Improving artificial intelligence models accuracy for monthly streamflow forecasting using grey Wolf optimization (GWO) algorithm

Journal Pre-proofs Research papers Improving Artificial Intelligence Models Accuracy for Monthly Streamflow Forecasting Using Grey Wolf Optimization (GWO) Algorithm Yazid Tikhamarine, Doudja Souag-Gamane, Ali Najah Ahmed, Ozgur Kisi, Ahmed El-Shafie PII: DOI: Reference:

S0022-1694(19)31170-9 https://doi.org/10.1016/j.jhydrol.2019.124435 HYDROL 124435

To appear in:

Journal of Hydrology

Received Date: Revised Date: Accepted Date:

17 October 2019 2 December 2019 3 December 2019

Please cite this article as: Tikhamarine, Y., Souag-Gamane, D., Najah Ahmed, A., Kisi, O., El-Shafie, A., Improving Artificial Intelligence Models Accuracy for Monthly Streamflow Forecasting Using Grey Wolf Optimization (GWO) Algorithm, Journal of Hydrology (2019), doi: https://doi.org/10.1016/j.jhydrol.2019.124435

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© 2019 Published by Elsevier B.V.

11. Introduction 2

Broad scale of applications in water resources management and planning require

3

an imperative development to obtain an optimum model for streamflow prediction.

4

It is essential to know that the precise estimation of streamflow is considered a

5

significant stochastic feature in environmental modeling and it has attracted a big

6

deal of importance in different kinds of applications in such as activities of water

7

operation, maintenance, management, agricultural and irrigation management,

8

drought and flood alert systems, etc. as reported in(Bruins, 1990; Amiri, 2015;

9

Vogel et al., 2015). From a practical point of view, the performance of streamflow

10

model necessitates a specific period of time to function, which can either be a long

11

time period (e.g., weekly, monthly and seasonal) or short time period (e.g., hourly

12

and daily), and this matter was the main focus of the researchers in water resources

13

field over the past twenty years(Zealand et al., 1999; Wu et al., 2009a, 2009b;

14

Ismail et al., 2012; Terzi and Ergin, 2014; Zia et al., 2015). Generally, it is difficult

15

to attain a precise and dependable streamflow model because the chaotic properties

16

of streamflow (El-Shafie et al., 2007), including non-linearity, stochasticity and

17

non-stationarity,

18

complexity(Bayazit, 2015). Moreover, several factors are also influencing the

19

stream flow for instance environment climate variability (Narsimlu et al. 2015),

20

local and seasonal patterns, heterogeneity in temperature at the local and regional

21

scale and the frequent of the precipitations, temporal and spatial variability of

22

watershed, the properties of catchments as well as the activities of human

23

beings(Danandeh Mehr et al., 2013; Dehghani et al., 2014; Maity and Kashid,

24

2011; Singh and Cui, 2015).

govern

the

behavior

of

streamflow

and

cause

its

25

Despite of all, water resources scholars are still making marvelous efforts to develop

26

a reliable and precise model for streamflow forecasting(Angelakis and Gikas, 2014),

27

those scholars thrive to develop and acquire original and innovative systems to beat

28

the complexities in their models for stream flow estimating.

291.1. Background 30

The hydrologists are vitally depends on streamflow forecasting in production of

31

sustainable theoretical basis of water infrastructures, in flood measurement control in

32

monitoring the operational projects through observation of river behavior. However,

33

the conventional regression based models are incapable to estimate the streamflow

34

data sufficiently because the relationship between the output and input variables are

35

inherently non-linear. Based on that, it is substantial to improve the capability of the

36

model employed for streamflow data prediction by effectively evaluate and extract the

37

non-linear manner of the relationship between the predictor-predicted.

38

Originally, the conventional statistical approaches mainly the ones that based on

39

theoretical techniques, such as regression methods and Box-Jenkin time series

40

approaches, were applied widely to analyze and model the hydrological time series,

41

mainly the estimation of streamflow (Amisigo et al., 2008; Box and Jenkins, 1970;

42

Chua and Wong, 2011; Valipour et al., 2013). During the last two decades, there are

43

abundant of researches have been developed to investigate the potential of using data-

44

driven models such as Artificial Intelligence (AI) methods. The purpose of data-

45

driven modeling is to use the techniques of artificial intelligence (AI) for extraction of

46

documented data pattern in past to predict streamflow data in future, and it has proven

47

to be highly prevalent and favorable forecasting tool, by generating estimated

48

streamflow data that effectively represent the actual streamflow data (Chen et al.,

49

2015; Deo and Şahin, 2016; Fathian et al., 2019; Zhang et al., 2015).

50

A variety of AI-based approaches, (which involve different variables related to

51

streamflow such as evaporation, drought and temperature) have been applied in a

52

plethora of forecasting studies for wide range of reasons. For instance the models can

53

be employed for local scale processes (e.g. irrigation or farms), they show a relative

54

competitive performance with low complexity comparing to the common physically-

55

based hydrological models, additionally, the data inexpensive nature of AI based

56

models and they can be used easily designing of forecasting models and related

57

applications (Ahmed et al., 2019; Afan et al., 2017). Several of AI models have been

58

established using Artificial Neural Networks (ANN) technique (Bai et at., 2016; El-

59

Shafie et al., 2009; Bahrami et al., 2016; Valipour et al., 2012), fuzzy logic and

60

Adaptive Neuro-Fuzzy Inference System (ANFIS) (El-Shafie et al., 2007b; Kisi,

61

2015; Sharma et al., 2015; Zounemat-Kermani and Teshnehlab, 2008), genetic

62

programing (Danandeh Mehr et al., 2014; Makkeasorn et al., 2008; Turan and

63

Yurdusev, 2014), algorithms of regression and support vector machine (Guo et al.,

64

2011; Rasouli et al., 2012; W. C. Wang et al., 2009).In general, compared to the

65

traditional auto regression and other regression based methods, the performance of

66

AI-based approaches for forecasting of streamflow is proven to be more reliable and

67

effective (Yaseen et al., 2015a; Zaher Mundher Yaseen et al., 2016).

68

Recently, major efforts have been made to review and explore the weakness and

69

merits of different types of AI methods in water resources field (Nourani et al., 2014;

70

Yaseen et al., 2015b).It was concluded that there is no “absolute” AI model

71

appropriate for all kinds of modelling (such as estimation, forecasting, classification,

72

optimization, etc.) and generally there was no individual machine learning approach

73

appropriate for all definite problems. Nevertheless, the precision of AI models (with

74

no data pre-processing techniques) can be improved by using hybrid AI models

75

(consist of coupled models) with data pre-processing methods. The common used

76

method for preprocessing of data in water resources applications is wavelet

77

transformation (WT)(Okkan and Ali Serbes, 2013; Parmar and Bhardwaj, 2014;

78

Pramanik et al., 2010; Wei et al., 2013).

79

From this perspective, it is evident that the certainty of the forecasted streamflow data

80

can be effectively enhanced by the integration of two or more techniques to model

81

and assimilate the data patterns and this is defined as hybrid approach (Fahimi et al.,

82

2016). The incorporation of different optimization models implanted into a separate

83

AI-based model leads to a noticeable improvement in the efficiency of streamflow

84

forecasting model which is demonstrated by reduction of the computational time and

85

by inference of an ideal solution for the valuation problem (Ch et al., 2014; Kavousi-

86

Fard et al., 2014a, 2014b).

87

Although the accuracy of forecasting model can be improved by hybrid models,

88

researchers are still conducting many experiments to generate a sufficient approach

89

that can deduce the optimal solutions in the area of forecasting through the utilization

90

of AI models with preprocessing methods. In general, it has been proved that for

91

engineering optimization application, the achievement of global optima is a vital step

92

in the successfulness of the prediction/forecasting models. A few researches showed

93

that the meta-heuristic algorithms, such as Genetic Algorithm (GA), Particle Swarm

94

Optimization (PSO), Harmony Search (HS), Ante Colony Optimization (ACO), have

95

excellent searching aptitudes to achieve the global optima and elude the local optima

96

compared to the classical optimization algorithms. These meta-heuristic algorithms

97

such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Harmony

98

Search (HS), Ante Colony Optimization (ACO). This is due to the fact that these

99

meta-heuristics algorithms might have the nature behave which tolerates the

100

stagnations in the local optima and rapidly switch to searching mode again to achieve

101

the global optima position. In fact, the mentioned optimization algorithms have

102

several advantages and disadvantages. The major advantage in these algorithms is

103

their ability to be adjusted to include several nonlinear systems in parallel or series

104

with different constraints and objective functions. The major disadvantage is the

105

difficulty of addressing the stochastic pattern of the system parameters, slow

106

convergence and lack of ability to distinguish the optimal global solutions. In this

107

context, this study introduces an improved meta-heuristic algorithm to address the

108

stochastic pattern of the prediction methods’ internal parameters and improve the

109

ability of the search procedure to return global optima with relatively faster

110

convergence. Furthermore, the GWO algorithm has the capability of balance between

111

exploration and exploitation efficiently and much better. This avoids a large number

112

of local solutions and provides an assessment of both accuracy and convergence

113

speed. On the other hand, the GWO algorithm with ANN model has been utilized to

114

estimate reference evapotranspiration (ET0) and compared with other four

115

optimization algorithms (PSO, ALO, WOA and MVO) (Tikhamarine et al., 2019a).

116

The results obtained demonstrate the superiority of the GWO algorithm over other

117

optimization algorithms in all cases. Toward this end, this study proposes the Grey

118

Wolf Optimization (GWO) algorithm to be integrated with the prediction model to

119

optimally search for the optimal modelling parameters. The fact that most of the

120

engineering applications of optimization, especially those applications for

121

prediction/forecasting desired variable, the entire searching space is not initially

122

recognized and has quite great numbers of local optima, and hence, the meta-heuristic

123

algorithms could help overcoming such challenges. As a consequence, this work aims

124

to integrate relatively new meta-heuristic optimization algorithm namely; Grey Wolf

125

Optimization (GWO) algorithm(Mirjalili et al., 2014)with ANN models to develop

126

robust integrated systems, as well as to examine the applicability of these hybrid

127

schemes in streamflow forecasting. The effectiveness of the proposed GWO

128

algorithm is suggested to be examined for ANN and AR methods as well.

1291.2. Objective 130

The foremost aim of this research paper is to create an effective hybrid system for

131

streamflow forecasting on monthly timescales by integrating a new meta-heuristic

132

optimization algorithm namely; Grey Wolf Optimization (GWO) algorithm with AI

133

models. In this study, the proposed GWO has been integrated with Support Vector

134

Machine (SVM), Multi-Layer Perceptron Neural Network (MLP-NN) and Auto-

135

Regression (AR) for comparative analysis purposes. Historical natural streamflow for

136

130 years on monthly basis have been collected at Aswan High Dam (AHD) on the

137

Nile River, Egypt to evaluate the performance of the proposed modelling technique.

138

Hydrological and statistical analysis for the collected dataset will be carried out to

139

investigate the practicality of the proposed integrated scheme for forecasting

140

streamflow at semi-arid zone. Moreover, quantitative performance indicators were

141

calculated to evaluate the validity of the integrated models, also the predicted and the

142

observed streamflow date were comprehensively analyzed.

1432. Case study and data description 144

Before introducing the theory and the development of the proposed model, it is

145

necessary to introduce the nature of the selected case study to evaluate the model. In

146

this section, information about the historical natural streamflow of the Nile River at

147

Aswan High Dam (AHD) will be presented. The importance of the choice of this case

148

study to examine the proposed model is due to the fact that AHD is considered as the

149

controller and the regulator for the water resources supply (almost 95%) for Arab

150

Republic of Egypt. In addition, the average amount of the annual streamflow of the

151

Nile River that crossing the AHD is relatively large and equal to 84 Billion Cubic

152

Meter (BCM). Furthermore, the Nile River is considered as one of the longest river

153

and largest catchment in the world and its streamflow is believed as highly stochastic

154

and non-linear, and hence, it is an advantageous and trustworthy illustration for

155

examining any potential method for streamflow forecasting model. Finally, it is vital

156

to accurately develop a model for streamflow forecasting in order to generate a proper

157

operational rule for AHD to have optimal water release policy that meet the water

158

demand downstream the dam.

159

Fortunately, 130 years of monthly natural streamflow at AHD is available during the

160

period between 1871 and 2000 at the Nile River Sector, Ministry of Water Resources

161

and Irrigation, Egypt. Figure 1 shows the natural monthly streamflow at AHD in

162

BCM. With careful visualization for Figure 1, it could be depicted that during these

163

130 years, there are two major seasons for hydrological regime for streamflow, wet

164

season during months between July and December while dry season during the period

165

between January and June. In addition, the fluctuation in the streamflow for the same

166

month along these 130 years showed that the streamflow is greatly stochastic and

167

extremely non-linear. For detailed analysis, it could be noticed that the maximum

168

annual streamflow was happened in 1877-78 water-year (an amount is almost equal to

169

150.33 BCM) and the annual minimum streamflow has been experienced in 1913-14

170

water year with an amount of 42.09 BCM.

171

It is essential to assure the reliabilty of the recorded data before the year 1900 because

172

during this time the natural streamflow were recorded by using manual measurment.

173

During this time, the streamflow records have been recorded by monitoring the water

174

level, and hence, using the rating curve to calculate the coresponding streamflow. In

175

order to verify the streamflow records before 1900, Hurst et al(1966)has been

176

reviewed. It has been concluded in Hurst et al (1966)that during this period, double

177

verification with another monitoring station namely, Halfa gauge station (south of

178

AHD) has been acquired and examined to assure the matching with those streamflow

179

records at AHD. In addition, it has been found out that the recent streamflow records

180

during the period between 1900 and 1960 maintain the same means and standard

181

deviation as the period before 1900, Hurst et al. (1966).

182

Generally, one of the major step that should be carried out before using such historical

183

data is to assure its reliability. In this context, prior statistical analysis of the collected

184

data have been accomplished. To substantiate the realibilty and the accuracy of the

185

collected data, two main indices namely; the mean and the standard deviation have

186

been evaluated for different 30 sets of 100 annual streamflow records for these 130

187

years data. It has been found that the mean values are ranged between 82.1 and 84,2

188

BCM while the standard deviation is ranged between 17.95 and 19.25 BCN. With this

189

pre-analysis for the collected data that showed a narrow range of the mean and the

190

standard deviation of the collected data could result in high confidenant of the

191

collected data that will be used in the current study.

192

Further analysis for the collected data have been performed at the monthly basis to

193

show that the data is highly frequent from one month to another and its wide range for

194

each month to demonstrate the difficulty modelling it. Table 1 shows the simple

195

statistical indices such as the minimum, maximum and the mean monthly streamflow

196

for each month during the period between the 1871 and 2000. As it could be noticed

197

in Table 1, the maximum and the maximum average streamflow have been

198

experienced in September with an amount of 32.79 BCM and 21.51 BCM,

199

respectively. . In addition, it could be depicted that the minimum streamflow has been

200

occurred in May with an amount equal to 0.80 BCM. In addition, the maximum and

201

the minimum standard deviation is ranged between 0.91 and 5.24 BCM for March and

202

September, respectively. Finally, in order to demonstrate how the range of the

203

streamflow is wide for each month, it is obvious that the largest range is occurred in

204

July (the maximum streamflow is almost six times the minimum streamflow), while

205

the smallest range is experienced in November (the maximum streamflow is almost

206

three times the minimum streamflow).

207

In order to recognize the wide range of the possible streamflow at the AHD for each

208

month, the streamflow data for each month has been discretized into different classes.

209

If the data for each month has been individually analyzed, it could be determined that

210

the streamflow for each month could categorized into three different classes high,

211

medium and low streamflow. In order to carry out this analysis, a straightforward

212

process has been performed to the experienced range of the streamflow for each

213

month (the difference between the maximum value of the streamflow events and the

214

minimum value during these 130 years records). The upper limit of the range of the

215

high class for any month is the maximum streamflow record while the lower limit is

216

70% of the difference between the minimum and the maximum streamflow records.

217

The lower limit of the range of the low class is the minimum streamflow record while

218

the upper limit is 30% of the difference between the minimum and the maximum

219

streamflow records. Finally, the upper limit of the range of the medium class is the

220

lower limit of the high class and lower limit is the upper limit of the lower class range.

221

It could be observed that there is a quite high variation of the ranges of the streamflow

222

class for each month, which reflects the highly stochastic of the streamflow pattern, as

223

shown in Figure 2.

224 225 2263. Methodology 2273.1. Methods 2283.1.1. Support Vector Regression (SVR) 229

Support vector regression (SVR) is a kind of regression intelligent model based on

230

support vector machine (SVM) which is developed by Smola (1996). Vapnik (1995)

231

developed the SVM technique for the first time on the basis of statistical learning

232

theory and structural risk minimization principle. The main objective of SVR is to

233

identify a function f (x) that uses all pairs (training data xi, / observed targets yi) with

234

the most minimum (ε) precision and became to be as linear as possible(Smola, 1996).

235

The SVR regression function is declaredas:

236

𝑓(𝑥) = 𝑤 × ∅(x) + 𝑏

237

Where; b is the bias term w is the weight vector in the feature space, ϕ is the transfer

238

function.

239

In order to find a suitable SVR function f (x), the problem of regression can be

240

expressed:

241

Minimize

1

2 2‖𝑤‖

(1)

𝑁

+ 𝐶∑𝑖 = 1(𝜉𝑖 + 𝜉𝑖∗ )

{

𝑦𝑖 ― f(x) ≤ ε + 𝜉𝑖 f(x) ― 𝑦𝑖 ≤ ε + 𝜉𝑖∗ 𝜉𝑖,𝜉𝑖∗ ≥ 0, 𝑖 = 1,2,3,….,𝑁

(2)

242

Subject to the condition:

(3)

243

Where C > 0 is a penalty parameter, 𝜉𝑖 and 𝜉𝑖∗ are the two slack variables to specify

244

the distance from observed values to the ε that corresponding boundary values. By

245

using the Lagrangian multipliers, the optimization issue is largely converted into a

246

quadratic programming and the solution of the nonlinear regression function can be

247

given as follows:

248

𝑓(𝑥) = ∑𝑖 = 1(𝛼𝑖 ― 𝛼𝑖∗ )𝐾(𝑥,𝑥𝑖) + 𝑏

249

Where K(x, xi) is the Kernel function, αi, αi* ≥ 0 are dual variables.

250

Steve Gunn (1998) describes the details of the use of SVM and SVR techniques in the

251

technical report: Vector Machinery Support for Classification and Regression.

252

The kernel function can be selected by different options including: linear, polynomial,

253

sigmoid and radial basis function (RBF) kernels. However, choosing the appropriate

254

kernel function is an important step in using SVR model. In this research, the RBF

255

was adopted as the kernel function because of its performance compared with other

256

kernel functions and the most popular used in literatures (Liong and Sivapragasam,

257

2002; Asefa et al., 2006; Kisi and Cimen, 2011; Tikhamarine et al., 2019b; He et al.,

258

2014).

259

The RBF kernel is defined as:

260

𝐾(𝑥,𝑥𝑖) = exp ( ― γ‖𝑥𝑖 ― 𝑥‖2)

261

Where γ is the kernel parameter, which means that C, γ, and ε are the three parameters

262

that are responsible for the SVR performance. In this study, SVR models were

263

implemented using Matlab software and LIBSVM (version 3.23) developed by Chang

264

and Lin (2011) and the default parameters were selected as follows (C = 1, γ = 0.01

265

and ε = 0.001) which are used in standard SVR.

𝑁

(4)

(5)

2663.1.2. Multiple linear regression (MLR) 267

Multiple linear regression (MLR) is a simple regression equation and one of the most

268

common methods used to solve classical regression problems in statistical analysis

269

(Tabari et al., 2011). In general, MLR is used to find an appropriate relationship

270

between the dependent variable (Y) and one or a set of independent variables (Xi).

271

MLR can generate a dependent relationship by building a linear equation calculated

272

based on the following formula:

273

𝑌 = 𝛼0 + 𝛼1𝑋1 + 𝛼2𝑋2 + 𝛼3𝑋3 +… + 𝛼𝑁𝑋𝑁

274

Where Y is the dependent variable (Qt) and Xi are the independent variables; and α0

275

to αN are the regression coefficients of MLR.

(6)

2762.3.1 Artificial neural network (ANN) 277

Artificial neural network (ANN) is a mathematical model inspired by the function of

278

the biological nervous system of the human being. The ANN has been presented for

279

the first time by McCulloch and Pitts (1943) and can be considered as a mathematical

280

model to solve the complex relationships between variables (Haykin, 1994).

281

The most commonly used ANN model is the multilayer perceptron neural network

282

(MLP) consisting of input and output layers and only one hidden layer in the middle

283

connected to each other with weights and biases. Standard ANN can be reported as an

284

MLP with Levenberg-Marquardt (LM) training algorithm.

285

The explicit mathematical expression to calculate the predicted streamflow can be

286

expressed using the artificial neural network as follows:

287

𝑌 = 𝐹2 ∑𝑖 = 1𝑊𝑘𝑗 × 𝐹1(𝐴𝑗) + 𝑏𝑜

288

𝐴𝑗 = ∑𝑖 = 1𝑋𝑖𝑊𝑗𝑖 + 𝑏𝑗

[

𝑛

𝑚

]

(7) (8)

289 290

Where;𝑌 is the output variable which calculated by the ANN model (the streamflow

291

predicted reported as Y), Xi is the input variable, 𝐴𝑗is the summation of the inputs and

292

their weights represented by equation (8).F1 is the activation function for the hidden

293

layer represented by equation (9), F2 is the activation function for the output layer Wij

294

is the weight between the input i and the hidden node j. bj is the bias of the hidden

295

neuron j, Wjk is the weight of connection of neuron j in the hidden layer to neuron k in

296

the output layer and bo is the bias of the output node k.

297

𝐹1(𝐴𝑗) = 1 + exp ( ― 𝐴𝑗)

298

Because there is no fixed way for selecting the appropriate number of hidden nodes in

299

ANN and in order to avoiding drawback in the large numbers of trial and error

300

process from other side the number of hidden nodes was calculated based on the

301

equation (10) used in literatures (Mirjalili, 2015; Faris et al., 2016; Aljarah et al.,

302

2018, 2019).

303

𝑚=2∗𝑛+1

304

Where; m is the number of neurons; n is the number of inputs.

1

(9)

(10)

305 3.1.4. Grey wolf optimizer (GWO) algorithm 306

The GWO algorithm is a new intelligent algorithm mimics the hierarchy and social

307

hunting of grey wolves proposed by Mirjalili et al.(2014). Generally, the pack of

308

wolves are divided into four groups; Alpha (α), Beta (β), Delta (δ) and the rest of

309

wolves are the Omega (ω). The most dominated wolf is Alpha and can be considered

310

as a leader of the pack. The domination level decrease from alpha to omega is shown

311

in Figure 4 (a).The GWO mechanism is carried out by splitting a set of solutions to

312

the given optimization problem into four groups. The first three solutions are the best

313

α, β and δ. The remaining solutions belong to ω wolves. To implement this

314

mechanism, the hierarchy in each iteration is updated is updated based on the three

315

best solutions. The illustration of the update location is shown in Figure 4 (b).

316

The main principal in GWO algorithm is searching, encircling, hunting, and attacking

317

the prey.

318

Before hunting process, the grey wolves are encircling the prey. The following

319

equations represent encircling behaviour of grey wolves:

320

𝑋(𝑡 + 1) = 𝑋𝑃(𝑡) ― 𝐴 ⋅ 𝐷

321

Where;𝑋(𝑡 + 1) is the next location of any wolf,𝑋𝑃(𝑡)is position vector of the grey

322

wolf, t is the current iteration, 𝐴 is matrix coefficient and 𝐷 is the distance separating

323

the grey wolf and the prey which can be estimated as follows:

324

𝐷 = |𝐶 ⋅ 𝑋𝑃(𝑡) ― 𝑋(𝑡)|

(12)

325

𝐴 = 2𝑎 ⋅ 𝑟1 ― 𝑎

(13)

326

𝐶 = 2 ⋅ 𝑟2

(14)

327

Where; 𝑟1and 𝑟2 are randomly generated from (0 to 1).

328

The previous equations permit a solution to relocate around the prey in a hyper-sphere

329

form (figure 4 (b)). This is not sufficient, nevertheless, to simulate the social

330

intelligence of grey wolves. In order to simulate the prey, the best solution obtained so

331

far considered as the alpha wolf is closer to the prey position, but the global optimal

332

solution is unknown, so it is assumed that the top three solutions have a good idea of

333

their location, therefore other wolves should be obliged to update their locations by

334

using the following equations:

335

𝑋(𝑡 +1) = 3𝑋1 + 3𝑋2 + 3𝑋3

336

Where;𝑋1,𝑋2 and 𝑋3 are calculated using the following equations:

337

𝑋1 = 𝑋𝛼(𝑡) ― 𝐴1 ∗ 𝐷𝛼

(16)

338

𝑋2 = 𝑋𝛽(𝑡) ― 𝐴2 ∗ 𝐷𝛽

(17)

339

𝑋2 = 𝑋𝛽(𝑡) ― 𝐴2 ∗ 𝐷𝛽

(18)

340

Where; 𝐷𝛼, 𝐷𝛽 and𝐷𝛿are given by:

341

𝐷𝛼 = |𝐶1 ⋅ 𝑋𝛼(𝑡) ― 𝑋|

1

1

1

(11)

(15)

(19)

342

𝐷𝛽 = |𝐶2 ⋅ 𝑋𝛽(𝑡) ― 𝑋|

(20)

343

𝐷𝛿 = |𝐶3 ⋅ 𝑋𝛿(𝑡) ― 𝑋|

(21)

344

The prey encircling and attacking are repeated until an optimum solution is obtained

345

or it reaches the maximum number of iterations.

3463.2. Inputs selection and model development 3473.2.1. Inputs selection 348

Choosing proper input variables is very important for the development of SVR, ANN

349

and MLR models since it gives the essential information about the designed system.

350

In the present study, eight different input combinations including the previous values

351

of streamflow have been selected based on simple autocorrelation function (ACF) and

352

partial autocorrelation function (PCF). ACF and PCF are frequently used to determine

353

the appropriate inputs in the time series prediction field. The ACF and PCF were

354

utilized to identify delays that clarify the variance in the predicted streamflow. Figure

355

3 represents the ACF and PACF curves for the Nile River at Aswan High Dam. The

356

lag that shows the great correlation have been chosen to be an input to the selected

357

model. The optimal input sets derived examining ACF and PACF in terms of model

358

number, inputs and output for each model are reported in Table 3.

3593.2.2. Model development 360

The optimization algorithm used in this study is the grey wolf optimize(Mirjalili et al.,

361

2014), because it is one of the modern swarm intelligence algorithm and used

362

successfully in the engineering field (Yu and Lu, 2018; Al Shorman et al., 2019;

363

Tikhamarine et al., 2019a; Maroufpoor et al., 2019).

364

Support vector regression and artificial neural network are belongs to the models of

365

artificial intelligence and shows high performance accuracy in modeling the nonlinear

366

relationship between predictors and predictors (Asefa et al., 2006; Lin et al., 2006;

367

W.-C. Wang et al., 2009). However, the performance of the SVR depends on its

368

parameters and the choice of kernel function. As with the artificial neural network, the

369

performance of the ANN is also depends on the correct selection of weights and

370

biases. The MLR is a linear regression models used to find an appreciated relationship

371

between variables. Nevertheless, the accuracy of MLR also depends on the correct

372

choice of regression coefficients (αi). Consequently, the correct selection of

373

parameters can be considered an optimization problem and need a high optimization

374

algorithm to resolve this problem. Therefore, SVR was coupledwith GWO to build

375

the SVR-GWO model, ANN was coupled with GWO to build the ANN-GWO and the

376

MLR embedded with GWO to construct the MLR-GWO model to predict the

377

streamflow for the Nile River at Aswan High Dam. The SVR-GWO, ANN-GWO and

378

MLR-GWO models were trained and tested for each of eight combination. The

379

flowcharts of the proposed hybrid models are illustrated in Fig. 4: (a) for the hybrid

380

ANN-GWO, (b) for MLR-GWO and (c) for SVR-GWO.

381

For the MLR-GWO, the search agent number is the number of regression coefficients

382

(equation 6) while in ANN-GWO the search agent number can be obtained using the

383

following equation:

384

Number of wights and bieses = (n × m) + (1 × m) + (m × 1) + 1 (22)

385

Where; the search agent number is the number of weights and biases for ANN model,

386

n is the number of inputs and m is the number of hidden neurons in the hidden layer.

387

Besides, as prescribed in literatures, we utilized the most parameter settings for GWO

388

algorithm. Table 4 gives initial parameters of GWO algorithm in detail for the SVR,

389

ANN and MLR models.

3903.3. Performance indicators

391

In this study, the performance criteria used to assess the model’s performance are:

392

Root mean squared error (RMSE):

393

RMSE =

394

Mean absolute error (MAE):

395

MAE = 𝑁∑𝑖 = 1|QO,𝑖 ― QP, 𝑖|

396

Coefficient of correlation (R):

1 𝑁

1

𝑁

∑𝑖 = 1(QP, 𝑖 ― QO,𝑖)2

𝑁

(0 < RMSE < ∞)

(23)

(0 < MAE < ∞)

(24)

(–1 < R < 1)

(25)

𝑁

∑𝑖 = 1(QO,𝑖 ― QO )(QP,𝑖 ― QP)

397

R=

398

Nash-Sutcliffe efficiency coefficient (NSE):

2 𝑁

𝑁

∑𝑖 = 1(QO,𝑖 ― QO ) ∑𝑖 = 1(QP,𝑖 ― QP)

[

𝑁

2

]

∑𝑖 = 1(QO, 𝑖 ― QP, 𝑖)2

(–∞ < NSE < 1)

399

NSE = 1 ―

400

Willmott Index (WI) (Willmott, 1981 and 1984):

[

𝑁

∑𝑖 = 1(QO, 𝑖 ― QO )

𝑛

2

∑𝑖 = 1(QO,𝑖 ― QP,𝑖)2

]

(0< WI < 1)

(26)

(27)

401

WI = 1 ―

402

Where; QO, 𝑖is the value of the streamflow observed in the current observed (i),QP, 𝑖is

403

the predicted value, QO is the average value of observations andQP is the average value

404

of the predictions and N is the number of the data.

𝑛

∑𝑖 = 1(|QO,𝑖 ― QO | + |QP,𝑖 ― QP|)2

4054. Results and discussions 406

In this research, the abilities of hybrid ANN-GWO, SVR-GWO and MLR-GWO data

407

driven methods are investigated in monthly inflow forecasting and their results are

408

compared with those of standard ANN and SVR approaches. Before applying the

409

mentioned methods, whole data are split in two data sets, 70% for training and 30%

410

for testing. Table 2 sums up the brief statistical parameters of the utilized data sets. It

411

is clear that both periods have generally similar characteristics and test data have a

412

little more skewed distribution compared to that of training. Previous inflow values

413

were utilized as inputs to the implemented methods and optimal inputs were decided

414

by applying correlation analysis. Autocorrelation function (ACF) and partial

415

autocorrelation function (PACF) are demonstrated in Figure 2 and the optimal input

416

sets derived examining ACF and PACF are reported in Table 3. Totally 8 input

417

combinations were selected considering the correlation magnitude of inflow data and

418

as can be observed, all previous values up to 13 lags were also utilized as input to

419

better see the usefulness of such analysis in modeling stage of data driven methods.

420

Table 4givesinitial parameters of GWO algorithm in detail for the SVR, ANN and

421

MLR methods. Some control parameters were decided considering the past literature.

422

The performance of the SCR-GWO is summarized in Table 5 for both training and

423

test stages. The optimized SVR parameters utilizing meta heuristic GWO algorithm

424

are also provided in the table for each input case. Table 5 clearly shows that the

425

simulation (training stage) and forecasting (testing stage) accuracy of the SVR-GWO

426

method varies with respect to input combination and the lowest training and testing

427

RMSE (1.5578 m3/s and 2.0570 m3/s), MAE (1.0762 m3/s and 1.2005 m3/s) and the

428

highest training and testing R (0.9682 and 0.9363), NSE (0.9566 and 0.8728) and WI

429

(0.9886 and 0.9671) belong to the third model with input of Qt-1, Qt-2, Qt-11, Qt-12, Qt-

430

13.Test

431

RMSE from 2.7808 m3/s to 2.0570 m3/s and increase NSE from 0.7676 to 0.8728) and

432

then decrease till 5th input combination (increase in RMSE from 2.1937 m3/s to

433

2.2880m3/s and decrease in NSE from 0.8554 to 0.8427). As evident from the results,

434

including high number inputs do not guarantee better forecasting accuracy (e.g.,

435

Zhang et al. 2019; Adnan et al. 2019; Shi et al. 2012) because increasing inputs

436

number may increase variance and add more complexity to the implemented model

accuracy of the SVR-GWO method increases up to 3rd input case (decrease in

437

and this deteriorate the model accuracy in forecasting. The other important issue

438

which is worthy to mention here that the input combination recommended by the

439

correlation analysis should not be directly applied to the data driven methods and it is

440

better to try several input cases by considering ACF and PACF.

441

Table 6sums up the performance of hybrid ANN-GWO method in training and testing

442

stages of different input cases. The optimal model structures are also provided in the

443

table. In the structure column, first number indicates input number, second number is

444

hidden node number and last number indicates the output Qt-1 (inflow value of current

445

month). For example, the ANN with 2-5-1structure has 2 inputs (Qt-1 and Qt-2), 5

446

hidden nodes and one output. Similar to the SVR-GWO, among the ANN-based

447

models, the third one having 5-11-1 structure with the input of Qt-1, Qt-2, Qt-11, Qt-12, Qt-

448

13has

449

(1.0168m3/s and 1.2999m3/s) and the highest training and testing R (0.9717 and

450

0.9314), NSE (0.9441 and 0.8561) and WI (0.9854 and 0.9636). An increase in

451

accuracy (decrease in RMSE from 3.1324 m3/s to 2.1883 m3/s and increase NSE from

452

0.7051 to 0.8561) from 1st input to 3rd input cases and then an increase in accuracy

453

(increase in RMSE from 2.3782m3/s to 2.3850m3/s and decrease in NSE from 0.8300

454

to 0.8290) is seen from 4th input to 5th input cases for the testing stage of ANN-GWO

455

method. For this method, also less number of inputs provides better accuracy

456

(compare the M3 with M2, M5, M6, M7 and M8).

457

Tables 7-8 give the accuracy of standard SVR and ANN methods in forecasting

458

monthly inflows. Alike to hybrid SVR-GWO and SVR-GWO, among the SVR-based

459

models, the third SVR-based model has the lowest performance in both training

460

(RMSE: 1.8253 m3/s, MAE: 0.9780 m3/s, R: 0.9724,NSE: 0.9404, WI: 0.9834) and

461

testing (RMSE: 2.2248 m3/s, MAE: 1.3190 m3/s, R: 0.9244, NSE: 0.8512, WI:

the lowest training and testing RMSE (1.7663m3/s and 2.1883m3/s), MAE

462

0.9605)stages (Table 7). Similarly, among the ANN-based models, the third ANN-

463

based model has the lowest performance in training (RMSE: 1.3110m3/s, MAE:

464

0.8006m3/s, R: 0.9845, NSE: 0.9692, WI: 0.9921) and testing (RMSE: 2.4214m3/s,

465

MAE: 1.3105m3/s, R: 0.9118, NSE: 0.8238, WI: 0.9539) stages (Table 8).In both

466

SVR and ANN methods, accuracy of the models increase from 1st input to 3rd input

467

and decrease from 4th to 5th input similar to hybrid versions. Also, for these methods,

468

better accuracy can be obtained utilizing the models with a smaller number of inputs.

469

The accuracy of hybrid MLR-GWO method is provided in Table 9 for training and

470

test periods. Parallel results were also obtained from this method so that the third

471

model had the best accuracy (RMSE: 2.0366 m3/s and2.3883 m3/s, MAE: 1.1255m3/s

472

and 1.3370 m3/s, R: 0.9623 and 0.9121, NSE: 0.9257 and 0.8286, WI: 0.9806 and

473

0.9538 for training and testing, respectively) in both periods and accuracy variation

474

with respect to input cases is similar to previous methods.

475

Comparison of hybrid methods (Table 5, 6 and 9) reveals that the both hybrid SVR-

476

GWO and ANN-GWO methods have better approximation accuracy in training stage

477

and performs superior to the MLR-GWOin forecasting (testing) stage in all input

478

cases. The best model is SVR-GWO-based model with 3rd input combination and

479

followed by the ANN-GWO-based model with the same input. This indicates the

480

nonlinearity of the investigated phenomenon and therefore, linear MLR-GWO-based

481

models cannot adequately map monthly inflows.The other advantages of hybrid SVR

482

and ANN methods are involving much more model parameters compared to hybrid

483

MLR approach. This can be clearly seen from the Tables 10-11 which present the

484

optimized parameters of ANN and MLR methods.High number of parameters adds

485

flexibility to the data driven methods but if they are adequately calibrated. The

486

relative RMSE, MAE, R, NSE and WI differences between the optimal hybrid SVR-

487

GWO (model M3) and ANN-GWO (model M3) in the testing stage are 6%, 7.6%,

488

0.5%, 2%, 0.4%, respectively.

489

Comparison of hybrid SVR-GWO and standard SVR methods (Table 5 and 7)

490

indicates that the hybrid SVR-GWO method can make better approximation in

491

training stage and produce better forecasts than the single SVR in forecasting monthly

492

inflow in all input cases. This indicates the superiority of GWO meta-heuristic

493

algorithm compared to training algorithm of standard SVR. The relative RMSE,

494

MAE, R, NSE and WI differences between the optimal hybrid SVR-GWO (model

495

M3) and SVR (model M3) in the testing stage are 7.5%, 9%, 1.3%, 2.5%, 0.7%,

496

respectively. Similarly, comparison of hybrid ANN-GWO and standard ANN

497

methods (Table 6 and 8) indicates that the GWO algorithm makes better

498

approximation in training stage and provides better forecasts than the single ANN in

499

forecasting inflow in all input cases. This reveals the superiority of GWO meta-

500

heuristic algorithm compared to the training algorithm (Levenberg-Marquardt) of

501

standard ANN. The relative RMSE, MAE, R, NSE and WI differences between the

502

optimal hybrid ANN-GWO (M3) and ANN (M3) in the testing stage are 10.7%,

503

0.8%, 2.1%, 3.8% and 1%, respectively. From the results obtained from Table 5-8, we

504

can say that the SVR method performs superior to the ANN method in forecasting

505

monthly inflows.Kernel function add flexibility to the SVR and thus it can implicitly

506

simulate the input data to a higher dimensional (HD) space. Linear solutions of the

507

problems in the feature space of HD correspond to non-linear solutions in the original

508

lower dimensional input space. These characteristicsmake SVR a feasible alternative

509

for the solution of several problems in water resources and hydrology, which are

510

naturally non-linear (Naganna and Deka, 2014).

511

Figure 6 illustrates time variation graph of observed and forecasted monthly inflows

512

by the optimal models. It also includes detail graphs of two sections. It is clearly seen

513

from detail graphs that the forecasts of the SVR-GWO-3 model are closer to the

514

corresponding observed inflow values. The forecasts of the optimal models are

515

compared in Figure 7 in the scatter plot form. It is apparent from the graphs that the

516

SVR-GWO-3 (model M3) has less scattered forecasts compared to ANN-GWO-3,

517

MLR-GWO-3, ANN-3 and SVR-3 models. The time variation of relative errors is

518

shown in Figure 8 for each method. As observed from the figure, the range of relative

519

errors is less for the SVR-GWO-3 model in comparison to the other models.

5205. Conclusion 521

A new integrated model, which combines the Grey Wolf Optimization (GWO)

522

algorithm and Support Vector Regression, was proposed to predict monthly

523

streamflow in Aswan High Dam. GWO was developed for optimizing the hyper-

524

parameters of SVR such as the global solution parameters and then it compared with

525

two more integrated algorithms such as Artificial Neural Network (ANN) and

526

Multiple Linear Regression (MLR). The results reveal, the integrated AI models with

527

GWO were proven to be more accurate and effective compared with the standard AI.

528

Moreover, the integrated SVR-GWO outperformed, in terms of performance criteria

529

convergence, ANN-GWO and MLR-GWO in forecasting the monthly streamflow.

530

This reveals the superiority of GWO meta-heuristic algorithm in optimizing the

531

parameter of the standard SVR to improve its accuracy in forecasting the streamflow.

532

In future researches, the improved revised algorithm of Grey Wolf Optimization can

533

be introduced to enhance the hyper-parameters optimization in order to avoid trapping

534

in local optima. In addition, in a few cases studies, there is a need to develop real-time

535

forecasting model, and hence, the time consuming for the model execution should be

536

lessen. Therefore, there is a need to improve the searching algorithm for the GWO to

537

achieve fast convergence procedure as well. Moreover, other advanced meta-heuristic

538

algorithms could be investigated to improve streamflow forecasting.

539 540

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800 801

802 803 35.00

AUG.

SEP.

OCT.

NOV.

DEC.

FEB.

MAR.

APR.

MAY

JUN.

JAN.

Natural Inflow (BCM)

30.00 25.00 20.00 15.00 10.00 5.00

1872-73 1876-77 1880-81 1884-85 1888-89 1892-93 1896-97 1900-01 1904-05 1908-09 1912-13 1916-17 1920-21 1924-25 1928-29 1932-33 1936-37 1940-41 1944-45 1948-49 1952-53 1956-57 1960-61 1964-65 1968-69 1972-73 1976-77 1980-81 1984-85 1988-89 1992-93 1996-97 2000-1

0.00

Year

804 805 806

Fig. 1 Monthly Natural Nile river flow at Aswan High Dam for the period between 1871 and 2000

807 808 809 810 35

High Inflow Mediam Inflow Low Inflow

30

Inflow (BCM)

25 20 15 10 5 0

811 812 813 814

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Fig. 2 Monthly hydrological characteristics at AHD. a) Monthly natural inflow classes (box-and-whisker plot)

815 816 817 818 819 820 821 822

Lag (Month)

Lag (Month)

823 824 825 826 827

Fig. 3 Autocorrelation and partial autocorrelation functions of Aswan station time series

828

a

829 830 831 832 833 834 835 836 837 838 839 840 841 842

843

b

844 845 846 847

Fig. 4 (a) The social hierarchy of grey wolves, (b) Illustration of position updating mechanism of ω wolves according to positions of α, β and δ wolves (Source Al Shorman et al., 2019).

848 849 850 851 852

Fig. 5 Flowchart of the proposed hybrid models. a: hybrid ANN-GWO. b: hybrid MLR-GWO and c: hybrid SVR-GWO

853

854 855

Fig. 6 Tim e vari atio n grap hs of the obs erve d and esti mat ed stre amf low s by

856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882

-3

883 884

Fig. 7 Scatterplots for the developed models in the testing period.

885 886 100 887 R el 50 888 at 0 889 iv -50 e 890 er -100 ro 891 0 r

SVR-GWO-3 Test

50

100

150

200

250

300

350

400

450

500

100

892 R el 50 893 at 0 iv 894-50 e er -100 895 ro 0 r 896

ANN-GWO-3 Test

50

100

150

200

250

300

350

400

100

897 R el 50 898 at 0 iv 899-50 e er -100 900 ro 0 901 r

450

500

MLR-GWO-3 Test

50

100

150

200

250

300

350

400

450

500

100

902 R el 50 903 at 0 iv 904-50 e -100 er 905 ro 0 906 r

SVR-3 Test

50

100

150

200

250

300

350

400

450

500

100

907 R el 50 908 at 0 iv 909-50 e er -100 ro 0 r

ANN-3 Test

50

100

150

200

250

300

350

400

450

500

910

Statistical index Min. Mean

Max.

Streamflow classes (Median value) High Medium Low

Month

(BCM/month)

(BCM/month)

(BCM/month)

(BCM/month)

(BCM/month)

(BCM/month)

Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul

29.10 32.79 27.40 14.40 11.00 7.70 6.04 5.81 5.26 4.72 5.16 11.03

6.50 7.31 5.97 4.12 2.83 1.72 1.15 1.07 0.95 0.80 0.90 1.74

18.96 21.51 14.52 8.11 5.54 4.31 3.16 2.75 2.52 2.33 2.25 5.24

27.5 31 21.2 10.9 6.5 4.8 3.7 3.5 2.7 2.5 2.8 7.7

20.4 24.05 15.6 7.3 4.3 3.15 1.95 1.7 1.15 1.35 1.65 4.75

15.05 18.55 11.3 4.75 2.7 1.9 0.8 0.55 0.3 0.65 0.9 2.8

911

Variable Sets Minimum Maximum (BCM/month) (BCM/month)

Training Testing Whole data

0.80 2.09 0.80

32.00 32.79 32.79

Statistical parameters Mean (BCM/month) Standard deviation 7.76 7.16 7.58

Skewness Kurtosis

7.48 5.77 7.01

1.25 1.79 1.39

0.37 2.52 0.84

912 913 914 915

Fig. 8 Relative error between observed and predicted models in the testing period.

916 917 918 919 920 921 922 923 924 925 926

Table 1. Statistical Summary and streamflow classes for the natural streamflow at AHD based the period between 1871 and 2000 Table 2. Statistical parameters of training and testing datasets for study station

Table 3. Different input combinations for forecasting future river inflow Models

INPUTS

OUTPUT

M1 M2 M3 M4 M5 M6 M7 M8

Qt-1, Qt-2 Qt-1, Qt-2, Qt-10, Qt-11, Qt-12, Qt-13 Qt-1, Qt-2, Qt-11, Qt-12, Qt-13 Qt-1, Qt-11, Qt-12, Qt-13 Qt-1, Qt-4, Qt-5 Qt-6, Qt-7, Qt-8, Qt-11, Qt-12, Qt-13 Qt-1, Qt-5 Qt-6, Qt-7, Qt-11, Qt-12, Qt-13 Qt-1, Qt-2, Qt-4, Qt-5, Qt-6, Qt-7, Qt-8, Qt-11, Qt-12, Qt-13 Qt-1, Qt-2, Qt-3, Qt-4, Qt-5 Qt-6, Qt-7, Qt-8, Qt-9, Qt-10, Qt-11, Qt-12, Qt-13

Qt Qt Qt Qt Qt Qt Qt Qt

927

Table 4. Initial parameters of the GWO meta-heuristic algorithm

928 Models

Number of search agent

SVR-GWO

03

ANN-GWO

MLR-GWO

929

Equation (23)

Equation (6)

Parameters

Value

Α

Min = 0 and max = 2

Number of agents

100

Iterations number

1000

Range partitions (C)

[10-5, 105] (Sudheer et al., 2014)

Range partitions (γ)

[10-5, 101] (Sudheer et al., 2014)

Range partitions (ε)

[10-5, 101] (Sudheer et al., 2014)

Α

Min = 0 and max = 2

Number of agents

100

Iterations number

1000

Range partitions (αi)

[-3, +3]

Α

Min = 0 and max = 2

Number of agents

100

Iterations number

1000

Range partitions (αi)

[-10-3, 103]

Table 5. RMSE, MAE, R, and NSE values during training and testing periods of

930

SVR-GWO parameters

Training 70% (1083 Samples)

Testing 30% (464 Samples)

γ

C

ε

RMSE (m3/s)

MAE (m3/s)

R

NSE

WI

RMSE (m3/s)

MAE (m3/s)

R

NSE

WI

M1

0.06151

2.978

0.01187

1.8960

1.0762

0.9682

0.9356

0.9828

2.7808

1.8187

0.8910

0.7676

0.9420

M2

0.00060

110.960

0.31953

1.6010

0.8881

0.9769

0.9541

0.9880

2.0756

1.1559

0.9332

0.8705

0.9649

M3

0.00328

6.706

0.43790

1.5578

0.8521

0.9782

0.9566

0.9886

2.0570

1.2005

0.9363

0.8728

0.9671

M4

0.00560

1.870

0.04146

1.7769

0.9753

0.9719

0.9435

0.9849

2.1937

1.2751

0.9269

0.8554

0.9613

M5

0.00046

24.147

0.25284

1.8461

0.9870

0.9693

0.9390

0.9837

2.2880

1.3038

0.9189

0.8427

0.9561

M6

0.00461

3.417

0.08112

1.6949

0.9113

0.9744

0.9486

0.9863

2.2376

1.3184

0.9245

0.8495

0.9583

M7

0.00068

39.065

0.02235

1.6395

0.8651

0.9758

0.9519

0.9874

2.1126

1.2318

0.9317

0.8659

0.9632

M8

0.00201

4.332

0.06883

1.5849

0.8495

0.9775

0.9550

0.9882

2.1315

1.2830

0.9303

0.8635

0.9637

SVR-GWO model

931 932 933 934 935 936 937

Table 6. RMSE, MAE, R, and NSE values during training and testing periods of

938

Training 70% (1083 Samples)

Testing 30% (464 Samples)

ANN Structure

RMSE (m3/s)

MAE (m3/s)

R

NSE

WI

RMSE (m3/s)

MAE (m3/s)

R

NSE

WI

M1

2-5-1

2.4293

1.7153

0.9458

0.8944

0.9711

3.1324

2.3454

0.8565

0.7051

0.9192

M2

6-13-1

1.8122

1.0781

0.9702

0.9412

0.9846

2.2625

1.3041

0.9246

0.8461

0.9608

M3

5-11-1

1.7663

1.0168

0.9717

0.9441

0.9854

2.1883

1.2999

0.9314

0.8561

0.9636

M4

4-9-1

1.9651

1.1333

0.9648

0.9309

0.9818

2.3782

1.3947

0.9118

0.8300

0.9520

M5

9-19-1

1.9280

1.0994

0.9662

0.9335

0.9825

2.3850

1.5014

0.9167

0.8290

0.9556

M6

7-15-1

1.8104

1.0648

0.9702

0.9413

0.9847

2.2560

1.3595

0.9228

0.8470

0.9598

M7

10-21-1

1.8680

1.1041

0.9683

0.9375

0.9836

2.2126

1.3393

0.9278

0.8529

0.9626

M8

13-27-1

1.8485

1.1337

0.9689

0.9388

0.9839

2.3845

1.5187

0.9177

0.8291

0.9556

Model

ANN-GWO

SVR-GWO

Model

939

ANN-GWO model

940 941

Table 7. RMSE, MAE, R, and NSE values during training and testing periods of SVR model

SVR parameters

SVR

Model

Training 70% (1083 Samples)

Testing 30% (464 Samples)

γ

C

ε

RMSE (m3/s)

MAE (m3/s)

R

NSE

WI

RMSE (m3/s)

MAE (m3/s)

R

NSE

WI

M1

0.01

1

0.001

3.1180

1.6409

0.9178

0.8260

0.9481

3.2319

1.9335

0.8361

0.6861

0.9099

M2

0.01

1

0.001

1.8603

0.9934

0.9718

0.9380

0.9826

2.3335

1.3235

0.9148

0.8363

0.9543

M3

0.01

1

0.001

1.8253

0.9780

0.9724

0.9404

0.9834

2.2248

1.3190

0.9244

0.8512

0.9605

M4

0.01

1

0.001

1.8635

1.0213

0.9702

0.9378

0.9829

2.2403

1.3216

0.9239

0.8492

0.9587

M5

0.01

1

0.001

2.1084

1.0996

0.9654

0.9204

0.9770

2.5427

1.4786

0.9111

0.8057

0.9405

M6

0.01

1

0.001

2.0247

1.0763

0.9670

0.9266

0.9791

2.4034

1.4161

0.9167

0.8264

0.9491

M7

0.01

1

0.001

2.0979

1.0703

0.9668

0.9212

0.9770

2.4938

1.4211

0.9090

0.8131

0.9432

M8

0.01

1

0.001

2.1878

1.1012

0.9641

0.9143

0.9748

2.7126

1.5005

0.8927

0.7788

0.9271

Table 8. RMSE, MAE, R, and NSE values during training and testing periods of ANN model

942 943

ANN

Model

ANN Structure

Training 70% (1083 Samples) RMSE MAE R NSE WI (m3/s) (m3/s)

Testing 30% (464 Samples) RMSE MAE R NSE (m3/s) (m3/s)

M1

2-5-1

1.9242

1.2552

0.9663

0.9337

0.9826

3.2161

2.2878

0.8640

0.6891

0.9230

M2

6-13-1

1.2031

0.7478

0.9870

0.9741

0.9934

2.4954

1.4227

0.9103

0.8128

0.9529

M3

5-11-1

1.3110

0.8006

0.9845

0.9692

0.9921

2.4214

1.3105

0.9118

0.8238

0.9539

M4

4-9-1

1.4532

0.8774

0.9809

0.9622

0.9903

2.5653

1.4181

0.9019

0.8022

0.9485

M5

9-19-1

1.1254

0.6876

0.9886

0.9773

0.9942

2.8792

1.7276

0.8776

0.7508

0.9348

M6

7-15-1

1.2462

0.7592

0.9860

0.9722

0.9929

2.8793

1.6991

0.8747

0.7508

0.9304

M7

10-21-1

1.0304

0.6328

0.9905

0.9810

0.9952

2.5973

1.5167

0.9149

0.7972

0.9526

M8

13-27-1

0.7334

0.4738

0.9952

0.9904

0.9976

3.5392

2.0917

0.8605

0.6235

0.9170

Table 9. RMSE, MAE, R, and NSE values during training and testing periods of MLR-GWO model

944 945

Training 70% (1083 Samples) Model

M1 M2

MLR-GWO

WI

M 3 M4 M5 M6 M7 M8

RMS E (m3/s) 4.011 9 2.170 9 2.036 6 2.087 4 3.963 0 2.844 7 2.067 7 2.751

Testing 30% (464 Samples)

MAE (m3/s)

R

NSE

WI

2.729 7 1.275 9 1.125 5 1.164 2 2.682 3 1.749 3 1.180 3 1.816

0.843 7 0.957 2 0.962 3 0.960 2 0.848 2 0.925 1 0.961 0 0.929

0.711 9 0.915 6 0.925 7 0.922 0 0.718 8 0.855 1 0.923 5 0.864

0.909 0 0.977 9 0.980 6 0.979 4 0.915 2 0.960 3 0.979 8 0.962

RMS E (m3/s) 3.916 5 2.518 1 2.388 3 2.445 0 4.066 4 3.046 7 2.417 6 2.952

MAE (m3/s)

R

NSE

WI

2.741 7 1.459 7 1.337 0 1.359 2 2.761 0 1.694 0 1.365 0 1.815

0.751 1 0.902 9 0.912 1 0.907 7 0.721 7 0.853 5 0.909 6 0.861

0.539 0 0.809 4 0.828 6 0.820 3 0.503 0 0.721 0 0.824 3 0.738

0.856 3 0.948 9 0.953 8 0.951 5 0.836 2 0.920 2 0.952 5 0.923

1

7

8

5

4

4

2

6

0

8

946

Table 10. Optimal MLR-GWO Parameters.

947

Model

MLR Equation Qt = 1.17417 *Qt-1 - 0.62788 Qt-2 + 3.50692

M1

M2 Qt = 0.66807*Qt-1 + 0.03843*Qt-2 + 0.0651*Qt-10 - 0.08724*Qt-11 + 0.97596*Qt-12 - 0.62635*Qt-13 0.61387 M3

Qt = 0.73655*Qt-1- 0.0.05822*Qt-2+ 0.07324*Qt-11 + 0. 79472*Qt-12 - 0. 54969*Qt-13 - 0.06750

M4

Qt = 0. 67595*Qt-1+ 0. 01034*Qt-11+ 0. 92619*Qt-12+ 0. 61721*Qt-13 -17546

MLR-GWO

M5 -0.03010 0.60628

-0.12549 0.12141 0.09536

-0.15357 0.09885

-0.02450

-1.42590 0.59310 Qt = -02450*Qt-1+0. 09885*Qt-4 -0. 15357*Qt-5 +0. 09536*Qt-6-0. 12141*Qt-7 -0.12549*Qt-8 0.60628*Qt-11 -03010*Qt-12 + 59310*Qt-13 -1.42590 M6

Qt = -0. 00085*Qt-1+0. 02141*Qt-5 -0. 02274*Qt-6 +0. 01284*Qt-7 +0.04802*Qt-11 +87295*Qt-12 +0.05226*Qt-13 -0.00142

M7

Qt =0. 68348*Qt-1 -0. 01288*Qt-2 - 0. 03845*Qt-4+ 0. 03824*Qt-5 -0. 04732*Qt-6+ 0. 01018*Qt-7+ 0. 00571*Qt-8- 0. 00068*Qt-11+ 0. 89217*Qt-12 -0. 60036*Qt-13+ 0. 53098

M8

Qt =0. 0000003*Qt-1 +0. 03549*Qt-2 -0. 08575*Qt-3+ 0. 01157*Qt-4 -0. 04458*Qt-5- 0. 05618*Qt6+

0. 01988*Qt-7- 0. 07905*Qt-8+ 0. 02019*Qt-9- 0. 10697*Qt-10+ 0. 09547*Qt-11+ 0. 79328*Qt-12+ 0. 00602*Qt-13+ 3.01200

948 949 950

Table 11. ANN Parameters versus ANN-GWO parameters for the best model (M3).

951

ANN

Model

Parameters IW1_1 =[0.645548352588935 0.490849210105060 -0.636814115741003 -0.197023018606021 1.15095937712121;-0.374741951566058 2.09602351895915 -0.948572086483014 0.448172347550920 -0.390243232966560;-0.508664569672024 1.42473219848372 1.81346308646462 -0.422220219074745 0.307797907505390;1.69099385336773 1.91762647812312 -1.37118240411381 0.991561905462959 -0.0645409927642741;1.16667356534635 1.03770562438892 -0.655038460261841 0.365195799970887 1.39859811955597;-0.436086805954856 -0.402680235576198 -0.674915728267535 0.573879996013445 0.970907264566725;1.40043825812518 -0.894001252195594 0.387858261897225 -0.235609833166580 1.22505735374624;0.157337536509532 0.440907790543129 2.09832896723230 -3.05274752078594 1.20295125926541;1.94877786008619 0.653535998773184 0.747141370532021 1.24502310208271 0.0407935886488361;0.995475357888400 0.663088707022791 -1.23122998045925 0.850028887499989 -0.420747514818101;0.507805643987633 1.25112356595343 0.347455862427703 -1.53293583422575 -0.415921708344004]

b1=[0.364567778992709;-0.253277415388278;0.0776162521275670;-0.335489901278488;1.49190695238374;1.23261751911969;0.766095010968687;-1.12821463276292;1.53241532465276;0.262077228973190;-1.24795736214157] LW2_1= [-1.27781252764868 0.750943129520385 -1.16056523012868 0.711855144075440 1.66920057928637 -1.07462179886860 -1.81513806044960 -0.706292165764873 1.35623649635319 1.27555964001351 0.568715308486404]

ANN-GWO

b2= [-0.523926338451804] IW1_1=[0.679633208818835 -0.0598736973233613 0.0290671638159691 -0.118898973732041 1.19996763002510;0.00969730241328533 -0.119191575741640 0.0194353130324328 1.07549721717923 0.652667379286810;-4.98948316200092e-05 -0.285414929657196 -1 0.346887454209408 0.722057669039339;-0.00385812223558482 -0.704007857206319 0.00969361579187186 0.00202473907503717 0.102333284563727;-0.0530813631767349 0.00874038005562050 0.940658603673713 1.88808217020703 0.558951907069030;1.45027460136395 0.0369561267210984 0.0938558513359091 0.0202778143417594 -0.0287463129026476;-0.129114564501272 -0.999595557587754 0.559480176726186 1.93021116801354 -0.0427165998042683;-0.0432015511228707 0.425699113581515 0.0390928538728479 -0.0269531954052187 0.653111091749518;0.0208566152618470 0.198693391722621 -0.0272336852909195 0.000128014264935471 0.691879316989853;0.927288538383699 -0.486089339990834 0.00749547491297705 0.532970735519229 -0.805370703363221;-0.455084554781621 0.157062590076071 -0.0106175418622636 -0.568214708185782 -0.0186313139069768] b1=[-0.999544626417181;-0.0257826944723780;1.54804395315099;0.419551508628818;0.201266019663968;1.33194412312976;-0.223980124713737;-0.355168069400448;0.0536968092305320;-0.138026029294201;-0.999752148603694] LW2_1=[0.519446674804311 -0.308173183218543 0.214442030218500 -0.0186771941202023 0.0485869393744300 0.0251913808893511 0.243068491697808 -0.0542094341107475 0.184388522291043 0.581590936122574 -0.895940329672519] b2=[-0.564570091202907]

952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971

Where: IW1_1: is the weight matrix between the input layer and the hidden layer. b1: is the bias vector between the input layer and the hidden layer. LW2_1 is the weight vector between the hidden layer and the output layer. b2: is the bias vector between the hidden layer and the output layer.

Improving Artificial Intelligence Models Accuracy for Monthly Streamflow Forecasting Using Grey Wolf Optimization (GWO) Algorithm Yazid Tikhamarine1, Doudja Souag-Gamane1, Ali Najah Ahmed2, Ozgur Kisi3 and Ahmed El-Shafie4 1LEGHYD Laboratory, Department of Civil Engineering, University of Scienceand Technology Houari Boumediene, BP 32, BabEzzouar, Algiers, Algeria 2Institute for Energy Infrastructure (IEI), Universiti Tenaga Nasional (UNITEN), Kajang 43000, Selangor Darul Ehsan, Malaysia 3School of Technology, Ilia State University, Tbilisi, Georgia 4Department of Civil Engineering, Faculty of Engineering, University of Malaya (UM), 50603 Kuala Lumpur, Malaysia

Abstract

972

Monthly streamflow forecasting is required for short- and long-term water resources

973

management especially in extreme events such as flood and drought. Therefore, there

974

is need to develop a reliable and precise model for streamflow forecasting. The

975

precision of Artificial Intelligence (AI) models can be improved by using hybrid AI

976

models which consist of coupled models. Therefore, the chief aim of this study is to

977

propose efficient hybrid system by integrating Grey Wolf Optimization (GWO)

978

algorithm with Artificial Intelligence (AI) models. 130 years of monthly historical

979

natural streamflow data will be used to evaluate the performance of the proposed

980

modelling technique. Quantitative performance indicators will be introduced to

981

evaluate the validity of the integrated models; in addition to that, comprehensive

982

analysis will be conducted between the predicted and the observed streamflow.The

983

results show the integrated AI with GWO outperform the standard AI methods and

984

can make better forecasting during training and testing phases for the monthly inflow

985

in all input cases. This finding reveals the superiority of GWO meta-heuristic

986

algorithm in improving the accuracy of the standard AI in forecasting the monthly

987

inflow.

988

Keywords: Streamflow, Estimation, Evolutionary algorithms, Aswan High Dam

989 990 991



Monthly streamflow forecasting is required for short- and long-term water resources management especially in extreme events such as flood and drought.

992 993 994 995



Grey Wolf Optimization (GWO) algorithm has been integrated to Support Vector Machine (SVM), Artificial Neural Network (ANN) and Multiple

996

Linear Regression (MLR) in order to compare the accuracy of the proposed

997

model.

998 999



Integrated AI with GWO outperform the standard AI methods and can make

1000

better forecasting during training and testing phases for the monthly inflow in

1001

all input cases.

1002 1003



The findings reveal the superiority of GWO meta-heuristic algorithm in

1004

optimizing the parameter of the standard SVR to improve its accuracy in

1005

forecasting the streamflow.

1006 1007

Yazid Tikhamarine : Methodology, Software, Visualization.

1008

Doudja Souag-Gamane: Conceptualization, Supervision, Writing- Reviewing and Editing

1009

Ali Najah Ahmed: Data curation, Writing- Original draft preparation, Validation.

1010

Ozgur Kisi: Writing- Original draft preparation, Writing- Reviewing and Editing,

1011

Supervision.

1012

Ahmed El-Shafie: Writing- Original draft preparation, Writing- Reviewing and Editing,

1013

Supervision.

1014 1015 1016

Declaration of interests

1017 1018 1019

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

1020 1021 1022 1023

☒The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

The authors declare no conflict of interest.

1024 1025 1026 1027 1028