Equivalence between Time Series Predictability and Bayes Error Rate

TL;DR

This paper proves that time series predictability is mathematically equivalent to the Bayes error rate, bridging two fields and improving predictability estimation accuracy through theoretical models.

Contribution

It establishes a rigorous equivalence between predictability and Bayes error rate, enabling more accurate predictability assessments in time series analysis.

Findings

Proves predictability equals Bayes error rate mathematically.

Shows Bayes error rate can improve predictability estimation.

Validates the approach with three theoretical models.

Abstract

Predictability is an emerging metric that quantifies the highest possible prediction accuracy for a given time series, being widely utilized in assessing known prediction algorithms and characterizing intrinsic regularities in human behaviors. Lately, increasing criticisms aim at the inaccuracy of the estimated predictability, caused by the original entropy-based method. In this brief report, we strictly prove that the time series predictability is equivalent to a seemingly unrelated metric called Bayes error rate that explores the lowest error rate unavoidable in classification. This proof bridges two independently developed fields, and thus each can immediately benefit from the other. For example, based on three theoretical models with known and controllable upper bounds of prediction accuracy, we show that the estimation based on Bayes error rate can largely solve the inaccuracy problem of predictability.