A neuro-fuzzy computing technique for modeling hydrological time series

Abstract

Intelligent computing tools such as artificial neural network (ANN) and fuzzy logic approaches are proven to be efficient when applied individually to a variety of problems. Recently there has been a growing interest in combining both these approaches, and as a result, neuro-fuzzy computing techniques have evolved. This approach has been tested and evaluated in the field of signal processing and related areas, but researchers have only begun evaluating the potential of this neuro-fuzzy hybrid approach in hydrologic modeling studies. This paper presents the application of an adaptive neuro fuzzy inference system (ANFIS) to hydrologic time series modeling, and is illustrated by an application to model the river flow of Baitarani River in Orissa state, India. An introduction to the ANFIS modeling approach is also presented. The advantage of the method is that it does not require the model structure to be known a priori, in contrast to most of the time series modeling techniques. The results showed that the ANFIS forecasted flow series preserves the statistical properties of the original flow series. The model showed good performance in terms of various statistical indices. The results are highly promising, and a comparative analysis suggests that the proposed modeling approach outperforms ANNs and other traditional time series models in terms of computational speed, forecast errors, efficiency, peak flow estimation etc. It was observed that the ANFIS model preserves the potential of the ANN approach fully, and eases the model building process.

Introduction

Time series modeling for either data generation or forecasting of hydrologic variables is an important step in the planning and operational analysis of water resources. Traditionally, autoregressive moving average (ARMA) models have been used for modeling water resource time series because such models are accepted as a standard representation of stochastic time series (Maier and Dandy, 1997). However, such models do not attempt to represent the non-linear dynamics inherent in the hydrologic process, and may not always perform well (Tokar and Johnson, 1999). Time series analysis requires mapping complex relationships between input(s) and output(s), since the forecasted values are mapped as a function of observed patterns in the past. Owing to the difficulties associated with non-linear model structure identification and parameter estimation, very few truly non-linear system theoretic hydrologic models have been reported (e.g. Jacoby, 1966, Amorocho and Brandstetter, 1971, Ikeda et al., 1976). In most cases, linearity or piecewise linearity has been assumed (Natale and Todini, 1976a, Natale and Todini, 1976b). Recently a growing interest in the modeling of nonlinear relationships has developed and a variety of test procedures for detecting the nonlinearities have evolved (Anders and Korn, 1999). If the aim of analysis is prediction, however, it is not sufficient to uncover the nonlinearities. One needs to describe them through an adequate nonlinear model. Unfortunately, for many applications the theory does not guide the model building process by suggesting the relevant input variables or the correct functional form. This particular difficulty makes it attractive to consider an ‘atheoretical’ but flexible class of statistical models (Anders and Korn, 1999).

Artificial neural networks (ANN) are essentially semi-parametric regression estimators and are well suited for this purpose, as they can approximate virtually any (measurable) function up to an arbitrary degree of accuracy (Hornik et al., 1989). A significant advantage of the ANN approach in system modeling is that one need not have a well-defined physical relationship for systematically converting an input to an output. Rather, all that is needed for most networks is a collection of representative examples (input–output pairs) of the desired mapping. The ANN then adapts itself to reproduce the desired output when presented with training sample input. The emergence of neural network technology has provided many promising results in the field of hydrology and water resources simulation. A comprehensive review of the application of ANN to hydrology can be found in the findings of the ASCE Task Committee, 2000a, ASCE Task Committee, 2000b.

Another soft computing technique, which has very recently received attention in hydrology, is the fuzzy-rule based approach in modeling. First introduced by Zadeh (1965), fuzzy logic and fuzzy set theory are employed to describe human thinking and reasoning in a mathematical framework. Fuzzy-rule based modeling is a qualitative modeling scheme where the system behavior is described using a natural language (Sugeno and Yasukawa, 1993). The last decade has witnessed a few applications of a fuzzy logic approach in water resources forecasting (Fujita et al., 1992, Zhu and Fujita, 1994, Zhu et al., 1994, Stuber et al., 2000, See and Openshaw, 2000, Hundecha et al., 2001, Xiong et al., 2001).

These intelligent computational methods offer real advantages over conventional modeling, including the ability to handle large amounts of noisy data from dynamic and nonlinear systems, especially when the underlying physical relationships are not fully understood. Each of these techniques is proven to be effective when used on their own. However, when combined together, the individual strengths of each approach can be exploited in a synergistic manner for the construction of powerful intelligent systems. In recent years, the integration of neural networks and fuzzy logic has given birth to new research into neuro-fuzzy systems. Neuro-fuzzy systems have the potential to capture the benefits of both these fields in a single framework. Neuro-fuzzy systems eliminate the basic problem in fuzzy system design (obtaining a set of fuzzy if-then rules) by effectively using the learning capability of an ANN for automatic fuzzy if-then rule generation and parameter optimization. As a result, those systems can utilize linguistic information from the human expert as well as measured data during modeling. Such applications have been developed for signal processing, automatic control, information retrieval, database management, computer vision and data classification (e.g. Jang, 1993). However, there is little discussion in the literature of more pragmatic hydrologic applications of this hybrid computing system.

The major objective of this paper is to investigate the potential of neuro-fuzzy systems in modeling hydrologic time series and to assess its performance relative to ANN and other traditional time series modeling techniques such as ARMA. The underlying principle and the neuro-fuzzy computing architecture are also discussed. The applicability of the method is demonstrated by modeling river flow for an Indian basin.