Hold-out estimates of prediction models for Markov processes

TL;DR

This paper analyzes the theoretical properties of the hold-out method for selecting prediction models in Markov processes, providing bounds and inequalities under ergodicity assumptions.

Contribution

It extends the understanding of hold-out model selection to Markovian time series, establishing generalization bounds and oracle inequalities.

Findings

Hold-out method is effective for Markov processes under ergodicity.

Generalization bounds are established for out-of-sample prediction.

The method adapts to noise conditions in Markovian models.

Abstract

We consider the selection of prediction models for Markovian time series. For this purpose, we study the theoretical properties of the hold-out method. In the econometrics literature, the hold-out method is called out-of-sample and is the main method to select a suitable time series model. This method consists of estimating models on a learning set and picking up the model with minimal empirical error on a validation set of future observations. Hold-out estimates are well studied in the independent case, but, as far as we know, this is not the case when the validation set is not independent of the learning set. In this paper, assuming uniform ergodicity of the Markov chain, we state generalization bounds and oracle inequalities for such method; in particular, we show that the out-of-sample selection method is adaptative to noise condition.