Abstract
In the prediction of (stochastic) time series, it has been common to suppose that an individual predictive method – for instance, an Auto-Regressive Integrated Moving Average (ARIMA) model – produces residuals like a white noise process. However, mainly due to the structures of auto-dependence not mapped by a given individual predictive method, this assumption may easily be violated, in practice, as pointed out in Firmino et al. (2015). In order to correct it (and accordingly to produce more forecasts with more accuracy power), this paper puts forward a Wavelet Hybrid Forecaster (WHF) that integrates the following numerical techniques: wavelet decomposition; ARIMA models; Artificial Neural Networks (ANNs); and linear combination of forecasts. Basically, the proposed WHF can map simultaneously linear – by means of a linear combination of ARIMA forecasts – and non-linear – through a linear combination of ANN forecasts – auto-dependence structures exhibited by a given time series. Differently of other hybrid methodologies existing in literature, the WHF forecasts are produced carrying into account implicitly the information from the frequency presenting in the underlying time series by means of the Wavelet Components (WCs) obtained by the wavelet decomposition approach. All numerical results show that WHF method has achieved remarkable accuracy gains, when comparing with other competitive forecasting methods already published in specialized literature, in the prediction of a well-known annual time series of sunspot.
Highlights
Over the years, several forecasting methods have been proposed with the aim of producing increasingly accurate predictions of time series
The collection of single predictive methods might yet be regrouped into two exclusive classes: the statistical one (here it lies, for instance, the AutoRegressive Integrated Moving Average (ARIMA) models), and the machine learning one (here it lies the Artificial Neural Networks (ANNs), as in HAYKIN, 2001)
Zhang (2003) shows in his numerical experiments – on which three very popular time series were forecasted by employing an hybrid forecaster – that the residuals produced by the ARIMA models could be efficiently predicted by using non-linear forecasters (namely, the Multi-Layer Perceptron ANNs (MLP-ANNs), as in HAYKIN, 2001))
Summary
Several forecasting methods have been proposed with the aim of producing increasingly accurate predictions of (stochastic) time series. The collection of single predictive methods might yet be regrouped into two exclusive classes: the statistical one (here it lies, for instance, the (linear) ARIMA models), and the machine learning one (here it lies the (non-linear) Artificial Neural Networks (ANNs), as in HAYKIN, 2001). In the numerical experiments in Firmino et al (2015), each time series was modelled by means of several plausible ARIMA models whose forecasts were added by the predictions of their residuals produced by different ANNs; the results achieved exhibit a remarkable accuracy gain in the hybrid forecasts when compared with other traditional methods, in all adherence statistics
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