Probabilistic Predictability of Stochastic Dynamical Systems

TL;DR

This paper introduces an epsilon-logarithm score for evaluating probabilistic predictions of stochastic dynamical systems, analyzing predictability based on neighborhood size, noise entropy, and system dimension, with theoretical and numerical insights.

Contribution

It proposes a generalized scoring rule for SDS predictability, providing theoretical characterizations and approximations, and analyzing trajectory score convergence behaviors.

Findings

Score convergence to expected score under i.i.d. noise

Error in score approximation scales with epsilon

Trajectory score converges at rate T^{-1/2}

Abstract

To assess the quality of a probabilistic prediction for stochastic dynamical systems (SDSs), scoring rules assign a numerical score based on the predictive distribution and the measured state. In this paper, we propose an $\epsilon$-logarithm score that generalizes the celebrated logarithm score by considering a neighborhood with radius $\epsilon$. We characterize the probabilistic predictability of an SDS by optimizing the expected score over the space of probability measures. We show how the probabilistic predictability is quantitatively determined by the neighborhood radius, the differential entropies of process noises, and the system dimension. Given any predictor, we provide approximations for the expected score with an error of scale $\mathcal{O}(\epsilon)$. In addition to the expected score, we also analyze the asymptotic behaviors of the score on individual trajectories. Specifically, we prove that the score on a trajectory can converge to the expected score when the process noises are independent and identically distributed. Moreover, the convergence speed against the trajectory length $T$ is of scale $\mathcal{O}(T^{-\frac{1}{2}})$ in the sense of probability. Finally, numerical examples are given to elaborate the results.