随机动力系统的概率可预测性
Probabilistic Predictability of Stochastic Dynamical Systems
AI总结：
本文提出ε-对数分数，通过优化期望分数刻画随机动力系统的概率可预测性，并给出其与邻域半径、噪声熵和系统维度的定量关系及轨迹收敛速度，附数值验证。
摘要：
为了评估随机动力系统（SDSs）概率预测的质量，评分规则基于预测分布和测量状态分配一个数值分数。在本文中，我们提出了一种ε-对数分数，它通过考虑半径为ε的邻域来推广著名的对数分数。我们通过在概率测度空间上优化期望分数来刻画随机动力系统的概率可预测性。我们展示了概率可预测性如何由邻域半径、过程噪声的微分熵以及系统维度定量决定。对于任意给定的预测器，我们提供了期望分数的近似，其误差量级为O(ε)。除了期望分数，我们还分析了单个轨迹上分数的渐近行为。具体来说，我们证明了当过程噪声独立同分布时，轨迹上的分数可以收敛到期望分数。此外，在概率意义上，收敛速度相对于轨迹长度T的量级为O(T^{-1/2})。最后，给出了数值例子来详细说明这些结果。
英文摘要：
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.
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