Mixture Models for Prediction from Time Series, with Application to Energy Use Data

Abstract

This paper aims to use mixture models to produce predictions from time series data. Given data of the form (ti, yi), i = 1, . . . , T , we propose a mix- ture model localized at time point tT with the k-th component as yi = mk (ti) + εik with mixing proportions πk (ti) such that 0 ≤ πk (ti) ≤ 1 and ∑K πk (ti) = 1, where K is the number of components. The k (·) are smooth unspecified regression functions, and the errors εik ∼ N(0, σ 2) are independently distributed. Estimation of this model is achieved through a kernel-weighted version of the EM-algorithm, using exponential kernels with different bandwidths (neighbour- hood sizes) hk as weight functions. By modelling a mixture of local regressions at a target time point tT but with different bandwidths hk , the estimated mixture probabilities are informative for the amount of information available in the data set at the scale of resolution corresponding to each bandwidth. Nadaraya- Watson and local linear estimators are used to carry out the localized estimation step. For prediction at time point tT +1, adequate methods are provided for each local method, and compared to competing forecasting routines. The data under study give the energy use for Bolivia, Lebanon, and Greece from 1971 to 2011.