Discriminant and cluster analysis for Gaussian stationary processes: local linear fitting approach

Abstract

This article is concerned with discrimination and clustering of Gaussian stationary processes. The problem of classifying a realization X _n  = (X ₁, …, X _n ) ^t from a linear Gaussian process X into one of two categories described by their spectral densities f ₁ (λ) and f ₂ (λ) is considered first. A discrimination rule based on a general disparity measure between every f _i (λ), i = 1, 2, and a nonparametric spectral density estimator fˆ _n (λ) is studied when local polynomial techniques are used to obtain fˆ _n (λ). In particular, three different local linear smoothers are considered. The discriminant statistic proposed here provides a consistent classification criterion for all three smoothers in the sense that the misclassification probabilities tend to zero. A simulation study is performed to confirm in practice the good theoretical behavior of the discriminant rule and to compare the influence of the different smoothers. The disparity measure is also used to carry out cluster analysis of time series and some examples are presented and compared with previous works.

Keywords:

• Discriminant analysis
• Spectral disparity measure
• Local polynomial fit
• Linear Gaussian process
• Cluster analysis

Acknowledgements

This research was partially supported by MCyT Grant BMF2002-00265 (European FEDER support included) and by XUGA Grants PGIDT01PXI10505PR and PGIDT03PXIC10505PN. The authors are pleased to acknowledge the helpful comments of the reviewer.