A novel hybrid water quality time series prediction method based on cloud model and fuzzy forecasting

Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

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Chemometrics and Intelligent Laboratory Systems journal homepage: www.elsevier.com/locate/chemolab

A novel hybrid water quality time series prediction method based on cloud model and fuzzy forecasting Weihui Deng, Guoyin Wang ⁎, Xuerui Zhang Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China

a r t i c l e

i n f o

Article history: Received 23 August 2015 Received in revised form 22 September 2015 Accepted 27 September 2015 Available online 3 October 2015 Keywords: Water quality time series prediction Cloud model Fuzzy time series Approximate periodicity

a b s t r a c t Accurate water quality time series prediction can provide support to early warning of water pollution as well as decision-making for water resource management. Due to the uncertainty of the water quality data including randomness, fuzziness, imprecision, and nonstationary, the prediction accuracy of the traditional models has been limited. In this paper, a multi-factor water quality time series prediction model is proposed, based on Heuristic Gaussian cloud transformation, the approximate periodicity of water quality parameter and fuzzy time series model. The proposed model uses the Heuristic Gaussian cloud transformation algorithm to extract the uncertain numerical time series into Gaussian clouds, and constructs the training dataset by calculating the length of the approximate periodicity, which can greatly reduce the noise data. Then, it applies the fuzzy time series model to do the prediction. The proposed model is tested for DO, CODMn, water temperature and EC prediction. The experimental results show that the proposed method significantly improved the prediction accuracy compared with the existing time series prediction models for water quality prediction. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Water is not only an indispensable natural resource for mankind's survival and society development, but also an important part of the ecological environment. Apart from drinking and personal hygiene, water still is a necessary condition for agricultural production, industrial and manufacturing processes, hydroelectric power generation, contamination dissolution releasing, biotransformation, and so on. However, because of humanity's inappropriate productive mode and living habits, it receives the greatest pollution load. In recent years, as water pollution incident happens more frequently, water quality assessment and prediction have gradually become the heated issue that many countries' environmental management department pay close attention to. Accurate water quality parameter predictions could provide support to early warning of water pollution and save time for decision-making. So far, two kinds of approaches have been extensively employed to predict water quality parameters. One kind of approach is time series prediction while the other predicts water quality parameters without considering time ahead. The latter method means predicting the values of the parameter at time t using the other parameters in the input structure at time t [1–6]. However, these methods just utilize the correlations between the predicted parameter and other parameters, not using the

⁎ Corresponding author at: Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, No.266 Fangzheng Avenue, Shuitu Hi-tech Industrial Park, Shuitu Town, Beibei District, Chongqing, 400714, China. E-mail addresses: [email protected] (W. Deng), [email protected] (G. Wang).

http://dx.doi.org/10.1016/j.chemolab.2015.09.017 0169-7439/© 2015 Elsevier B.V. All rights reserved.

information of previous weeks or days. In addition, they cannot predict the future values of the parameter. In this paper, we mainly discuss the water quality time series prediction methods, which predict the values of the parameter at time t by utilizing the preceding time series of the same parameter and others. Over the past several decades, many statistical analyses and artificial intelligence modeling methods have been successfully applied for water quality time series prediction. Jayawardena and Lai [7] applied the statistic time series model to analyze the water quality in Pearl River of China. Synthetic water quality data were generated by using the probability distribution of the independent residuals, and forecasting of future water quality data was done by using Auto-Regressive Integrated Moving Average (ARIMA) model. Ahmad et al. [8] used the multiplicative ARIMA model to predict electrical conductivity, chlorides and BOD from the river Ganges in India. Kurunc et al. [9] developed the seasonal ARIMA and TF models using monthly water quality and streamflow time series from 1984 to 1996 for the Yesilirmak River, Turkey. Parmar [10] utilized statistical, fractal and time series analysis method to model BOD, AMM and TKN in Yamuna river, India. Arya and Zhang [11] applied order series method (OSM) to fulfill the normality assumption and then used time series analysis approach to model and predict univariate dissolved oxygen and temperature time series for four water quality assessment stations at Stillaguamish River located in the state of Washington. Due to the fact that most of the statistical-based water quality time series models are linear and distributed normally, they cannot handle the nonlinear prediction problem. In the past few decades, many artificial intelligent approaches have been used to address the problem of water quality time series prediction, e.g. Artificial Neural Network

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W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

(ANN), Support Vector Machine (SVM) and some hybrid approaches. French et al. [12] applied the ANNs to predict algal blooms from water quality databases. Maier and Dandy [13–16] have done a series of researches about the application of ANNs in the field of water quality prediction. Xu et al. [17] and Alizadeh et al. [18] developed wavelet neural network (WNN) model to predict dissolved oxygen and temperature. Burchard-Levine et al. [19] examined the ability of GA-ANN model for ammonia-nitrogen (NH3-N), chemical oxygen demand in manganese (CODMn) and total organic carbon (TOC) prediction. Chung et al. [20] applied ANN coupled with Gamma Test (ANN–GT) to predict NH3-N base on water quality, hydrological and economic data. Besides the ANNs, many other data driven models have also been successfully applied for water quality time series prediction [21–25]. Partalas et al. [26] studied the greedy ensemble selection family of algorithm for ensembles of regression models and applied it to water quality time series prediction. Hatzikos et al. [27] discussed the problem of predicting future values for a number of water quality parameters. They investigated the ability to predict future values for a varying number of days ahead and the effect of including values from a varying number of past values. Liu et al. [28] proposed a hybrid approach of support vector regression (SVR) with genetic algorithm optimization for aquaculture water quality prediction. Faruk et al. [29] analyzed the advantages of the linear model and nonlinear model, and developed a hybrid neural network and ARIMA model to predict boron, dissolved oxygen and water temperature. However, the aforementioned models cannot obtain good prediction performance when the water quality data possess uncertain characteristics caused by instruments or sensors. There is still a burning need to develop models that can handle the uncertainty inherent in certain water quality data, such as inaccuracies in measurements, incomplete sets of observations, or difficulties in obtaining the measurements. In recent years, granular computing and fuzzy time series prediction models, as two popular uncertain information processing approaches, have been applied to various time series problem domains, such as stock prices forecasting [30–32] and inventory demand [33,34]. The main objective of this study is to develop a multi-factor water quality time series prediction model, which cannot only gain higher prediction accuracy but also handle the uncertain dataset efficiently, based on Gaussian cloud transformation, the approximate period of water quality parameter and fuzzy time series model. Firstly, the hybrid model utilizes the Gaussian cloud transformation algorithm to discretize historical data and abstract the water quality time series into concepts at different granularities. Then, the length of approximate periodicity for the forecasted parameter is calculated to construct the training dataset. Thirdly, the multi-factor fuzzy time series model is adopted to predict the water quality parameter. At last, we use the adaptive expectation model (AEM) to modify the predicted value further. In order to test the prediction performance of the proposed method, two water quality time series datasets were collected to do the numerical experiment. The first dataset consisted of dissolve oxygen (DO) and chemical oxygen demand in manganese (CODMn), covered the period from the first week of January, 2004 to the last week of December, 2013 in the upper reaches of Yangtze River, China. It was used to examine the performance of DO and CODMn prediction. We performed the water temperature and electric conductivity prediction on the second dataset, which consisted of water temperature and electric conductivity at three stations in the River Beas, Deep Bay Water Control Zone, Hong Kong. The dataset included 348 samples which were recorded regularly each month at three stations over a period of 29 years from 1986 to 2014. The experimental results showed that the proposed method significantly improved the prediction accuracy compared to some existing time series prediction models (ARMA, RBF-NN, NAR, SVM, ANN–GT and OSM) mentioned above for single-step-ahead water quality time series prediction.

The rest of this paper is organized as follows. Section 2 briefly reviews some basic concepts of fuzzy time series and cloud model. In Section 3, a hybrid water quality time series prediction model based on Gaussian cloud transformation, the approximate period of water quality parameter and fuzzy time series model is presented. The experimental design and experimental results are discussed in Section 4. And the last section summarizes the conclusions. 2. Preliminaries 2.1. Fuzzy time series In this section, we briefly review some basic concepts and definitions of fuzzy time series. Fuzzy time series model was first introduced by Song and Chissom [35], which was used to forecast the enrollments in the University of Alabam. Chen [36] improved the model by simplifying the union operation. Later, many researchers have studied fuzzy time series model both in theoretical and practical application [30–34]. Some definitions of fuzzy time series are described as follows: Definition 1. Let U be the universe of discourse, U = {u1, u2, ⋯, un}. A fuzzy set A in U can be defined as A = fA(u1)/u1 + fA(u2)/ u2 + ⋯ + fA(un)/un, where fA is the membership function of the fuzzy set A and fA : U → [0, 1], fA(ui) denotes the membership degree of ui in the fuzzy set A, 1 ≤ i ≤ n, the symbol “+” denotes the operation of union. Definition 2. Let Y(t)(t = ⋯, 0, 1, 2, ⋯) be the universe of discourse in which fuzzy sets fi (t)(i = 1, 2, ⋯) are defined. Assume that F(t) = {f 1(t), f2(t), ⋯}. Then, F(t) is called a fuzzy time series on Y(t)(t = ⋯, 0, 1, 2, ⋯). Definition 3. Let F(t) and F(t − 1) be fuzzy sets in time series. Assume that F(t) is caused by F(t − 1), then the fuzzy logical relationship can be represented as F(t − 1) → F(t). If F(t − 1) = Ai and F(t) = Aj, then the fuzzy logical relationship can be represented as Ai → Aj, where “Ai” and “Aj”are called “the current state” and “the next state”. Definition 4. Let F(t) is caused by (F1(t − 1), F2(t − 1), ⋯, Fn(t − 1)), then the multi-factor fuzzy logical relationship can be expressed as (F1(t − 1), F2(t − 1), ⋯, Fn(t − 1)) → F(t), where F1(t − 1) is called the main factor and Fi(t − 1)(2 ≤ i ≤ n) are called the secondary factors. Let F1(t − 1) = A1,i, F2(t − 1) = A2,i, ⋯ , Fn(t − 1) = An,i, and F(t) = Aj, then the multi-factor fuzzy logical relationship can be expressed as A1,i, A2,i, ⋯, An,i → Aj, where “A1,i, A2,i, ⋯, An,i” are called “the current state” and “Aj” is called “the next state”.

2.2. Gaussian cloud model The cloud model, as a new cognition model of uncertainty, was proposed by Li [37] based on the probability theory and fuzzy sets theory. It is a significant approach to realize the bidirectional cognitive transformation between the qualitative concepts and the quantitative description. In the cloud model theory, it is possible to measure the deviation of a random phenomenon from a normal distribution when the random phenomenon does not satisfy strictly a normal distribution [38]. Meanwhile, cloud models can formally describe the inherent relation between randomness and fuzziness. Definition 5. Let U be an universal set described by precise numbers, and C be a qualitative concept on U. If there is a number x ∈ U, where x is a random realization of the concept C, and the certainty degree of x for C, i.e., μ(x) ∈ [0, 1], is a random value with stabilization tendency. Then the distribution of x is called a cloud on the universe of discourse and each x is called a cloud drop. Cloud models use three numerical characteristics to describe the intension of a concept, namely Ex (Expectation), En (Entropy), He (Hyper

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

41

Entropy), where Ex is the expected sample of a concept with membership degree 1, En is used to depict the uncertainty of samples in the concept, and He is used to depict the uncertainty of the membership degree. The Gaussian cloud model based on Gaussian distribution function and Gaussian membership function is the most important cloud model, which can be defined as follows: Definition 6. Let U be an universal set described by precise numbers, and C(Ex, En, He) be a qualitative concept on U. If a quantitative number x(x ∈ U) is a random realization of the concept C, and x is subjected to the Gaussian distribution x ~ N(Ex, En′2), where En′ is a random realiza2 tion of the Gaussian distribution n En′ ~2 oN(En, He ), and the certainty ðx−ExÞ , then the distribution of x on degree of x on U is μðxÞ ¼ exp − 2ðEn0 Þ2 U is a Gaussian cloud.

3. The proposed model In this section, a multi-factor water quality time series prediction model (namely GCT-FTS) based on Heuristic Gaussian cloud transformation, the approximate periodicity of water quality parameter and fuzzy time series model is presented, which can be divided into four stages clearly differentiated as Fig. 1. In the proposed model, the parameter used for prediction is called the main factor, and others are called the elementary secondary factors. Fig. 2 shows the flowchart of the proposed model. All the steps composing the hybrid model are going to be described in the following subsection.

Fig. 2. Flowchart of the proposed hybrid water quality time series prediction model.

Let X{xi|i = 1, 2, ⋯, N} be the quantitative data sample, assume that the number of Gaussian cloud model is M and the threshold value of error is ε, then the steps composing HGCT can be described as follows: Step 1 Count the frequency distribution of data sample X{xi|i = 1, 2, ⋯, N}

3.1. Heuristic Gaussian cloud transformation Heuristic Gaussian cloud transformation (HGCT) is a method based on Gaussian mixture model (GMM), which can transform a data set in problem domain into multiple Gaussian clouds (or concepts) with different granularities in cognition domain. It provides a soft partitioning method about transforming the quantitative data into multiple qualitative concepts. The Confusion Degree (CD), CD = 3He/En, is used to measure the overlap extent between concepts generated by HGCT. The value of CD can be changed by controlling the input parameter (the number of Gaussian clouds) of HGCT.

  h y j ¼ pðxi Þ; i ¼ 1; 2; ⋯; N; j ¼ 1; 2; ⋯; N0

where y denotes the universe of discourse of data sample. Step 2 Initialize the expectation uk, the standard deviation σk and the amplitude ak of the kth(k = 1, ⋯, M) Gaussian distribution as follows: uk ¼

k  maxðX Þ Mþ1

ð2Þ

σ k ¼ maxðX Þ Stage 1: Heuristic Gaussian cloud transformation For each factor Raw data

Perform Gaussian Transfor -mation method to generate Gaussian distributions

Transform the Gaussian distributions into Gaussian clouds

ak ¼ Gaussian clouds

Current state

Training dataset, Gaussian clouds

Construct the training dataset

Training dataset

( N0 X

hðyi Þ  ln

i¼1

) M  X   ak g yi ; uk ; σ 2k

ð5Þ

k¼1

where ðyi −uk Þ2

Stage 3: Fuzzy time series model

−   1 g yi ; uk ; σ 2k ¼ pffiffiffiffiffiffi e 2πσ k

Fuzzify the historical data

Define fuzzy sets

Establish FLR Construct FLRG

Defuzzy and predict the value

Predicted value

Stage 4: Modifying the predicted value Current state Predicted value

ð4Þ

Step 3 Calculate the objective function J(θ), as shown by: J ðθÞ ¼

Calculate the length of approximate periodicity for the main factor

ð3Þ

1 M

Stage 2: Constructing the training dataset Raw data

ð1Þ

Establish adaptive expectation model

Modify the predicted value

ð6Þ

Step 4 For the kth(k = 1, ⋯, M) Gaussian distribution, update the parameters based on the maximum likelihood estimation as follows: N X

Output

uk ¼ Fig. 1. The detailed four stages of the proposed water quality time series prediction model.

2σ 2 k

Lk ðxi Þxi

i¼1 N X i¼1

ð7Þ Lk ðxi Þ

42

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49 N X

σ 2k ¼

coarse-grained time unit, then construct the occurrence vector Vw of wave trough, shown as follows:

Lk ðxi Þðxi −uk ÞT ðxi −uk Þ

i¼1 N X

ð8Þ fT 1 ; T 2 ; ⋯T i ; ⋯; T N gð1≤T i ≤M; 1≤i ≤NÞ

Lk ðxi Þ

i¼1 N 1X ak ¼ L ðxi Þ N i¼1 k

ð9Þ

where   ak g xi ; uk ; σ 2k M  X   an g xi ; un ; σ 2n

Lk ðxi Þ ¼

ð10Þ

n¼1

Step 5 Calculate the new value of objective function Jð~θÞ, as shown by: ( ) N0 M    X X   2 ~ hðyi Þ  ln ak g yi ; uk ; σ k J θ ¼

Thus, the length of the approximate periodicity can be calculated as follows: L = ⌊STD/2⌋ × 2, where STD denotes the standard variation of the occurrence vector Vw. For example, in this paper, the frequency of data collection for dissolved oxygen(DO) is 1 time/week, the initial training dataset covered a period from 2004 to 2012 totally 9 years, and the wave trough of year appeared sequential in the 28th week, the 34th week, the 41th week, the 31th week, the 18th week, the 34th week, the 19th week, the 10th week, and the 20th week, then we can construct the occurrence vector Vw = {28, 34, 41, 31, 18, 34, 19, 10, 20}. Thus, the length of the approximate periodicity L can be calculated by L = ⌊STD/ 2⌋ × 2 = 8, where the standard variation of the occurrence vector STD = 8.3183.

uk−1 þ 3α 1 σ k−1 ¼ uk −3α 1 σ k

ð12Þ

uk þ 3α 2 σ k ¼ ukþ1 −3α 2 σ kþ1

ð13Þ

3.2.2. Constructing the training dataset Assume that the next fine-grained time unit needed to predict is the tth time unit. Then the “current state” is the data sample in the (t − 1)th time unit and the data samples in the period of [t − 1 − L/2, t − 1 + L/2] in each coarse-grained time unit can be selected to construct the training dataset. Following the example in Section 3.2.1, assume that we need to predict the value of DO in the 6th week, 2013. Then the “current state” is the data sample in the 5th week, 2013. Therefore, the data samples from the first week to the ninth week in every year (2004–2012) will be selected as the training samples.

α k ¼ minðα 1 ; α 2 Þ

ð14Þ

3.3. Fuzzy time series model

i¼1

ð11Þ

k¼1

If jJð~θÞ−JðθÞjbε, then go to Step 6; otherwise, go to Step 3. Step 6 For the kth(k = 1, ⋯, M) Gaussian distribution, calculate the standard deviation's convergent–divergent ratio αk as follows:

th

Step 7 Calculate the final parameters of the k (k = 1, ⋯, M) Gaussian cloud, as shown by: Exk ¼ uk

ð15Þ

Enk ¼ ð1 þ α k Þ  σ k =2

ð16Þ

Hek ¼ ð1−α k Þ  σ k =6

ð17Þ

CDk ¼ ð1−α k Þ=ð1 þ α k Þ

ð18Þ

In this paper, we use the HGCT algorithm to generate different numbers of Gaussian clouds for different water quality parameters. E.g. for the ith factor, the historical numerical time series can be extracted into mi Gaussian clouds Ci, j(Exi, j, Eni, j, Hei, j)(1 ≤ i ≤ p, 1 ≤ j ≤ mi), where p is the number of all the water quality parameters used for prediction. 3.2. Constructing the training dataset It is well known that many water quality parameters exist the approximate periodicity over time. E.g. for dissolved oxygen, the wave crest of its yearly concentration curve appears in winter while the wave trough appears in summer, or the wave crest of its daily concentration curve appears in the evening while the wave trough appears at noon. In this study, we utilize the approximate periodicity to construct the training dataset, and the process can be shown in the following two subsections. 3.2.1. Calculating the length of approximate periodicity For the main factor(the parameter needed to predict), assume that the period of the historical water quality dataset covered N coarsegrained time units (e.g. year), which can be divided into M finergrained time units (e.g. week), and assume that the wave trough (or the wave crest) appeared at the T i th fine-grained time unit in the ith

The typical fuzzy time series model consists of four steps: (1) determine and partition the universe of discourse into intervals; (2) define fuzzy sets and fuzzify the historical samples; (3) establish the fuzzy logical relationships (FLR) and group the FLRs into fuzzy logical relationship groups (FLRG); (4) defuzzify and calculate the predicted outputs. In this paper, as the historical numerical time series have been extracted into Gaussian clouds in Section 3.1, the proposed model can omit the first step above and define fuzzy sets based on the Gaussian clouds. The remaining three steps are going to be depicted in the subsequence subsections. 3.3.1. Defining fuzzy sets and fuzzifying the historical samples Define mi fuzzy sets for the ith factor based on the mi Gaussian clouds Ci, j(Exi, j, Eni, j, Hei, j) obtained in Section 3.1, shown as follows: Ai;1 ¼ 1=C i;1 þ 0:5=C i;2 þ 0=C i;3 þ 0=C i;4 þ ⋯ þ 0=C i;mi −1 þ 0=C i;mi Ai;2 ¼ 0:5=C i;1 þ 1=C i;2 þ 0:5=C i;3 þ 0=C i;4 þ ⋯ þ 0=C i;mi −1 þ 0=C i;mi Ai;3 ¼ 0=C i;1 þ 0:5=C i;2 þ 1=C i;3 þ 0:5=C i;4 þ ⋯ þ 0=C i;mi −1 þ 0=C i;mi ⋮ Ai;mi ¼ 0=C i;1 þ 0=C i;2 þ 0=C i;3 þ 0=C i;4 þ ⋯ þ 0:5=C i;mi −1 þ 1=C i;mi If i = 1, then A1;1 ; A1;2 ; ⋯; A1;m1 are the fuzzy sets for the main factor; otherwise, Ai;1 ; Ai;2 ; ⋯; Ai;m1 are the fuzzy sets for the ith secondary factor. Fuzzify the historical samples based on the fuzzy sets and Gaussian clouds. E.g. for the ith factor, assume that the observed value is xi,t, then calculate the certainty degree (ui, j) of xi,t for all Gaussian clouds Ci, j(Exi, j, Eni, j, Hei, j) (1 ≤ j ≤ mi), respectively. Assume that Ci,max is the Gaussian cloud that the maximum certainty degree ui,max related to, then the observed value xi,t can be fuzzified into the fuzzy set Ai,max. 3.3.2. Establishing fuzzy logical relationships Establish multi-factor fuzzy logical relationships (FLRs) by searching all fuzzy sets in time series with the pattern (F1(t − 1), F2(t − 1), ⋯, Fp(t − 1)) → F(t). For example, assume that the fuzzy sets for all factors at time t − 1 are A1,i1, A2,i2, ⋯, and Ap,ip, respectively, and the

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

fuzzy sets for the main factor at time t is A1,k. Then, we can establish the FLR: A1,i1, A2,i2, ⋯, Ap,ip → A1,k. Construct the fuzzy logical relationships groups(FLRG) based on FLRs and the training dataset. In the training dataset, all FLRs having the same “current state” will be grouped into the same FLRG. For example, if “A1,i1, A2,i2, ⋯, Ap,ip” is the “current state” of one FLR in the training dataset and there are r FLRs in the training dataset as follows:

Table 1 The statistical summary of DO and CODMn dataset. Parameters

Stations

Skewness

Std

Mean

Max

Min

DO

Station 1 Station 2 Station 3 Station 1 Station 2 Station 3

0.4540 0.8450 0.9100 2.0988 1.9703 1.3755

1.3594 1.3187 0.7479 0.8880 1.0207 1.1779

8.4761 8.8865 8.7898 2.0048 2.2733 1.6023

13.0 14.4 13.9 7.6 9.8 6.9

5.88 5.07 6.94 0.7 0.5 0

CODMn

A1;i1 ; A2;i2 ; ⋯; Ap;ip →A1;k1 ; A1;i1 ; A2;i2 ; ⋯; Ap;ip →A1;k2 ; ⋯; A1;i1 ; A2;i2 ; ⋯; Ap;ip →A1;kr Then the r FLRs can be grouped into the same FLRG, shown as follows: A1;i1 ; A2;i2 ; ⋯; Ap;ip →A1;k1 ; A1;k2 ; ⋯; A1;kr 3.3.3. Defuzzifying and calculating the predicted value Assume that the “current state” at time t − 1 is “A1,i1, A2,i2, ⋯, Ap,ip”, then we can defuzzify and calculate the predicted value at time t by following the rule: Rule 1 If there is only one FLR in the FLRG, shown as follows: A1;i1 ; A2;i2 ; ⋯; Ap;ip →A1;k1 then, the predicted value P(t) at time t can be calculated as follows:   P ðt Þ ¼ 1=2  Ex1;k1 þ Sðt−1Þ where Ex1,k1 is the expectation of the Gaussian cloud C1,k1 corresponding to A1,k1, S(t − 1) denotes the observed value at time t − 1 for the main factor. Rule 2 If there are r FLRs in the FLRG, shown as follows: A1;i1 ; A2;i2 ; ⋯; Ap;ip →A1;k1 ; A1;k2 ; ⋯; A1;kr then, the predicted value P(t) at time t can be calculated as follows: P ðt Þ ¼ 1=2 



n1  Ex1;k1 þ n2  Ex1;k2 þ ⋯ þ nr  Ex1;kr þ Sðt−1Þ n1 þ n2 þ ⋯ þ nr

where ni is the number of A1,ki appearing in the FLRG, Ex1,ki is the expectation of the Gaussian cloud C1,ki corresponding to A1,ki, 1 ≤ i ≤ r, S(t − 1) denotes the observed value at time t − 1 for the main factor. Rule 3 If there is no FLR in the FLRG, shown as follows: A1;i1 ; A2;i2 ; ⋯; Ap;ip →# where the symbol “#” denotes an unknown value. Then, the predicted value P(t) at time t can be calculated as follows:   P ðt Þ ¼ 1=2  Ex1;i1 þ Sðt−1Þ where is the expectation of the Gaussian cloud Ex1,i1 corresponding to A1,i1, S(t − 1) is the observed value C1,i1 at time t − 1 for the main factor. 3.4. Modifying the predicted value To optimize the model's prediction performance, we utilize the adaptive expectation model (AEM) to modify the predicted value, shown as follows: FP ðt Þ ¼ Sðt−1Þ þ h  ðP ðt Þ−Sðt−1ÞÞ

43

where FP(t) denotes the final predicting output at time t, h is the weight coefficient, P(t) is the predicted value obtained in Section 3.3.3, and S(t − 1) denotes the observed value at time t − 1. 4. Experimental results and discussion 4.1. Data source In numerical experiments, two water quality time series datasets coming from the real world are used. Both of them are collected from three water quality monitoring stations (namely Station 1, Station 2 and Station 3) with upstream–downstream relationship. The first dataset is used to predict DO and CODMn, while the second dataset is utilized to do the water temperature and EC prediction. The statistical summary of two datasets is demonstrated in Tables 1 and 2, respectively. More generally, we consider here: (1) DO and CODMn time series dataset The dataset derives from three water quality monitoring stations in the upper reaches of the Yangtze River Mainstream, China, (http://datacenter.mep.gov.cn/report/getCount-Graph.do?type= runQianWater). Station 1 locates in the downstream of Station 2 with about 250 km apart, while Station 3 locates in the upstream of Station 2 with about 700 km apart. The dataset consists of DO and CODMn totaly 520 samples which were monitored regularly each week at three stations over a period of ten years from 2004 to 2013. The weekly DO and CODMn dataset covered from 2004 to 2012, totally 468 samples, are selected as the initial training dataset, while the remaining data are used as the testing data. (2) Water temperature and EC time series dataset The dataset, collected regularly each month at three stations in the River Beas, Deep Bay Water Control Zone, Hong Kong (http://epic. epd.gov.hk/EPICRIVER/river/display/search/), includes water temperature and Electric Conductivity totaly 348 samples over a period of 29 years from 1986 to 2014. The monthly water temperature and EC dataset covered from 1986 to 2012, totally 324 samples, are selected as the initial training dataset, while the remaining data are used as the testing data. 4.2. Experiment setup and Performance criteria design In this paper, we have selected DO, CODMn, water temperature and EC to evaluate the models performance. Each parameter has been designed to an independent experiment, namely DO prediction, CODMn prediction, water temperature prediction and EC prediction. In order to make full use of the relationships between the upstream stations Table 2 The statistical summary of water temperature and EC dataset. Parameters

Stations

Skewness

Std

Mean

Max

Min

Water temperature

Station 1 Station 2 Station 3 Station 1 Station 2 Station 3

−0.4763 −0.4075 −0.4459 5.0627 2.3953 2.7348

4.9121 4.6019 4.7803 292.18 256.36 329.71

24.8129 23.6687 23.9790 316.25 362.76 456.10

34.4 33.4 34.2 3600 1730 2700

7.8 7.5 8.0 59 65 86

EC

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W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

and the downstream stations, we not only utilized the water quality parameter at upstream stations as a part of the inputs to predict the parameter at downstream stations, but also selected the water quality parameter at downstream stations as a part of the inputs to predict the parameter at upstream stations. For example, in the experiment of DO prediction, the inputs of the model consisted of the preceding time series of DO at all three stations(Station 1, Station 2 and Station 3). For the sake of comparing the proposed hybrid models prediction accuracy, some existing water quality time series prediction models (ARMA, RBF-NN, NAR, SVM, ANN–GT and OSM) were performed on the dataset used in this study. The ARMA, NAR and OSM model predicted the values of the parameter at Station t(t = 1, 2, 3) only by utilizing the preceding time series of the same parameter at Station t, while the inputs of the RBF-NN and SVM consisted of the same parameters at three Stations. For ANN–GT, all the water quality parameters at three stations are selected to do the key factor selection through the Gamma Test. The performances of the models were evaluated according to four statistical indexes: the Mean Square Error (MSE), the Mean Absolute Percentage Error (MAPE), the Nash–Sutcliffe Coefficient of Efficiency (CE), and the Pearson product–moment correlation coefficient (R). All the performance criteria can be computed as follows: MSE ¼

n  2 1X y −yp;i n i¼1 m;i

ð19Þ

   n  1X ym;i −yp;i    n i¼1  ym;i 

ð20Þ

MAPE ¼

n  2 X ym;i −yp;i

CE ¼ 1− i¼1 n  X 2 ym;i −ym

ð21Þ

i¼1 n   X  ym;i −ym yp;i −yp i¼1 ffi R ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  n  2 X 2 X ym;i −ym yp;i −yp i¼1

ð22Þ

i¼1

where n denotes the number of data needed to predict, ym,i and yp,i are the measured value and the predicted value, respectively, while ym and yp are the mean of the measured value and the predicted value, respectively. 4.3. Dissolve oxygen (DO) prediction The dataset used to predict DO consists of the DO time series at all three stations in dataset 1. The proposed multi-factor water quality time series prediction model (GCT-FTS) performed DO prediction at each station, and the three-factors GCT-FTS can be express as follows: DO−N ¼ f GCT− FTS ðDO−1; DO−2; DO−3Þ

ð23Þ

where DO–N is the dissolve oxygen at Station N(N = 1, 2, 3) and fGCT − FTS is the predictor constructed by GCT-FTS model. Generally, the water quality pollution of a downstream station is affected by the discharge of local area from the upstream station. On the contrary, the water quality in downstream is a reflection of the water quality in upstream. In order to investigate the relationship between the stations, the two-factor GCT-FTS models and the one-factor GCT-FTS model were performed on the dataset. The experimental results are shown in Tables 3, 4, and 5, which imply that the inputs consisting of DO-1 and DO-2 can obtain the best prediction performance for DO prediction at Station 1, that the inputs consisting of DO-2 and DO-3 can obtain the best prediction performance for DO prediction at

Table 3 DO prediction of the proposed model at Station 1. Input parameters

Numbers of cloud models

MSE

MAPE(%)

CE

R

DO-1 DO-1, DO-2 DO-1, DO-3 DO-1, DO-2, DO-3

m1 = 12 m1 = 11, m2 = 6 m1 = 12, m3 = 8 m1 = 18, m2 = 16, m3 = 8

0.1339 0.1317 0.1397 0.1339

3.1888 3.1740 3.2774 3.2435

0.8479 0.8504 0.8414 0.8473

0.9213 0.9230 0.9189 0.9245

Table 4 DO prediction of the proposed model at Station 2. Input parameters

Numbers of cloud models

MSE

MAPE(%)

CE

R

DO-2 DO-2, DO-1 DO-2, DO-3 DO-2, DO-1, DO-3

m2 = 6 m2 = 19, m1 = 12 m2 = 6, m3 = 6 m2 = 6, m1 = 8, m3 = 19

0.2332 0.2221 0.2105 0.2135

4.3700 4.2443 4.1792 4.2635

0.7547 0.7663 0.7785 0.7753

0.8801 0.8882 0.8917 0.8873

Station 2, and that the inputs consisting of DO-1, DO-2, and DO-3 can obtain the best prediction performance for DO prediction at Station 3. In other word, the relative importance of DO-2 is higher than DO-3 for DO-1 prediction, the relative importance of DO-3 is higher than DO-1 for DO-2 prediction, and both DO-1 and DO-2 are very important to predict DO-3. In addition, all the results demonstrate that the performance of GCT-FTS model for DO prediction is good as the mean of MSE is 0.1349, the mean of MAPE is 3.1663, the mean of CE is 0.8188, and the mean of R is 0.9090. The proposed model is compared with the some existing water quality time series prediction models (ARMA, RBF-NN, NAR, SVM, ANN–GT and OSM). The accuracy of the MSE, MAPE, CE, and R of the different models for DO prediction at each station is shown in Figs. 3–5. Fig. 3 shows that the MSE and MAPE of the GCT-FTS model are obviously smaller than the other models while the CE and R are bigger than the other models. The average prediction accuracy of the MSE, MAPE, CE, and R for traditional models at Station 1 is 0.1573, 3.4024, 0.8211, and 0.9079, respectively. It means that the proposed model achieves a remarkable average improvement with respect to the MSE with a value of 16.21%, MAPE with a value of 3.02%, CE with a value of 3.56%, and R with a value of 1.67%. Fig. 4 illustrates that the average values of the MSE, MAPE, CE, and R for traditional models at Station 2 are 0.2414, 4.3577, 0.7461, and 0.8694, respectively. With the application of the GCT-FTS model, there is an average decrease of 12.80% and 4.10% for MSE and MAPE as well as an average increase of 4.35% and 2.57% for CE and R. In Fig. 5, all the models obtain a higher prediction accuracy with respect to the MSE with an average value of 0.0548, MAPE with an average value of 2.0780, CE with an average value of 0.8289, and R with an average value of 0.9157. The proposed model still increases the prediction accuracy with the average improvements of 13.48%, 5.94%, 2.48%, and 0.71% for MSE, MAPE, CE and R, respectively. Figs. 3–5 indicate that the proposed model has achieved a higher prediction accuracy, which obtained the average improvements of

Table 5 DO prediction of the proposed model at Station 3. Input parameters

Numbers of cloud models

MSE

MAPE(%)

CE

R

DO-3 DO-3, DO-1 DO-3, DO-2 DO-3, DO-1, DO-2

m3 = 15 m3 = 10, m1 = 5 m3 = 21, m2 = 8 m3 = 7, m1 = 12, m2 = 16

0.0513 0.0505 0.0513 0.0474

1.9584 1.9514 2.0649 1.9546

0.8372 0.8396 0.8370 0.8495

0.9168 0.9166 0.9168 0.9223

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

45

Fig. 3. Comparisons of prediction accuracy for DO prediction at Station 1.

Fig. 4. Comparisons of prediction accuracy for DO prediction at Station 2.

14.16%, 4.35%, 3.46%, and 1.65% for MSE, MAPE, CE and R, respectively, compared with the traditional models. That is to say the GCT-FTS model is an efficient time series model for DO prediction.

multi-factor water quality time series prediction model (GCT-FTS) performed prediction at each station, and the three-factors GCT-FTS can be express as follows:

4.4. Chemical oxygen demand in manganese (CODMn) prediction

CODMn −N ¼ f GCT− FTS ðCODMn −1; CODMn −2; CODMn −3Þ

The dataset used to predict CODMn consists of the time series of the CODMn time series at all three stations in dataset 1. The proposed

where CODMn-N is the CODMn at Station N(N = 1, 2, 3) and fGCT − FTS is the predictor constructed by GCT-FTS model.

Fig. 5. Comparisons of prediction accuracy for DO prediction at Station 3.

ð24Þ

46

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

Table 6 CODMn prediction of the proposed model at Station 1. Input parameters

Numbers of cloud models

MSE

MAPE(%)

CE

R

CODMn-N(N = 1) CODMn-N(N = 1,2) CODMn-N(N = 1,3) CODMn-N(N = 1,2,3)

m1 = 17 m1 = 25, m2 = 6 m1 = 20, m3 = 23 m1 = 22, m2 = 6, m3 = 23

0.1339 0.0996 0.1144 0.0938

14.3343 13.6980 13.2556 12.6279

0.7431 0.8090 0.7806 0.8200

0.8629 0.9003 0.8850 0.9059

Table 7 CODMn prediction of the proposed model at Station 2. Input parameters

Numbers of cloud models

MSE

MAPE(%)

CE

R

CODMn-N(N = 2) CODMn-N(N = 1,2) CODMn-N(N = 2,3) CODMn-N(N = 1,2,3)

m2 = 5 m2 = 5, m1 = 6 m2 = 5, m3 = 16 m2 = 5, m1 = 9, m3 = 10

0.1853 0.1899 0.1726 0.1729

11.4677 11.4282 10.6766 10.7295

0.7092 0.7020 0.7292 0.7286

0.8442 0.8408 0.8631 0.8595

Table 8 COD Mn prediction of the proposed model at Station 3. Input parameters

Numbers of cloud models

MSE

MAPE(%)

CE

R

CODMn-N(N = 3) CODMn-N(N = 1,3) CODMn-N(N = 2,3) CODMn-N(N = 1,2,3)

m3 = 13 m3 = 25, m1 = 17 m3 = 9, m2 = 10 m3 = 10, m1 = 8, m2 = 5

0.0692 0.0545 0.0638 0.0537

16.9065 15.4549 16.1203 15.4875

0.6888 0.7548 0.7133 0.7584

0.8488 0.8883 0.8600 0.8821

Fig. 6. Comparisons of prediction accuracy for CODMn prediction at Station 1.

Fig. 7. Comparisons of prediction accuracy for CODMn prediction at Station 2.

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

47

Fig. 8. Comparisons of prediction accuracy for CODMn prediction at Station 3.

The experimental results of all multi-factor GCT-FTS models (including one-factor model, two-factors models, and three-factors model) for CODMn prediction at each station are shown in Tables 6, 7, and 8. It is clear that both CODMn-1, CODMn-2 and CODMn-3 are very important to predict CODMn-1 and CODMn-3, while the relative importance of CODMn-3 is higher than CODMn-1 for CODMn-2 prediction. These results illustrate that the GCT-FTS model performs well for CODMn prediction with the average of MSE is 0.1160, the average of MAPE is 13.5156, the average of CE is 0.7448, and the average of R is 0.8701. We made a comparison of the performance between the proposed model and some existing models (ARMA, RBF-NN, NAR, SVM, ANN–GT and OSM). The accuracy of the MSE, MAPE, CE, and R of the different models for CODMn prediction at each station is shown in Figs. 6–8. Fig. 6 gives the value of MSE, MAPE, CE, and R for five different models used in this study. The average MSE, MAPE, CE, and R of the traditional models can be calculated, where the average MSE is equal to 0.1303, the average MAPE is equal to 15.08, the average CE is equal to 0.7337, and the average R is equal to 0.8649. Thus, the average improvements of the proposed model for the MSE, MAPE, CE, and R are 31.59%, 16.30%, 11.77% and 4.74%, respectively. In Fig. 7, we can see that the average MSE, MAPE, CE, and R for traditional models at Station 2 are 0.2074, 12.0155, 0.6762, and 0.8477, respectively. Applying the GCT-FTS model, there is a decrease of 16.77% and 11.14% for MSE and MAPE as well as an increase of 7.84% and 1.81% for CE and R. Fig. 8 shows that the GCT-FTS model outperforms the traditional models which obtain the average prediction accuracy of 0.0685, 16.6953, 0.6875, and 0.8429 for the MSE, MAPE, CE, and R, respectively. With the application of the proposed model, the average improvements are 21.63%, 7.23%, 10.30%, and 4.65% for the MSE, MAPE, CE, and R, respectively. Figs. 6–8 indicate that a remarkable improvement in CODMn prediction is achieved with the proposed GCT-FTS model. The average improvements are 23.33%, 11.55%, 9.97%, and 3.73% for CODMn for the MSE, MAPE, CE, and R, respectively. The experimental results demonstrate that the GCT-FTS model performs well for adequate predicting CODMn.

where Tmp-i is the water temperature at Station i(i = 1, 2, 3) and fGCT ‐ FTS is the predictor constructed by GCT-FTS model. The experimental results are shown in Table 9. It is clear that Tmp-3 is much more important than Tmp-2 for calculating the predicting MSE, while the relative importance of Tmp-2 is higher than Tmp-3 for calculating the predicting MAPE. Overall, the relative importance of Tmp-2 is a litter higher than Tmp-3 for water temperature prediction. These results demonstrate that the GCT-FTS model performs well for water temperature prediction with the average of MSE is 5.3441 and the average of MAPE is 6.2536. We made a comparison of the performance between the proposed model and some existing models (ARMA, RBF-NN, NAR, SVM, ANN–GT and OSM). The accuracy of the MSE and MAPE of the different models for Water Temperature prediction at Station 1 is shown in Fig. 9. The average MSE and MAPE of the existing models can be calculated, where the average MSE is equal to 7.0480 and the average MAPE is equal to 8.6934. Thus, the average improvements of the proposed model for the MSE and MAPE are 34.68% and 24.42%, respectively. 4.6. Electric conductivity(EC) prediction The dataset used to predict EC consists of the electric conductivity time series at all three stations in dataset 2. Due to the measured value of electric conductivity range from 59 to 3600, the natural logarithm of EC is taken to preprocess the dataset. The proposed multi-factor water quality time series prediction model (GCT-FTS) performed prediction at Station 1, and the three-factor GCT-FTS can be express as follows: EC‐1 ¼ f GCT‐ FTS ðEC‐1; EC‐2; EC‐3Þ

ð26Þ

where EC-i is the electric conductivity at Station i(i = 1, 2, 3) and fGCT ‐ FTS is the predictor constructed by GCT-FTS model. Table 10 shows the experimental results, which implies that EC-2 is much more important than EC-3 to predict Electric Conductivity. In addition, these results illustrate that the GCT-FTS model has a better performance for Electric Conductivity prediction with the average of MSE is 0.0667 and the average of MAPE is 2.9878.

4.5. Water temperature prediction The dataset used to predict water temperature consists of the water temperature time series at all three stations in dataset 2. The proposed multi-factor water quality time series prediction model (GCT-FTS) performed prediction at Station 1, and the three-factor GCT-FTS can be express as follows: Tmp−1 ¼ f GCT‐ FTS ðTmp−1; Tmp−2; Tmp−3Þ

ð25Þ

Table 9 Water temperature prediction of the proposed model at Station 1. Input parameters

Numbers of cloud models

MSE

MAPE(%)

Tmp-1 Tmp-1, Tmp-2 Tmp-1, Tmp-3 Tmp-1, Tmp-2, Tmp-3

m1 = 8 m1 = 6, m2 = 14 m1 = 5, m2 = 5 m1 = 6, m2 = 12, m3 = 12

5.1692 5.3268 4.9182 5.9224

6.3751 5.6787 6.7498 6.2110

48

W. Deng et al. / Chemometrics and Intelligent Laboratory Systems 149 (2015) 39–49

Fig. 9. Comparisons of prediction accuracy for Water Temperature prediction at Station 1.

periodicity of water quality parameter and fuzzy time series model. The novelty of the proposed model can be summarized as follows:

Table 10 EC prediction of the proposed model at Station 1. Input parameters

Numbers of cloud models

MSE

MAPE(%)

EC-1, EC-1, EC-2 EC-1, EC-3 EC-1, EC-2, EC-3

m1 = 21 m1 = 20, m2 = 13 m1 = 21, m2 = 5 m1 = 21, m2 = 7, m3 = 5

0.0692 0.0631 0.0671 0.0673

3.0030 2.8229 3.0481 3.0772

The proposed model is compared with the some existing water quality time series prediction models (ARMA, RBF-NN, NAR, SVM, ANN–GT and OSM) and the accuracy of the MSE and MAPE of the different models Electric Conductivity prediction at Station 1 is shown in Fig. 10. The average MSE and MAPE of the existing models can be calculated, where the average MSE is equal to 0.9 and the average MAPE is equal to 3.5403. With the application of the GCT-FTS model, there is an average decrease 29.88% and 20.26% for MSE and MAPE, respectively.

5. Conclusions Numerous studies have been conducted to improve the accuracy of water quality time series prediction, but few studies have attempted to use the cloud model theory and fuzzy time series model to handle the uncertain dataset, which extracted the numerical time series into cloud models and represented it by linguistic value (fuzzy sets). In this paper, we proposed a multi-factor water quality time series prediction model based on Heuristic Gaussian cloud transformation, the approximate

• The numerical time series data are extracted into different numbers of Gaussian clouds for different factors by using the Heuristic Gaussian cloud transformation algorithms. It can achieve the soft partitioning of uncertain numerical data, which the traditional discretization methods can not realize. • This study constructed the training dataset by using the approximate periodicity of the water quality parameters, which can greatly reduce the noise data and obtain high quality training dataset. Then, the prediction accuracy can be significantly improved. • Fuzzy time series prediction model was applied to generate the computation rule and calculate the predicted value. Compared with the traditional models, it represented the historical numerical data as linguistic value and predicted the water quality parameter based on the fuzzy logical relationship groups, which can better handle the uncertain dataset. • To optimize the model's prediction performance, the proposed model applied the adaptive expectation model to modify the predicted value and increase the final prediction performance. DO prediction, CODMn prediction, water temperature prediction and electric conductivity prediction from the proposed model were compared with those obtained from the existing time series models (ARMA, NAR, RBF-NN, SVM, ANN–GT and OSM). Owing to its ability in handling the uncertain dataset, the accuracy measures MSE, MAPE, CE and R demonstrated that the proposed model provided much better

Fig. 10. Comparisons of prediction accuracy for EC prediction at Station 1.

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accuracy over the existing models mentioned above for water quality predictions. Although the proposed hybrid method has greatly improved the traditional water quality time series methods both in data and model, there are still many limitations needed to improved such as the efficient method of employing the approximate periodicity of the water quality parameters. In the future, we will study how to utilize the approximate periodicity more efficient as well as apply the proposed model to provide decision-making supports for water resource management in the Yangtze River. Acknowledgments This work is supported by the National Science and Technology Major Project (2014ZX07104-006) and Hundred Talents Program of CAS (Y21Z110A10). We would like to express thanks to the anonymous reviewers for their invaluable comments and suggestions. Conflicts of interest Authors declare that there are no conflict of interest. References [1] L.-M.L. He, Z.-L. He, Water quality prediction of marine recreational beaches receiving watershed baseflow and stormwater runoff in southern California, USA, Water Res. 42 (10) (2008) 2563–2573. [2] H.R. Maier, A. Jain, G.C. Dandy, K.P. Sudheer, Methods used for the development of neural networks for the prediction of water resource variables in river systems: current status and future directions, Environ. Model Softw. 25 (8) (2010) 891–909. [3] N. Basant, S. Gupta, A. Malik, K.P. Singh, Linear and nonlinear modeling for simultaneous prediction of dissolved oxygen and biochemical oxygen demand of the surface water: a case study, Chemom. Intell. Lab. Syst. 104 (2) (2010) 172–180. [4] A.A. Najah, A. El-Shafie, O.A. Karim, O. Jaafar, Integrated versus isolated scenario for prediction dissolved oxygen at progression of water quality monitoring stations, Hydrol. Earth Syst. Sci. 15 (8) (2011) 2693–2708. [5] H.G. Han, Q.L. Chen, J.F. Qiao, An efficient self-organizing RBF neural network for water quality prediction, Neural Netw. 24 (7) (2011) 717–725. [6] A. Najah, A. El-Shafie, O. Karim, A.H. El-Shafie, Performance of ANFIS versus MLP-NN dissolved oxygen prediction models in water quality monitoring, Environ. Sci. Pollut. Res. 21 (3) (2014) 1658–1670. [7] A.W. Jayawardena, F. Lai, Time series analysis of water quality data in Pearl River, china, J. Environ. Eng. 115 (3) (1989) 590–607. [8] S. Ahmad, I.H. Khan, B.P. Parida, Performance of stochastic approaches for forecasting river water quality, Water Res. 35 (18) (2001) 4261–4266. [9] A. Kurunç, K. Yürekli, O. Çevik, Performance of two stochastic approaches for forecasting water quality and streamflow data from Yeşilırmak River, Turkey, Environ. Model Softw. 20 (9) (2005) 1195–1200. [10] K.S. Parmar, R. Bhardwaj, Statistical, time series, and fractal analysis of full stretch of river Yamuna (India) for water quality management, Environ. Sci. Pollut. Res. 22 (1) (2015) 397–414. [11] F.K. Arya, L. Zhang, Time series analysis of water quality parameters at Stillaguamish River using order series method, Stoch. Env. Res. Risk A. 29 (1) (2015) 227–239. [12] F. Recknagel, M. French, P. Harkonen, K.I. Yabunaka, Artificial neural network approach for modelling and prediction of algal blooms, Ecol. Model. 96 (1) (1997) 11–28.

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