studies confirmthat the multilayer perceptron (MLP) trained by the back-propagation algorithm is one of the most frequently studiedANNmodels since it is a universal approximator [10ndash15]

The important direction in ANN research in waterresources is the development and application of hybrid and

integrated neural networkmodels [16ndash19] For example Sham-seldin and OrsquoConnor [20] used the feedforward MLP for up-dating the outflow forecast Huo et al [16] developed the twoversions of integrated ANN models and successfully appliedthem on monthly outflow forecast The first version of inte-grated ANN models uses several outputs from several MLPmodels as inputs to final MLP the second aggregates thoseoutputs from severalMLPmodels into one input to finalMLP

The main aims of the presented paper are to develop andapply several models of integrated neural networks for fore-casting of drought indices and compare the integrated ANNmodels with the currently known models based onMLPTherest of the paper is organized as follows Section 2 describesdrought indices architecture of tested neural network mod-els model performance measures training method anddatasets Section 3 shows the results and discussion Section 4concludes the paper

2 Material and Methods

21 Drought Indices The studied droughts were describedusing the two drought indices the standardized precipitation

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 3868519 17 pageshttpdxdoiorg10115520163868519

2 Computational Intelligence and Neuroscience

DIMSE DIMSE

DIdMSE DIdMSE

DItPI DItPI

DIPID DIPID

DIOF

sANN

Inpu

t set

sIn

put s

ets

Inpu

t set

sIn

put s

ets

Inpu

t set

sIn

put s

ets

Inpu

t set

sIn

put s

ets

OF MSE

OF dMSE

OF tPI

OF PID

OF MSE

OF dMSE

OF tPI

OF PID

hANN

OF oneof four used OF

OF objective functionused for singleobjective optimization

Figure 1 The scheme of tested ANN architectures sANN is based on single MLP hANN is integrated ANN

index SPI index [21ndash23] and the standardized precipitationevapotranspiration index SPEI index [24 25]

The SPI index is based on the evaluation of precipitationdata The precipitation data are linked to the selected prob-ability distribution which is further standardized using thenormal distribution with zero mean and standard deviationof one It is often expressed as ameteorological drought index[11] and it is used for the assessment of agricultural andhydrological droughts [21]

The estimation of SPI consists of the determination ofprobability distribution of analyzed precipitation data thecalculation of probabilities for measured precipitation datafrom cumulative distribution function of fitted probabilitydistribution and the application of the inverse of distributionfunction of normalized normal distribution on probabilities[21 22]

The SPEI drought index is based on the precipitation andpotential evapotranspiration dataThe information about thepotential evapotranspiration temperature is mostly derivedusing the temperature data The SPEI index is expressedusing the differences between precipitation and potentialevapotranspiration Its calculation technically follows thederivation of SPI index the only difference is that instead ofthe precipitation time series the time series of the abovemen-tioned differences are used [24 25]

The estimation of SPI and SPEI drought indices wasmadeusing the R package [26] The probability distribution ofSPEI was expressed using the three-parameter log-logisticprobability distribution the SPI probability distribution wascalculated using the Gamma distribution The parameterswere identified using the method of unbiased probabilityweighted moments [24 25]

The SPI indicates the extremity of droughts The SPIvalues split the range into extremely dry (SPI le minus2) severelydry (minus2 lt SPI le minus15) moderately dry (minus15 lt SPI le minus1)and near neutral conditions (minus10 lt SPI le 10) [22 23]

22 sANN Model The architecture of the first analyzed neu-ral network model was based on the feedforward multilayerperceptron with one hidden layer of neurons sANN (thesingle ANNmodel see Figure 1)This type of neural networkarchitecture has been already applied on drought indicesforecast [10 11 27]

The sANN model has the following mathematical for-mula

DI119891 = Vout0 +

119873hd

sum119895=1

Vout119895 119891(Vhd0119895 +

119873in

sum119894=1

Vhd119895119894 119909119894) (1)

where DI119891 is a network output that is drought index forecastfor a given time interval 119909119894 is network input for input layerneuron 119894 normalized on the interval (0 1)119873in is the numberof MLP inputs Vhd119895119894 is the weight of the connection betweeninput 119894 and hidden layer neuron 119895 119891( ) is the activationfunction for all hidden layer neurons 119873hd is the number ofhidden neurons Vout119895 is the weight of the connection betweenthe hidden neuron 119895 and output neuron and Vhd0119895 V

out0 are

biases of neurons [14 28 29]The type of activation function of neurons in hidden layer

was the RootSig [30 31] Its form is

119910 (119886) =119886

1 + radic1 + 1198862 (2)

with 119886 = sum119873in119894=1 V119895119894119909119894 + V0119895

Computational Intelligence and Neuroscience 3

Since the neurons weights of sANN are unknown realparameters their values were estimated using training algo-rithms and calibration and validation dataset of analyzedtime series of drought indices The number of hidden layerneurons was selected according to the current experiencewith drought indices forecast using the ANN models [1011 27] The presented analysis was focused on testing threesets of ANN models with different numbers of hidden layerneurons119873hd = 4 6 8

23 hANNModel Thenewly proposed hANN integrates fiveMLP (sANN) models Figure 1 shows its scheme The hANNis formed from two layers of sANN models The first layerconsists of four sANN The second layer is formed from onesANN Outputs of the first layer of sANN are inputs to thesANN in the second layerThe final forecast of the time seriesof selected drought index is obtained from the output fromthe last MLP The tested architecture of hANN model wasbased on the integrated neural network model of Huo et al[16]

The main enhancement lies in the inputs of the last MLPmodelThe inputs are obtained from four outputs from sANNmodels which were trained according to the different neuralnetwork performance statistics MSE dMSE tPI and CI(see Section 24) Suggested approach combines the specificaspects of training sANNusing different performance indicesin one hybrid neural network The last MLP is an errorcorrection model or static updating model of drought indexforecast [20 32]

The unknown parameters of hANN are the real values ofall sANNweights Values of weights were estimated using theglobal optimization algorithm and calibration and validationdatasets We tested only those hANN models for which allfive sANNs had the same number of neurons in the hiddenlayers The training of hANN is explained in Section 25 Theanalyzed hANN models used the same inputs sets on foursANN in the first layer of hANN

24 The Performance of ANN Models The evaluations ofANN simulations of time series of drought indices for train-ing and for validation datasets were based on the followingstatistics [33ndash35]

Mean Absolute Error (MAE)

MAE = 1119899

119899

sum119905=1

10038161003816100381610038161003816DI119900 [119905] minus DI119891 [119905]

10038161003816100381610038161003816 (3)

Mean Squared Error (MSE)

MSE = 1119899

119899

sum119905=1

(DI119900 [119905] minus DI119891 [119905])2 (4)

Means Squared Error in Derivatives (dMSE)

dMSE = 1119899 minus 1

119899

sum119905=2

(dDI119900 [119905] minus dDI119891 [119905])2 (5)

Nash-Sutcliffe (NS) Efficiency

NS = 1 minussum119899119894=119905 (DI119900 [119905] minus DI119891 [119905])

2

sum119899119905=1 (DI119900 [119905] minus DI119900)

2 (6)

Transformed Persistency Index (tPI)

tPI =sum119899119905=1 (DI119900 [119905] minus DI119891 [119905])

2

sum119899119905=1 (DI119900 [119905] minus DI119900 [119905 minus LAG])

2 (7)

Persistency Index (PI)

PI = 1 minus tPI (8)

Combined Index (CI)

CI = 085tPI + 015dMSE (9)

Persistency Index 2 (PI2)

PI2 = 1 minussum119899119905=1 (DI119900 [119905] minus DIANN1 [119905])

2

sum119899119905=1 (DI119900 [119905] minus DIANN2 [119905])

2 (10)

119899 represents the total number of time intervals to be pre-dicted DI119900 is the average of observed drought index DI119900dDI119900[119905] = DI119900[119905] minus DI119900[119905 minus 1] and dDI119891[119905] = DI119891[119905] minusDI119891[119905 minus 1] LAG is the time shift describing the last observeddrought index DI119900[119905 minus LAG] and LAG is equal to two inthe presented analysis PI2 was applied on the comparison offorecast DIANN

1

with DIANN2

made by two different neuralnetwork models

25 The ANN Training Method The training of tested sANNwas based on solving inverse problems using the globaloptimization algorithmThe values of sANNparameters werefound according to the minimization of performance indicesMSE dMSE tPI and CI The performance indices wereestimated on times series of analyzed drought indices AllsANNs were trained in batch mode Only the single objectiveoptimization methods were used [13 36]

The training of hANN consisted of two steps The firststep was related to the training of four sANN models EachsANN model had been trained using one of the four mainobjective functionsMSE dMSE tPI and CIThe second stepwas based on the training of the last sANN The fifth sANNwas trained using one of the four objective functions andglobal optimization algorithm in batchmode Training of onehANN was built on solving five single objective optimizationproblems [16]

The adaptive differential evolution JADEwas applied as amain global optimization algorithm [37] JADE is an adaptiveversion of differential evolution which was developed byStorn and Price [38] It is a nature inspired heuristics Theoptimization process is based on the iterative work with pop-ulation of models Each population member is representedby the vector of its parameters The differential evolutioncombines the mutation crossover and selection operators[39 40]


The used adaptive mutation operator has the followingformula

V119896 [119894]hdout= V119896 [119894 minus 1]

hdout

+ 119865119894 (Vhdout119901-best minus V119896 [119894 minus 1]

hdout)

+ 119865119894 (V1199031 [119894 minus 1]hdoutminus V1199032 [119894 minus 1]

hdout)

(11)

The value of ANN weight Vhdout119896

is changed during the 119894thiteration using Vhdout

119901-best parameter which is randomly selectedfrom 119901 top models of population Vhdout1199031 and Vhdout1199032 areweights of randomly selected models from population and119865119894 is the mutation factor which is adaptively adjusted usingthe Cauchy distribution The top models in populationsare those which have the best values of analyzed objectivefunction in a given generation The binomial crossoveroperator is controlled by the crossover probability CR119894 whichis automatically updated using the normal distribution Thedetailed explanation of JADE parameter adaptation togetherwith selection of Vhdout

119901-best is presented in the work of Zhang andSanderson [37]

26TheDataset Description Weused for the drought indicesneural network prediction the data obtained from twowatersheds The data were part of large dataset preparedwithin the MOPEX experiment framework [41 42] TheMOPEX dataset provides the benchmark hydrological andmeteorological data which were explored in large number ofenvironmentally oriented studies [43ndash47]

The first basin was Leaf River near Collins Mississippiwith area 192436 km2 USGS ID-02472000 and the sec-ond was the Santa Ysabel Creek near Ramona California29007 km2 USGS ID-11025500 both catchments are locatedin the USA The original daily records were aggregatedinto the monthly time scale We used the records from theperiod 1948ndash2002 The calibration period was formed fromthe period 1948ndash1975 the validation dataset consisted of therecords from the period 1975ndash2002 The standard length ofanalyzed benchmark dataset was used in the presented study[42]

Table 1 shows the inputs for tested neural networkmodelson both catchments The forecasted output was DI[119905] forboth SPI and SPEI drought indices 119889119879 was the monthlymean of averages of differences between daily maximum andminimum temperatures

Although there were several derived automatic linear andnonlinear procedures of input selection for ANNmodels theapplied input selection procedure was iterative Estimationsof cross-correlation and autocorrelation of input time serieswere used for making the decision about the final testedinput sets [48ndash50] Since ANN models are capable of cap-turing the nonlinearities between the input and output datawe compared the ANN simulations which were obtainedusing several combinations of different input variables withdifferent memories Table 1 presents the final list of the testedinputs

Table 1 The inputs for all tested neural network models

Input set Number of inputs Input variables3-119873hd-1 3 DI[119905 minus 119894] for 119894 = 2 3 46-119873hd-1 6 DI[119905 minus 119894] and 119889119879[119905 minus 119894] for 119894 = 2 3 49-119873hd-1 9 DI[119905 minus 119894] for 119894 = 2 3 10

The nonlinear transformation was applied on all ANNdatasets Its form was

119863trans = 1 minus exp (minus120574 (119863orig + 12min (119863orig))) (12)

with original data119863orig transformed119863trans andminimum ofuntransformed data min(119863orig) This nonlinear transforma-tion emphasises the low values which are connected to severedrought events

3 Results and Discussion

In our experiment we analyzed for each inputs set 3 sANNarchitectures They were formed by three different numbersof hidden neurons 119873hd = 4 6 8 All sANN and hANNmodels were calibrated 25 times using 4 objective functionsMSE tPI CI and dMSE and JADE optimizer All five sANNmodels in one hANN had the same value of119873hd In total wetested 1800 sANNmodels and 1800 hANNmodels

The initial settings of JADE hyperparameters were similarfor all ANNmodel runs The population of models consistedof 20 times number of weights in sANN or hANN the Vhdout

119901-best wasrandomly selected from 45 percent of the best models inpopulation The best models in given generation were thosewhich were sorted according to the values of objective func-tion The number of generations was 40 The selected valuesbalanced the exploration and exploitation during the searchprocess and helped to avoid the premature convergence of thepopulation of the models The hyperparameter 120574 of (12) wasset to 015

The results of ANNmodels trained using the dMSE wereomitted in our presentation since all models provided theworst results However outputs generated by sANN trainedby the dMSEwere inputs to the last sANN in all tested hANNmodels

Since the Persistency Index (PI) is sensitive to timingerror of forecast and enables the comparison of the simulationof drought indices with the naive model formed by the lastknown information about drought index [33] we selected itas a main reference index

31The Forecast of SPI Index Theresults ofmedians ofmodelperformance indices on SPI forecast are presented in Tables2 3 and 4 The medians were calculated for each set of 25simulation runs The SPI results are in several aspects similarto those of SPEI forecast

When comparing the results of hANN with the resultsof sANN formed from single multilayer perceptron theresults of hANN were superior in terms of the medians ofMAE dMSE MSE and PI on calibration period for bothcatchments


Table 2 The medians of performance metrics for MLP architecture 3-119873hd-1 on SPI forecast for sANN TA and hANN TA TA shows thetraining error function and the best models are marked with bold fonts

Calibration period Validation periodMAE MSE dMSE NS PI MAE MSE dMSE NS PI

Leaf River3-4-1sANN MSE 388119864 minus 01 256119864 minus 01 185119864 minus 01 745119864 minus 01 minus757119864 minus 02 440119864 minus 01 305119864 minus 01 168119864 minus 01 636119864 minus 01 minus155119864 minus 01

sANN tPI 390119864 minus 01 257119864 minus 01 190119864 minus 01 745119864 minus 01 minus772119864 minus 02 432119864 minus 01 299119864 minus 01 172119864 minus 01 644119864 minus 01 minus132119864 minus 01

sANN PID 386119864 minus 01 250119864 minus 01 177119864 minus 01 751119864 minus 01 minus495119864 minus 02 431119864 minus 01 293119864 minus 01 167119864 minus 01 650119864 minus 01 minus112119864 minus 01

hANN MSE 366119864 minus 01 227119864 minus 01 155119864 minus 01 738119864 minus 01 480119864 minus 02 400119864 minus 01 255119864 minus 01 147119864 minus 01 640119864 minus 01 335119864 minus 02

hANN tPI 367119864 minus 01 226119864 minus 01 150119864 minus 01 744119864 minus 01 515119864 minus 02 396119864 minus 01 249119864 minus 01 144E minus 01 627119864 minus 01 549119864 minus 02hANN PID 364119864 minus 01 225119864 minus 01 169119864 minus 01 739119864 minus 01 555119864 minus 02 394119864 minus 01 248119864 minus 01 152119864 minus 01 620119864 minus 01 615119864 minus 02

3-6-1sANN MSE 408119864 minus 01 271119864 minus 01 202119864 minus 01 731119864 minus 01 minus136119864 minus 01 424119864 minus 01 283119864 minus 01 169119864 minus 01 662119864 minus 01 minus747119864 minus 02



hANN MSE 364119864 minus 01 223E minus 01 162119864 minus 01 751119864 minus 01 639119864 minus 02 388119864 minus 01 242119864 minus 01 154119864 minus 01 630119864 minus 01 824119864 minus 02hANN tPI 363E minus 01 223E minus 01 148E minus 01 746119864 minus 01 632119864 minus 02 386119864 minus 01 234E minus 01 145119864 minus 01 625119864 minus 01 114E minus 01hANN PID 364119864 minus 01 224119864 minus 01 159119864 minus 01 751119864 minus 01 594119864 minus 02 389119864 minus 01 239119864 minus 01 154119864 minus 01 633119864 minus 01 933119864 minus 02

3-8-1sANN MSE 374119864 minus 01 232119864 minus 01 207119864 minus 01 760119864 minus 01 271119890 minus 02 410119864 minus 01 266119864 minus 01 184119864 minus 01 683E minus 01 minus675119864 minus 03sANN tPI 376119864 minus 01 240119864 minus 01 208119864 minus 01 761119864 minus 01 minus760119864 minus 03 414119864 minus 01 273119864 minus 01 185119864 minus 01 675119864 minus 01 minus340119864 minus 02


hANN MSE 363E minus 01 223119864 minus 01 171119864 minus 01 761119864 minus 01 656E minus 02 393119864 minus 01 243119864 minus 01 156119864 minus 01 659119864 minus 01 802119864 minus 02hANN tPI 364119864 minus 01 224119864 minus 01 167119864 minus 01 762119864 minus 01 593119864 minus 02 383E minus 01 236119864 minus 01 155119864 minus 01 654119864 minus 01 105119864 minus 01hANN PID 363E minus 01 225119864 minus 01 175119864 minus 01 763E minus 01 571119864 minus 02 385119864 minus 01 238119864 minus 01 160119864 minus 01 667119864 minus 01 964119864 minus 02

Santa Ysabel Creek3-4-1sANN MSE 341119864 minus 01 228119864 minus 01 174119864 minus 01 548119864 minus 01 368119864 minus 02 475119864 minus 01 383119864 minus 01 199119864 minus 01 666119864 minus 01 minus288119864 minus 01

sANN tPI 350119864 minus 01 235119864 minus 01 161119864 minus 01 534119864 minus 01 774119864 minus 03 436119864 minus 01 352119864 minus 01 185119864 minus 01 693119864 minus 01 minus183119864 minus 01

sANN PID 345119864 minus 01 230119864 minus 01 167119864 minus 01 543119864 minus 01 275119864 minus 02 475119864 minus 01 385119864 minus 01 201119864 minus 01 664119864 minus 01 minus295119864 minus 01

hANN MSE 303119864 minus 01 210119864 minus 01 141119864 minus 01 529119864 minus 01 113119864 minus 01 359119864 minus 01 286119864 minus 01 161E minus 01 564119864 minus 01 375119864 minus 02hANN tPI 309119864 minus 01 209119864 minus 01 136E minus 01 517119864 minus 01 116119864 minus 01 357119864 minus 01 283119864 minus 01 162119864 minus 01 630119864 minus 01 473119864 minus 02hANN PID 297E minus 01 208119864 minus 01 144119864 minus 01 536119864 minus 01 120119864 minus 01 338E minus 01 284119864 minus 01 165119864 minus 01 558119864 minus 01 439119864 minus 02

3-6-1sANN MSE 342119864 minus 01 229119864 minus 01 167119864 minus 01 546119864 minus 01 331119864 minus 02 449119864 minus 01 357119864 minus 01 189119864 minus 01 689119864 minus 01 minus200119864 minus 01




hANN tPI 306119864 minus 01 206E minus 01 150119864 minus 01 558119864 minus 01 128119864 minus 01 355119864 minus 01 278119864 minus 01 176119864 minus 01 620119864 minus 01 658119864 minus 02hANN PID 304119864 minus 01 208119864 minus 01 159119864 minus 01 556119864 minus 01 122119864 minus 01 355119864 minus 01 276119864 minus 01 182119864 minus 01 639119864 minus 01 716E minus 02



sANN PID 327119864 minus 01 222119864 minus 01 190119864 minus 01 560119864 minus 01 632119864 minus 02 422119864 minus 01 327119864 minus 01 223119864 minus 01 714E minus 01 minus100119864 minus 01hANN MSE 304119864 minus 01 206E minus 01 168119864 minus 01 562119864 minus 01 129119864 minus 01 367119864 minus 01 285119864 minus 01 193119864 minus 01 642119864 minus 01 424119864 minus 02hANN tPI 302119864 minus 01 206E minus 01 157119864 minus 01 563E minus 01 131E minus 01 354119864 minus 01 277E minus 01 180119864 minus 01 630119864 minus 01 666119864 minus 02hANN PID 307119864 minus 01 206E minus 01 155119864 minus 01 554119864 minus 01 129119864 minus 01 382119864 minus 01 291119864 minus 01 181119864 minus 01 636119864 minus 01 199119864 minus 02




Leaf River6-4-1sANN MSE 576119864 minus 01 525119864 minus 01 271119864 minus 01 478119864 minus 01 minus120119864 + 00 592119864 minus 01 543119864 minus 01 270119864 minus 01 352119864 minus 01 minus106119864 + 00

sANN tPI 591119864 minus 01 553119864 minus 01 292119864 minus 01 450119864 minus 01 minus132119864 + 00 618119864 minus 01 609119864 minus 01 302119864 minus 01 274119864 minus 01 minus131119864 + 00

sANN PID 578119864 minus 01 528119864 minus 01 253119864 minus 01 475119864 minus 01 minus122119864 + 00 620119864 minus 01 594119864 minus 01 240119864 minus 01 291119864 minus 01 minus125119864 + 00

hANN MSE 402119864 minus 01 281119864 minus 01 150119864 minus 01 613119864 minus 01 minus179119864 minus 01 446119864 minus 01 324119864 minus 01 153119864 minus 01 432119864 minus 01 minus230119864 minus 01

hANN dMSE 281119864 + 00 890119864 + 00 110E minus 01 minus617119864 + 03 minus363119864 + 01 319119864 + 00 110119864 + 01 115E minus 01 minus732119864 + 03 minus408119864 + 01hANN tPI 391119864 minus 01 253119864 minus 01 148119864 minus 01 609119864 minus 01 minus622119864 minus 02 439119864 minus 01 306119864 minus 01 150119864 minus 01 427119864 minus 01 minus159119864 minus 01

hANN PID 399119864 minus 01 261119864 minus 01 146119864 minus 01 598119864 minus 01 minus973119864 minus 02 445119864 minus 01 308119864 minus 01 151119864 minus 01 389119864 minus 01 minus166119864 minus 01

6-6-1sANN MSE 566119864 minus 01 512119864 minus 01 231119864 minus 01 491119864 minus 01 minus115119864 + 00 622119864 minus 01 584119864 minus 01 238119864 minus 01 303119864 minus 01 minus122119864 + 00



hANN MSE 395119864 minus 01 262119864 minus 01 141119864 minus 01 616E minus 01 minus101119864 minus 01 421119864 minus 01 274119864 minus 01 154119864 minus 01 436E minus 01 minus404119864 minus 02hANN tPI 391119864 minus 01 257119864 minus 01 136119864 minus 01 605119864 minus 01 minus768119864 minus 02 419119864 minus 01 270119864 minus 01 140119864 minus 01 424119864 minus 01 minus243119864 minus 02





hANN MSE 391119864 minus 01 256119864 minus 01 151119864 minus 01 507119864 minus 01 minus755119864 minus 02 399E minus 01 251119864 minus 01 144119864 minus 01 286119864 minus 01 480E minus 02hANN tPI 383E minus 01 247E minus 01 146119864 minus 01 511119864 minus 01 minus368E minus 02 430119864 minus 01 290119864 minus 01 149119864 minus 01 257119864 minus 01 minus982119864 minus 02hANN PID 386119864 minus 01 247E minus 01 155119864 minus 01 511119864 minus 01 minus372119864 minus 02 398119864 minus 01 249E minus 01 156119864 minus 01 312119864 minus 01 566119864 minus 02

Santa Ysabel Creek6-4-1sANN MSE 500119864 minus 01 404119864 minus 01 201119864 minus 01 197119864 minus 01 minus710119864 minus 01 708119864 minus 01 781119864 minus 01 215119864 minus 01 318119864 minus 01 minus163119864 + 00

sANN tPI 524119864 minus 01 427119864 minus 01 195119864 minus 01 153119864 minus 01 minus805119864 minus 01 709119864 minus 01 775119864 minus 01 206119864 minus 01 323119864 minus 01 minus161119864 + 00

sANN PID 528119864 minus 01 451119864 minus 01 219119864 minus 01 104119864 minus 01 minus909119864 minus 01 773119864 minus 01 923119864 minus 01 225119864 minus 01 194119864 minus 01 minus210119864 + 00

hANN MSE 350119864 minus 01 242119864 minus 01 124119864 minus 01 201119864 minus 01 minus244119864 minus 02 409119864 minus 01 328119864 minus 01 147119864 minus 01 minus721119864 minus 02 minus103119864 minus 01

hANN tPI 326119864 minus 01 226119864 minus 01 125119864 minus 01 215119864 minus 01 459119864 minus 02 415119864 minus 01 331119864 minus 01 142119864 minus 01 126119864 minus 01 minus113119864 minus 01

hANN PID 345119864 minus 01 230119864 minus 01 124119864 minus 01 164119864 minus 01 283119864 minus 02 426119864 minus 01 353119864 minus 01 146119864 minus 01 minus138119864 minus 01 minus189119864 minus 01

6-6-1sANN MSE 511119864 minus 01 422119864 minus 01 196119864 minus 01 163119864 minus 01 minus782119864 minus 01 617119864 minus 01 603119864 minus 01 221119864 minus 01 473119864 minus 01 minus103119864 + 00



hANN MSE 342119864 minus 01 222119864 minus 01 126119864 minus 01 335E minus 01 610119864 minus 02 408119864 minus 01 331119864 minus 01 151119864 minus 01 330119864 minus 01 minus113119864 minus 01hANN tPI 303E minus 01 221119864 minus 01 117E minus 01 318119864 minus 01 640119864 minus 02 358E minus 01 294E minus 01 141E minus 01 265119864 minus 01 106E minus 02hANN PID 336119864 minus 01 219119864 minus 01 118119864 minus 01 306119864 minus 01 734119864 minus 02 401119864 minus 01 307119864 minus 01 143119864 minus 01 284119864 minus 01 minus336119864 minus 02

6-8-1sANN MSE 515119864 minus 01 438119864 minus 01 278119864 minus 01 132119864 minus 01 minus850119864 minus 01 617119864 minus 01 594119864 minus 01 292119864 minus 01 481E minus 01 minus999119864 minus 01sANN tPI 524119864 minus 01 445119864 minus 01 272119864 minus 01 117119864 minus 01 minus881119864 minus 01 621119864 minus 01 644119864 minus 01 291119864 minus 01 438119864 minus 01 minus117119864 + 00


hANN MSE 336119864 minus 01 218E minus 01 140119864 minus 01 267119864 minus 01 784E minus 02 423119864 minus 01 333119864 minus 01 161119864 minus 01 322119864 minus 01 minus120119864 minus 01hANN tPI 333119864 minus 01 225119864 minus 01 137119864 minus 01 274119864 minus 01 507119864 minus 02 404119864 minus 01 328119864 minus 01 153119864 minus 01 365119864 minus 01 minus104119864 minus 01

hANN PID 331119864 minus 01 231119864 minus 01 141119864 minus 01 250119864 minus 01 230119864 minus 02 375119864 minus 01 311119864 minus 01 168119864 minus 01 203119864 minus 01 minus473119864 minus 02




Leaf River9-4-1

sANN MSE 377119864 minus 01 235119864 minus 01 171119864 minus 01 751119864 minus 01 742119864 minus 01 454119864 minus 01 338119864 minus 01 189119864 minus 01 605119864 minus 01 731119864 minus 01

sANN tPI 379119864 minus 01 236119864 minus 01 198119864 minus 01 750119864 minus 01 741119864 minus 01 460119864 minus 01 344119864 minus 01 216119864 minus 01 598119864 minus 01 726119864 minus 01

sANN PID 386119864 minus 01 245119864 minus 01 152119864 minus 01 741119864 minus 01 732119864 minus 01 462119864 minus 01 350119864 minus 01 175119864 minus 01 591119864 minus 01 722119864 minus 01


hANN tPI 344119864 minus 01 202119864 minus 01 153119864 minus 01 766119864 minus 01 386119864 minus 02 416119864 minus 01 283119864 minus 01 162119864 minus 01 579119864 minus 01 775119864 minus 01

hANN PID 347119864 minus 01 201119864 minus 01 136E minus 01 757119864 minus 01 779119864 minus 01 408119864 minus 01 274119864 minus 01 157119864 minus 01 575119864 minus 01 782119864 minus 019-6-1sANN MSE 375119864 minus 01 229119864 minus 01 200119864 minus 01 757119864 minus 01 749119864 minus 01 444119864 minus 01 323119864 minus 01 214119864 minus 01 623119864 minus 01 743119864 minus 01



hANN MSE 342E minus 01 198119864 minus 01 157119864 minus 01 768119864 minus 01 784119864 minus 01 408119864 minus 01 279119864 minus 01 169119864 minus 01 588119864 minus 01 779119864 minus 01hANN tPI 344119864 minus 01 199119864 minus 01 159119864 minus 01 770119864 minus 01 782119864 minus 01 403E minus 01 268E minus 01 174119864 minus 01 603119864 minus 01 787E minus 01hANN PID 346119864 minus 01 202119864 minus 01 144119864 minus 01 753119864 minus 01 779119864 minus 01 410119864 minus 01 273119864 minus 01 150E minus 01 587119864 minus 01 783119864 minus 01

9-8-1sANN MSE 372119864 minus 01 229119864 minus 01 191119864 minus 01 757119864 minus 01 749119864 minus 01 441119864 minus 01 319119864 minus 01 215119864 minus 01 627E minus 01 746119864 minus 01sANN tPI 379119864 minus 01 238119864 minus 01 196119864 minus 01 748119864 minus 01 739119864 minus 01 454119864 minus 01 336119864 minus 01 218119864 minus 01 607119864 minus 01 733119864 minus 01


hANN MSE 345119864 minus 01 196E minus 01 157119864 minus 01 775E minus 01 785E minus 01 404119864 minus 01 269119864 minus 01 171119864 minus 01 607119864 minus 01 786119864 minus 01hANN tPI 345119864 minus 01 197119864 minus 01 164119864 minus 01 775E minus 01 784119864 minus 01 408119864 minus 01 274119864 minus 01 171119864 minus 01 594119864 minus 01 782119864 minus 01hANN PID 344119864 minus 01 198119864 minus 01 149119864 minus 01 766119864 minus 01 783119864 minus 01 407119864 minus 01 277119864 minus 01 163119864 minus 01 604119864 minus 01 780119864 minus 01

Santa Ysabel Creek9-4-1sANN MSE 397119864 minus 01 288119864 minus 01 227119864 minus 01 556119864 minus 01 764119864 minus 01 537119864 minus 01 450119864 minus 01 212119864 minus 01 625119864 minus 01 645119864 minus 01





hANN PID 367119864 minus 01 262119864 minus 01 176119864 minus 01 551119864 minus 01 785119864 minus 01 504119864 minus 01 397119864 minus 01 165119864 minus 01 470119864 minus 01 687119864 minus 01

9-6-1sANN MSE 406119864 minus 01 288119864 minus 01 243119864 minus 01 556119864 minus 01 764119864 minus 01 542119864 minus 01 455119864 minus 01 220119864 minus 01 620119864 minus 01 641119864 minus 01



hANN MSE 356119864 minus 01 248119864 minus 01 196119864 minus 01 578E minus 01 796119864 minus 01 473119864 minus 01 359119864 minus 01 183119864 minus 01 577119864 minus 01 716119864 minus 01hANN tPI 358119864 minus 01 249119864 minus 01 202119864 minus 01 574119864 minus 01 796119864 minus 01 446119864 minus 01 334119864 minus 01 186119864 minus 01 572119864 minus 01 736119864 minus 01

hANN PID 375119864 minus 01 260119864 minus 01 175E minus 01 523119864 minus 01 787119864 minus 01 535119864 minus 01 434119864 minus 01 161E minus 01 441119864 minus 01 657119864 minus 019-8-1sANN MSE 408119864 minus 01 291119864 minus 01 268119864 minus 01 552119864 minus 01 761119864 minus 01 516119864 minus 01 413119864 minus 01 241119864 minus 01 656E minus 01 674119864 minus 01sANN tPI 404119864 minus 01 290119864 minus 01 249119864 minus 01 554119864 minus 01 762119864 minus 01 538119864 minus 01 438119864 minus 01 239119864 minus 01 634119864 minus 01 654119864 minus 01


hANN MSE 354E minus 01 246E minus 01 190119864 minus 01 574119864 minus 01 798E minus 01 441E minus 01 355119864 minus 01 186119864 minus 01 547119864 minus 01 720119864 minus 01hANN tPI 356119864 minus 01 249119864 minus 01 201119864 minus 01 578E minus 01 795119864 minus 01 442119864 minus 01 331E minus 01 194119864 minus 01 528119864 minus 01 738E minus 01hANN PID 366119864 minus 01 261119864 minus 01 175E minus 01 544119864 minus 01 786119864 minus 01 534119864 minus 01 434119864 minus 01 161E minus 01 476119864 minus 01 657119864 minus 01


Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(b)

Figure 2The calibration results of SPI forecast using hANN 9-8-1 trained onMSE the blue line observations the green line forecasted SPImedians and the dotted lines simulation error bounds

hANN models with nine inputs provided better SPIforecast according to the values of medians of PI index thanmodels with six and three inputs Similar recommendationson input datasets were confirmed in [10 51]

The best models according to the medians of PI valueswere hANN models with 9-8-1 architecture with the lastsANN trained by the MSE on Santa Ysabel Creek calibrationdataset (PI = 080) The best values of median of PersistencyIndex (PI = 079) were obtained from hANN forecast onvalidation dataset using Leaf River dataset 9-6-1 architectureand tPI for optimization of last sANN (see Table 4)

Figures 2 and 3 respectively show the forecast of SPIin calibration and validation which were obtained usingthe hANN models with the highest values of medians ofPersistency Index

Results of PI2 index for the models with 9-119873hd-1 archi-tecture show that 9-8-1 architecture was superior for SPIforecast on validation datasets for both analyzed catchments(see Table 5) The comparison of calibration results showsthat the 9-6-1 architecture has the highest values of PI2 onSanta Ysabel Creek dataset and 9-8-1 has highest values ofPI2 on Leaf River calibration dataset

Table 6 shows the results of PI2 index which were cal-culated using the results of hANN with 9-119873hd-1 architectureValues of PI2 enable us to compare the tested ANN modelsaccording to the performance of the optimization function

which were applied on training of the last MLP The hANNmodels with the last sANN trained by the tPI were superiorto hANN with last sANN trained using MSE or CI oncalibration and validation Leaf River datasets

The Santa Ysabel Creek datasets show that the best resultswere obtained by the tPI optimization for calibration of thelast sANN of hANN models according to PI2 The valuesof PI2 for validation dataset show that hANN with the lastsANN trained using the MSE provided better simulationresults than the remaining hANNmodels (see Table 6)

32 The Forecast of SPEI Index The mean performance ofneural network models was explained using the mediansof model evaluation metrics The medians were obtainedfrom the results of 25 runs on each basin for each ANNmodel architecture Tables 7 8 and 9 show the results ofthe evaluations of SPEI forecasts using the medians of MAEMSE dMSE NS and PI

The integrated hANN models were superior to singlemultilayer perceptron models sANN in terms of the bestvalues of medians of performance indicesMAEMSE dMSEand PI for both catchments One of the exceptions can befound in Leaf River dataset the NS of the single MLP for 3-8-1 trained using MSE on calibration and validation periodsHowever this sANN model with the highest values of NSdid not produce the highest values of PI (see the calibration


Table 5 The values of PI2 on architectures 9-119873hd-1 for SPI forecast with hANN

Calibration period Validation period9-4-1 9-6-1 9-8-1 9-4-1 9-6-1 9-8-1

Leaf River9-4-1 000119864 + 00 minus152119864 minus 02 minus291119864 minus 02 000119864 + 00 minus631119864 minus 03 minus364119864 minus 02

9-6-1 150119864 minus 02 000119864 + 00 minus137119864 minus 02 627119864 minus 03 000119864 + 00 minus299119864 minus 02

9-8-1 283119864 minus 02 135119864 minus 02 000119864 + 00 351119864 minus 02 290119864 minus 02 000119864 + 00

Santa Ysabel9-4-1 000119864 + 00 minus692119864 minus 03 minus477119864 minus 03 000119864 + 00 213119864 minus 02 minus183119864 minus 03

9-6-1 688119864 minus 03 000119864 + 00 213119864 minus 03 minus218119864 minus 02 000119864 + 00 minus236119864 minus 02

9-8-1 475119864 minus 03 minus214119864 minus 03 000119864 + 00 182119864 minus 03 231119864 minus 02 000119864 + 00

Table 6 The values of PI2 on training functions for SPI forecast with hANN

Calibration period Validation periodMSE tPI PID MSE tPI PID

Leaf RiverMSE 000119864 + 00 minus294119864 minus 03 164119864 minus 02 000119864 + 00 minus525119864 minus 03 244119864 minus 02

tPI 294119864 minus 03 000119864 + 00 193119864 minus 02 522119864 minus 03 000119864 + 00 295119864 minus 02

PID minus167119864 minus 02 minus197119864 minus 02 000119864 + 00 minus250119864 minus 02 minus304119864 minus 02 000119864 + 00

Santa YsabelMSE 000119864 + 00 minus107119864 minus 03 561119864 minus 02 000119864 + 00 151119864 minus 03 207119864 minus 01

tPI 106119864 minus 03 000119864 + 00 571119864 minus 02 minus151119864 minus 03 000119864 + 00 206119864 minus 01


Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)

Figure 3 The validation results of SPI forecast using hANN Leaf River 9-6-1 trained on PI Santa Ysabel Creek 9-8-1 trained on PI The blueline observations the green lines forecasted SPI medians and the dotted line simulation error bounds


Table 7 The medians of performance metrics for MLP architecture 3-119873hd-1 on SPEI forecast for sANN TA and hANN TA TA shows thetraining error function and the best models are marked with bold fonts







hANN PID 351119864 minus 01 210119864 minus 01 143E minus 01 748119864 minus 01 507119864 minus 02 405119864 minus 01 261119864 minus 01 152E minus 01 591119864 minus 01 505119864 minus 023-6-1sANN MSE 379119864 minus 01 236119864 minus 01 179119864 minus 01 762119864 minus 01 minus707119864 minus 02 437119864 minus 01 300119864 minus 01 186119864 minus 01 655119864 minus 01 minus908119864 minus 02





hANN PID 355119864 minus 01 209119864 minus 01 152119864 minus 01 763119864 minus 01 541119864 minus 02 395E minus 01 253119864 minus 01 160119864 minus 01 630119864 minus 01 812119864 minus 023-8-1sANN MSE 371119864 minus 01 224119864 minus 01 202119864 minus 01 775E minus 01 minus134119864 minus 02 420119864 minus 01 278119864 minus 01 208119864 minus 01 680E minus 01 minus118119864 minus 02sANN tPI 372119864 minus 01 228119864 minus 01 175119864 minus 01 771119864 minus 01 minus312119864 minus 02 424119864 minus 01 286119864 minus 01 183119864 minus 01 671119864 minus 01 minus381119864 minus 02


hANN MSE 351119864 minus 01 209119864 minus 01 157119864 minus 01 769119864 minus 01 525119864 minus 02 395E minus 01 248119864 minus 01 166119864 minus 01 659119864 minus 01 983119864 minus 02hANN tPI 350E minus 01 208119864 minus 01 164119864 minus 01 773119864 minus 01 596119864 minus 02 398119864 minus 01 252119864 minus 01 168119864 minus 01 655119864 minus 01 828119864 minus 02hANN PID 350E minus 01 207E minus 01 162119864 minus 01 768119864 minus 01 629E minus 02 390119864 minus 01 246E minus 01 173119864 minus 01 643119864 minus 01 104E minus 01




hANN MSE 350119864 minus 01 266119864 minus 01 189119864 minus 01 545119864 minus 01 122119864 minus 01 362119864 minus 01 286119864 minus 01 171E minus 01 626119864 minus 01 441119864 minus 02hANN tPI 344119864 minus 01 263E minus 01 195119864 minus 01 528119864 minus 01 130E minus 01 371119864 minus 01 281119864 minus 01 179119864 minus 01 639119864 minus 01 593119864 minus 02hANN PID 354119864 minus 01 265119864 minus 01 180E minus 01 539119864 minus 01 125119864 minus 01 376119864 minus 01 284119864 minus 01 165119864 minus 01 570119864 minus 01 503119864 minus 02





hANN tPI 351119864 minus 01 264119864 minus 01 187119864 minus 01 550119864 minus 01 127119864 minus 01 362119864 minus 01 277E minus 01 176119864 minus 01 652119864 minus 01 721E minus 02hANN PID 352119864 minus 01 264119864 minus 01 193119864 minus 01 552119864 minus 01 127119864 minus 01 362119864 minus 01 278119864 minus 01 176119864 minus 01 659119864 minus 01 705119864 minus 02



sANN PID 382119864 minus 01 283119864 minus 01 239119864 minus 01 558119864 minus 01 637119864 minus 02 432119864 minus 01 330119864 minus 01 220119864 minus 01 716E minus 01 minus104119864 minus 01hANN MSE 349119864 minus 01 264119864 minus 01 200119864 minus 01 560119864 minus 01 126119864 minus 01 391119864 minus 01 290119864 minus 01 180119864 minus 01 651119864 minus 01 304119864 minus 02


hANN PID 339E minus 01 263E minus 01 208119864 minus 01 563E minus 01 129119864 minus 01 358E minus 01 281119864 minus 01 190119864 minus 01 653119864 minus 01 612119864 minus 02




Leaf River6-4-1

ANN MSE 599119864 minus 01 566119864 minus 01 241119864 minus 01 431119864 minus 01 minus156119864 + 00 640119864 minus 01 613119864 minus 01 261119864 minus 01 295119864 minus 01 minus123119864 + 00

ANN tPI 576119864 minus 01 532119864 minus 01 234119864 minus 01 465119864 minus 01 minus141119864 + 00 634119864 minus 01 614119864 minus 01 249119864 minus 01 294119864 minus 01 minus123119864 + 00

ANN PID 583119864 minus 01 524119864 minus 01 224119864 minus 01 473119864 minus 01 minus137119864 + 00 622119864 minus 01 581119864 minus 01 236119864 minus 01 332119864 minus 01 minus111119864 + 00


hANN tPI 392119864 minus 01 256119864 minus 01 126E minus 01 575119864 minus 01 minus160119864 minus 01 442119864 minus 01 308119864 minus 01 144E minus 01 356119864 minus 01 minus119119864 minus 01hANN PID 384119864 minus 01 252119864 minus 01 132119864 minus 01 556119864 minus 01 minus141119864 minus 01 445119864 minus 01 316119864 minus 01 151119864 minus 01 351119864 minus 01 minus149119864 minus 01

6-6-1ANN MSE 577119864 minus 01 512119864 minus 01 228119864 minus 01 485119864 minus 01 minus132119864 + 00 645119864 minus 01 644119864 minus 01 229119864 minus 01 259119864 minus 01 minus134119864 + 00



hANN MSE 381119864 minus 01 243119864 minus 01 126E minus 01 590119864 minus 01 minus100119864 minus 01 436119864 minus 01 305119864 minus 01 145119864 minus 01 382119864 minus 01 minus109119864 minus 01hANN tPI 390119864 minus 01 242119864 minus 01 148119864 minus 01 616119864 minus 01 minus960119864 minus 02 417119864 minus 01 278119864 minus 01 168119864 minus 01 380119864 minus 01 minus950119864 minus 03





hANN MSE 370E minus 01 235119864 minus 01 138119864 minus 01 580119864 minus 01 minus630119864 minus 02 423119864 minus 01 279119864 minus 01 156119864 minus 01 397119864 minus 01 minus153119864 minus 02hANN tPI 370E minus 01 229E minus 01 135119864 minus 01 612119864 minus 01 minus349E minus 02 416E minus 01 275E minus 01 154119864 minus 01 364119864 minus 01 622E minus 04hANN PID 378119864 minus 01 241119864 minus 01 132119864 minus 01 634E minus 01 minus897119864 minus 02 423119864 minus 01 281119864 minus 01 149119864 minus 01 434E minus 01 minus232119864 minus 02

Santa Ysabel Creek6-4-1ANN MSE 562119864 minus 01 501119864 minus 01 222119864 minus 01 218119864 minus 01 minus657119864 minus 01 673119864 minus 01 678119864 minus 01 192119864 minus 01 416119864 minus 01 minus127119864 + 00

ANN tPI 564119864 minus 01 511119864 minus 01 289119864 minus 01 203119864 minus 01 minus688119864 minus 01 683119864 minus 01 780119864 minus 01 265119864 minus 01 328119864 minus 01 minus161119864 + 00

ANN PID 556119864 minus 01 486119864 minus 01 218119864 minus 01 242119864 minus 01 minus605119864 minus 01 712119864 minus 01 741119864 minus 01 200119864 minus 01 362119864 minus 01 minus148119864 + 00

hANN MSE 378119864 minus 01 295119864 minus 01 175119864 minus 01 271119864 minus 01 247119864 minus 02 433119864 minus 01 343119864 minus 01 160119864 minus 01 243119864 minus 01 minus146119864 minus 01


6-6-1ANN MSE 531119864 minus 01 485119864 minus 01 289119864 minus 01 243119864 minus 01 minus604119864 minus 01 588119864 minus 01 557119864 minus 01 260119864 minus 01 520119864 minus 01 minus865119864 minus 01

ANN tPI 544119864 minus 01 487119864 minus 01 248119864 minus 01 240119864 minus 01 minus611119864 minus 01 588119864 minus 01 552119864 minus 01 236119864 minus 01 525E minus 01 minus845119864 minus 01ANN PID 542119864 minus 01 501119864 minus 01 258119864 minus 01 218119864 minus 01 minus656119864 minus 01 657119864 minus 01 637119864 minus 01 234119864 minus 01 451119864 minus 01 minus113119864 + 00

hANN MSE 367E minus 01 278119864 minus 01 166119864 minus 01 397E minus 01 827119864 minus 02 380E minus 01 318119864 minus 01 154119864 minus 01 469119864 minus 01 minus628119864 minus 02hANN tPI 381119864 minus 01 280119864 minus 01 173119864 minus 01 389119864 minus 01 760119864 minus 02 414119864 minus 01 328119864 minus 01 159119864 minus 01 493119864 minus 01 minus972119864 minus 02

hANN PID 371119864 minus 01 276E minus 01 166119864 minus 01 387119864 minus 01 882E minus 02 418119864 minus 01 315119864 minus 01 154119864 minus 01 461119864 minus 01 minus542119864 minus 026-8-1ANN MSE 597119864 minus 01 578119864 minus 01 260119864 minus 01 981119864 minus 02 minus910119864 minus 01 722119864 minus 01 826119864 minus 01 235119864 minus 01 288119864 minus 01 minus177119864 + 00





hANN PID 379119864 minus 01 294119864 minus 01 172119864 minus 01 349119864 minus 01 275119864 minus 02 389119864 minus 01 312E minus 01 158119864 minus 01 421119864 minus 01 minus446E minus 02




Leaf River9-4-1






hANN PID 348119864 minus 01 205119864 minus 01 142E minus 01 748119864 minus 01 775119864 minus 01 418119864 minus 01 292119864 minus 01 151E minus 01 582119864 minus 01 768119864 minus 019-6-1sANN MSE 380119864 minus 01 232119864 minus 01 211119864 minus 01 754119864 minus 01 745119864 minus 01 441119864 minus 01 316119864 minus 01 221119864 minus 01 631119864 minus 01 750119864 minus 01




hANN tPI 342E minus 01 201119864 minus 01 151119864 minus 01 772E minus 01 780119864 minus 01 404119864 minus 01 261E minus 01 162119864 minus 01 613119864 minus 01 793E minus 01hANN PID 345119864 minus 01 202119864 minus 01 145119864 minus 01 764119864 minus 01 778119864 minus 01 415119864 minus 01 287119864 minus 01 158119864 minus 01 614119864 minus 01 772119864 minus 01




hANN MSE 343119864 minus 01 198119864 minus 01 156119864 minus 01 770119864 minus 01 783119864 minus 01 395119864 minus 01 262119864 minus 01 168119864 minus 01 632E minus 01 792119864 minus 01hANN tPI 344119864 minus 01 194E minus 01 162119864 minus 01 767119864 minus 01 787E minus 01 394E minus 01 261E minus 01 168119864 minus 01 605119864 minus 01 793E minus 01hANN PID 345119864 minus 01 200119864 minus 01 146119864 minus 01 758119864 minus 01 781119864 minus 01 414119864 minus 01 285119864 minus 01 155119864 minus 01 602119864 minus 01 774119864 minus 01


sANN tPI 407119864 minus 01 294119864 minus 01 230119864 minus 01 547119864 minus 01 759119864 minus 01 551119864 minus 01 453119864 minus 01 214119864 minus 01 621E minus 01 643119864 minus 01sANN PID 417119864 minus 01 311119864 minus 01 191119864 minus 01 520119864 minus 01 744119864 minus 01 638119864 minus 01 601119864 minus 01 180119864 minus 01 498119864 minus 01 526119864 minus 01


hANN tPI 359119864 minus 01 256119864 minus 01 192119864 minus 01 580E minus 01 790119864 minus 01 463119864 minus 01 352119864 minus 01 181119864 minus 01 543119864 minus 01 722119864 minus 01hANN PID 373119864 minus 01 264119864 minus 01 172E minus 01 547119864 minus 01 783119864 minus 01 527119864 minus 01 422119864 minus 01 161E minus 01 488119864 minus 01 667119864 minus 01




hANN MSE 355119864 minus 01 249119864 minus 01 193119864 minus 01 570119864 minus 01 795119864 minus 01 428E minus 01 328E minus 01 185119864 minus 01 595119864 minus 01 741E minus 01hANN tPI 353119864 minus 01 250119864 minus 01 193119864 minus 01 568119864 minus 01 794119864 minus 01 442119864 minus 01 334119864 minus 01 183119864 minus 01 564119864 minus 01 736119864 minus 01

hANN PID minus372119864 minus 01 259119864 minus 01 175119864 minus 01 551119864 minus 01 787119864 minus 01 515119864 minus 01 409119864 minus 01 167119864 minus 01 445119864 minus 01 677119864 minus 01




hANN MSE 355119864 minus 01 248E minus 01 204119864 minus 01 576119864 minus 01 796E minus 01 451119864 minus 01 341119864 minus 01 189119864 minus 01 498119864 minus 01 731119864 minus 01hANN tPI 351E minus 01 248E minus 01 206119864 minus 01 579119864 minus 01 796E minus 01 437119864 minus 01 330119864 minus 01 196119864 minus 01 549119864 minus 01 740119864 minus 01hANN PID 365119864 minus 01 257119864 minus 01 178119864 minus 01 532119864 minus 01 789119864 minus 01 506119864 minus 01 396119864 minus 01 169119864 minus 01 330119864 minus 01 688119864 minus 01


Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)

Figure 4The calibration results of SPEI forecast using hANN the blue line observations the salmon line forecasted SPEI medians and thedotted lines simulation error bounds

period for 3-8-1 hANN and 3-8-1 sANN for Leaf River inTable 7)

hANN model results obtained from nine SPEI inputswere superior to the results obtained from ANN modelswith three and six inputs Simulation results from hANNmodels with three inputs were superior in terms of PI tohANN models with six inputs in the first layer of sANNIncorporation of other information into the ANN inputs didnot improve the SPEI forecast

The hANN models with 9-8-1 and 9-6-1 sANN archi-tectures trained on tPI index were superior in terms of thevalues of PI for both catchments for calibration results Thecalibration results of the best hANNmodels trained on tPI forboth catchments are shown in Figure 4 The best simulationresults according to the medians of PI were obtained forhANN on sANN architectures 9-8-1 9-6-1 trained on tPI andMSE for both basins The time series are shown in Figure 5

When comparing hANN architectures with 9 inputs thebest models according to the PI2 were those with 6 hiddenlayer neurons in calibration of Leaf River dataset while onvalidation data the best PI2 values were obtained from hANNmodes with eight hidden layer neurons On Santa Ysabeldatasets the models with best PI2 indices were hANN with8 hidden layer neurons for calibration while hANN modelswith 6 hidden neurons were superior to the validation dataset(see Table 10)

Table 11 shows the comparison of the influence of differ-ent optimization functions on the calibration and validationof hANNmodels with nine inputsTheoptimization based onMSE was capable of providing better hANNmodels than theoptimizations which used tPI and CI on Leaf River datasetThe tPI optimization function enabled us to find hANNmodels which had had better PI2 values in Santa Ysabeldatasets Note that the differences between PI2 for tPI andMSE are very small

33 Discussion The results of our computational experimentshow the high similarities of values of SPI and SPEI droughtindices The values of correlation coefficients between theSPEI and SPI values were 098 for Leaf River dataset and099 for Santa Ysabel dataset Small differences between bothdrought indices reflect the fact that the temperatures trendswere not apparent in both analyzed datasets [24 25]

We used the single multilayer perceptron model as amain benchmark model for hANN This model showed itssimulation abilities and was compared with other forecastingtechniques for example ARIMA models [11 13 27]

The comparison of ANN and hANN clearly confirmsthe finding which was made by Shamseldin and OrsquoConnor[20] and Goswami et al [32] It shows the benefits of newlytested neural network model The updating of simulatedvalues of drought indices using the additional MLP was in


Table 10 The values of PI2 on architectures 9-119873hd-1 on SPEI forecast with hANN



9-6-1 221119864 minus 02 000119864 + 00 251119864 minus 03 159119864 minus 02 000119864 + 00 minus197119864 minus 02


Santa Ysabel9-4-1 000119864 + 00 minus136119864 minus 02 minus156119864 minus 02 000119864 + 00 minus657119864 minus 02 minus933119864 minus 03

9-6-1 134119864 minus 02 000119864 + 00 minus199119864 minus 03 617119864 minus 02 000119864 + 00 529119864 minus 02

9-8-1 154119864 minus 02 198119864 minus 03 000119864 + 00 924119864 minus 03 minus559119864 minus 02 000119864 + 00

Table 11 The values of PI2 on training functions for SPEI forecast with hANN


Leaf RiverMSE 000119864 + 00 417119864 minus 04 231119864 minus 02 000119864 + 00 766119864 minus 04 448119864 minus 02

tPI minus417119864 minus 04 000119864 + 00 227119864 minus 02 minus766119864 minus 04 000119864 + 00 440119864 minus 02


Santa YsabelMSE 000119864 + 00 minus189119864 minus 03 514119864 minus 02 000119864 + 00 minus173119864 minus 02 181119864 minus 01



Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)

Figure 5 The validation results of SPEI forecast using hANN the blue line observations the salmon line forecasted SPEI medians and thedotted lines simulation error bounds


all cases successful in terms of the improvement of SPEIand SPI drought forecast according to PI values Similaroverall benefits of the integrated neural networkmodels wereconfirmed by [16] on simulations of monthly flows

Our computational exercise also confirms the improve-ments of hANN drought forecast in terms of correctingthe time shift error [11 34 35] The tested hybrid neuralnetwork models decreased the overall time shift error interms of dMSE values However the differences betweenhANN models trained on CI designed to correct the timeshift error did not show significant improvements over thehANN trained on MSE or tPI

The increased accuracy of drought index forecast wasalso influenced by using a model with the higher number ofparameters The simplest sANN with 3 inputs and 4 hiddenlayer neurons had 21 parameters while the hANN with 9inputs and 8 hidden layer neurons had 405 parameters Thehigh number of parameters may limit the application ofhANN model in the case where other parsimonious modelswith similar simulation performances are available

4 Conclusions

We analyzed the forecast of two drought indices SPEI andSPI using two types of neural network models The firstmodel was based on the feedforward neural network withthree layers of neuronsThe secondone integrates the droughtforecasts from five single multilayer perceptrons trained bythe four different performance measures into the hybridintegrated neural network

The SPEI and SPI neural network forecast was based onthe data obtained from the period 1948ndash2002 from two USwatersheds The analyzed data were collected under MOPEXframework

When evaluating the ANN models performance theresults of four fromfivemodel performance indices show thathybrid ANNmodels were superior to the singleMLPmodels

When comparing three different input sets on the SPEIand SPI forecast the input sets with nine lagged monthlyvalues of SPEI and SPI indices were superior Adding theother types of inputs did not improve the results of neuralnetwork forecast

The tested hANN and sANN models were trained usingadaptive differential evolution The nature inspired globaloptimization algorithm was capable of successfully trainingneural networks models The optimization was based onfour functional relationships describing model performanceMSE dMSE tPI and CI indices The worst training resultswere obtained with ANN models based on dMSE

When comparing hANN models according to the num-ber of neurons in the hidden layer two neural networkarchitectures 9-6-1 and 9-8-1 generated the highest valuesof PI2 on SPEI and SPI forecast Also when evaluatingthe influence of different optimization functions on hANNperformance using PI2 the tPI and MSE neural networkperformance functions were superior to dMSE and CI

Although SPEI and SPI indices are using the precipitationdata and have some degree of similarity the best predictions

were obtained using the different combination of neuralnetwork model and training and training criteria

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Dai ldquoDrought under global warming a reviewrdquo WileyInterdisciplinary Reviews Climate Change vol 2 pp 45ndash652011

[2] E R Cook D M Meko D W Stahle and M K CleavelandldquoDrought reconstructions for the continental United StatesrdquoJournal of Climate vol 12 no 4 pp 1145ndash1163 1999

[3] A Dai ldquoIncreasing drought under global warming in observa-tions and modelsrdquo Nature Climate Change vol 3 no 1 pp 52ndash58 2013

[4] A K Mishra and V P Singh ldquoDrought modelingmdasha reviewrdquoJournal of Hydrology vol 403 no 1-2 pp 157ndash175 2011

[5] H K Ntale and T Y Gan ldquoDrought indices and their applica-tion to East Africardquo International Journal of Climatology vol 23no 11 pp 1335ndash1357 2003

[6] S Morid V Smakhtin and M Moghaddasi ldquoComparison ofseven meteorological indices for drought monitoring in IranrdquoInternational Journal of Climatology vol 26 no 7 pp 971ndash9852006

[7] H Wu M J Hayes A Weiss and Q Hu ldquoAn evaluation ofthe standardized precipitation index the China-Z index and thestatistical Z-scorerdquo International Journal of Climatology vol 21no 6 pp 745ndash758 2001

[8] J Keyantash and J A Dracup ldquoThe quantification of droughtan evaluation of drought indicesrdquo Bulletin of the AmericanMeteorological Society vol 83 no 8 pp 1167ndash1180 2002

[9] K Haslinger D Koffler W Schoner and G Laaha ldquoExplor-ing the link between meteorological drought and stream-flow effects of climate-catchment interactionrdquoWater ResourcesResearch vol 50 no 3 pp 2468ndash2487 2014

[10] S Morid V Smakhtin and K Bagherzadeh ldquoDrought forecast-ing using artificial neural networks and time series of droughtindicesrdquo International Journal of Climatology vol 27 no 15 pp2103ndash2111 2007

[11] A Belayneh J Adamowski B Khalil and B Ozga-ZielinskildquoLong-term SPI drought forecasting in the Awash River Basinin Ethiopia using wavelet neural networks and wavelet supportvector regression modelsrdquo Journal of Hydrology vol 508 pp418ndash429 2014

[12] J C Ochoa-Rivera ldquoProspecting droughts with stochasticartificial neural networksrdquo Journal ofHydrology vol 352 no 1-2pp 174ndash180 2008

[13] A K Mishra and V R Desai ldquoDrought forecasting using feed-forward recursive neural networkrdquo Ecological Modelling vol198 no 1-2 pp 127ndash138 2006

[14] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[15] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986


[16] Z Huo S Feng S Kang G Huang F Wang and P GuoldquoIntegrated neural networks for monthly river flow estimationin arid inland basin of Northwest Chinardquo Journal of Hydrologyvol 420-421 pp 159ndash170 2012

[17] A O Pektas and H K Cigizoglu ldquoANN hybrid model versusARIMA and ARIMAX models of runoff coefficientrdquo Journal ofHydrology vol 500 pp 21ndash36 2013

[18] V Nourani A Hosseini Baghanam J Adamowski and O KisildquoApplications of hybrid wavelet-artificial Intelligence models inhydrology a reviewrdquo Journal of Hydrology vol 514 pp 358ndash3772014

[19] W Wang P H A J M V Gelder J K Vrijling and JMa ldquoForecasting daily streamflow using hybrid ANN modelsrdquoJournal of Hydrology vol 324 no 1ndash4 pp 383ndash399 2006

[20] A Y Shamseldin andKMOrsquoConnor ldquoA non-linear neural net-work technique for updating of river flow forecastsrdquo Hydrologyand Earth System Sciences vol 5 no 4 pp 577ndash598 2001

[21] M J Hayes M D Svoboda D AWilhite andO V VanyarkholdquoMonitoring the 1996 drought using the standardized precipita-tion indexrdquo Bulletin of the American Meteorological Society vol80 no 3 pp 429ndash438 1999

[22] N B Guttman ldquoComparing the palmer drought index andthe standardized precipitation indexrdquo Journal of the AmericanWater Resources Association vol 34 no 1 pp 113ndash121 1998

[23] A Cancelliere G D Mauro B Bonaccorso and G RossildquoDrought forecasting using the standardized precipitationindexrdquoWater Resources Management vol 21 no 5 pp 801ndash8192007

[24] S M Vicente-Serrano S Beguerıa and J I Lopez-MorenoldquoA multiscalar drought index sensitive to global warming thestandardized precipitation evapotranspiration indexrdquo Journal ofClimate vol 23 no 7 pp 1696ndash1718 2010

[25] S Beguerıa S M Vicente-Serrano F Reig and B LatorreldquoStandardized precipitation evapotranspiration index (SPEI)revisited parameter fitting evapotranspiration models toolsdatasets and drought monitoringrdquo International Journal ofClimatology vol 34 no 10 pp 3001ndash3023 2014

[26] S Beguerıa and S M Vicente-Serrano ldquoSPEI Calculation ofthe Standardised PrecipitationmdashEvapotranspiration Indexrdquo Rpackage version 16

[27] T-W Kim and J B Valdes ldquoNonlinear model for droughtforecasting based on a conjunction of wavelet transforms andneural networksrdquo Journal of Hydrologic Engineering vol 8 no6 pp 319ndash328 2003

[28] C M Bishop Neural Networks for Pattern Recognition OxfordUniversity Press New York NY USA 1995

[29] R D Reed and R J Marks Neural Smithing SupervisedLearning in Feedforward Artificial Neural Networks MIT PressCambridge Mass USA 1998

[30] G S D S Gomes T B Ludermir and L M M R LimaldquoComparison of new activation functions in neural networkfor forecasting financial time seriesrdquo Neural Computing ampApplications vol 20 no 3 pp 417ndash439 2011

[31] P Maca P Pech and J Pavlasek ldquoComparing the selectedtransfer functions and local optimization methods for neuralnetwork flood runoff forecastrdquoMathematical Problems in Engi-neering vol 2014 Article ID 782351 10 pages 2014

[32] M Goswami K M OrsquoConnor K P Bhattarai and A Y Sham-seldin ldquoAssessing the performance of eight real-time updatingmodels and procedures for the Brosna Riverrdquo Hydrology andEarth System Sciences vol 9 no 4 pp 394ndash411 2005

[33] P K Kitanidis and R L Bras ldquoReal-time forecasting witha conceptual hydrologic model 2 Applications and resultsrdquoWater Resources Research vol 16 no 6 pp 1034ndash1044 1980

[34] C W Dawson R J Abrahart and L M See ldquoHydroTest aweb-based toolbox of evaluation metrics for the standardisedassessment of hydrological forecastsrdquo Environmental Modellingamp Software vol 22 no 7 pp 1034ndash1052 2007

[35] CWDawson R J Abrahart and LM See ldquoHydroTest furtherdevelopment of a web resource for the standardised assessmentof hydrologicalmodelsrdquo EnvironmentalModelling and Softwarevol 25 no 11 pp 1481ndash1482 2010

[36] A P Piotrowski and J J Napiorkowski ldquoOptimizing neuralnetworks for river flow forecastingmdashevolutionary computationmethods versus the levenberg-marquardt approachrdquo Journal ofHydrology vol 407 no 1ndash4 pp 12ndash27 2011

[37] J Zhang andA Sanderson ldquoJade adaptive differential evolutionwith optional external archiverdquo IEEE Transactions on Evolution-ary Computation vol 13 pp 945ndash958 2009

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K V Price R M Storn and J A Lampinen DifferentialEvolutionmdashA Practical Approach to Global Optimization Nat-ural Computing Springer Berlin Germany 2006

[40] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011

[41] Q Duan J Schaake V Andreassian et al ldquoModel ParameterEstimation Experiment (MOPEX) an overview of science strat-egy and major results from the second and third workshopsrdquoJournal of Hydrology vol 320 no 1-2 pp 3ndash17 2006

[42] J Schaake Q Duan V Andreassian S Franks A Hall andG Leavesley ldquoThe model parameter estimation experiment(MOPEX)rdquo Journal of Hydrology vol 320 no 1-2 pp 1ndash2 2006

[43] T Ao H Ishidaira K Takeuchi et al ldquoRelating BTOPMCmodel parameters to physical features of MOPEX basinsrdquoJournal of Hydrology vol 320 no 1-2 pp 84ndash102 2006

[44] A YeQDuan X Yuan E FWood and J Schaake ldquoHydrologicpost-processing of MOPEX streamflow simulationsrdquo Journal ofHydrology vol 508 pp 147ndash156 2014

[45] J Chen F P Brissette D Chaumont and M Braun ldquoPer-formance and uncertainty evaluation of empirical downscal-ing methods in quantifying the climate change impacts onhydrology over two North American river basinsrdquo Journal ofHydrology vol 479 pp 200ndash214 2013

[46] P A Troch G F Martinez V R N Pauwels et al ldquoClimate andvegetation water use efficiency at catchment scalesrdquo Hydrologi-cal Processes vol 23 no 16 pp 2409ndash2414 2009

[47] M Huang Z Hou L R Leung et al ldquoUncertainty analysisof runoff simulations and parameter identifiability in thecommunity landmodel evidence fromMOPEXbasinsrdquo Journalof Hydrometeorology vol 14 no 6 pp 1754ndash1772 2013

[48] E Mutlu I Chaubey H Hexmoor and S G Bajwa ldquoCom-parison of artificial neural network models for hydrologicpredictions at multiple gauging stations in an agriculturalwatershedrdquo Hydrological Processes vol 22 no 26 pp 5097ndash5106 2008

[49] R J May H R Maier G C Dandy and T M K G FernandoldquoNon-linear variable selection for artificial neural networksusing partial mutual informationrdquo Environmental Modelling ampSoftware vol 23 no 10-11 pp 1312ndash1326 2008


[50] R J May G C Dandy H R Maier and J B NixonldquoApplication of partial mutual information variable selection toANN forecasting of water quality in water distribution systemsrdquoEnvironmentalModellingamp Software vol 23 no 10-11 pp 1289ndash1299 2008

[51] U G Bacanli M Firat and F Dikbas ldquoAdaptive Neuro-Fuzzy inference system for drought forecastingrdquo StochasticEnvironmental Research and Risk Assessment vol 23 no 8 pp1143ndash1154 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



DIMSE DIMSE

DIdMSE DIdMSE

DItPI DItPI

DIPID DIPID

DIOF

sANN

Inpu

t set

sIn

put s

ets

Inpu

t set

sIn

put s

ets

Inpu

t set

sIn

put s

ets

Inpu

t set

sIn

put s

ets

OF MSE

OF dMSE

OF tPI

OF PID

OF MSE

OF dMSE

OF tPI

OF PID

hANN

OF oneof four used OF

OF objective functionused for singleobjective optimization

Figure 1 The scheme of tested ANN architectures sANN is based on single MLP hANN is integrated ANN

index SPI index [21ndash23] and the standardized precipitationevapotranspiration index SPEI index [24 25]

The SPI index is based on the evaluation of precipitationdata The precipitation data are linked to the selected prob-ability distribution which is further standardized using thenormal distribution with zero mean and standard deviationof one It is often expressed as ameteorological drought index[11] and it is used for the assessment of agricultural andhydrological droughts [21]

The estimation of SPI consists of the determination ofprobability distribution of analyzed precipitation data thecalculation of probabilities for measured precipitation datafrom cumulative distribution function of fitted probabilitydistribution and the application of the inverse of distributionfunction of normalized normal distribution on probabilities[21 22]

The SPEI drought index is based on the precipitation andpotential evapotranspiration dataThe information about thepotential evapotranspiration temperature is mostly derivedusing the temperature data The SPEI index is expressedusing the differences between precipitation and potentialevapotranspiration Its calculation technically follows thederivation of SPI index the only difference is that instead ofthe precipitation time series the time series of the abovemen-tioned differences are used [24 25]

The estimation of SPI and SPEI drought indices wasmadeusing the R package [26] The probability distribution ofSPEI was expressed using the three-parameter log-logisticprobability distribution the SPI probability distribution wascalculated using the Gamma distribution The parameterswere identified using the method of unbiased probabilityweighted moments [24 25]

The SPI indicates the extremity of droughts The SPIvalues split the range into extremely dry (SPI le minus2) severelydry (minus2 lt SPI le minus15) moderately dry (minus15 lt SPI le minus1)and near neutral conditions (minus10 lt SPI le 10) [22 23]

22 sANN Model The architecture of the first analyzed neu-ral network model was based on the feedforward multilayerperceptron with one hidden layer of neurons sANN (thesingle ANNmodel see Figure 1)This type of neural networkarchitecture has been already applied on drought indicesforecast [10 11 27]

The sANN model has the following mathematical for-mula

DI119891 = Vout0 +

119873hd

sum119895=1

Vout119895 119891(Vhd0119895 +

119873in

sum119894=1

Vhd119895119894 119909119894) (1)

where DI119891 is a network output that is drought index forecastfor a given time interval 119909119894 is network input for input layerneuron 119894 normalized on the interval (0 1)119873in is the numberof MLP inputs Vhd119895119894 is the weight of the connection betweeninput 119894 and hidden layer neuron 119895 119891( ) is the activationfunction for all hidden layer neurons 119873hd is the number ofhidden neurons Vout119895 is the weight of the connection betweenthe hidden neuron 119895 and output neuron and Vhd0119895 V

out0 are

biases of neurons [14 28 29]The type of activation function of neurons in hidden layer

was the RootSig [30 31] Its form is

119910 (119886) =119886

1 + radic1 + 1198862 (2)

with 119886 = sum119873in119894=1 V119895119894119909119894 + V0119895








MAE = 1119899

119899

sum119905=1

10038161003816100381610038161003816DI119900 [119905] minus DI119891 [119905]

10038161003816100381610038161003816 (3)


MSE = 1119899

119899

sum119905=1

(DI119900 [119905] minus DI119891 [119905])2 (4)



119899

sum119905=2

(dDI119900 [119905] minus dDI119891 [119905])2 (5)



2

sum119899119905=1 (DI119900 [119905] minus DI119900)

2 (6)



2


2 (7)



Combined Index (CI)

CI = 085tPI + 015dMSE (9)



2


2 (10)


1

with DIANN2







V119896 [119894]hdout= V119896 [119894 minus 1]

hdout


hdout)


hdout)

(11)




















































































Leaf River9-4-1



























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(b)




























Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)



































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










MAE = 1119899

119899

sum119905=1

10038161003816100381610038161003816DI119900 [119905] minus DI119891 [119905]

10038161003816100381610038161003816 (3)


MSE = 1119899

119899

sum119905=1

(DI119900 [119905] minus DI119891 [119905])2 (4)



119899

sum119905=2

(dDI119900 [119905] minus dDI119891 [119905])2 (5)



2

sum119899119905=1 (DI119900 [119905] minus DI119900)

2 (6)



2


2 (7)



Combined Index (CI)

CI = 085tPI + 015dMSE (9)



2


2 (10)


1

with DIANN2







V119896 [119894]hdout= V119896 [119894 minus 1]

hdout


hdout)


hdout)

(11)




















































































Leaf River9-4-1



























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(b)




























Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)



































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





V119896 [119894]hdout= V119896 [119894 minus 1]

hdout


hdout)


hdout)

(11)




















































































Leaf River9-4-1



























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(b)




























Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)



































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in

































































Leaf River9-4-1



























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(b)




























Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)



































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPI

(b)




























Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)



































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




















Leaf River

minus2

minus1

0

1

2

SPI

1980 1985 1990 1995 20001975Date(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPI

(b)



































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




































Leaf River6-4-1
































Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Leaf River9-4-1





























Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Leaf River

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1955 1960 1965 1970 19751950Date

minus2

minus1

0

1

2

SPEI

(b)



























Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




















Leaf River

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(a)

Santa Ysabel

1980 1985 1990 1995 20001975Date

minus2

minus1

0

1

2

SPEI

(b)






4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







4 Conclusions











References





























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in
















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article forecasting spei and spi drought indices...

Documents