Dynamic Adaptive Partitioning for Nonlinear Time Series

Peter Bühlmann

April 1998

Abstract

We propose a dynamic adaptive partitioning scheme for nonparametric analysis of stationary nonlinear time series with values in $\R^d$ ($d \ge 1$). We use information from past values to construct adaptive partitioning in a {\em dynamic} fashion which is then different from the more common static schemes in the regression set-up. The idea of dynamic partitioning is novel. We make it constructive by proposing an approach based on quantization of the data and adaptively modelling partition cells with a parsimonious Markov chain. The methodology is formulated in terms of a model class, the so-called quantized variable length Markov chains (QVLMC). We discuss when and why such a QVLMC partitioning scheme is in some sort natural and canonical. We explore asymptotic properties and give some numerical results which reflect the finite sample behavior.

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