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Probabilistic Predictability of Stochastic Dynamical Systems

Tao Xu, Yushan Li, Jianping He

TL;DR

This work introduces the ε-logarithm score, a neighborhood-based generalization of the logarithm scoring rule, to quantify probabilistic predictability in stochastic dynamical systems (SDSs). It characterizes the optimal expected score by optimizing over predictive distributions and shows that the maximum scales with $d_x \log(2\epsilon)$ and the entropy structure of the process, with an explicit deconvolution relation for the optimal predictor. The authors develop a partition-based framework to approximate the ε-logarithm score with error $O(\epsilon)$ and prove that, for SDSs with i.i.d. process noises, the trajectory score converges to the expected score at a rate $O_p(T^{-1/2})$. Numerical simulations on a linear SDS with uniform noise corroborate the theory, illustrating how predictability depends on the neighborhood radius, noise entropy, and system dimension, and providing practical, trajectory-based evaluation methods.

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 $ε$-logarithm score that generalizes the celebrated logarithm score by considering a neighborhood with radius $ε$. 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 $\mathcal{O}(ε)$. 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.

Probabilistic Predictability of Stochastic Dynamical Systems

TL;DR

This work introduces the ε-logarithm score, a neighborhood-based generalization of the logarithm scoring rule, to quantify probabilistic predictability in stochastic dynamical systems (SDSs). It characterizes the optimal expected score by optimizing over predictive distributions and shows that the maximum scales with and the entropy structure of the process, with an explicit deconvolution relation for the optimal predictor. The authors develop a partition-based framework to approximate the ε-logarithm score with error and prove that, for SDSs with i.i.d. process noises, the trajectory score converges to the expected score at a rate . Numerical simulations on a linear SDS with uniform noise corroborate the theory, illustrating how predictability depends on the neighborhood radius, noise entropy, and system dimension, and providing practical, trajectory-based evaluation methods.

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 -logarithm score that generalizes the celebrated logarithm score by considering a neighborhood with radius . 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 . 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 is of scale in the sense of probability. Finally, numerical examples are given to elaborate the results.
Paper Structure (20 sections, 7 theorems, 62 equations, 1 figure)

This paper contains 20 sections, 7 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Motivations
  4. Contributions
  5. Related Works
  6. Preliminaries and Problem Formulation
  7. Preliminaries
  8. System and Predictor Model
  9. Problems of Interest
  10. Probabilistic Predictability of SDSs
  11. Approximation and Convergence Analysis
  12. Simulation
  13. Simulation Setup
  14. Results and Analysis
  15. Conclusion
  16. ...and 5 more sections

Key Result

Theorem 1

Given a neighborhood radius $\epsilon \geq 0$ and a trajectory length $T \geq 1$, i) the optimal expected $\epsilon$-logarithm score for predicting the trajectory $\mathbf{x}_{1:T}$ of an SDS $\Phi$ is where $\tilde{p}^\star_{\mathbf{x}_{1:T}}$ satisfies the following equations $\forall\,k \in [1,T]$, ii) The optimal predictor $p^\star_{\mathbf{x}_k \mid \mathbf{x}_{1:k-1}}$ attaining the predic

Figures (1)

  • Figure 1: $\epsilon$-logarithm scores $\bar{\mathcal{L}}_\epsilon(p_{\mathbf{x}_{1:T}}, x_{1:T}^{(n)})$ v.s. the time step $T$: given $\epsilon$ and trajectories $\{x_{1:T}^{(n)}\}_{n=1}^{10^5}$ of $\Phi$, i) score curves on three individual trajectories are randomly chosen for presentation; ii) the expected scores (blue solid line) and 95% confidence intervals (blue transparent area) at each step are calculated from the scores on $100,000$ trajectories; iii) the predictability (red dotted line) is evaluated by Theorem \ref{['thm:predictability']} and the expected score's approximation (blue dotted line) is evaluated by Theorem \ref{['thm:approx_elog_traj']}.

Theorems & Definitions (12)

  • Definition 1: $\epsilon$-logarithm score
  • Definition 2: $\epsilon$-logarithm score for SDSs
  • Theorem 1: Probabilistic predictability
  • Remark 1
  • Definition 3: Unifrom grid partition
  • Proposition 1: Evaluation of $\mathcal{L}_{\Sigma^\ell}$
  • Lemma 1
  • Theorem 2: Evaluation of $\mathcal{L}_\epsilon$
  • Lemma 2: Approximation of $\mathcal{L}_\epsilon(\hat{p}_\mathbf{x}, p_\mathbf{x})$
  • Theorem 3: Approximation of $\bar{\mathcal{L}}_\epsilon(\hat{p}_{\mathbf{x}_{1:T}}, p_{\mathbf{x}_{1:T}})$
  • ...and 2 more