On Consistent Nonparametric Order Determination and Chaos

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1992 · Volume 54 · Issue 2

DOI: 10.1111/j.2517-6161.1992.tb01890.x

Abstract

We give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models. We argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos. Setting the stochastic models within the framework of non-linear autoregression, we introduce the notion of a generalized partial autocorrelation and an order. We approach the estimation of the embedding dimension via order determination of an unknown non-linear autoregression by cross-validation, and give justification by proving its consistency under global boundedness. As a by-product, we provide a theoretical justification of the final prediction error approach of Auestad and Tjøstheim. Some illustrations based on the Hénon map and several real data sets are given. The bias of the residual sum of squares as essentially a noise variance estimator is quantified.

Keywords

attractors, bandwidth, bias correction, binary shift map, canadian lynx, chaos, consistency, cross-validation, dimension, double windows, embedding dimension, epidemics, experimental data, final prediction error, fractals, generalized partial autocorrelation function, global boundedness, hénon map, kernel estimation, limit cycles, limit points, local instability, lyapunov exponent, map reconstruction, measles, nonlinear autoregression, order determination, order of non-linear autoregression, predictive residuals, residual sum of squares, skeleton, u-statistics, wolf's sunspot numbers

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