论文标题
随机动力学系统的概率可预测性
Probabilistic Predictability of Stochastic Dynamical Systems
论文作者
论文摘要
为了评估随机动力学系统（SDSS）概率预测的质量，评分规则分配了基于预测分布和测量状态的数值得分。在本文中，我们提出了一个$ε$ -logarithm分数，该得分通过考虑一个半径$ε$的社区来概括著名的对数得分。我们通过优化概率度量空间的预期得分来表征SD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，过程噪声的差分熵和系统维数定量确定。给定任何预测指标，我们提供了预期分数的近似值，并具有比例$ \ MATHCAL {O}（ε）$的误差。除了预期的分数外，我们还分析了单个轨迹的分数的渐近行为。具体而言，我们证明，当过程噪声独立并且分布相同时，轨迹上的分数可以收敛到预期得分。此外，相对于轨迹长度$ t $的收敛速度是$ \ mathcal {o}（t^{ - \ frac {1} {2}}}）$。最后，给出数值示例以详细说明结果。
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.
