Applications of Universal Source Coding to Statistical Analysis of Time Series

TL;DR

This paper demonstrates how universal data compression codes can be applied to various statistical analysis tasks for time series, providing practical and often more powerful methods for estimation, prediction, and hypothesis testing.

Contribution

It introduces a unified approach using universal codes for statistical problems in time series analysis, bridging data compression and classical statistical methods.

Findings

Universal codes enable effective estimation of probabilities and densities.

The proposed methods often outperform traditional statistical tests.

Applications include prediction, regression, classification, and hypothesis testing.

Abstract

We show how universal codes can be used for solving some of the most important statistical problems for time series. By definition, a universal code (or a universal lossless data compressor) can compress any sequence generated by a stationary and ergodic source asymptotically to the Shannon entropy, which, in turn, is the best achievable ratio for lossless data compressors. We consider finite-alphabet and real-valued time series and the following problems: estimation of the limiting probabilities for finite-alphabet time series and estimation of the density for real-valued time series, the on-line prediction, regression, classification (or problems with side information) for both types of the time series and the following problems of hypothesis testing: goodness-of-fit testing, or identity testing, and testing of serial independence. It is important to note that all problems are considered in the framework of classical mathematical statistics and, on the other hand, everyday methods of data compression (or archivers) can be used as a tool for the estimation and testing. It turns out, that quite often the suggested methods and tests are more powerful than known ones when they are applied in practice.