Optimal prediction of data with unknown abrupt change points
We develop a novel methodology for predicting time series under unknown abrupt changes in data generating distributions. Based on Kolmogorov and Tikhomirov's e entropy, we propose a concept called e-predictability that quantifies the size of a model class and the maximal number of structural changes that allows the achievability of asymptotic optimal prediction. To predict under abrupt changes, our basic idea is to apply ϵ-net to discretize a nonparametric or parametric model class with an appropriately chosen e, and then apply a kinetic model averaging over the quantizers. Under reasonable assumptions, we prove that the average predictive performance is asymptotically as good as the oracle, i.e. when all the data generating distributions are known in advance. We show that the assumptions hold for a rather wide class of time variations. The results also address some puzzles related to the 'prediction-inference dilemma' in the context of change point analysis.
