Journal of Water Resource and Protection, 2011, 3, 50-59 doi:10.4236/jwarp.2011.31006 Published Online January 2011 (http://www.scirp.org/journal/jwarp) Time Series Modeling of River Flow Using Wavelet Neural Networks B. Krishna, Y. R. Satyaji Rao, P. C. Nayak Deltaic Regional Center, National Institute of Hydrology Kakinada, Andhra Pradesh, India E-mail: [email protected] Received September 24, 2009; revised October 18, 2010; accepted November 19, 2010 Abstract A new hybrid model which combines wavelets and Artificial Neural Network (ANN) called wavelet neural network (WNN) model was proposed in the current study and applied for time series modeling of river flow. The time series of daily river flow of the Malaprabha River basin (Karnataka state, India) were analyzed by the WNN model. The observed time series are decomposed into sub-series using discrete wavelet transform and then appropriate sub-series is used as inputs to the neural network for forecasting hydrological variables. The hybrid model (WNN) was compared with the standard ANN and AR models. The WNN model was able to provide a good fit with the observed data, especially the peak values during the testing period. The benchmark results from WNN model applications showed that the hybrid model produced better results in estimating the hydrograph properties than the latter models (ANN and AR). Keywords: Time Series, River Flow, Wavelets, Neural Networks 1. Introduction place in the system being modeled. Time series analysis requires mapping complex relationships between input(s) Water resources planning and management requires, and output(s), since the forecasted values are mapped as output from hydrological studies. These are mainly in a function of observed patterns in the past. On the other the form of estimation or forecasting of the magnitude of hand, classical non-linear techniques typically require hydrological variables like precipitation, stream flow and large amounts of exogenous data, which are not always groundwater levels using historical data. Forecasting of available [3]; some non-linear approaches outperform hydrological variables especially stream flow is utmost the linear techniques, as periodic gamma autoregressive important to provide a warning of the extreme flood or processes (PGAR) for instance [4], but they can only be drought conditions and helps to develop operation sys- applied to a single site, i.e., they are limited to be uni- tems of multipurpose reservoirs. Similarly, groundwater variate models. Owing to the difficulties associated with level modeling is useful for consumptive use studies for non-linear model structure identification and parameter better watershed management. estimation, very few truly non-linear system theoretic Time series modeling for either data generation or fo- hydrologic models have been reported [5-7]. It seems recasting of hydrological variables is an important step necessary that nonlinear models such as neural networks, in the planning and management of water resources. which are suited to complex nonlinear problems, be used Most of the time series modeling procedures fall within for the time series modeling of river flow. the framework of multivariate Auto Regressive Moving Artificial Neural Networks (ANN) has been used suc- Average (ARMA) model [1]. Traditionally, ARMA cessfully to overcome many difficulties in time series models have been widely used for modeling water re- modeling of river flow. Time series modeling using sources time-series modeling [2]. The time series models ANN has been a particular focus of interest and better are used to describe the stochastic structure of the time performing models have been reported in a diverse set of sequence of a hydrological variable measured over time. fields that include rainfall-runoff modeling [8-12] and Time-series models are more practical than conceptual groundwater level prediction [13-16]. It is also reported models because it is not essential to understand the in- that ANN models are not very satisfied in precision for ternal structure of the physical processes that are taking forecasting because it considered only few aspects of Copyright © 2011 SciRes. JWARP B. KRISHNA ET AL. 51 time series property [17]. In order to raise the forecasted formance of WNN model was compared with the ANN precision and lengthen the forecasted time, an alternative and AR models. model should be envisaged. In this paper, a new hybrid model called wavelet neural network model (WNN), 2. Methods of Analysis which is the combination of wavelet analysis and ANN, has been proposed. The advantage of the wavelet tech- 2.1. Wavelet Analysis nique is that it provides a mathematical process for de- composing a signal into multi-levels of details and anal- The wavelet analysis is an advance tool in signal proc- ysis of these details can be done. Wavelet analysis can essing that has attracted much attention since its theo- effectively diagnose signals of main frequency compo- retical development [33]. Its use has increased rapidly in nent and abstract local information of the time series. In communications, image processing and optical engi- the past decade, wavelet theory has been introduced to neering applications as an alternative to the Fourier signal processing analysis. In recent years, the wavelet transform in preserving local, non-periodic and multis- transforms has been successfully applied to wave data caled phenomena. The difference between wavelets and analysis and other ocean engineering applications Fourier transforms is that wavelets can provide the exact [18,19]. locality of any changes in the dynamical patterns of the In recent years, wavelet theory has been introduced in sequence, whereas the Fourier transforms concentrate the field of hydrology [17,20-22]. Wavelet analysis has mainly on their frequency. Moreover, Fourier transform recently been identified as a useful tool for describing assume infinite-length signals, whereas wavelet trans- both rainfall and runoff time-series [21,23-26]. By cou- forms can be applied to any kind and size of time series, pling the wavelet method with the traditional AR model, even when these sequences are not homogeneously sam- the Wavelet-Autoregressive model (WARM) is devel- pled in time [34]. In general, wavelet transforms can be oped for annual rainfall prediction [27]. Coulibaly [28] used to explore, denoise and smoothen time series, aid in used the wavelet analysis to identify and describe vari- forecasting and other empirical analysis. ability in annual Canadian stream flows and to gain in- Wavelet analysis is the breaking up of a signal into sights into the dynamical link between the stream flows shifted and scaled versions of the original (or mother) and the dominant modes of climate variability in the wavelet. In wavelet analysis, the use of a fully scalable Northern Hemisphere. Due to the similarity between modulated window solves the signal-cutting problem. wavelet decomposition and one hidden layer neural net- The window is shifted along the signal and for every work, the idea of combining both wavelet and neural net- position the spectrum is calculated. Then this process is work has resulted recently in formulation of wavelet neu- repeated many times with a slightly shorter (or longer) ral network, which has been used in various fields [29]. window for every new cycle. In the end, the result will Results show that, the training and adaptation efficiency be a collection of time-frequency representations of the of the wavelet neural network is better than other net- signal, all with different resolutions. Because of this col- works. Dongjie [30] used a combination of neural net- lection of representations we can speak of a multiresolu- works and wavelet methods to predict ground water levels. tion analysis. By decomposing a time series into time- Aussem [31] used a Dynamical Recurrent Neural Network frequency space, one is able to determine both the do- (DRNN) on each resolution scale of the sunspot time se- minant modes of variability and how those modes vary ries resulting from the wavelet decomposed series with the in time. Wavelets have proven to be a powerful tool for Temporal Recurrent Back propagation (TRBP) algorithm. the analysis and synthesis of data from long memory Partal [32] used a conjunction model (wavelet-neuro- processes. Wavelets are strongly connected to such fuzzy) to forecast the Turkey daily precipitation. The ob- processes in that the same shapes repeat at different or- served daily precipitations are decomposed to some sub ders of magnitude. The ability of the wavelets to simul- series by using Discrete Wavelet Transform (DWT) and taneously localize a process in time and scale domain then appropriate sub series are used as inputs to neuro- results in representing many dense matrices in a sparse fuzzy models for forecasting of daily precipitations. form. In this paper, an attempt has been made to forecast the time series of Daily River flow by developing wavelet 2.2. Discrete Wavelet Transform (DWT) neural network (WNN) models. Time series of river flow was decomposed into wavelet sub-series by discrete The basic aim of wavelet analysis is both to determine the wavelet transform. Then, neural network model is con- frequency (or scale) content of a signal and to assess and structed with wavelet sub-series as input, and the origi- determine the temporal variation of this frequency content. nal time series as output. Finally, the forecasting per- This property is in complete contrast to the Fourier analy- Copyright © 2011 SciRes. JWARP 52 B. KRISHNA ET AL. sis, which allows for the determination of the frequency tion components (Figure 1). content of a signal but fails to determine frequency time- dependence. Therefore, the wavelet transform is the tool 2.3. Mother Wavelet of choice when signals are characterized by localized high frequency events or when signals are characterized by a The choice of the mother wavelet depends on the data to large numbers of scale-variable processes. Because of its be analyzed.