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Dynamic factor analysis and artificial neural networkfor estimating pan evaporation at multiple stations innorthern TaiwanF.J. Chang a , W. Sun a & C.H. Chung aa Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei,10617, TaiwanVersion of record first published: 25 Mar 2013.

To cite this article: F.J. Chang , W. Sun & C.H. Chung (2013): Dynamic factor analysis and artificial neuralnetwork for estimating pan evaporation at multiple stations in northern Taiwan, Hydrological Sciences Journal,DOI:10.1080/02626667.2013.775447

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1Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 2013http://dx.doi.org/10.1080/02626667.2013.775447

Dynamic factor analysis and artificial neural network for estimatingpan evaporation at multiple stations in northern Taiwan

F.J. Chang, W. Sun and C.H. Chung

Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 10617, Taiwanchangfj@ntu.edu.tw

Received 21 November 2011; accepted 8 October 2012; open for discussion until 1 November 2013

Editor D. Koutsoyiannis; Associate editor C.Y. Xu

Citation Chang, F.J., Sun, W., and Chung, C.H., 2013. Dynamic factor analysis and artificial neural network for estimating panevaporation at multiple stations in northern Taiwan. Hydrological Sciences Journal, 58 (4), 1–13.

Abstract Evaporation is an important reference for managers of water resources. This study proposes a hybridmodel (BD) that combines back-propagation neural networks (BPNN) and dynamic factor analysis (DFA) tosimultaneously precisely estimate pan evaporation at multiple meteorological stations in northern Taiwan throughincorporating a large number of meteorological data sets into the estimation process. The DFA is first used toextract key meteorological factors that are highly related to pan evaporation and to establish the common trendof pan evaporation among meteorological stations. The BPNN is then trained to estimate pan evaporation withthe inputs of the key meteorological factors and evaporation estimates given by the DFA. The BD model suc-cessfully inherits the advantages from the DFA and BPNN, and effectively enhances its generalization ability andestimation accuracy. The results demonstrate that the proposed BD model has good reliability and applicability insimultaneously estimating pan evaporation for multiple meteorological stations.

Key words pan evaporation; artificial neural network; dynamic factor analysis; back-propagation neural network; meteorolog-ical stations

Analyse factorielle dynamique et réseaux de neurones artificiels pour l’estimation des évapora-tions de bac de plusieurs stations dans le nord de TaïwanRésumé L’évaporation est une référence importante pour les gestionnaires des ressources en eau. Cette étudepropose un modèle hybride (BD) qui combine des réseaux de neurones à rétropropagation (RNRP) et l’analysefactorielle dynamique (AFD) pour estimer avec précision l’évaporation de bac simultanément pour de multi-ples stations météorologiques dans le nord de Taïwan, en intégrant un grand nombre de séries de donnéesmétéorologiques dans le processus d’estimation. L’AFD est d’abord utilisée pour extraire les principaux fac-teurs météorologiques qui sont fortement liés à l’évaporation de bac, et pour établir la tendance commune del’évaporation de bac entre les stations météorologiques. Le RNRP est ensuite entrainé à estimer l’évaporation debac avec les entrées des principaux facteurs météorologiques et des estimations d’évaporation données par l’AFD.Le modèle BD hérite avec succès des avantages de l’AFD et du RNRP, et améliore efficacement sa capacité degénéralisation et sa précision d’estimation. Les résultats montrent que le modèle BD proposé a une bonne fiabilitéet applicabilité pour l’estimation de l’évaporation de bac simultanément sur de multiples stations météorologiques.

Mots clefs évaporation de bac; réseaux de neurones artificiels; analyse factorielle dynamique; réseau de neurones àrétropropagation; stations météorologiques

INTRODUCTION

Evaporation is an important factor that affects waterresources. An accurate estimation of evaporation isessential to the management of agricultural irriga-tion, water balance, water supply and land resources

planning. Complex interactions between land andatmosphere systems prevent effective and accurateestimations of evaporation. The amount of evapo-ration may be influenced by several meteorologicalfactors, such as temperature, atmospheric pressure,wind speed and solar radiation. A number of direct

© 2013 IAHS Press

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and indirect methods have been applied to evapo-ration measurement and estimation in hydrologicalpractices. Pan evaporation is one of the direct meth-ods for evaporation measurement (Kohler et al. 1955,Farnsworth et al. 1982). Researchers have presentedseveral indirect methods, including empirical for-mulas, semi-empirical formulas, mass transfer andwater budget, in past decades (Warnaka and Pochop1988, Allen et al. 1998, Vallet-Coulomb et al. 2001,Trajkovic and Kolakovic 2010). Donohue et al. (2010)captured the dynamics in evaporative demand withina changing climate through different evaporation for-mulations. Many researchers have raised the needfor accurate estimates of evaporation (Sudheer et al.2003, Szilagyi and Jozsa 2009, Chang et al. 2010).The accuracy of evaporation estimation relies highlyon precise and reliable meteorological data. Empiricalformulas require a wide range of data types, whichmakes it difficult to obtain measurements such as radi-ation. Choosing a suitable model that fits the widerange of data types well is another critical issue.Establishing proper models would help overcomenonlinear problems.

Artificial neural networks (ANNs) were createdto simulate the nervous system and brain activity.In recent decades, a number of neural networks, suchas the back-propagation neural network (Rumelhartet al. 1986), the recurrent neural network (Williams1989) and the fuzzy neural network (Nie and Linkens1994), were developed to solve a wide variety of prob-lems (Schalkoff 1997, Chang et al. 2008, Chen andChang 2009, Chiang and Chang 2009, Ozkan et al.2011). Researchers tried to estimate evaporation byfitting the relationship between meteorological fac-tors (Burman 1977, Coulomb et al. 2001, Gavin andAgnew 2004, Kisi, 2006). Nevertheless, evaporationdistribution is highly nonlinear. Fortunately, ANNsare a suitable tool to deal with nonlinear problems.The application of ANNs to the fields of evapo-ration and evapotranspiration has been proposed inmany studies (Bruton et al. 2000, Trajkovic et al.2000, Odhiambo et al. 2001, Shiri and Kisi 2001,Keskin and Terzi 2006a, 2006b, Kisi and Ozturk2007, Parasuraman et al. 2007, Gonzales-Camachoet al. 2008, Kim and Kim 2008, Kisi 2009a). Tabariet al. (2010) estimated daily pan evaporation in asemi-arid region using different artificial neural net-works and multivariate nonlinear regression. Sudheeret al. (2002) applied the back-propagation neuralnetwork to predict pan evaporation. Kisi (2007)used the feed-forward ANN technique incorporatedwith the Levenberg-Marquardt algorithm to model

evapotranspiration. Kisi (2009b) reported that multi-layer perceptrons and radial basis neural networks aresuitable for the estimation of daily pan evaporation.Trajkovic (2009) applied the radial basis functionnetwork to pan evaporation for evapotranspirationconversions. Chang et al. (2010) presented that theself-organizing map neural network could produce asignificant map to relate climatic features to evapora-tion in different seasons for a single meteorologicalstation. Nourani and Sayyah Fard (2012) conductedsensitivity analysis on the artificial neural networkoutputs for evaporation estimation at several clima-tologic regions. Chung et al. (2012) applied a spatialneural fuzzy network for estimating pan evaporationat ungauged sites. In addition, pan evaporation hasbeen widely used as an index of evapotranspiration(Kohler et al. 1955, Christiansen 1966). Choudhuryet al. (1994), Xu et al. (2006), McVicar et al.(2007) discussed the relationship between referenceevapotranspiration and pan evaporation to obtain pancoefficients.

Dynamic factor analysis (DFA) was originallydeveloped for econometric (Geweke 1978) and psy-chological fields (Molenaar 1985), and is a useful toolfor dimension reduction, especially for time series.The DFA extracts the common trends between multi-variate time series and the relationships between theseries and the selected explanatory variables (Zuuret al. 2003a, 2003b, Zuur and Pierce 2004). The DFAwas used to recognize the common trends of ground-water levels (Markus et al. 1999); to estimate ground-water arsenic trends (Kuo and Chang 2009) andgroundwater quality trends (Munoz-Carpena et al.2005); and to identify the effects of hydrological fac-tors on local groundwater nitrate concentration (Ritteret al. 2007). To the best of our knowledge, the DFA isseldom applied to evaporation estimation.

In the past, evaporation estimation has beenmainly performed at individual stations separately,and it is more difficult to make estimations at mul-tiple stations simultaneously The traditional modelsusually need to calibrate their parameters for a spe-cific station (Keskin and Terzi 2006a, Yeh et al. 2008),and therefore those models might well estimate evap-oration at this station, while getting poor estimatesat the other stations. Therefore, there is a need for amodel that can provide the estimation for multiple sta-tions simultaneously with satisfactory performance.The primary objective of this study is to suggest apossible method that uses meteorological variables toestimate pan evaporation at multiple meteorologicalstations in a specific region.

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DFA and ANN for estimating pan evaporation at multiple sites 3

METHODOLOGIES

A hybrid BPNN (back-propagation neural network)and DFA model (hereinafter called the BD model) forestimating pan evaporation (Epan) at multiple meteo-rological stations simultaneously, is proposed in thisstudy. The concept is to use the DFA for extractingthe key meteorological factors in a region and thecommon trend of pan evaporation among meteoro-logical stations, then to use the BPNN for modellingthe nonlinear relationship between key meteorologi-cal factors and DFA estimates, and consequently toobtain accurate pan evaporation. The BD model isthen compared with traditional BPNN.

Dynamic factor analysis (DFA)

The DFA, a method developed for multivariate time-series analysis, can be used to analyse the commontrends of time series. The DFA can be expressed interms of a linear combination of common trends,explanatory variables, an intercept and noise, as fol-lows (Zuur et al. 2003a): N time series = linearcombination of M common trends + level parameters+ explanatory variables + noise.

Despite the fact that more trends can lead to amore accurate fit, the DFA will choose the smallestM , while obtaining a reasonable fit, because a largerM will increase model complexity. The DFA can esti-mate M common trends instead of N common trends,and ideally M is much smaller than N .

The mathematical formulation for the DFAmodel is given by:

Sn(t) =M∑

m=1

γm,nαm(t) + μn +K∑

k=1

βk,nxk(t)

+ εn(t)

(1)

αm(t) = αm(t − 1) + ηm(t) (2)

where Sn(t) is the value of the nth response variableat time t, which represents the evaporation at the nthmeteorological station at time t in this study; γm,n(t)is the factor loading and αm(t) is the mth commontrend at time t;

∑Mm=1 γm,nαm(t) is the linear combi-

nation of common trends; μn is the nth constant levelparameter which is the intercept term in the regres-sion; βk,n are the regression coefficients for the Kexplanatory variables xk(t), which are the meteoro-logical variables used in this study; εn(t) and ηm(t)

are error components, which are assumed to be inde-pendent of each other, homogeneous and normallydistributed with mean zero and a known covariancematrix. Each time series represents the estimation ofevaporation at one meteorological station.

Akaike’s information criterion (AIC) (Akaike1974) is used to determine suitable DFA models,which can balance the complication and estimationaccuracy of the models. In general cases, the math-ematical formulation for AIC is given by:

AIC = 2K − 2 ln(L) (3)

where K is the number of parameters, and L isthe maximum value of the likelihood function forthe estimated model. The DFA results presented inthis study were obtained by the Brodgar statisticalsoftware (Highland Statistics Ltd., UK, http://www.brodar.com). A complete and detailed description ofthe DFA is given in Zuur et al. (2003a) and Ritter et al.(2007).

Back-propagation neural networks (BPNN)

The BPNN, developed by Rumelhart et al. (1986),has been widely applied in the fields of estimationand prediction. The BPNN has a feed-forward archi-tecture, in which its basic framework consists of aninput layer, a hidden layer and an output layer (Fig. 1).When executing the BPNN, data are generally dividedinto three subsets for use in the training, validationand testing phases, respectively. The goal of the super-vised BPNN in the training phase is to minimizethe global error. A complete and detailed descrip-tion of the standard BPNN algorithm can be foundin Rumelhart et al. (1986).

Fig. 1 Architecture of the proposed BD model.

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BD model—a hybrid model of BPNN and DFA

The framework of the BD model is illustrated inFig. 1, and the procedure is described below.

Key factors extractions The first approach forthe BD model is to determine the key factors bythe DFA with the AIC as the determination criterion.Candidate models consist of different input combina-tions under n meteorological factors for the BPNN,and the best input combination will be the modelwith the minimum AIC value. The available climaticdata measured at meteorological stations of Taiwan’sCentral Weather Bureau consist of wind speed,sunshine hours, global solar radiation, temperature,humidity, pressure, rainfall and pan evaporation.Previous studies applied different combinations of cli-matic variables to the estimation of pan evaporation(Tabari et al. 2010, Shiri and Kisi 2011). Referringto the variables used in our previous research on panevaporation (Chang et al. 2010, Chung et al. 2012),this study focuses on investigating the effects of windspeed, sunshine hours, radiation, temperature, humid-ity and dew point on pan evaporation. Models withfive out of six factors were evaluated in the first roundof AIC calculation; therefore there are six candidatemodels in total. The model with the smallest AICvalue becomes the evaluation target in the secondround. Models with four factors out of five factors areevaluated in the second round; there are five models

in total. Again, the model with the smallest AICvalue becomes the evaluation target in the third round.This procedure terminates when big differences occurbetween the AIC values of two consecutive roundsor when the round with only one factor completes.The best input combination that contains the desiredkey factors is achieved when the minimum AIC valueoccurs. In this study, the key factors determined bythe DFA are wind speed, radiation, temperature anddew point (see Table 1). The DFA also establishes onecommon trend as an input to the BPNN.

Neural network construction The BPNN withone or two hidden layers could generally convergebetter because more than two hidden layers wouldresult in slow convergence and many local minima(Chiang et al. 2004). Therefore, one hidden layer isdesigned for this case study. In addition, too manyhidden neurons may lead each hidden neuron tomemorize a single input pattern and thereby reducethe generalization capability of the BPNN, whichwill easily cause over-fitting problems. There is nostandard way to determine the number of hidden neu-rons; a common way is by trial-and-error. The finalarchitecture is determined based on RMSE in thevalidation set. In this case study, one hidden layerand 10 hidden neurons are determined for the BPNNthrough an intensive trial-and-error procedure basedon 1–15 neurons. Finally, the constructed network isused to estimate pan evaporation.

Table 1 Determination of the best input combination to the BPNN by the DFA with AIC.

Model Input combination∗ AIC value

Wind speed Sunshine hours Radiation Temperature Humidity Dew point

6 factors • • • • • • 11453

1st round: 5 factors(1–1) • • • • • 11564(1–2) • • • • • 11455(1–3) • • • • • 11890(1–4) • • • • • 11496(1–5) • • • • • 11447(1–6) • • • • • 11466

2nd round: 4 factors(2–1) • • • • 11565(2–2)∗ • • • • 11450(2–3) • • • • 11889(2–4) • • • • 11721(2–5) • • • • 11535

3rd round: 3 factors(3–1) • • • 18206(3-2) • • • 19253(3–3) • • • 18368(3–4) • • • 18189

Note: ∗The best combination of key factors determined by the DFA.

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DFA and ANN for estimating pan evaporation at multiple sites 5

In short, the DFA is good at extracting key fac-tors and providing the common trend of evaporationthat illustrates the evaporation characteristics in thestudy area, whereas the BPNN is good at provid-ing accurate evaporation estimation, but is unableto effectively identify or explicitly provide impor-tant factors that influence evaporation. Therefore, theproposed hybrid BD model takes the advantages ofthe DFA and BPNN to improve the accuracy of panevaporation estimation.

Performance criteria Three criteria, root meansquare error (RMSE), coefficient of efficiency (CE)and mean absolute error (MAE), are used to evalu-ate model performance. The RMSE (equation (4)) isa useful measure for illustrating the predictive capa-bility of a model; and CE (equation (5)) is widelyused in assessing the performance of hydrological andwater quality models and ranges between −∞ and1. Both MAE (equation (6)) and RMSE are quantita-tive statistics adopted to measure how close the modelsimulations are to the observations:

RMSE =

√√√√√n∑

i=1[oi − ei]

2

n(4)

CE = 1 −

n∑i=1

(oi − ei)2

n∑i=1

(oi − _o)2

(5)

MAE =

n∑i=1

|oi − ei|n

(6)

where oi,_o and ei are the observed, average of

observed and predicted values, respectively, and n isthe number of data.

APPLICATION

Study area and data set

In this study, meteorological data were collectedfrom six meteorological stations in northwesternTaiwan (Fig. 2). Stations Taipei, Banqiao, Keelungand Hsinchu are located within urban areas, whilestations Anbu and Jutzhu are located in mountain-ous areas (the elevation of the six stations is shownin Table 2). The land uses in this case study includeregions with high population density (Taipei and

Fig. 2 Locations with elevation (m a.s.l.) of meteorologicalstations in the study area.

Banqiao), agricultural land (Hsinchu), a harbour area(Keelung) and nature reserves (Anbu and Jutzhu).The complicated topographic features and land useshave resulted in variations in evaporation over thisstudy area. In general, each meteorological station inTaiwan is equipped with a sheathed thermometer, apropeller anemometer, a piston mercury barometer, apyranometer, a tipping-bucket raingauge, a Class Apan, a hair hygrometer, a solar-cell sunshine recorderand a psychrometer for measuring temperature, windspeed, pressure, global solar radiation, rainfall, panevaporation, humidity, sunshine hours and humidity,respectively. More details can be found at the web site(http://www.cwb.gov.tw/V7e/index_home.htm) of theCentral Weather Bureau, Taiwan.

The meteorological factors used in this study arepan evaporation (mm d-1), wind speed (m s-1), sun-shine hours (h), radiation (MJ m-2 d-1), temperature(◦C), humidity (%) and dew point (◦C) (Table 2).Evaporation is measured by Class A pan. In total,5983 daily meteorological data sets were collectedat six meteorological stations between 1 January2007 and 30 September 2010. The statistical param-eters of the factors for each data set are listed inTable 2, and include mean, standard deviation (SD),coefficient of variation (CV), maximum (max), mini-mum (min) and correlation coefficient (CC). The CCis used to investigate the correlation of evaporationwith meteorological variables. Table 2 reflects the

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Table 2 Daily statistical parameters of each data set (see text for explanation of abbreviations).

Station (Elevation) Data set Unit Mean SD CV (SD/mean) Max Min CC

Taipei Evaporation∗ (mm d-1) 3.28 1.98 0.60 7.90 0.10 1.00(5.3 m) Wind speed (m s-1) 2.43 1.24 0.51 5.60 0.50 0.05

Sunshine hour (h) 4.48 3.79 0.84 11.80 0.00 0.60Radiation (MJ m-2 d-1) 10.93 6.47 0.59 23.50 0.00 0.69Temperature (◦C) 23.39 5.47 0.23 31.70 9.30 0.55Humidity (%) 74.09 8.84 0.11 95.00 52.00 −0.57Dew point (◦C) 18.29 5.17 0.28 25.00 3.10 0.34

Banqiao Evaporation (mm d-1) 2.67 1.74 0.65 7.80 0.10 1.00(9.7 m) Wind speed (m s-1) 2.08 1.22 0.58 5.60 0.30 0.08

Sunshine hour (h) 4.73 3.95 0.83 12.30 0.00 0.74Radiation (MJ m-2 d-1) 11.58 7.21 0.62 25.26 0.00 0.81Temperature (◦C) 23.12 5.42 0.23 31.10 8.80 0.67Humidity (%) 74.55 7.15 0.9 93.00 56.00 −0.55Dew point (◦C) 18.18 5.13 0.28 25.20 3.80 0.51

Keelung Evaporation (mm d-1) 3.04 2.14 0.70 9.40 0.00 1.00(26.7 m) Wind speed (m s-1) 2.92 1.39 0.47 8.70 0.50 −0.15

Sunshine hour (h) 3.97 4.00 1.00 12.00 0.00 0.72Radiation (MJ m-2 d-1) 10.76 8.04 0.74 23.55 0.00 0.77Temperature (◦C) 22.75 5.32 0.23 31.50 9.90 0.63Humidity (%) 76.33 8.51 0.11 96.00 55.00 −0.55Dew point (◦C) 18.21 5.15 0.28 25.20 3.60 0.44

Hsinchu Evaporation (mm d-1) 3.32 2.00 0.60 8.50 0.10 1.00(26.9 m) Wind speed (m s-1) 2.02 1.03 0.50 6.80 0.60 −0.07

Sunshine hour (h) 5.46 4.09 0.74 12.70 0.00 0.69Radiation (MJ m-2 d-1) 12.63 6.84 0.54 25.25 0.13 0.82Temperature (◦C) 22.95 5.57 0.24 30.90 9.10 0.73Humidity (%) 74.25 7.95 0.10 90.00 54.00 −0.49Dew point (◦C) 17.92 5.39 0.30 25.20 3.00 0.56

Anbu Evaporation (mm d-1) 1.49 1.19 0.79 4.90 0.10 1.00(825.8 m) Wind speed (m s-1) 2.81 1.29 0.45 9.10 0.80 −0.17

Sunshine hour (h) 2.94 3.18 1.08 11.40 0.00 0.74Radiation (MJ m-2 d-1) 10.90 7.55 0.69 30.95 0.00 0.81Temperature (◦C) 17.07 5.40 0.31 24.80 2.60 0.52Humidity (%) 88.11 8.01 0.90 100.00 51.00 −0.56Dew point (◦C) 14.96 5.21 0.34 21.70 1.20 0.36

Jutzhu Evaporation (mm d-1) 1.92 1.43 0.74 9.10 0.10 1.00(607.1 m) Wind speed (m s-1) 1.72 1.29 0.75 6.40 0.20 −0.28

Sunshine hour (h) 3.66 3.42 0.93 11.10 0.00 0.60Radiation (MJ m-2 d-1) 7.59 4.45 0.58 16.66 0.00 0.71Temperature (◦C) 19.63 5.50 0.28 27.30 3.60 0.43Humidity (%) 83.37 7.08 0.80 99.00 60.00 −0.54Dew point (◦C) 16.64 5.39 0.32 24.30 0.20 0.29

Note: ∗Evaporation is measured by Class A pan.

distinguishing characteristics for urban and mountain-ous areas, respectively. There are five findings:

(a) the humidity at all stations is high and reachesover 70%;

(b) the sunshine hours at all stations show very highvariations that exceed 0.8 generally (refer to CVvalues in Table 2);

(c) radiation and sunshine hours have higher corre-lations with pan evaporation;

(d) wind speed seems to have no relationship withpan evaporation due to low CC values; and

(e) the pan evaporation at all stations is <10 mm d-1.It is important to notice that the data sets reflecthigh variability.

For example, the means of CV values for pan evap-oration, radiation and wind speed are 68%, 62% and54%, respectively, which are apparently higher thanthose (50%, 40% and 30% accordingly) of the datasets presented in Kisi (2009b).

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DFA and ANN for estimating pan evaporation at multiple sites 7

Fig. 3 (a) Common trend and (b) corresponding canonical coefficients for the best DFA model. Dotted line indicates thethreshold for the weak correlation between the common trend and the meteorological stations.

Figure 3 indicates that the DFA can identifythe differences between two groups (station Taipeiversus the other five stations) based on the com-mon trend analysis, which implies that some unknownfactors affecting evaporation do exist. Therefore, it isessential to include DFA estimates as inputs to theBPNN, because the inherent advantages of the com-mon trend information are deemed as an implicationof unknown factors. In other words, wind speed isselected as a key factor for pan evaporation mod-elling by the DFA, even though wind speed shows no

linear relationship with pan evaporation based on itscorrelation coefficient in Table 2.

All 5983 data sets were allocated into training,validation and test subsets for the BPNN: 2965 datasets from five stations (except Jutzhu) in 2007 and2009 were used for training; 1380 data sets from fivestations (except Jutzhu) in 2008 were used for val-idation; and 1638 data sets from all six stations in2010 were used for testing. To verify model reliability,data sets from station Jutzhu were arranged for useonly in the testing phase. The BD model is expected

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to be more suitable for urban areas than for moun-tainous areas, because four out of six meteorologicalstations are located in urban areas.

RESULTS

The first approach of the BD model is to extract keyfactors by the DFA, with AIC as the determination cri-terion, through examining various input combinationsof meteorological factors that are wind speed, sun-shine hours, radiation, temperature, humidity and dewpoint (Table 1). The factor “humidity” is deleted in thefirst round, and the factor “sunshine hours” is deletedin the second round. For the combinations with lessthan four factors, the AIC value becomes larger than18 000 (see the third round in Table 1). Therefore, thebest DFA model consists of four key factors—windspeed, radiation, temperature and dew point—eventhough factors “sunshine hours” and “humidity” areoften regarded as important factors for evaporation.The reasons for the exclusion of factors “sunshinehours” and “humidity” from the key factor list are thatsunshine and radiation are potential energy sourcesfor evaporation, and that humidity is always high andhas similar statistical properties in northern Taiwan,and thus contributes very little to the evaporationestimation in the proposed BD model. The four keyfactors (wind speed, radiation, temperature and dewpoint) and six estimates given by the best DFA modelwill be adopted as inputs to the BPNN of the BDmodel.

Figure 3 shows the common trend of evaporationand the corresponding canonical coefficients of thebest DFA model, where the dashed line indicates thethreshold of a weak correlation between the meteoro-logical stations and the common trend. The DFA candetect the unexplained variations in data and establishthe common trend of the evaporation time-series data.There are six meteorological stations in the studyarea, and therefore only one common trend is estab-lished because too many common trends may lead toover-fitting problems. The result shows that this com-mon trend is highly related to station Taipei but isless related to the other five meteorological stations(canonical coefficients <0.3). Although the physicalexplanation of the common trend is still unknown,the six meteorological stations can be clearly distin-guished into two groups according to the threshold:one group consists of only Taipei station, and theother group consists of the remaining five meteorolog-ical stations. When analysing the statistical features,differences between Taipei station and the others

are not obvious (Table 2). Therefore, it is essentialto include DFA estimates as inputs to the BPNN,because their inherent advantages for the commontrend information are deemed as an implication ofunknown factors.

Two comparative models are presented, whichare the BP (four key factors) and BP (six factors)models—referred to as BP4 and BP6, hereafter. Boththe BD and BP4 models have the same network struc-ture (one hidden layer and 10 hidden neurons). TheBP6 also consists of one hidden layer and 10 neu-rons through an intensive trial-and-error procedure.Comparison results are shown in Table 3. The resultsshow that: (a) BP4 performs better than BP6 in termsof RMSE and MAE, which reveals the extraction ofkey factors by the DFA can effectively improve theestimation accuracy; and (b) the RMSE and MAE val-ues of Anbu station (located in a mountainous area)are smaller than those of the stations located in urbanareas. However, the Anbu station has the lowest CEvalue compared with the stations located in urbanareas. For Jutzhu station (in a mountainous area), itsperformance is in general not as good as the stationsin urban areas. Table 2 reveals that the mean evap-oration at Anbu station is about a half of those atthe stations in urban areas. Therefore, it is concludedthat the BD model performs better (in term of largerCE values) in urban areas than in mountainous areas.

Table 3 Performance of the BD and BP models at meteo-rological stations in the testing phase.

Criteria Station BD BP4 BP6

RMSE Taipei 0.88 1.04 1.18Banqiao 1.03 1.00 1.06Keelung 1.06 1.25 1.33Hsinchu 1.01 1.08 1.18Anbu 0.81 1.32 1.42Jutzhu 1.03 1.15 1.27

MAE Taipei 0.67 0.78 0.92Banqiao 0.76 0.76 0.79Keelung 0.82 0.98 1.07Hsinchu 0.77 0.81 0.92Anbu 0.59 1.07 1.25Jutzhu 0.69 0.83 0.98

CE Taipei 0.74 0.64 0.54Banqiao 0.64 0.66 0.63Keelung 0.77 0.69 0.65Hsinchu 0.74 0.70 0.65Anbu 0.51 −0.28 −0.48Jutzhu 0.47 0.34 0.20

Notes: BP4: BP (4 key factors); BP6: BP (6 factors).RMSE and MAE are quantitative statistics adopted to measurehow close the model simulations are to the observations.CE: Coefficient of efficiency is used to assess the predictivepower of hydrological models.

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Table 4 Performance of the BD and BP models in dailyand monthly scales in the testing phase.

RMSE MAE CE

Day Month Day Month Day Month

BD 0.97 11.72 0.72 8.62 0.72 0.91BP4 1.14 18.51 0.87 13.92 0.62 0.78BP6 1.24 21.30 0.99 17.19 0.55 0.72

The BD model is considered as a suitable model forestimating at multiple stations simultaneously.

Table 4 illustrates the test performance of theBD, BP4 and BP6 models in daily and monthlyscales. Because most of the daily evaporation obser-vations are less than 10 mm d-1, daily performancecannot be shown clearly. When evaporation is esti-mated on monthly scales, the estimation accuracyof the BD model is much better than that of theBP4 and BP6 models. Compared with the BP4 andBP6 models, the BD model has improvement rates of15% and 22%, respectively, in terms of daily RMSE.In general, the BD model outperforms the BP4 andBP6 models.

Figure 4 illustrates the test performance of theBD and BP4 models in monthly scales based on theobservation values of daily pan evaporation. Resultsshow that the BD model performs better than the BPmodel in most cases because the BD model fits theobservation lines very well and has slight errors atthe extreme values for all stations. It indicates that theBD model gives good estimations on the evaporationtrend and extreme values. Even though the BP modelcan catch the evaporation trends, its performance isnot as good as that of the BD model at the extremevalues. From Fig. 4 it can be seen that less evapora-tion occurs at stations Jutzhu and Anbu (mountainousareas). The results of BP models are inconsistent atthose two stations, and the BD model fits the obser-vation lines better than the other models. In summary,the proposed BD model can be a reliable estimationtool for multiple stations.

DISCUSSION

The above results demonstrate that the novel esti-mation model (BD) is a reliable tool for estimat-ing pan evaporation at multiple stations simultane-ously. The DFA successfully extracts key factorsand provides the common trend of evaporation thatillustrates the evaporation characteristics in the studyarea. Evaporation at six meteorological stations is

affected by both key factors and unknown factorsimplied in the common trend. In fact, the mechanismthat drives evaporation is very complicated. The driv-ing key factors may be different from area to area,and factors that produce a disturbance may not bethe key factors in that area. In this study area (north-west Taiwan), both sunshine hours and humidity arenot important factors when estimating evaporation.However, the estimation accuracy of the BP6 model isworse than that of the BD model and BP4, because theBP6 model still uses the factors “sunshine hours” and“humidity”. Table 2 reveals that Keelung and Hsinchustations have the highest standard deviation values forsunshine hours. From the sensitivity analysis (Beven1979), the estimation given by the empirical formulais the most sensitive to RN, which is related to sun-shine hours and radiation. However, the BD modelexcludes the factor “sunshine hours”, which indicatesthat the latter is not a key factor in this study area. TheDFA successfully extracts the key factors and analy-ses the phenomenon of evaporation. The mechanismof evaporation is nonlinear and can be handled by theBPNN with key factors and DFA estimates as inputs.The proposed BD model is expected to be moresuitable in urban areas than in mountainous areasbecause four out of the six meteorological stationsare located in urban areas. The study results indi-cate the BD model can be suitably applied to a regioncovering both urban and mountainous areas. In sum-mary, the main contribution of the proposed BDmodel is:

(a) the DFA can significantly identify key factors forevaporation in complex regions, such as north-western Taiwan, and effectively improves theestimation accuracy of the BPNN; and

(b) it is a reliable estimation model for multiplestations.

Evapotranspiration data are crucial to watermanagement, agricultural engineering and vegeta-tion science (Xu and Singh 2005, Aytek et al.2008, Trajkovic 2010, Yang et al. 2011). Referenceevapotranspiration can be measured directly bylysimeters or estimated indirectly from meteorolog-ical data or pan evaporation. However, direct mea-surements are rarely available. In contrast, pan evap-oration data are easily accessible. Therefore manystudies have tried to recruit pan evaporation datato estimate evapotranspiration (Grismer et al. 2002,Snyder et al. 2005, Abdel-Wahed et al. 2008, Xinget al. 2008, Trajkovic 2009, Liang et al. 2011).Raghuwanshi and Wallender (1998) also noted that

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Fig. 4 Estimation of monthly pan evaporation at six meteorological stations in the testing phase.

more than 50 methods were proposed to estimate ref-erence evapotranspiration based on different assump-tions, among which pan evaporation received themost attention when comparing various methods ofevapotranspiration estimation. This indicates a clearneed for pan evaporation estimation. This studyintends to provide a hybrid mechanism (machinelearning technique coupled with factor analysis) toaccurately estimate pan evaporation based on selectedmeteorological variables at multiple gauging stations.

This approach can be extensively applied to othermeteorological estimation cases of interest.

CONCLUSIONS

Evaporation is an important factor that affects waterresources. Due to the complex topological terrainsand land uses, evaporation in different areas showsdifferent features; consequently, evaporation estima-tion requires models to be nonlinear and flexible in

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DFA and ANN for estimating pan evaporation at multiple sites 11

determining the coefficients of parameters. In thisstudy, the BD model incorporates a large numberof data obtained from six meteorological stations innorthern Taiwan into an estimation process and pro-vides the estimation of pan evaporation for multiplestations. For identifying the performance of the BDmodel, the traditional BPNN (BP) is used for com-parison purposes. The results show that ANNs (BDand BPNN) can not only extract area-dependent char-acteristics to construct appropriate estimation mod-els, but they also have good generalization ability.However, BPNNs may not always estimate evapora-tion well due to the lack of effective identification ofimportant input factors. In this situation, the incor-poration of the DFA into the BD model constructionprocess can provide key factors and a common trendso that the constructed BD model can estimate evap-oration more accurately than the BPNN model.

In summary, the BD model is a reliable esti-mation model for multiple stations; the estimationaccuracy of the BD model can be improved by incor-porating the DFA, which can extract the key factors,obtain the common trend, and provide crucial infor-mation; the BD model successfully inherits the advan-tages from the DFA and BPNN which effectivelyenhances its generalization ability and estimationaccuracy; and the BD model is suitable and effectivefor solving high-dimensional and/or nonlinear prob-lems for multiple sites. This novel BD model can beapplied not only to evaporation estimation, but alsoto cases whose phenomena are complicated due tounknown factors.

Acknowledgements This study was partially sup-ported by the National Science Council (NSC),Taiwan, ROC (Grant no. NSC 97-2313-B-002-013-MY3). The authors are indebted to the Editor,Associate Editor and reviewers for their valuablecomments and suggestions.

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