On Predictability of Time Series

TL;DR

This paper critically examines the method for estimating time series predictability, revealing overestimations caused by ambiguities and inaccuracies in previous approaches, and clarifies the correct estimation process for researchers.

Contribution

It identifies and corrects misunderstandings in the original predictability estimation method, providing a clearer framework for accurate time series predictability analysis.

Findings

Demonstrates overestimation of predictability in various time series

Shows the deviation of Lempel-Ziv estimator for random data

Provides guidelines for correct predictability estimation

Abstract

The method to estimate the predictability of human mobility was proposed in [C. Song \emph{et al.}, Science {\bf 327}, 1018 (2010)], which is extensively followed in exploring the predictability of disparate time series. However, the ambiguous description in the original paper leads to some misunderstandings, including the inconsistent logarithm bases in the entropy estimator and the entropy-predictability-conversion equation, as well as the details in the calculation of the Lempel-Ziv estimator, which further results in remarkably overestimated predictability. This paper demonstrates the degree of overestimation by four different types of theoretically generated time series and an empirical data set, and shows the intrinsic deviation of the Lempel-Ziv estimator for highly random time series. This work provides a clear picture on this issue and thus helps researchers in correctly estimating the predictability of time series.