A New Approach for Parameter Estimation of Autoregressive Models Using Adaptive Network-Based Fuzzy Inference System (ANFIS) Hamid R. Safavi, Mohammad Hossein Golmohammadi, Maryam Zekri & Samuel Sandoval-Solis Iranian Journal of Science and Technology, Transactions of Civil Engineering ISSN 2228-6160 Iran J Sci Technol Trans Civ Eng DOI 10.1007/s40996-017-0068-x 1 23 Your article is protected by copyright and all rights are held exclusively by Shiraz University. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Iran J Sci Technol Trans Civ Eng DOI 10.1007/s40996-017-0068-x RESEARCH PAPER A New Approach for Parameter Estimation of Autoregressive Models Using Adaptive Network-Based Fuzzy Inference System (ANFIS) 1 1 2 Hamid R. Safavi • Mohammad Hossein Golmohammadi • Maryam Zekri • Samuel Sandoval-Solis3 Received: 24 June 2016 / Accepted: 7 July 2017 Ó Shiraz University 2017 Abstract Time series modeling plays an important role in new driven method from ANFIS shows that this system can different fields of science and engineering such as be employed as a parameter estimator for time series hydrology and water resources management. The proper models such as AR models. estimation of the parameters in time series models is one of the essential steps of modeling. Yule–Walker, least square, Keywords Parameter estimation Á Autoregressive models Á Burge and forward–backward approaches are known, and Hydrologic time series Á Adaptive network-based fuzzy common methods of parameter estimation are used in inference system (ANFIS) Á Zayandehrud dam many time series studies. Recently, intelligent techniques such as adaptive network-based fuzzy inference system (ANFIS) have been used for time series modeling. Review 1 Introduction of previous researches, especially in the field of hydro- logical time series, shows that these systems are often used Modeling of hydrological processes resulting from the as intelligent forecasting systems; indeed, they were con- interaction of different variables is an important step in the sidered as a black box. In this study, using ANFIS and its water resources planning and management. Nonlinear and basic concepts, a new approach is devised for parameter dynamic properties of hydrological processes and uncer- estimation of autoregressive (AR) models. Performance of tainties of data are the main reasons of applying the time this approach is evaluated through the Akaike information series modeling. On the other hand, time series models can criterion; also its application has been surveyed in time be used for design and operation of water resource systems, series forecasting by naturalized inflow of the Zayandehrud according to the temporal and spatial statistics records by dam located in central Iran. Results show that the proposed predicting hydrological variables such as river flow, rain- approach has a good and effective performance for fall, humidity and temperature. Time series analysis is one parameter estimation of AR models which can be depicted of the most common methods of forecasting and data as a new ‘‘intelligent approach.’’ In addition, this capa- generation of hydrological processes, especially for oper- bility of ANFIS in parameter estimation is a new applica- ation of water resource systems such as dams and rivers as tion of ANFIS that was not addressed in the past. Also, the surface water resources and aquifers as groundwater resources as well as conjunctive use systems (Safavi 2014). Research on hydrologic time series has been aimed at & Hamid R. Safavi studying the main statistical characteristics, providing [email protected] physical justification to some stochastic models, develop- 1 Department of Civil Engineering, Isfahan University of ing new and/or alternative models, improving the estimates Technology, Isfahan, Iran of model parameters, developing new or improving exist- 2 Department of Electrical and Computer Engineering, Isfahan ing modeling procedures, improving tests of goodness of University of Technology, Isfahan, Iran fit, developing procedures on dealing with model and 3 Department of Land, Air and Water Resource, University of parameter uncertainties and studying the sensitivity of California, Davis, Davis, CA, USA 123 Author's personal copy Iran J Sci Technol Trans Civ Eng models and model parameters in applied hydrology (Salas systems. Neuro-fuzzy systems have the potential to capture et al. 1997). the benefits of both these fields in a single framework In time series analysis and modeling, the relationships (Nayak et al. 2004). Adaptive network-based fuzzy infer- between inputs and outputs are mapped as a function of ence system (ANFIS), which consists of the neural net- observed patterns in the past. Conventional time series works and fuzzy logic methods, has been used in many methods including autoregressive (AR), autoregressive hydrologic applications such as rainfall-runoff process for moving average (ARMA) and autoregressive integrated predicting daily runoff at multiple gauging stations moving average (ARIMA) models (Box and Jenkins 1976) (Nourani and Komasi 2013) and improving rainfall fore- have been used for hydrologic modeling. However, such casting efficiency (Akrami et al. 2013), reservoir operation models do not attempt to represent inherent nonlinear and (Valizadeh and El-Shafie 2013), decision support systems dynamic characteristics of the hydrologic process and may (Petrovic et al. 2006), discharge routing (Khatibi et al. not always perform well (Tokar and Johnson 1999; Nayak 2011), evapotranspiration estimation (Cobaner 2011), river et al. 2004). Anyway, the conventional time series mod- streamflow and dams inflow forecasting (Sanikhani and eling methods have served the scientific community for a Kisi 2012; El-Shafie et al. 2007), and water demand fore- long time (Zounemat-Kermani and Teshnelab 2008). In casting (Tabesh and Dini 2009). ANFIS eliminates the time series modeling, it is necessary to determine param- basic problem of fuzzy systems design (obtaining a set of eters of each model to develop time series models (Sang fuzzy if–then rules) using the learning capability of an 2012, 2013; Dutta et al. 2012). One of the most familiar ANN, effectively, for automatic fuzzy if–then rule gener- methods of parameter estimation of AR models which is ation and parameter optimization. referred to as Yule–Walker (YW) method (Yule 1927; Aforementioned researches are examples of many Walker 1931) is based directly on the linear relationship researches which have investigated the applications of between the co-variances and the AR parameters (Stoica ANFIS in hydrologic filed, especially in time series mod- and Moses 2005; Hipel and McLeod 1994). Another eling and forecasting. In these researches, ANFIS is con- method, the so-called least-squares (LSs) method is based sidered as a black box which means that after training and on a least-squares solution of AR parameters using the testing, it can be used as an intelligent model to simulate or time-domain equation (Stoica and Moses 2005). Burg predict the uncertain future. So, it was not used to estimate (1975) expressed the problems of LS method and devel- various parameters using inner parameters of ANFIS such oped a new method for AR parameter estimation that is as weights and output of membership functions of fuzzy based on LS method to improve the mentioned problems inference system (FIS). In this study, based on the basic and is depicted as Burg’s method (Burg 1975). Details of concepts of ANFIS, a new approach for parameter esti- these methods are provided in his PhD dissertation. For- mation of AR models is devised which is a novel technique ward–backward (FB) method is another method that esti- for estimating parameters of time series such as AR time mates AR parameters by minimizing the sum of a least- series models. Based on the hybrid method for training of squares criterion for a forward model and the analogous ANFIS, the new approach can be considered as a combi- criterion for a time-reversed model (Marple 1987). FB nation of LS, FB and Burg’s approaches with emphasis that approach has the same order of computational complexity the new approach is derived from ANFIS and it is a new as the popular Burg algorithm. Marple (1987) concluded capability of this system. Performance of the models that the LS algorithm is an attractive alternative to the Burg developed by new approach in prediction is surveyed by algorithm for AR spectral estimation. The functions of mean squared relative error (MSRE), the coefficient of these approaches can be used in some mathematics soft- efficiency (CE) and mean absolute error (MAE) in com- ware such as MATLAB. parison of prediction with models developed by YW, LS, Recently, artificial intelligence techniques such as arti- Burg and FB approaches. On the other hand, presented ficial neural networks (ANNs) and fuzzy logic have been approach shows a new application of ANFIS. The appli- used as efficient alternative tools for the modeling and cability and performance of this approach have been sur- forecasting of complex hydrologic systems and time series veyed by the Zayandehrud dam inflow as case study. (Jeong et al. 2012; Kim and Singh 2013; Awan and Bae 2014). These methods are able to execute parallel com- putations and simulate nonlinear system which is hard to describe by traditional physical modeling. These intelligent 2 Case Study: Zayandehrud Dam systems have provided a wide range of applications in hydrology and water resources management (Safavi et al.