Novel approach for streamflow forecasting using a ...

Journal of Hydrology 554 (2017) 263–276

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Research papers

Novel approach for streamflow forecasting using a hybrid ANFIS-FFA model Zaher Mundher Yaseen a,b,⇑, Isa Ebtehaj c, Hossein Bonakdari c, Ravinesh C. Deo d, Ali Danandeh Mehr e, Wan Hanna Melini Wan Mohtar a, Lamine Diop f,g, Ahmed El-shafie h, Vijay P. Singh i a

Civil and Structural Engineering Department, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia Dams and Water Resources Department, College Of Engineering, University of Anbar, Ramadi, Iraq Department of Civil Engineering, Razi University, Kermanshah, Iran d School of Agricultural, Computational and Environmental Sciences, Institute of Agriculture and Environment (IAg & E), University of Southern Queensland, Springfield, QLD 4300, Australia e Department of Civil Engineering, Near East University, 99138, Nicosia, North Cyprus, Mersin 10, Turkey f UFR S2ATA « Sciences Agronomiques, d’Aquaculture et des Technologies Alimentaires », Université Gaston Berger (UGB) BP 234-Saint Louis, Senegal g Department of Food, Agricultural and Biological Engineering, The Ohio State University, 590 Woody Hayes Dr., Columbus, OH 43210, USA h Civil Engineering Department, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia i Department of Biological and Agricultural Engineering and Zachry Department of Civil Engineering, Texas A&M University, 2117 TAMU, College Station, TX 77843-2117, USA b c

a r t i c l e

i n f o

Article history: Received 21 July 2017 Received in revised form 2 September 2017 Accepted 5 September 2017 Available online 8 September 2017 Keywords: Streamflow forecasting ANFIS-FFA Antecedent seasonal variations Tropical environment

a b s t r a c t The present study proposes a new hybrid evolutionary Adaptive Neuro-Fuzzy Inference Systems (ANFIS) approach for monthly streamflow forecasting. The proposed method is a novel combination of the ANFIS model with the firefly algorithm as an optimizer tool to construct a hybrid ANFIS-FFA model. The results of the ANFIS-FFA model is compared with the classical ANFIS model, which utilizes the fuzzy c-means (FCM) clustering method in the Fuzzy Inference Systems (FIS) generation. The historical monthly streamflow data for Pahang River, which is a major river system in Malaysia that characterized by highly stochastic hydrological patterns, is used in the study. Sixteen different input combinations with one to five time-lagged input variables are incorporated into the ANFIS-FFA and ANFIS models to consider the antecedent seasonal variations in historical streamflow data. The mean absolute error (MAE), root mean square error (RMSE) and correlation coefficient (r) are used to evaluate the forecasting performance of ANFIS-FFA model. In conjunction with these metrics, the refined Willmott’s Index (Drefined), NashSutcliffe coefficient (ENS) and Legates and McCabes Index (ELM) are also utilized as the normalized goodness-of-fit metrics. Comparison of the results reveals that the FFA is able to improve the forecasting accuracy of the hybrid ANFIS-FFA model (r = 1; RMSE = 0.984; MAE = 0.364; ENS = 1; ELM = 0.988; Drefined = 0.994) applied for the monthly streamflow forecasting in comparison with the traditional ANFIS model (r = 0.998; RMSE = 3.276; MAE = 1.553; ENS = 0.995; ELM = 0.950; Drefined = 0.975). The results also show that the ANFIS-FFA is not only superior to the ANFIS model but also exhibits a parsimonious modelling framework for streamflow forecasting by incorporating a smaller number of input variables required to yield the comparatively better performance. It is construed that the FFA optimizer can thus surpass the accuracy of the traditional ANFIS model in general, and is able to remove the false (inaccurately) forecasted data in the ANFIS model for extremely low flows. The present results have wider implications not only for streamflow forecasting purposes, but also for other hydro-meteorological forecasting variables requiring only the historical data input data, and attaining a greater level of predictive accuracy with the incorporation of the FFA algorithm as an optimization tool in an ANFIS model. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction ⇑ Corresponding author at: Civil and Structural Engineering Department, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia; Dams and Water Resources Department, College Of Engineering, University of Anbar, Ramadi, Iraq. E-mail address: [email protected] (Z.M. Yaseen). http://dx.doi.org/10.1016/j.jhydrol.2017.09.007 0022-1694/Ó 2017 Elsevier B.V. All rights reserved.

1.1. Streamflow modeling overview Streamflow forecasting is a vital task for hydrologists in providing a sustainable conceptual design of water infrastructures, flood

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control measures and examining the river behavior for operational purposes. Streamflow is influenced by the temporal and spatial variability of watershed, variability of climate, seasonal patterns and local to regional-scale heterogeneity in precipitation and temperature patterns (Maity and Kashid, 2011; Danandeh Mehr et al., 2013; Dehghani et al. 2014; Singh and Cui, 2015). The incorporation of only the relevant input variables into a predictive model is essential to ensure a high accuracy level of the estimated streamflow data. However, the inherent non-linear relationships between input and output variables in streamflow forecasting tasks is proven to be an ongoing scientific challenge as the conventional regression-based approaches are unable to model the streamflow data accurately. Therefore, an integration of the objective (or forecasting) model with an optimizer algorithm can help analyze and extract the non-linear behaviors between the predictorpredictand relationships more effectively, and thus enhance the modelling capability of a standalone forecasting model applied for an estimation of streamflow data. Data-driven modeling which aims to apply artificial intelligence (AI) techniques to extract the data patterns in historical variables in order to forecast the future events, has proven to be a very popular and successful forecasting and prediction tool. Most recently, a massive development accomplished by several researchers in the field of hydrology; for instance, sediment transport modeling (Afan et al., 2017; Himanshu et al., 2017; Zounemat-Kermani et al., 2016), water level (Chang et al., 2014; Khatibi et al., 2014), ground water simulation (Chang et al., 2016, 2015; Luo et al., 2016; Singh et al., 2014), pan evaporation prediction (Kisi, 2015; Wang et al., 2017), rainfall pattern analysis (Chang and Tsai, 2016), water irrigation prediction (Zhang et al., 2017), On the hand and particularly for the studied hydrological process (e.g., streamflow), AI models exhibited an outstanding advancement results with a good level of agreement with the observed streamflow data (Chen et al., 2015; Deo and S ß ahin, 2016; Huang et al., 2014; Kasiviswanathan et al., 2016; Meshgi et al., 2015; Yaseen et al., 2016a, Yaseen et al., 2016b; Zhang et al., 2015). A plethora of forecasting studies performed on streamflow-related variables (e.g. drought, temperature and evaporation) have been presented using different AI-based models for a variety of reasons, such as the easiness of the forecasting model design and the relevant application, data inexpensive nature of the models, lower complexity of the models compared to the typical hydrological (physically-based) models, local-scale applications (e.g. farms or irrigations) and their relative competitive performance in relation to physically-based forecasting models (Abbot and Marohasy, 2014, 2012). As such, various AIapproaches have been widely adopted in streamflow forecasting since the non-stationarities contained in the input-target data (i.e. those driven by seasonal pattern, jumps and temporal trends) are implicitly considered in the internal structure of an AI model (Maier and Dandy, 2000). In general, streamflow forecasting using AI-based methods provided better estimation compared to the conventional autoregressive and other regression-based techniques (Yaseen et al., 2015; Yaseen et al., 2016a, Yaseen et al., 2016b). Based on historical research of streamflow modeling using AI methodologies, semantic-based fuzzy neural architecture has elevated the modeling of streamflow to a level of high accuracy in term of forecasting and prediction. Significant improvements in the accuracy of streamflow forecasts have been observed for the case of Adaptive Neuro Fuzzy Inference System (ANFIS) technique when applied as a comparison to the other AI approaches, even at extreme flow events (Ashrafi et al., 2017; Hipni et al., 2013; Lee et al., 2013; Nayak et al., 2007; Sharma et al., 2015; Valizadeh and El-Shafie, 2013). In general, the ANFIS model has great ability to integrate the power of a fuzzy logic system with the numeric

power of a neural system adaptive network in modeling numerous processes. Therefore, ANFIS-based models take advantage of the basic fuzzy systems which are able to deal with imprecision and vagueness in the inflow (predictor) dataset (Zadeh, 1994). In another context, hybrid approaches utilizing the integrating of two or more data assimilation and modelling technique has led to the increase in the preciseness of forecasted streamflow data (Fahimi et al., 2016). The performance of a streamflow model has exhibited an improvement by an integration of various optimization algorithms that are embedded into a standalone AI-based model as this allows modelers to deduce an optimal solution for the estimation problem, whilst simultaneously reducing the computational time (Ch et al. 2014; Kavousi-Fard et al. 2014). Nature-inspired metaheuristic optimization algorithms have received considerable research attention, enabling the researchers to attain enhanced performance of standalone (non-optimized) artificial intelligence models. Optimizer-based models includes the utilization of several ‘wrapper’ type algorithms, such as the ant colony optimization (ACO), particle swarm optimization (PSO), cuckoo search (CS) and the genetic algorithm (GA), which are known to significantly improve the performance of models applied in the streamflow forecasting (Sudheer et al., 2013; Taormina and Chau, 2015a). On the same standpoint, a recently developed swarm based, firefly algorithm (FFA) is gaining significant research attention with a number of studies reporting a favorable enhancement in their modelling accuracy (Ebtehaj et al., 2017; Ebtehaj and Bonakdari, 2016). This technique mimics the flashing behavioral patterns of fireflies, where they interact based on their flashing characteristics including the brightness, frequency and duration (which in fact, represents an objective function of the optimizer). The FFA is found to be more robust compared to the other optimization methods as both the global and local optima in the predictor data can be simultaneously and effectively resolved (Kavousi-Fard et al., 2014; Lukasik and Zak, 2009; Xiong et al., 2014; Yang, 2010). However, the application of the FFA method for streamflow forecasting is yet to be explored in greater detail. There is compelling evidence that hybrid modelling technique based on Support Vector Machines (SVM) as a predictive model and the FFA as the optimizer algorithm yields a superior performance compared to an ANN, Genetic Programming (GP) and Auto-Regressive Moving Average (ARMA) techniques in hydrological research, materials and medical sciences (Gocic´ et al. 2015; Cekaite 2016; Moghaddam et al. 2016; Amiri et al. 2013; Shamshirband et al. 2016). In a recent study, the FFA optimizer has also been incorporated into a radial basis function which was proven to yield better forecasting accuracy of water levels compared to a set of non-hybrid models (Soleymani et al., 2016). It is also noteworthy that several recent works have utilized an SVM model integrated with an FFA optimizer method, but a comparison of this with the ANFIS model showed the latter yielded higher forecasting accuracy (although the SVM resulted in modestly faster computation for multi-input variables) (He et al., 2014; Jothiprakash and Magar, 2012; Najafi et al., 2009). 1.2. Hydrological aspect and research motivation During the previous century, hydrologists accomplished enormous researches efforts in order to answer the following question ‘‘What happens to rain?” by focusing on different hydrologic cycle components individually incorporating, but not limited to, evaporation, rainfall, groundwater, infiltration and streamflow. Due to the fact that the hydrologic process for those components are embedded with high non-linearity, non-stationery and redundant, it may be unachievable to fully address the above mentioned complex question. In this context, the hydrologists focused on

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elucidating the hydrological process for these components based on the theory of developing hydrological mathematical models. In a broader perspective, hydrological mathematical models could be categorized into i) mechanistic models, and ii) data-driven models (Yaseen et al., 2016a, Yaseen et al., 2016b). Developing mechanistic models for hydrological process aspects is devious that requires high-level erudite mathematical procedure and significant amount of calibration data to assure acceptable level of model accuracy. As a result, the hydrologists paid attention to utilize the data-driven models rather than the mechanistic models as a suitable alternative to model the complex hydrological process. This is due to the fact that the data-driven model has the aptitude to mimic the input-output dynamics without bearing in mind an insightful understanding of the fundamental physical process. Autoregressive Moving Average (ARMA) and Autoregressive Moving Average with Exogenous input (ARMAX) are considered as the most common conventional data-driven models for hydrological process and time series (Box and Jenkins, 1970). These two models are basically structured for linear time series and utilize the classical statistics to analyze the data. With comprehending that, the hydrological process exhibited high nonlinearity between the input and the output variables, the feature of the nonlinear dynamics in the data is unachievable. Over the past three decades, the above mentioned methods have been slowly replaced by advanced soft computing modeling that showed the ability to detect the nonlinearity dynamics in the data and reliability in identifying and estimating the dependencies in the historical data of the hydrological variables (Solomatine and Ostfeld, 2008). The appearance of the soft computing approaches and the promising results achieved in the hydrological modeling, leading to introduce a new term namely Neuro-Hydrology which is now is considered as a new chapter in ASCE Task Committee on hydrology (ASCE, 2000). On the other hand, such approach still has negative side as the soft computing is pure black-box models and unpaid attention for the fundamental physics. Demonstrating that, the soft computing models are impure black-box model and showed that there is a possibility for such models to extract and mimic partially the physics involved in the hydrological process and thus it is considered as a trend for the hydrologists to enhance its suitability for hydrological model. Until recently, hydrologists have directed their researches to identifying the mechanism by which soft computing modeling approach learn the embedded physics in the hydrological patterns in the input-output data. Taking all the together, in this study, further step for understanding the embedded mechanism in the streamflow patterns by introducing the evolutionary algorithm called FFA hybridized with classical ANFIS model for monthly time series forecasting. This is enhancing the accuracy of soft computing modeling procedure as it offers a comprehensive understanding to adjust and optimize the modeling parameters and wider selection of the membership function to represent the appropriate system model for streamflow modeling. 1.3. Research objectives Taking into account the superior performance of an ANFIS over an SVM model that used continuous time-series data, this study aims to develop and investigate the statistical performance of a novel hybrid ANFIS-FFA model. The performance accuracies of the proposed hybrid model are compared with a traditional ANFIS model which is trained by the Fuzzy c-means (FCM) clustering. In order to practically evaluate the applicability of ANFIS-FFA model, we test it for streamflow forecasting for a case study region in a tropical climatic zone. Therefore, the historical data for Pahang River, which is a major river system in Malaysia is utilized. The

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hydrological conditions for the case study site is known for high surface runoff during seasonal monsoons in April and December, with the lowest flows occurring in months of June. Frequent occurrences of devastating flood events, in particular during the monsoon season thus reinforces the importance of streamflow forecasting in this tropical region (Gasim et al., 2013). The rest of the paper is structured as follows: Section 2 presents a theory of the ANFIS, FFA and hybrid models adopted in this paper, followed by Section 3 where the materials are described in terms of the case study and model’s performance evaluation metrics. Section 4 presents the results with the assessment of the forecasting accuracy of the hybrid ANFIS-FFA model and the relevant discussions in respect to a traditional ANFIS model including the advantages, limitations and future improvements, while Section 5 presents the conclusion of this research work. 2. Theoretical overview 2.1. Adaptive Neuron fuzzy Inference system (ANFIS) ANFIS models combine fuzzy systems and the learning ability of neural networks (Ebtehaj and Bonakdari, 2014; Kurtulus and Razack, 2010; Tabari et al., 2012). There are mainly three types of ANFIS models, classified broadly as Mamdani, Sugeno, and Tsumoto. However, the Sugeon’s system is the most frequently used (Takagi and Sugeno, 1985). Fuzzy logic models utilize membership functions to convert input data into fuzzy values ranging between 0 and 1 (Ç ekmisß et al., 2014). The structure of an ANFIS model is composed by both nodes and rules. Nodes are functioning as membership functions (MFs) while the rules permit one to model the relationships between a predictor (input) and the predictand (output). In developing an ANFIS model, there are several types of membership functions (MF) (e.g. sigmoid, triangular, Gaussian, trapezoidal, etc.) that could be considered. Choice of the suitable MF in this paper followed earlier work that applied the Gaussian equation in their ANFIS models (Awadallah et al., 2009; Moosavi et al., 2013; Shirmohammadi et al., 2013). It is important to note that the Gaussian shape is the most popular MF for characterizing fuzzy systems because of its concise notation and smoothness (Gholami et al., 2017). This MF has its own advantages over the other equations, such as being non-zero, smooth and defined by two parameters only which are optimize through training process. Therefore, for this research paper, the Gaussian MF function has been applied following Eq. (1) (Awadallah et al., 2009).

U Ni ¼

expððx  ci Þ2 Þ 2r2j

ð1Þ

where UNi is the membership function and x represents the input at i node. rj and ci are the conditional parameters of the function. ANFIS requires feature extraction rules applied to the inputtarget data that are stocked in a fuzzy based rule system (i.e., ‘the IF- THEN’ rule). The rules are defined based on their antecedents (If part) and consequents (Then part). In a Sugeno system, a rule is constituted by weighted linear combination of the crisp inputs. Eqs. (2) and (3) present the rules for an ANFIS system with two inputs (x and y) and one output f.

Rule 1 : IF x is P 1 and y is Q 1 ; then f 1 ¼ p1 x þ q1 y þ r 1

ð2Þ

Rule 2 : IF x is P 2 and y is Q 2 ; then f 2 ¼ p2 x þ q2 y þ r 2

ð3Þ

Pi and Qi are fuzzy sets, fi represents the output within the fuzzy region, and pi, qi, and ri are the design parameters determined during the model’s training process (i = 1, 2). Fig. 1 shows an architecture of the ANFIS model with two inputs (x and y) and one output (f).

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Fig. 1. Architecture of an ANFIS with 2 inputs and 5 layers. Layer 1 (Input Fuzzy Rules); Layer 2 (Input MF); Layer 3 (Fuzzy Neurons); Layer 4 (Output MF); Layer 5 (Summation and Weights); w = weights; x, y = inputs; P1 and P2 = fuzzy rules; Q1, Q2 = fuzzy rules; x, y (with arrows = target-input in training phase. F = output.

In this study, an ANFIS model is adopted mainly due to its good capability of learning, constructing, expensing and classifying the input-target data. This model has the advantage of extraction patterns in the input data based on fuzzy rules to seek expert knowledge and adaptively construct a rule base. In a streamflow forecasting problem which is generally complicated due to the chaotic nature of the data itself, an ANFIS model can intelligently extract information and convert it to fuzzy systems, but this a larger time expended in training the model is necessary for accurate estimation (e.g., Chang et al., 2002; Ponnambalam et al., 2003; Chang and Chang, 2006). More detail about ANFIS structure can be found in Chang and Chang (2006). It is noteworthy also, that since forecasts are dependent on the optimally selected Gaussian membership function parameters in Eq. (1) (i.e. rj and cj). Therefore, in this study, a Firefly Optimization Algorithm (FFA) is applied in the modelling phase to determine the best set of parameters for the MF. 2.2. Firefly optimization algorithm (FFA) Firefly optimization algorithm (Yang, 2010), inspired by social behavior of fireflies, is based on their flashing characteristics. There are three essential rules in the FFA structure, that is based on the characteristics of real fireflies regardless of their gender: (i) Fireflies are assumed to be unisex, as a result each firefly has the ability to attract another; (ii) the intensity of the luminous determines the attractiveness of one firefly to another; and (iii) the brightness is proportional to the amount of light emitted by the firefly. Based on these rules, the objective function of the FFA model is represented by the brightness of the firefly and the intensity of the light emitted from firefly. The intensity and the attractiveness are presented in Eqs. (4) and (5), respectively. Each firefly has its typical attractiveness b which tells us how strong it attracts other members of the swarm (Yang, 2010).

I ¼ I0 ecr

2

wðrÞ ¼ w0 ecr

ð4Þ 2

ð5Þ

where I and w(r) represent the light intensity and the attractiveness at distance r from the firefly, respectively. I0 and W0 are the light intensity and attractiveness at distance r = 0 from the firefly and y is the coefficient of light absorption. The distance r between any two fireflies i and j is expressed by the Eq. (6) (Yang, 2010):

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d uX r ij ¼ kxi þ xj k ¼ t ðxi;k  xj;k Þ k¼1

ð6Þ

where xi and xj are the location of fireflies i and j in a Cartesian coordinate system. As mentioned, firefly is attracted by another and vice- versa, thus the movement of firefly a by another firefly b is represented in Eq. (7) (Yang, 2010).

Dxi ¼ b0 ecr ðxj  xi Þ þ aei 2

ð7Þ

where aei is the randomized term and b0 e ðxj  xi Þ is symbolized as the attraction term, aandei are the randomization coefficients and random number vector, respectively. a varies between 0 and 1. Based on Eq. (7), the next movement of firefly is its position at i + Dxi. To design an optimal ANFIS model integrated with the FFA, the basic parameters, namely the light absorption coefficient (c), attraction coefficient base (b0), and movement coefficient (a) should be adjusted. In this paper, the optimal values of c, b0 and a are considered to be 1.2, 2.0 and 0.3, which are determined by a trial and error process. For more details about the FFA procedure, readers can refer to the work of Yang (2010). cr 2

2.3. Hybrid model based on ANFIS-FFA In this study, the number of inputs in the training period for input combinations is varied from 1 to 16, which are presented in 16 different input combinations (Table 1) while all forecasting models have only one output (i.e., streamflow as the objective variable). The antecedent’s values were obtained using autocorrelation function statistical approach as shown in Fig. 2. The range of membership function (MF) parameters that required optimization was defined before the training process is applied on the ANFIS model. In the training phase, the root mean square error (RMSE) (as objective function) is monitored to evaluate the accuracy of the ANFIS-FFA model. In this procedure, the antecedent’s parameters of the ANFIS model which essentially consist of MF parameters, are optimized by the FFA scheme. The first iteration of the ANFIS-FFA scheme is initiated by a generation of an initial firefly population randomly so that each firefly is mapped onto the ANFIS set(s). The attractiveness of the fireflies is computed based on the light intensity of fireflies. Therefore, the fireflies’ attractiveness is compared collectively and firefly that yield the maximum brightness are selected and the other fireflies are naturally attracted toward to this firefly. Also, the RMSE which is considered as fitness function in the hybrid model is calculated to evaluate the performance of ANFISFFA modeling system. The training process are continued until we either reach a maximum number of iterations or the fitness function value is acceptable. Flowchart of the proposed hybrid ANFIS-FFA model is presented in Fig. 3.

Z.M. Yaseen et al. / Journal of Hydrology 554 (2017) 263–276 Table 1 Different input combinations of historical data considered for forecasting future streamflow. Model Designation

Input combination(s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

t t t t t t t t t t t t t t t t

– – – – – – – – – – – – – – – –

1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

t t t t t t t t t t t t t t t

– – – – – – – – – – – – – – –

2 3 6 12 2, t 2, t 2, t 3, t 3, t 2, t 2, t 2, t 2, t 2, t 2, t

– – – – – – – – – – –

3 6 12 6 12 3, t 3, t 6, t 3, t 3, t 3, t

–6 – 12 – 12 – 6, t –12 –12, t –24 – 6, t –12, t – 24

In developing a forecasting model, the training accuracy of the model is commonly determined the root mean square error (RMSE). As stimulated by (Chai and Draxler, 2014), this metric satisfies the triangle inequality that is required for a distance function metric for model evaluation. RMSE is preferred in data assimilation field where the sum of squared errors is often defined as the cost function to be minimized by adjusting model parameters. In such applications, which are similar to the modelling performed in this paper, penalizing large errors through the defined least-square terms proves to be very effective in improving model performance (Chai and Draxler, 2014). Moreover, studies performed on the sensitivity of forecasting models that use only RMSE found that a detailed interpretation is not critical in model development stage, since the variations of the same model will have similar error distributions, especially in the training phase. As a common practice, the RMSE is normally considered as the first order assessment of the trained model, and was therefore adopted in the present study. It is noteworthy to mention, there have been several researches predominantly performed in the hydrological application forecasting area that used this indicator as the objective criterion for model training and final selection (Abbot and Marohasy, 2012; Keshtegar et al., 2016; Liu et al., 2012; Wei, 2012). Therefore, we have adopted the RMSE as the objective metric to select the trained forecasting models.

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3. Materials 3.1. Case study area The case study area, Pahang River is located in the Malaysian Peninsular, coordinately between the latitude 2° 480 4500 – 3° 400 2400 N and longitude 101° 160 3100 – 103° 290 3400 E. The river system is a confluence of Jelai and Tembeling rivers from the upstream which join at Kuala Tembeling. This is the longest river in this area (of approximately 459 km) and its basin area covers a surface of about 27,000 km2. The topography of the region is mountainous with an elevation varying between 2,187 m at Mt. Tahan and 0 m at the river mouth. The stream-flow of Pahang River was measured at Station 3527410 (Lubok Paku), which is the most downstream stage station in the Pahang River basin. Fig. 4 shows a detailed map of the Pahang river basin. The upper part of the river is situated at the Main Range of Titiwangsa. Pahang River basin. This region is characterized by a hot and a wet climate that is essentially influenced by the climatic regime of Northeast and Southwest monsoonal winds varying in their direction according to the season. The mean annual temperature and relative humidity is about 26.4 °C and 86%, respectively, with an average annual rainfall ranging from about 2000 to 3000 mm. The largest part of the rainfall occurs between midOctober and mid-January; whereas, the months of June and July are considered to be the driest period in the basin (Harun et al., 2014). The mean measured flow rate of Pahang river at Lubok Paku is about 596 m3/s and the land use is characterized by an increase of urbanization that is expected to affect the Pahang river basin discharge rates. In this study, to forecast the streamflow data using an ANFISFFA hybrid model, different input combinations of historical streamflow data applied in the training period are considered, which are presented in Table 1. The number of input combinations is from 1 to 16 based on considering the statistically lagged effects of seasonal and annually changing patterns noted in the historical input dataset. 3.2. Performance evaluation criteria Considering the stochastic nature of hydrological variable, one must not rely on a single criteria when evaluating the performance of a statistical model (Dawson et al., 2007; Krause et al., 2005). This in this paper, both visual plots in conjunction with normalized

Fig. 2. Auto-correlation function (ACF) of the raw time series stream-flow data for Pahang River.

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Fig. 3. A flowchart of the proposed hybrid ANFIS-FFA model.

statistical metrics are used to establish the efficacy of an ANFIS and ANFIS-FFA hybrid model. We selected root mean square error (RMSE), mean absolute error (MAE) and correlation coefficient (r) (Krause et al., 2005) as first order statistics to evaluate the forecasting performance viz:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X RMSE ¼ t ðQ  Q i;f Þ2 N i¼1 i;o MAE ¼

ð8Þ

N 1X jQ  Q i;f j N i¼1 i;o

ð9Þ 1

0

N C B X B  o Þ ðQ  Q fÞ C ðQ i;o  Q C B i;f C B i¼1 C v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r¼B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC; 1 6 r 6 1 Bu u N C BuX N uX Bu  o Þ2 t Q  Q fC A @t ðQ i;o  Q i;f i¼1

i¼1

ð10Þ

where N is the number of datum points in the testing period (independent sample), Qi,o and Qi,fis the ith value of observed and forecasted time series, respectively. The RMSE and MAE deduce goodness-of-fit regardless of the sign of the difference between forecasted and observed streamflow but it must be noted that RMSE is appropriate for a set of normally distributed forecasting errors (Chai and Draxler, 2014). In reality, this criterion may not be satisfied by all forecasting models so in this paper, we applied the MAE that evaluates all deviations of the forecasted data from the observed values in an equal manner regardless of the sign (Deo et al., 2016a; Krause et al., 2005). The value of r, typically bounded by [1, 1] describes the covariance in the observed streamflow data that can be explained by the forecasting model but it comes at the expense of linear assumptions (Deo et al., 2016a; Krause et al., 2005). The r, RMSE and MAE can also be insensitive to outliers and the additive and proportional differences between the model forecasts and observations (Legates and Mccabe, 1999; Willmott and Matsuura, 2005). In general, a shift in forecasted values; for example, can lead to erroneous conclusions if the model is evaluated solely based on r.

Z.M. Yaseen et al. / Journal of Hydrology 554 (2017) 263–276

269

Fig. 4. Location of the stream-flow station and river map of the Pahang River basin.

Complementary information about the ANFIS-FFA model’s preciseness was gleaned from the Nash-Sutcliffe coefficient (ENS), refined Willmott’s Index (Drefined) and Legates and McCabes Index (ELM) as normalised goodness-of-fit measures viz (Nash and Sutcliffe, 1970; Legates and Mccabe, 1999):

2

ENS

3 N X 2 6 ðQ o;i  Q f ;i Þ 7 6 i¼1 7 7; 1 6 ENS 6 1 ¼16 6X 7 N 4 25  o;i Þ ðQ o;i  Q

ð11Þ

i¼1

Drefined

8 PN ðQ Q Þ > > Pi¼1N i;f i;o ; when > 1  ½ > > C jQ i;o Q o j > i¼1 > > PN > PN <  i¼1 ðQ i;f  Q i;o Þ 6 C i¼1 jQ i;o  Q o j ¼ ; PN  oj > C jQ Q > i¼1 i;o > P  1; when > N > ðQ i;f Q i;o Þ > > i¼1 > > PN : PN  i¼1 ðQ i;f  Q i;o Þ > C i¼1 jQ i;o  Q o j

0 6 Drefined 6 1

2

ELM

4. Results and discussion 4.1. Selection of the best input combinations

ð12Þ 3 N X jQ  Q j 6 o;i f ;i 7 7 6 i¼1 7; 0 6 ELM 6 1 ¼16 7 6X N 4  o;i j5 jQ o;i  Q

sitivity exhibited by ENS (Willmott, 1981) since the ratio of the model errors can be analyzed rather than the square of the model error difference (Willmott, 1984). The Legates and McCabe Index (0  ELM  1.0) was applied since this metric has an advantage over the WI when relatively high forecasted values are expected, even for a poorly fitted model (Willmott, 1984). When compared with WI, the outliers in the forecasted data are likely to lead to a relatively higher value of the WI due to the squaring of the differences between observed and forecasted data (Willmott, 1981). Consequently, we have used the ELM (Legates and McCabe, 1999) as an advancement of the WI where the model errors and the differences are given appropriate weighting, and as such, they are not inflated by their square values.

ð13Þ

i¼1

Note that in Eq. (12), the value for C is typically taken to be 2 (Willmott et al., 2012). The inspected ENS is able to assess the model’s ability to forecast the data that deviates from the mean values (Nash and Sutcliffe, 1970); however, this metric is sensitive to the differences between the observed and forecasted values (Legates and Mccabe, 1999). To further investigate the forecasting model’s ability, we have also employed the WI as an improvement over the RMSE, r and ENS. It is noted that WI measures the ratio of the mean square error and the potential error multiplied by the number of observations and then subtracted from unity. This index aims to assess the differences based on squared differences and so can overcome the insen-

Prior to the applying the proposed ANFIS-FFA model, we investigated the ability of a stand-alone ANFIS technique, as the bench mark model, to predict the monthly streamflow for the case study region (Pahang River). Then, the model was integrated with the FFA technique for each input combinations separately. Table 2 shows the efficiency results of the optimal evolved ANFIS and ANFIS-FFA models in terms of the correlation coefficient (r), RMSE, and MAE metrics acquired in the test period for each input combination. The efficiency of each combination is also compared in Table 3 with respect to the Nash-Sutcliffe’s coefficient (ENS), Legates and McCabe’s index (ELM) and the refined Willmott’s Index of agreement (DRefined) deduced in the testing period. The tables indicate excellent performances in terms of a lower value of the RMSE and MAE and a higher value of the ENS, ELM and Drefined metric for Models 16 and 15 that utilized ANFIS and ANFIS-FFA, respectively, to forecast the present streamflow data. The accuracy results given in both the tables also show that the ANFIS-FFA model is superior to the ANFIS model more or less in all the input combinations. The reason behind this may lie on the robustness of the FFA method that is linked to the ANFIS model,

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Z.M. Yaseen et al. / Journal of Hydrology 554 (2017) 263–276 Table 2 ANFIS and integrated-ANFIS with Firefly Optimizer algorithm results for sixteen models developed for forecasting streamflow. The best model for each case based on the lowest MAE/RMSE is indicated in red/blue.

ANFIS

ANFIS-FFA

Model

r

RMSE, m3/s

MAE, m3/s

Model

r

RMSE, m3/s

MAE, m3/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.594 0.849 0.830 0.787 0.853 0.858 0.894 0.923 0.805 0.870 0.955 0.962 0.973 0.976 0.997 0.998

38.758 25.563 27.054 30.098 25.267 24.940 21.843 18.652 28.948 23.940 14.524 13.243 11.116 10.569 3.792 3.276

23.004 16.611 16.675 16.796 12.902 16.309 13.567 11.018 16.418 13.782 8.784 7.593 6.427 5.281 1.879 1.553

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.671 0.911 0.861 0.816 0.952 0.933 0.897 0.963 0.852 0.949 0.973 0.980 0.993 0.991 1.000 0.999

35.746 19.964 24.667 28.212 14.829 17.511 21.621 13.025 25.520 15.249 11.342 9.567 5.743 6.452 0.984 2.641

21.152 11.993 11.731 14.332 8.397 11.201 14.024 5.944 13.211 6.506 5.392 5.028 2.838 2.471 0.364 1.166

Table 3 Same as Table 2 but performance metrics based on Nash-Sutcliffe’s coefficient (ENS), Legates and McCabe’s index (ELM) and the refined Willmott’s Index of agreement (DRefined) deduced in the testing period.

ANFIS Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

ENS 0.353 0.720 0.688 0.619 0.728 0.735 0.799 0.852 0.648 0.755 0.911 0.925 0.947 0.952 0.994 0.995

ANFIS-FFA

ELM 0.260 0.468 0.467 0.469 0.586 0.479 0.571 0.647 0.481 0.558 0.722 0.756 0.794 0.831 0.940 0.950

Drefined 0.630 0.734 0.734 0.735 0.793 0.740 0.786 0.823 0.741 0.779 0.861 0.878 0.897 0.915 0.970 0.975

which contributed to an optimization of the membership function parameters for each input combination. The performance of each input combination using the ANFIS and ANFIS-FFA models with respect to relative (%) mean absolute error within the testing period are also visually compared in Fig. 5. It is clearly seen that the ANFIS-FFA model not only significantly reduces the relative MAE of the optimal model, but also provides an accurate model with a lower number of input variables (see Table 1). Clearly, this is a desirable feature of the ANFIS-FFA model for potential practical applications in data sparse regions. 4.2. ANFIS-FFA vs. ANFIS The observed and forecasted streamflow hydrographs within the testing period using optimal ANFIS and the optimal ANFISFFA model for the best input combinations are shown in Figs. 6 and 7, respectively. In these hydrographs, extremely low flow

Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

ENS 0.450 0.829 0.741 0.665 0.906 0.869 0.804 0.928 0.726 0.901 0.946 0.961 0.986 0.982 1.000 0.997

ELM 0.319 0.616 0.625 0.547 0.731 0.642 0.557 0.809 0.583 0.791 0.830 0.839 0.909 0.921 0.988 0.963

Drefined 0.660 0.808 0.813 0.774 0.865 0.821 0.778 0.905 0.791 0.896 0.915 0.919 0.954 0.960 0.994 0.981

durations have also been magnified for further visual inspection where the ordinate scale of 1 to 15 is applied. Although various definitions of low flows exist in literature based on the occurrence periods such as winter for low flows and summer for low flows (Smakhtin, 2001), in this study we selected an exceedance probability of 25% (i.e. first quartile that approximated to about 15 m3/ s) as a threshold for the extremely low flows. Indeed, if an average flow in a month is less than 15 m3/s, it is considered as extremely low flow duration in this particular study region. In a similar way, Demirel et al. (2015) selected the exceedance probability of 75% (third quartile) as a threshold for the low flows in the Moselle River, France. The figures illustrate that the both ANFIS and ANFIS-FFA models are able to forecast the peak values of observational data in these hydrographs. They estimate the global and local maximum values quite satisfactorily. However, they do not provide such accurate estimates for extremely low flows. Inasmuch as we used the RMSE

Z.M. Yaseen et al. / Journal of Hydrology 554 (2017) 263–276

271

Fig. 5. Relative mean absolute error (MAE, %) for the ANFIS and ANFIS-integrated model with Firefly optimizer algorithm for all sixteen models considered in the testing period.

Fig. 6. Observational and forecasted streamflow hydrographs in the testing period for input combination #15 (top row); and the representative details for extremely low flow durations (bottom column).

as the objective metric to calibrate the ANFIS and the ANFIS-FFA models, this satisfactory estimation of peak flow values is as anticipated since the RMSE can show the minimum error in situations of high flows due to the squaring terms applied (Eq. (8)). Although both the ANFIS and ANFIS-FFA models forecast the peaks quite accurately, the ANFIS model does exhibit lower accuracy with negative forecasts for some of low observed flow values (see Fig. 5) whereas this is not the case for the ANFIS-FFA model.

The ANFIS and ANFIS-FFA models are illustrated in a scatter plot (Fig. 8) where the corresponding observed data are shown in the testing period. The figure shows both forecasting models are able to forecast the peak stream flow (Q > 100 m3/s) quite successfully. However, the ANFIS-FFA model exhibits greater accuracy than the ANFIS model as applied to forecast the medium and low flows, thus justifying its superiority over the non-optimized ANFIS model.

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Z.M. Yaseen et al. / Journal of Hydrology 554 (2017) 263–276

Fig. 7. Similar to Fig. 6 but for the combination #16.

Fig. 8. Scatter plots of ANFIS and ANFIS-FA forecasted streamflow a) Combination 15, b) combination 16 at the testing period.

Table 4 shows the distribution statistics of the forecasting errors for the ANFIS and integrated ANFIS-FFA model. It can be seen that, independent of the inputs combinations chosen, the integrated ANFIS-FFA performs better with the lowest relative error value and with a less sparse distribution. Taking individually, the input combination, Model 16 and 15 yielded the best results for the optimal ANFIS and ANFIS-FFA, respectively, with forecasting errors varying from 20.4 to 13.2 for the ANFIS model and between 10.7 and 15.3 for the integrated ANFIS-FFA. These results confirm the superiority of the integrated ANFIS-FFA model over the ANFIS standalone model. The integrated ANFIS-FFA forecast are able to better represent the low and the median streamflow values, which are also reflected by the smaller forecast error. Finally, the frequency (%) of magnitudes of the forecasted error are presented in Fig. 9. For very low magnitudes (from 0 to 0.5), the ANFIS exhibits lower frequency (59%) while the frequency reaches 84% for the integrated ANFIS- FFA. However, for the higher magnitudes, the frequency is very important for ANFIS model. This result shows that higher forecast errors are more presented in

the case of the ANFIS model, which reveals that the integrated ANFIS-FFA is more precise and accurate in forecasting streamflow compared to the classical ANFIS; 4.3. Advantages, Limitation and future improvement This study that utilised the ANFIS-FFA model relative to the standalone ANFIS model, was highly successful in enhancing the forecasting accuracy of streamflow data by a significant margin when applied to the case of Pahang River, a major river system in Malaysia. While this study has verified the better utility of an ANFIS-FFA model over a standalone ANFIS model, the case study reported only the data for one study location, with historical values for training the model. For its practical application, the veracity of the hybrid model can be evaluated for a range of hydrological sites, including additional hydro-meteorological data (e.g. precipitation, evaporation and temperatures) that can also have a moderating effect on streamflow prediction, and thus, could improve the proposed hybrid model. Although all of these data are difficult to acquire in geographically diverse regions, an inclusion of more

273

Z.M. Yaseen et al. / Journal of Hydrology 554 (2017) 263–276 Table 4 Distribution statistics of forecasting error for ANFIS and integrated-ANFIS with Firefly Optimizer algorithm.

Error Statistic Lower Quartile Upper Quartile Median Maximum Minimum

ANFIS Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -5.8

-4.4

-9.0

-5.7

-3.5

-5.1

-7.4

-3.9

-8.2

-5.6

-4.8

-2.5

-3.2

-1.4

-0.6

-0.3

18.4 4.3 69.1 -292.7

13.2 1.6 73.6 -111.8

12.9 1.3 90.6 -175.0

10.4 2.0 67.6 -239.8

7.4 1.5 111.7 -210.7

12.1 2.6 67.0 -118.6

7.4 0.0 74.3 -89.1

7.5 0.9 117.8 -66.6

12.9 0.5 90.8 -227.1

10.7 1.7 140.2 -124.3

5.5 0.4 40.3 -76.5

4.0 0.1 39.0 -60.4

4.1 0.0 41.6 -66.4

1.8 0.0 49.6 -55.5

0.8 0.0 18.5 -21.3

0.3 0.0 13.2 -20.4

ANFIS-FFA Lower Quartile Upper Quartile Median Maximum Minimum

-4.4

-3.6

-3.9

-4.4

-2.4

-5.1

-6.6

-1.4

-4.5

-0.9

-1.7

-1.4

-0.6

-0.2

0.0

-0.2

15.6 4.7 79.6 -257.3

8.0 0.5 70.3 -101.7

7.4 0.6 98.7 -202.2

8.8 0.9 79.6 -237.9

5.4 0.3 60.4 -73.9

8.3 0.7 56.9 -80.5

9.9 0.4 75.1 -85.0

3.0 0.0 94.4 -73.8

7.1 0.3 101.8 -208.7

2.2 0.0 106.7 -88.7

2.1 0.0 41.5 -62.4

1.8 0.0 42.4 -48.6

0.8 0.0 28.4 -32.0

0.3 0.0 28.0 -36.2

0.0 0.0 4.3 -6.8

0.2 0.0 15.3 -10.7

frequency, %

80

(a) 59

60 40 20 0

10 0

0.5

6 1

3 1.5

4

22 2

2.5

3

3 3.5

22 4

1 4.5

5

84 frequency, %

80

(b)

60 40 20 6

3 0

0

0.5

1

11

2

1.5 2 2.5 3 3.5 magnitude of forecasted error

11 4

4.5

5

Fig. 9. Histogram of percentage frequency of forecasted error in various error magnitudes for: (a) best ANFIS, (b) ANFIS-integrated model with firefly optimizer algorithm (ANFIS-FA). The actual frequency in each error bracket is indicated.

input variables nonetheless can allow a better pre-selection of inputs by novel algorithms (e.g. tree-based iterative input selection, evolutionary method such as coral reef optimisation and other statistical approaches) (Deo et al., 2016a; Galelli and Castelletti, 2013; López et al., 2005; Salcedo-Sanz et al., 2014; Taormina and Chau, 2015b) prior to the optimization of the proposed ANFIS model with the FFA method. In this paper the ANFIS model utilized the fuzzy concept with membership functions [0, 1], however, an improvement of the proposed model could also be made in a follow-up study by utilizing an interval-valued fuzzy linear-programming (IVFL) method with an incorporation of the interval parameters in an interval-valued fuzzy system (Lu et al., 2010; Wang et al., 2012). These parameters could then be directly optimized by the FFA method. In terms of its practicality, the benefit of IVFL could be the explicit addressing of individual uncertainty and dual uncertainties for the model to assess the associated risks for decision-making by using forecasted data. Moreover, as an alternative forecasting tool, one could enhance ANFIS-FFA model with a Bayesian Model Averaging

(BMA) technique that allows a better assessment model selection uncertainty (Kim et al., 2015; Rathinasamy et al., 2013; Wintle et al., 2003). Moreover, a single forecasting model with an optimized set of parameters does not yield information on the forecasted error bounds. This is because model forecasts can respond to differently to various model runs, so an ensemble-averaging approach used previously (e.g., Efron and Tibshirani, 1993; Tiwari and Chatterjee, 2010; Tiwari and Adamowski, 2013) can be adopted where the proposed ANFIS-FFA model can be statistically evaluated for parametric uncertainty (Tiwari and Adamowski, 2013; Tiwari and Chatterjee, 2011). For ensemble-based modelling, there also exist further opportunity to apply ensemble transform Kalman filter to study the relationship between the filter performance and ensemble size (Rasmussen et al., 2015). This adaptive method could allow an ensemble-based ANFIS-FFA model to be fed with a lesser number of ensemble members for any particular predictor variable from a multi-input dataset. In real life decisionsupport systems in hydrological forecasting, application of such an ensemble framework (Rasmussen et al., 2015; Tiwari and Adamowski, 2013; Tiwari and Chatterjee, 2010) incorporating the ANFIS-FFA method can help better understand the forecasting model’s error bound and also to attain an improved forecasting accuracy. In this paper we followed earlier works (Mayilvaganan and Naidu, 2011) and thus applied the Gaussian membership (MF) to construct ANFIS models in order to define how each point in input space is mapped to a membership value (i.e. degree of membership) between 0 and 1. While our results (Tables 2 and 3) and subsequent comparison of ANFIS models revealed good performance, further studies could should investigate several other MF (i.e. triangular, trapezoidal, sigmoid, generalized bell shaped, p-shaped, S-shaped etc) (Ali et al., 2015; Mayilvaganan and Naidu, 2011) with different partitioning of training and testing datasets. Off course, each MF in ANFIS model has its own merits and weaknesses. Sigmoidal MF, for example, was seen to offer better approximation of the shape of inputs and outputs where an advantages of former over the latter MF is include better shoulders and approximation through due to tapering edges (Kochhar et al., 2016). Bell shaped and Gaussian MF can have the advantage that they are smooth and non-zero at all points (Cortés et al., 2012), and thus will capture data features that possess such attributes more accurately than triangular and trapezoidal MF that are not smooth. Having

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said that, the optimality of any prescribed MF in an ANFIS model is expected to depend on the actual data features and the particular modelling problem at hand; hence, an independent study that compares the various MFs is warranted. Although this is a useful task, it was beyond the scope of the present study and thus, could be a useful subject of an independent research. With natural features in hydrological data series that encompass stochasticity and non-stationarity patterns (e.g. jumps, random fluctuation, seasonality and temporal trend), an improvement in the ANFIS-FFA model could be made via multiresolution data analysis techniques such as non-decimated maximum overlap wavelet transformation of the input data to better resolve the various (low and high) frequencies therein in order to enhance the model’s accuracy (Deo et al., 2016a,b; Nourani et al., 2014). An independent study could also benchmark a waveletbased ANFIS-FFA model with an Empirical Mode Decomposition (EMD) technique applied into an ANFIS-FFA model (Boudraa and Cexus, 2007; Fan et al., 2016; Wang et al., 2015). Since the EMD approach is a fully data-driven process that allows modelers to locally separate original and complex time-series into low and high frequency components, including the reduction of noise, its application in hydrological forecasting where complex input data are often encountered is an interesting research endeavor.

5. Concluding remarks Due to chaotic nature of streamflow, application of data-driven models for forecasting streamflow is a challenging task for hydrologists, and thus, requires novel optimization techniques integrated with standalone models to boost the accuracy by optimizing the model’s parameters. Adaptive Neuro Fuzzy Inference System (ANFIS) is a popular neural network model based on Takagi– Sugeno fuzzy inference system integrating neural networks with fuzzy logic to capture the benefits of both models as a single predictive framework. This paper proposed a new hybrid framework, namely the ANFIS-FFA model for a case study region to forecast monthly streamflow. This model was developed on the basis of integrating the FFA optimization technique with the ANFIS method to optimize the membership function variables in training process of the ANFIS model where the lowest root mean square error was sought for hybrid ANFIS-FFA model with input combinations of historical training data. The model was applied to forecast monthly streamflow at hydrological site of Pahang River, Malaysia, and evaluated with the stand-alone ANFIS model. The results showed that the ANFIS-FFA model was not only is superior to the ANFIS model but also behaved like a parsimonious model for streamflow prediction using lower number of input variables required to yield comparatively better performance. From the point of forecasting extremely low flow data, the ANFIS model, despite being a nonlinear modeling schemes, did not provide precise results due to the fact that the implemented objective function for the calibration of the model tends to minimize forecasting errors that mostly resulting peak values. To cope with this problem, the combination of FFA optimizer algorithm with the standalone ANFIS model showed that the new hybrid model can surpass the accuracy of ANFIS model in general and to remove the false (negatively) forecasted data in the ANFIS model for extreme low flows. Thus, it is construed that the ANFIS-FFA hybrid model can successfully be implemented as a robust alternative for monthly streamflow forecasting, as has been the numerous applications in hydrological forecasting (e.g., Chang et al. 2002; Ponnambalam et al. 2003; Chang and Chang, 2006) particularly when precise estimation of low flows is being investigated. This work can be extended for future research by investigating the other hydrological prospective ‘‘e.g., seasonal spatial-temporal forecasting”.

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