Identification and prediction of a class of non-Gaussian time-series via transformation

Whereas optimal prediction of Gaussian sequences requires the employment of a linear filter with consistently identifiable parameters and with Gaussian white noise input, the optimal predictor of non-Gaussian sequences is n nonlinear filter, having an independent noise input. Since the latter cannot be identified directly without prior knowledge of the non-linearity, the optimal linear predictor is usually identified where a non-Gaussian white noise input is considered and which is fully optimal only when that input turns out to be independent in all moments. However, if the non-Gaussian sequence is the outcome of a Gaussian sequence passed through a zero memory non-linearity or through non-linear measurement elements, a transformation of the non-Gaussian sequence into a Gaussian one is possible, such that optimal non-linear prediction may be approximated to any required degree, as is shown by the analysis of the present work. Furthermore, the parameters of that predictor may be consistently identified in the absence of any parameter information.