Forecasting non-stationary time series by wavelet process modelling

Many time series in the applied sciences display a time-varying second order structure. In this article, we address the problem of how to forecast these non-stationary time series by means of non-decimated wavelets. We first consider a model in which only the variance evolves with time. We define a predictor for this model and show an application to the Dow Jones index. Then, we generalise the definition to the case of time-varying covariance using the class of Locally Stationary Wavelet processes. We introduce a new predictor based on wavelets and derive the prediction equations as a generalisation of the Yule-Walker equations. We propose an automatic computational procedure for choosing the parameters of the forecasting algorithm. Finally, we apply the general prediction algorithm to a meteorological time series.