Author's Accepted Manuscript

Linear genetic programming application forsuccessive-station monthly streamflow pre-diction

Ali Danandeh Mehr, Ercan Kahya, CAHİTYerdelen

PII: S0098-3004(14)00101-0DOI: http://dx.doi.org/10.1016/j.cageo.2014.04.015Reference: CAGEO3373

To appear in: Computers & Geosciences

Received date: 9 November 2013Revised date: 22 March 2014Accepted date: 29 April 2014

Cite this article as: Ali Danandeh Mehr, Ercan Kahya, CAHİT Yerdelen, Lineargenetic programming application for successive-station monthly streamflowprediction, Computers & Geosciences, http://dx.doi.org/10.1016/j.cageo.2014.04.015

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1

Linear Genetic Programming Application for Successive-Station Monthly 1

Streamflow Prediction 2

Short Title: Streamflow prediction using LGP 3

4

ALI DANANDEH MEHRa, ERCAN KAHYAb, CAHİT YERDELENc 5

a Corresponding author. Istanbul Technical University, Civil Engineering Department, Hydraulics 6

Division, 34469, Maslak, Istanbul, Turkey, Phone: +90 553 417 8028 Fax: +90 212 285 65 87 Email: 7

danandeh@itu.edu.tr 8

b Istanbul Technical University, Civil Engineering Department, Hydraulics Division, Istanbul, Turkey. 9

E-mail: kahyae@itu.edu.tr 10

c Ege University, Civil Engineering Department, Hydraulics Division, Izmir, Turkey. E-mail: 11

cahit.yerdelen@ege.edu.tr 12

13

Abstract 14

In recent decades, artificial intelligence (AI) techniques have been pronounced as a branch of 15

computer science to model wide range of hydrological phenomena. A number of researches 16

have been still comparing these techniques in order to find more effective approaches in 17

terms of accuracy and applicability. In this study, we examined the ability of linear genetic 18

programming (LGP) technique to model successive-station monthly streamflow process, as 19

an applied alternative for streamflow prediction. A comparative efficiency study between 20

LGP and three different artificial neural network algorithms, namely feed forward back 21

propagation (FFBP), generalized regression neural networks (GRNN), and radial basis 22

function (RBF), has also been presented in this study. For this aim, firstly, we put forward six 23

different successive-station monthly streamflow prediction scenarios subjected to training by 24

2

LGP and FFBP using the field data recorded at two gauging stations on Çoruh River, Turkey. 25

Based on Nash-Sutcliffe and root mean square error measures, we then compared the 26

efficiency of these techniques and selected the best prediction scenario. Eventually, GRNN 27

and RBF algorithms were utilized to restructure the selected scenario and to compare with 28

corresponding FFBP and LGP. Our results indicated the promising role of LGP for 29

successive-station monthly streamflow prediction providing more accurate results than those 30

of all the ANN algorithms. We found an explicit LGP-based expression evolved by only the 31

basic arithmetic functions as the best prediction model for the river, which uses the records of 32

the both target and upstream stations. 33

34

Keywords: Artificial neural networks; linear genetic programming; streamflow prediction; 35

successive stations. 36

37

Introduction 38

Artificial neural networks (ANNs) are from popular artificial intelligence (AI) techniques 39

broadly used in various fields of geoscience. They are capable of using field data directly and 40

modelling the corresponding phenomena without prior knowledge of it. Successful results of 41

ANN application in geoscience particularly in hydrological predictions have been extensively 42

published in recent years (e.g. Minns and Hall 1996; Nourani et al. 2008, Besaw et al. 2010; 43

Piotrowski et al. 2014). Our review concerning the application of different ANN structures in 44

streamflow forecasting indicated that it has been received tremendous attention of research 45

(e.g. Dawson and Wilby 1998; Dolling and Varas 2002; Cannas et al. 2006; Kerh and lee 46

2006; Kisi and Cigizoglu 2007; Adamowski 2008; Kişi 2009; Shiri and Kisi 2010; Marti et 47

3

al. 2010; Nourani et al. 2011; Abrahart et al. 2012; Can et al. 2012; Krishna 2013; Kalteh 48

2013; Danandeh Mehr et al. 2013). 49

50

Minns and Hall (1996) introduced ANNs as rainfall-runoff models and demonstrated that 51

they are capable of identifying usable relationships between discharges and antecedent 52

rainfalls. Kerh and Lee (2006) applied an ANN-based model using information at stations 53

upstream of Kaoping River to forecast flood discharge at the downstream station which lacks 54

measurements. They found that the back-propagation ANN model performs relatively better 55

than conventional Muskingum method. Besaw et al. (2010) developed two different ANN 56

models using the time-lagged records of precipitation and temperature in order to forecast 57

streamflow in an ungauged basin in the US. The authors explained that ANNs forecasts daily 58

streamflow in the nearby ungauged basins as accurate as in the basin on which they were 59

trained. Can et al. (2012) used streamflow records of nine gauging stations located in Çoruh 60

River basin to model daily streamflow in Turkey. They compared the performance of their 61

ANN-based models with those of auto regressive moving average (ARMA) models and 62

demonstrated that the ANNs resulted in higher performance than ARMA. A comprehensive 63

review concerning the application of different ANN structures in river flow prediction has 64

been presented by Abrahart et al. (2012). 65

66

In spite of providing satisfactory estimation accuracy, all aforementioned ANN-based models 67

are implicit and often are criticized as ‘ultimate black boxes’ that are difficult to interpret 68

(Babovic 2005). Depending on the number of applied hidden layers, they may produce huge 69

matrix of weights and biases. Consequently, the necessity of additional studying in order to 70

develop not only explicit but also precise models still requires serious attention. 71

72

4

Genetic programming (GP) is a heuristic evolutionary computing technique (Koza 1992; 73

Babovic 2005) that has been pronounced as an explicit predictive modelling tool for 74

hydrological studies (Babovic and Abbott 1997a; Babovic and Keijzer 2002). The capability 75

of GP to model hydro-meteorological phenomena as well as its degree of accuracy are of the 76

controversial topics in recent hydroinformatic studies (e.g. Ghorbani et al. 2010; Kisi and 77

Shiri 2012; Yilmaz and Muttil 2013; Wang et al. 2014). 78

79

After Babovic and Abbott (1997b), who pronounced GP as an advanced operational tool to 80

solve wide range of hydrological modelling problems, GP and it’s variants/advancements 81

were considered broadly in different hydrological processes such as rainfall-runoff (Babovic 82

and Keijzer 2002; Khu et al. 2001; Liong et al. 2001; Whigham and Crapper 2001; Nourani et 83

al. 2012), sediment transport (Babovic 2000; Aytek and Kisi 2008; Kisi and Shiri 2012), sea 84

level fluctuation (Ghorbani et al. 2010), precipitation (Kisi and Shiri 2011), evaporation (Kisi 85

and Guven 2010), and others. 86

87

GP and its variants have also been received remarkable attention in the most recent 88

comparative studies among different AI techniques (e.g. Ghorbani et al. 2010; Kisi and 89

Guven 2010; Kisi and Shiri 2012). In the field of streamflow forecasting, Guven (2009) 90

applied linear genetic programming (LGP), an advancement of GP, and two versions of 91

neural networks for daily flow prediction in Schuylkill River, USA. The author demonstrated 92

that the performance of LGP was moderately better than that of ANNs. Wang et al. (2009) 93

developed and compared several AI techniques comprising ANN, neural-based fuzzy 94

inference system (ANFIS), GP, and support vector machine (SVM) for monthly streamflow 95

forecasting using long-term observations. Their results indicated that the best performance in 96

terms of different evaluation criteria can be obtained by ANFIS, GP and SVM. Londhe and 97

5

Charhate (2010) used ANN, GP, and model trees (MT) to forecast river flow one-day in 98

advance at two gauging stations in India’s Narmada Catchment. The authors concluded that 99

ANN and MT techniques perform almost equally well, but GP performs better than its 100

counterparts. Ni et al. (2010) applied GP to model the impact of climate change on annual 101

streamflow of the West Malian River, China. They compared the results of GP with those of 102

ANN and multiple linear regression models and indicated that GP provides higher accuracy 103

than the others. Yilmaz and Muttil (2013) used GP to predict river flows in different parts of 104

the Euphrates River basin, Turkey. They compared the results of GP with those of ANN and 105

ANFIS and demonstrated that GP are superior to ANN in the middle zone of the basin. 106

Among the most recent comparative studies between different AI techniques, Wang et al. 107

(2014) proposed the singular spectrum analysis (SSA) in order to modify SVM, GP, and 108

seasonal autoregressive (SAR) models. They applied the modified models to predict monthly 109

inflow for three Gorges Reservoirs and indicated that modified GP is slightly superior to 110

modified SVM at peak discharges prediction. Although there are some other comparative 111

studies between GP and different AI techniques, to the best of our knowledge, there is no 112

research examining the performance of LGP for successive-station monthly streamflow 113

prediction in comparison with different ANN structures/algorithms. 114

115

The main goals and motivation of our study are (i) to further enhance the available LGP 116

modelling tool to provide an explicit expression for successive-stations streamflow prediction 117

and (ii) for the first time, to compare the efficiency of LGP with three different ANN 118

algorithms for monthly streamflow prediction. In this way, at the first stage, we put forward 119

six different successive-station prediction scenarios structured by commonly used feed-120

forward back propagation neural network algorithm (FFBP). Then, using LGP technique a 121

new set of explicit expressions has been generated for these scenarios. We performed a 122

6

comparative performance analysis between the proposed LGP and FFBP models using Nash-123

Sutcliffe efficiency and root mean square error measures. As a consequence of the first stage 124

of the study, the best scenario was identified and discussed. In the second stage, two other 125

ANN algorithms, namely generalized regression neural networks (GRNN), and radial basis 126

function (RBF) neural networks were utilized to restructure the best prediction scenario. 127

Ultimately, we put forward a discussion about both accuracy and applicability of different 128

ANN and LGP models. 129

130

It is observed that some of gauging stations are closed down in all over the world where the 131

stations are no longer required or funding to support continued operation is limited. Using 132

successive-station prediction strategy, in case of developing a plausible model between a pair 133

of upstream-downstream stations, the model can be used as a substitute for the station(s) 134

which is at risk for discontinuation. In addition, since inputs of the successive-station 135

prediction models are only time-lagged streamflow observations, such models are also 136

considered more useful for the catchments with sparse rain gauge stations (Besaw et al., 137

2010). The successive-station strategy also tends to decrease the lagged prediction effect of 138

commonly proposed single-station runoff-runoff modes which has been mentioned by some 139

researchers (Chang et al. 2007; De Vos and Rientjes 2005; Muttil and Chau 2006; Wu et al 140

2009a). 141

142

Overview of FFBP, GRNN, and RBF networks 143

144 ANNs are from black-box regression methods which are commonly used to find out the 145

nonlinear systems attitude. FFBP networks are probably the most popular ANNs in 146

hydrological problems (Tahershamsi et al. 2012; Krishna 2013) which considered as general 147

7

nonlinear approximations (Hornik et al. 1989). The primary goal of this algorithm is to 148

minimize the estimation error by searching for a set of connection weights, synaptic weights, 149

which cause the network to produce outputs closer to the targets. They are typically 150

composed of three parts: a) input layer including a number of input nodes, b) one or more 151

hidden layers and c) a number of output layer nodes. The number of hidden layers and 152

relevant nodes are two of the design parameters of FFBP networks. A neural network with 153

too many nodes may overfit the data, causing poor generalization on data not used for 154

training, while too few hidden units may underfit the model (Fletcher et al. 1998).The input 155

nodes do not perform any transformation upon the input data sets. They only send their initial 156

weighted values to hidden layer nodes. The hidden layer nodes typically receive the weighted 157

inputs from the input layer or a previous hidden layer, perform their transformations on it, 158

and pass the output to the next adjacent layer which is generally another hidden layer or an 159

output layer. The output layer consists of nodes that receive the hidden layer outputs and send 160

it to the modeller. Initial synapses are progressively corrected during the training process that 161

compares predicted outputs with corresponding observations and back-propagates any errors 162

to minimize them. The design issues, training mechanisms and application of FFBP in 163

hydrological studies have been the subject of different studies (e.g. Abrahart et al. 2012; 164

Nourani et al. 2013). Therefore, to avoid duplication, we only introduced the main concepts 165

of the FFBP here. FFBP modelling attributes used in the proposed models will be stated in 166

the following. 167

168

RBF is a variant of ANN that uses radial basis functions as activation functions. with the 169

exception of a few researches (e.g. Azmathullah et al. 2005; Bateni et al. 2007; Kisi 2008; 170

Tahershamsi et al. 2012), RBF networks have not been used broadly in hydrological studies. 171

A typical RBF network has a feed forward structure consists of an input layer, a hidden layer 172

8

with a radial basis activation function, and a linear output layer. The hidden layer node 173

calculates the Euclidean distance between the centre of function and the network input layer 174

and then passes the result to the radial basis function. Thus the hidden layer performs a fixed 175

nonlinear transformation which maps the input space onto a new space. The output layer 176

implements a weighted sum of hidden layer outputs. In order to use RBF, it is required to 177

specify the number of layers, a radial activation function, and a criterion for network training. 178

More information on the application of RBF networks on streamflow prediction has been 179

provided by Kisi (2008). 180

181

GRNN is a kind of radial basis networks that only uses training data with BP algorithm to 182

derive estimation function. It typically comprises four layers including input, pattern, 183

summation, and output layers. The number of input units in the first layer is equal to the total 184

number of parameters. The first layer is fully connected to the pattern layer, where each unit 185

represents a training pattern and its output is a measure of the distance of the input from the 186

stored patterns. Each pattern layer is connected to the two neurons (i.e. S-summation and D-187

summation) in the summation layer. The S-summation neuron computes the sum of the 188

weighted outputs of the pattern layer whereas the D-summation neuron calculates the un-189

weighted outputs of the pattern neurons (Cigizoglu and Alp 2004). Successful application of 190

GRNN networks for daily flow forecasting in intermittent rivers has been reported by 191

Cigizoglu (2005). More details on the basics of the GRNN algorithm can be obtained in the 192

literature (Specht 1991). 193

194

195

196

197

9

198

199

Overview of GP and LGP 200

201

GP is a so-called symbolic regression technique (Babovic 2005) that automatically solves 202

problems without pre-specified form or structure of the solution in advance (Koza 1992). In 203

brief, GP is a systematic, domain-independent method for getting computers to solve 204

problems automatically starting from a high-level statement of what needs to be done (Poli et 205

al. 2008). Unlike ANN, GP is self-parameterizing that builds model’s structure without any 206

user tuning. 207

208

Individual solutions in GP are computer programs represented as parse trees (Fig. 1). The 209

population of initial generation is typically generated through a random process. However, 210

subsequent generations are evolved through genetic operators of selection reproduction, 211

crossover and mutation (Babovic and Keijzer 2002). The major inputs for a GP model are (1) 212

patterns for learning, (2) fitness function (e.g. minimizing the squared error), (3) functional 213

and terminal sets, and (4) parameters for the genetic operators like the crossover and mutation 214

probabilities (Sreekanth and Datta 2011). As shown in Fig. 1, in GP modelling, the functions 215

and terminals are chosen randomly from the user defined sets to form a computer model in a 216

tree-like structure with a root node and branches extending from each function and ending in 217

a leaf or a terminal. In many cases in GP leaves are the inputs to the program. More details on 218

GP can be obtained from Koza (1992) and Babovic and Keijzer (2000). 219

10

220

Fig. 1. Tree representing of function ((x .y - x/y)) 221

222

Besides the tree-based GP, which is also referred to as the traditional Koza-style GP, there 223

are new variants of GP such as linear, graph-based, probabilistic and multi objective GP (Poli 224

et al. 2008). LGP is a subset of GP that has emerged, recently (Brameier and Banzhaf 2007). 225

Comparing LGP to traditional GP, there are some main differences. The tree-based programs 226

used in GP correspond to expressions from a functional programming language. Functions 227

are located at root and inner nodes while the leaves hold input values or constants. In 228

contrast, LGP denotes a GP variant that evolves sequences of instructions from an imperative 229

programming language or machine language (Brameier and Banzhaf 2007). The term “linear” 230

refers to the imperative program representation. It does not mean that the method provides 231

linear solutions. An example of an LGP evolved program, which is the C code (with removed 232

introns) of a model developed in this study, is illustrated as follows: 233

234

L0: f [0] += Input001; 235 L1: f [0] += -1.0632883443450928f; 236 L2: f [0] *= f [0]; 237 L3: f [0] /= 1.216173887252808f; 238 L4: f [0] += -0.2360081672668457f; 239 L5: f [0] *= f [0]; 240 L6: f [0] *= Input000; 241 L7: f [0] += Input000; 242 243

where f[0] represent the temporary computation variable created in the program by LGP. 244

LGP uses such temporary computation variables to store values while performing 245

11

calculations. The variable f[0] is initialized to the value of “Input001” in this program and the 246

output is the value remaining in f[0] in the last line of the code. 247

248

Similar to pseudo-algorithm of any GP variants, LGP generally solves any problem through 249

the six steps: (i) generation of an initial population (machine-code functions) randomly by the 250

user defined functions and terminals; (ii) Selection of two functions from the population 251

randomly, Comparison of the outputs and designation of the function that is more fit as 252

winner_1 and less fit as loser_1; (iii) Selection of two other functions from the population 253

randomly and designation of the winner_2 and loser_2; (iv) Application of transformation 254

operators to winner_1 and winner_2 to create two similar, but different evolved programs 255

(i.e. offspring) as modified winners (v) replace The loser_1 and loser_2 in the population 256

with modified winners and (vi) Repetition of steps (i)-(v) until the predefined run termination 257

criterion. 258

259

Akin to GP, the user specified functions in LGP can consist of the arithmetic operations (+, - , 260

×, ÷), Boolean logic functions, conditionals, and any other mathematical functions such Sin, 261

Ln, EXP, and others. The choice of the functional set determines the complexity of the 262

evolved program. For example, a functional set with only addition and subtraction results in a 263

linear model whereas a functional set which includes exponential functions will result in a 264

highly nonlinear model. The terminal set contains the arguments for the functions and can 265

consist of numerical constants, logical constants, variables, etc. More details on the 266

application of LGP in predictive modelling can be obtained from Poli et al. (2008). The LGP 267

modelling attributes and its relevant parameters used in this study will be explained in the 268

following. 269

270

12

Data preparation and efficiency criteria 271

272

As shown in Fig. 2, the data used in this study were selected from two gauging stations, 273

namely station 2322 and station 2315, on Çoruh River located in eastern Black Sea region, 274

Turkey. The river springs from Mescit Mountains in Bayburt and reaches the Black Sea in 275

Batum City of Georgia after a course of 431 kms. Yearly mean flow of the river before 276

leaving the Turkey’s border is about 200m3/s. The applied data is composed of 348 277

observations of the mean monthly streamflow at each station. The statistical characteristics of 278

applied streamflow time series at 29-year period (1972-2000) are presented in Table1. 279

Because there are only 348-month observations in our data source, only training and 280

validation data sets was applied in the present study. The training data was used for model 281

fitting and the validation data was used for testing as well as evaluating different models. 282

Considering previous similar studies (e.g. Nourani et al. 2011; Kalteh 2013), the training and 283

validation data percentage were assumed as 75% and 25%, respectively (Fig.3). Therefore, 284

the entire data set were divided into two subsets. The statistical parameters of each set are 285

tabulated in Table 2. Details on data splitting methods can be found at (May et al. 2010; Wu 286

et al. 2012, 2013). 287

288

289

13

290

Fig. 2. Location of Study area (Çoruh River Basin) 291

292 Table1. The monthly statistical parameters of observed streamflow data 293

Statistical parameter

Station Upstream (2322) Downstream (2315)

raw normalized Raw normalized Number of data (X) 348 348 348 348 Xmax (m3/s) 867.4 0.932 1018 0.940 Xmin (m3/s) 33.5 0.132 48.9 0.140 Xmean (m3/s) 158.6 0.252 207.1 0.271 Standard Deviation (m3/s) 159.1 0.152 184.4 0.152

Coefficient of Skewness 1.727 1.727 1.700 1.700 294

Table2. Statistical parameters of subsets 295 Statistical parameter Entire

data Training

subset Validation

subset Number of data (X) 348 261 96 Xmax (m3/s) 1018 930 1018 Xmin (m3/s) 33.5 36 48 Xmean (m3/s) 158.6 158 223 Standard Deviation (m3/s) 159.1 156 204 Coefficient of Skewness 1.7271 1.71 1.71

296

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297

Fig. 3. Mean monthly streamflow observations at Çoruh River 298

299

Normalization is a standard operating procedure in any data-driven modelling technique. In 300

this study, before training of the ANN and LGP models, the normalization was applied for 301

the data, which made them dimensionless and confined within a certain range. After training 302

process, the model that yields the best results in terms of Nash-Sutcliffe efficiency (NSE) and 303

root mean squared error (RMSE) at validation period was selected as the most efficient 304

model. NSE is a normalized statistic that indicates how well the plot of observed versus 305

predicted data fits the 1:1 line (Eq. (1), Nash and Sutcliffe 1970). RMSE, Eq. (2), measures 306

the root average of the squares of the errors. The error is the amount by which the value 307

implied by the estimator differs from the target or quantity to be estimated. 308

nobs pre 2i i

i 1n

obs obs 2i mean

i 1

( X X )NSE 1

( X X )

=

=

−= −

−

∑∑

(1) 309

nobs pre 2i i

i 1

( X X )RMSE

n=

−=∑

(2) 310

15

where obsiX = observed value of X, pre

iX = predicted value obsmeanX = mean value of observed 311

data and n = number of observed data. Obviously, a high value for NSE (up to one) and a 312

small value for RMSE indicate high efficiency of the corresponding model. A number of 313

studies have indicated that a hydrological model can be sufficiently assessed by these two 314

statistics (Nourani et al. 2012). 315

316

Proposed models 317

318 In this section, successive-stations scenarios proposed for monthly streamflow prediction at 319

downstream station are explained. Two successive gauging stations on Çoruh River with 320

approximately 60km distance have been selected and six different scenarios (models (1) to 321

(6)) have been considered to train by AI techniques. In other words, the streamflow value at 322

downstream is assumed to be a function of finite sets of concurrent or antecedent streamflow 323

observations. The assumed scenarios can be expressed as follows: 324

325

Model (1) Dt = f (Dt-1) 326

Model (2) Dt = f (Ut) 327

Model (3) Dt = f (Ut, Dt-1) 328

Model (4) Dt = f (Ut-1, Ut, Dt-1) 329

Model (5) Dt = f (Ut-1, Dt-1) 330

Model (6) Dt = f (Ut-1, Ut, Dt-2, Dt-1) 331

where Dt and Ut represent downstream and upstream monthly streamflow, respectively and 332

indices t-1 and t-2 symbolize the one-month lag and two-months lag, respectively. 333

16

334

Application of FFBP network 335

336 At the first stage of the study, we used the commonly proposed three-layer feed forward back 337

propagation (FFBP) network to construct the FFBP-based ANN models for all of the 338

aforementioned scenarios. It has been proved that FFBP networks are satisfied for any 339

forecasting problem in hydrology (ASCE Task committee 2000; Nourani et al. 2008). In the 340

FFBP, any input node was multiplied by a proper weight initially then was shifted by a 341

constant value (i.e. bias), and finally was transformed using a predefined transfer function. 342

Sigmoid transfer function in the hidden layer and linear transfer function in the output layer 343

were adopted in our FFBP. The expression for an output value of a three-layered FFBP 344

networks is given in Eq. (3) (Nourani et al. 2011). 345

N NM N

C O kj h ji i jo koj 1 i 1

y f W .f W x W W= =

⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∑ ∑

(3) 346

where Wji is a weight in the hidden layer connecting the ith neuron in the input layer and the 347

jth neuron in the hidden layer, Wjo is the bias for the jth hidden neuron, fh is the activation 348

function of the hidden neuron, Wkj is a weight in the output layer connecting the jth neuron in 349

the hidden layer and the kth neuron in the output layer, Wko is the bias for the kth output 350

neuron, fo is the activation function for the output neuron, xi is ith input variable for input 351

layer and yc , yo are computed and observed output variables, respectively. NN and MN are the 352

number of neurons in the input and hidden layers, respectively. The weights are different in 353

the hidden and output layers, and their values can be changed during the process of the 354

training. 355

356

17

After definition of the three-layer structure, the training process was carried out using the 357

Levenberg–Marquardt (LM) algorithm written for MATLAB® software. The successful 358

implementation of LM algorithm to train ANN-based hourly, daily, and monthly streamflow 359

prediction models is frequently reported (e.g. Chau et al. 2005; Cannas et al. 2006; Kisi 2010; 360

Danandeh mehr et al. 2013). As it mentioned previously, a critical issue in ANN modelling is 361

to avoid likely under or overfitting problems. In the former situation, the network may not be 362

possible to fully detect all attributes of a complex data set, while overfitting may reduce the 363

model ability for generalization properties. Application of cross-validation or selection of 364

appropriate number of neurons in hidden layer using the trial-and-error procedure with a 365

confined training iteration number (epoch), were commonly suggested to prevent these 366

problems (Principe et al. 2000; Cannas et al. 2006; Nourani et al. 2008; Kisi 2008; Wu et al. 367

2009b; Elshorbagy et al. 2010; Krishna 2013). The trial-and-error method is implemented in 368

this study in order to avoid overfitting problem during the training process All of the 369

proposed models in this study, possess fixed input-nodes (varying from 1 to 4 for models (1) 370

to (6)), a single hidden layer with nodes vary from 1 to 10 and just 1 node in the output layer. 371

No significant improvement in model performance was found when the number of hidden 372

neurons was increased from the limit, which is similar to the experiences of other researchers 373

(Cheng et al. 2005; Cannas et al. 2006; Wu et al. 2009b). 374

375

Application of LGP 376

377

We applied Discipulus®, the LGP soft-ware package developed by Francone (1998) in order 378

to create the best programs for already proposed scenarios (i.e. models (1) to (6)). To 379

generate the best programs, several decisions are required to be made by modeller. The first 380

decision consists of choosing the sets of terminals and functions utilized for creation of the 381

18

initial population. The terminal set obviously is program’s external inputs consist of the 382

independent variables of each model. A series of random constants between -1 and 1 is also 383

assumed to be in our terminal set as part of potential solution. The choice of the appropriate 384

function set is not so clear; however, an appropriate guess will be helpful in order to include 385

all the necessary functions. In this study, we considered only four basic arithmetic functions 386

including addition, subtraction, multiplication and division in our function set. Other kind of 387

mathematical functions were ignored to avoid increasing complexity of the final solution. In 388

other words, the power of LGP is assessed for the nonlinear streamflow process modelling by 389

only basic arithmetic functions and confined program size. 390

391

Definition of the terminal and function sets by modeller, indirectly defines the search space 392

for LGP to generate initial population. In the next step, the generated programs (initial 393

population) must be ranked based on the fitness value and then new programs should be 394

created by using both crossover and mutation operators (Poli et al., 2008). Eventually, the 395

best program is selected from the new generations. We used RMSE measure of validation 396

period to select the best program for each scenario. Other adopted parameters for LGP setting 397

was tabulated in Table 3. 398

399

Table 3. Parameter settings for the LGP training 400

Parameter Value Initial Population (programs) 100 Mutation Rate % 5 Crossover Rate % 50 Initial Program Size 64 (Byte) Maximum Program Size 256 (Byte) Maximum Numbers of Runs 300 Generation Without Improvement 300 Maximum Generation Since Start 1000

401

19

In addition to parameter selection, one of the main concerns of LGP modelling as well as 402

ANN is overfitting problem. This is more likely to happen when a small data sets or a large 403

number of generations is used for LGP runs. In order to prevent overfitting problem in this 404

study, we firstly confined both maximum number of generations and maximum size of 405

programs to 1000 generation, and 256 byte, respectively (see Table 3). Then, as suggested by 406

Nourani et al. (2013), we monitored simultaneously error variations in both training and 407

validation sets in each LGP run in order to stop the run at the time that error of validation set 408

begins to rise. Stopping the run after a certain number of generations or certain length of 409

wall-clock time has also been previously suggested by Babovic and Keijzer (2000). 410

411

Application of GRNN and RBF networks 412

413

At the second stage of the present study, in order to generalize the results of comparative 414

study between LGB and ANN-based models for successive-station monthly streamflow 415

prediction, we considered GRNN and RBF algorithms to restructure the best scenario 416

designated in the first stage. For this aim, two different program codes were written in the 417

MATLAB® to generate the GRNN and RBF networks. In GRNN modelling, the training and 418

validation data sets were selected identical to those of FFBP. However, in RBF only training 419

part of data was used to structure the new ANN models. Determination of optimum value of 420

spread parameter in both GRNN and RBF was one of the important aspects of an efficient 421

network design in this phase. With respect to the RMSE and NSE measures of validation 422

step, we employed a trial-error method to select the optimum value of spread parameter as 423

suggested by Cigizoglu (2005). In order to optimize RBF model, different number of nodes in 424

the hidden layer (RBF centres) were also examined in each trial. For this aim, the utilized 425

training program adds neurons to the hidden layer of a RBF network until it meets the 426

20

specified mean squared error or maximum number of neurons. In order to make a fare 427

comparison with FFBP, the maximum number of hidden layer neurons within RBF networks 428

was also confined to 10 that already had been accepted as a threshold for FFBP networks. 429

430

Results and discussion 431

432

According to Models (1) to (6), six different streamflow prediction scenarios have been 433

modelled by both FFBP and LGP techniques and have been compared with each other using 434

RMSE and NSE values. The efficiency results of the best developed FFBP and LGP models 435

at validation period were tabulated in Table 4. The results indicated that Model (1) resulted in 436

the lowest performance level with respect to all of the scenarios. It might because of 437

insufficient inputs accepted in this scenario. Although LGP yielded the slightly higher 438

performance in terms of both RMSE and NSE measures, it is obvious that the single-step-439

ahead monthly streamflow prediction scenario could not provide a reliable prediction 440

(NSE=0.457) for Çoruh River. 441

442

Table 4. Performance comparison of LGP and FFBP results at validation period 443

FFBP LGP Model Prediction scenario NHL* RMSE NSE RMSE NSE

1 Dt = f (Dt-1) 3 0.131 0.355 0.119 0.457 2 Dt = f (Ut) 3 0.130 0.363 0.099 0.624 3 Dt = f (Ut, Dt-1) 4 0.079 0.766 0.064 0.845 4 Dt = f (Ut-1, Ut, Dt-1) 4 0.037 0.948 0.029 0.967 5 Dt = f (Ut-1, Dt-1) 5 0.039 0.943 0.029 0.968 6 Dt = f (Ut-1, Ut, Dt-2, Dt-1) 6 0.046 0.921 0.029 0.969

* Nodes in hidden layer 444 445

Based upon abovementioned successive-station prediction strategy, in the second scenario 446

(i.e. Model (2)) concurrent upstream flow was considered as the modelling input instead of 447

21

increasing in the lag time of downstream station records. Considering the remarkable 448

differences between NSE values of LGP and FFBP in this scenario (i.e. more than 40% 449

improvement), LGP is superior to FFBP. Compared to the Model (1), although significant 450

improvement in the performance of LGP Model (2) was resulted in this scenario, the model 451

does not provide a suitable streamflow prediction scenario yet (NSE=0.624). The reason 452

behind is apparently related to the fact that the monthly flow in the study reach generally has 453

a spatially increasing regime, which is also distinguishable in Fig. 3. The results of the first 454

two models indicated the necessity of additional forecasting lead time and/or more efficient 455

input variables. 456

457 The first successive-station scenario, Model (3), demonstrated a dramatic improvement in the 458

performance levels of both FFBP and LGP models. The reason lies under the fact that 459

existing sub-basins between the stations have considerable physical effects (i.e., increasing 460

drainage area) on the occurrence of flow at downstream station. This is the reason why we 461

intentionally keep downstream flow at time (t-1), Dt-1, in constructing Models (4)–(6). 462

Efficiency values of Model (3) imply that LGP is still remains superior to FFBP. 463

464 Models (4) to (6) represent three effective successive-station combinations with high 465

performance levels of both LGP and FFBP. In all cases, LGP shows slightly higher 466

performance than FFBP. Due to striking coherence between the observed and predicted 467

values in these models, there is a very strict competition among them to be chosen as an 468

optimum streamflow prediction model for the river. Models (4) and (5) use both flows at time 469

(t-1) in downstream and upstream stations; however, the former also includes streamflow at 470

time (t) in upstream station (Ut). Removing one of the input variables (i.e. Ut) from Model (4) 471

interestingly created inconsiderable effect in the efficiency of corresponding FFBP and LGP 472

models. Compared to Models (5), having two more inputs (i.e. Ut and Dt-2) in Model (6) 473

22

causes the FFBP network to reduce the efficiency of the model. However, LGP still provides 474

progressive performance. Diminishing results of FFBP in this case may relate to the fact that 475

ANN-based models are unsatisfactory in case of highly non-stationary phenomena (Cannas et 476

al. 2006) or highly noisy data (Nourani et al. 2011). 477

478

Considering all the aforementioned results and the concept of simplicity and applicability as 479

the main issue of modelling, Model (5) has been evaluated as the best scenario in this study. 480

The dimensionless explicit expression of the model resulting by LGP was given in Eq. (4). It 481

shows that LGP generated a quadratic equation in Dt-1-squared, scaled and shifted by 482

upstream flow for the previous month (Ut-1). 483

221

11 236.0217.1

)063.1(⎥⎦

⎤⎢⎣

⎡−

−×+= −

−−t

tttDUUD (4) 484

Apart from highly nonlinear relation between downstream flow (Dt) and its values at previous 485

month (Dt-1), the model (i.e. Eq. (4)) exposes a linear relation between Dt and Ut-1. Regarding 486

the linear and nonlinear terms of the equation, its plausibility was investigated through the 487

dyadic scatterplots presented in Fig. 4 and piecewise linear surface plot of Dt vs. Ut-1, Dt-1 488

with the observations superposed on it illustrated in Fig. 5. These figures prove a linear 489

correlation between Dt and Ut-1 and highly nonlinear relation between Dt and Dt-1. Three 490

dimensional surface plot of the proposed model based on a mesh grid produced by observed 491

Ut-1 and Dt-1 was also presented in Fig. 6. Owing to the physical interrelation between Ut-1 and 492

Dt-1, any point on this surface does not necessarily provide a physically acceptable prediction 493

for Dt. It is why some gaps can be seen within the surface (see Fig. 6b). According to the 494

model, one-month lag is enough for successive-station monthly streamflow prediction at our 495

study reach. 496

497

23

498

Fig. 4. Scatterplots of observed data used in the present study 499

500

501

Fig. 5. Three dimensional Scatterplots of the observation data 502

503

504 Fig. 6. Three dimensional surface plot, a) front view and b) top view perspective of Eq. 4. 505

506 507

a) b)

24

In order to assess the efficiency of Model (5) in detail, LGP and FFBP predicted time series 508

and their scatter plots compared to the corresponding observations for the validation period as 509

illustrated in Fig. 7. The figure shows both FFBP and LGP models are able to predict the low 510

and medium monthly streamflow (Dt<500 m3/s) successfully. The LGP acts more accurate 511

than FFBP to predict the global maximum, global minimum, local maxima, and local 512

minima, which warrants its superiority to FFBP in overall sense. 513

514

515

Fig. 7. Predicted and scatter plots of the proposed (a) FFBP and (b) LGP prediction models 516

517

As it mentioned previously, in order to investigate the efficiency of LGP technique in 518

comparison with different ANN algorithms, the selected scenario (i.e. Model (5)) was also 519

subjected for restructuring using RBF and GRNN. The efficiency results of the best RBF and 520

GRNN networks developed for the scenario was compared with those of FFBP and LGP as 521

tabulated in Table 5. Corresponding predictions of the validation period comparing to the 522

relevant observations were also depicted in Fig. 8. The results indicated that both RBF and 523

25

GRNN networks are able to produce monthly streamflow of the target station more precise 524

than FFBP. The RBF network predicts global maximum value more reliable than other ANN 525

networks. The proposed LGP model is still superior to both RBF and GRNN not only in term 526

of extreme values prediction but also in low and medium flows prediction. 527

528

Table 5. Efficiency results of LGP and different ANN models at validation period 529

Model RMSE NSE

LGP 0.029 0.968 FFBP 0.039 0.943 RBF 0.034 0.955

GRNN 0.036 0.950

530

All of the ANN algorithms assessed in this study not only provides implicit networks with 531

accuracy less than LGP technique but also they suffer from issues of optimum parameter 532

selection. It was observed that after each FFBP trail, different prediction values are obtained 533

by unique FFBP structure resulting in dynamic performance level. This drawback is mainly 534

due to the random assignment of synapses in the beginning of each trial. The efficiency of 535

each RBF network developed in this study was highly depended on the number of hidden 536

nodes, RBF centres, and spread constant chosen for the network. The appropriate selection of 537

the latter was also crucial to develop the most efficient GRNN network. 538

26

539

Fig. 8. Predicted and scatter plots of the (a) RBF and (b) GRNN models at validation period 540

541

Some of the training trials revealed the fact that awkward selection of RBF centres and 542

spread constant may provide unsatisfactory results. Therefore, modellers should be very 543

careful when they choose these parameters. Otherwise, implementation of an optimization 544

technique in parallel with training algorithm is inevitable. In contrast, in LGP-based 545

modelling, selected parameters (see Table 3) do not play significant role in the efficiency of 546

the best program evolved by LGP. Heuristics-based evolutionary optimization feature of LGP 547

lets modellers choose the required parameters in a variety of range. Evidently, a wisdom 548

selection, specifically for function and terminal sets, based upon physics of the studying 549

phenomenon is always suggested. 550

551

552

553

554

555

27

556

557

Summary and conclusion 558

559

In this study, based upon spatial and temporal features of the historical streamflow records on 560

Çoruh River, we investigated the ability of LGP to model the nonlinear successive-station 561

streamflow process and compared its efficiency with that of calculated by three different 562

ANN algorithms namely FFBP, RBF, and GRNN. Considering the successive-station 563

prediction strategy, we assumed that streamflow at our target station is a function of historical 564

streamflow records at the station and another one in the upstream. Therefore, we put forward 565

six different scenarios between the stations and then evaluated them as a candidate for 566

monthly streamflow prediction model for the river. 567

568

Our results demonstrated that the LGP and ANN are both able to handle the nonlinearity and 569

non-stationary elements of the successive-station streamflow process in general. With respect 570

to all of the scenarios examined here, LGP approach resulted in higher performance than all 571

of the ANN algorithms, even though only basic arithmetic functions (+, - , ×, ÷) were adopted 572

in the LGP function set. The proposed LGP model was superior to ANNs not only in extreme 573

flow prediction but also in low and medium flows (Dt<500 m3/s). It is also obtained that an 574

explicit LGP-based model, comprising one-month-lag records of both target and upstream 575

stations, provides the best prediction model for the target station. The results also revealed a 576

diminishing performance in FFBP model when the number of input variables increases to 577

four, whereas it is not the case in LGP model. This may be related to disability of ANNs for 578

modelling highly noisy data. 579

580

28

Since the programs evolved using LGP technique can be represented by the explicit 581

mathematical equations (e.g. Eq. (4)), they are preferential to ANNs not only for practical use 582

but also for mining the knowledge from the information contained in the field data. 583

Evidently, the empirical successive-station equation proposed in this study providing strong 584

expressivity underlies streamflow process more than the body of observation data between 585

the stations. However, this type of empirical equations sometimes shows such a complexity 586

that cannot be easily interpreted and this issue can be considered as major disadvantage of 587

LGP model, indicating a necessity for further studies to overcome such problems. 588

589

Acknowledgements 590

 591

The authors would like to thank Dr. Vasily Demyanov, Associate Editor Computers 592

and Geosciences, for his helpful suggestions during the revision of the initial manuscript. We 593

also would like to thank two anonymous reviewers for their fruitful critiques improving this 594

paper. 595

596

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828

34

• We compared FFBP, GRNN, RBF neural networks and LGP for successive-station 829 monthly streamflow prediction. 830

• Both ANNs and LGP models are more reliable in low and medium flow prediction. 831 • LGP is more capable of capturing extreme values than ANNs. 832 • LGP is superior to ANN in terms of overall accuracy and practical applicability. 833 • In contrast with implicit ANNs, LGP provided explicit equation for streamflow 834

prediction. 835