Wavelet-based Multiresolution Forecasting

In this report, we discuss results of modelling and forecasting nonstationary financial time series using a combination of the maximal overlap discreet wavelet transform (MODWT) and fuzzy logic. A financial time series is decomposed into an over complete, sh ift invariant wavelet representation. A fuzzy-rule base is created for each individual wavelet sub-series to p redict future values. To form the aggregate forecas t, the individual wavelet sub-series forecasts are recombi ned utilizing the linear reconstruction property of the wavelet multiresolution analysis (MRA). Results are present ed for IBM, NASDAQ and S&P 500 daily (adjusted) close values. I. INTRODUCTION The successful application of modern modelling tech niques like neural networks and fuzzy logic to financial time series requires a certain uniformity (stationarity) of the data. Financial time series data is inherently nonstationary and may be a superposition of many sources exhibiting different dynamics. Neural networks (employing nonlinear autoregressions) and fuzzy logic models (employing fuzzy-rule bases) can be termed as global approximators where only one model is used to characterize an entire process. Therefore, such techniques usually give be st results for stationary time series. Recently, there has been an increased interest in m ultiresolution decomposition techniques like the wavelet transform for elucidating complex relations hips in nonstationary financial time series (4). Th e wavelet transform can produce a good local represen tation of a signal in both time and frequency domain and is not restrained by the assumption of s tationarity (5). Moreover, the wavelet approach has formalized old notions of decomposing a financial t ime series into trend, seasonal and business cycle components (7). Motivated by the spatial frequency resolution property of the wavelet transform, several hybrid schemes (local models) have been dev eloped, for example (1), which combine wavelet analysis with machine learning approaches like neur al networks for time series prediction. In this report, we present results of predicting fi nancial time series with a fuzzy-wavelet hybrid sys tem that incorporates multiscale wavelet decompositions into a set of fuzzy-rule bases. The system employs a shift invariant wavelet transform called the maximal overlap discrete wavelet transform (MODWT) (6). Essentially, the so-called a trous filtering scheme (8) is applied to generate MODWT decompositions of a financial time series. A fuzzy- rule base is created to predict each wavelet decomposition separately. To generate a global fore cast, the prediction results of individual wavelet decompositions are combined directly using the linear reconstruction property of the wavelet multiresolution analysis (MRA).