Time-series Random Process Complexity Ranking Using a Bound on Conditional Differential Entropy

• URL: http://arxiv.org/abs/2510.20551v1
• Date: Thu, 23 Oct 2025 13:36:04 GMT
• Title: Time-series Random Process Complexity Ranking Using a Bound on Conditional Differential Entropy
• Authors: Jacob Ayers, Richard Hahnloser, Julia Ulrich, Lothar Sebastian Krapp, Remo Nitschke, Sabine Stoll, Balthasar Bickel, Reinhard Furrer,
• Abstract summary: We build on the information theoretic prediction error bounds established by Fang et al.<n>We show that conditional differential entropy textbf$h(X_k mid X_k-1,...,X_k-m)$ is upper bounded by a function of the determinant of next-step prediction errors.
• Score: 0.8666096694354596
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Conditional differential entropy provides an intuitive measure for relatively ranking time-series complexity by quantifying uncertainty in future observations given past context. However, its direct computation for high-dimensional processes from unknown distributions is often intractable. This paper builds on the information theoretic prediction error bounds established by Fang et al. \cite{fang2019generic}, which demonstrate that the conditional differential entropy \textbf{$h(X_k \mid X_{k-1},...,X_{k-m})$} is upper bounded by a function of the determinant of the covariance matrix of next-step prediction errors for any next step prediction model. We add to this theoretical framework by further increasing this bound by leveraging Hadamard's inequality and the positive semi-definite property of covariance matrices. To see if these bounds can be used to rank the complexity of time series, we conducted two synthetic experiments: (1) controlled linear autoregressive processes with additive Gaussian noise, where we compare ordinary least squares prediction error entropy proxies to the true entropies of various additive noises, and (2) a complexity ranking task of bio-inspired synthetic audio data with unknown entropy, where neural network prediction errors are used to recover the known complexity ordering. This framework provides a computationally tractable method for time-series complexity ranking using prediction errors from next-step prediction models, that maintains a theoretical foundation in information theory.

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