Wasserstein Gradients for the Temporal Evolution of Probability Distributions
Wasserstein Gradients for the Temporal Evolution of Probability Distributions
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DOI：
10.1214/21-ejs1883
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发表时间：
2018-09
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影响因子：
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通讯作者：
Yaqing Chen;H. Muller
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文献类型：
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作者：
Yaqing Chen;H. Muller
对概率测度的流动进行了许多研究，通常是在梯度流动方面。我们在这里介绍了一种新的方法，用于模拟经验观察到的分布流随时间的瞬时演变，其数据分析重点尚未被探索。该模型基于Wasserstein距离描述一维欧几里得空间$\mathbb{R}$随时间的分布流，利用最优运输图随时间的导数。最优交通地图的时间动态用时变分布数据来说明，这些数据包括年收入分布、历年死亡率的演变，以及来自纵向富Z生长研究的儿童年龄相关身高分布数据。
Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We introduce here a novel approach for the modeling of the instantaneous evolution of empirically observed distribution flows over time with a data-analytic focus that has not yet been explored. The proposed model describes the observed flow of distributions on one-dimensional Euclidean space $\mathbb{R}$ over time based on the Wasserstein distance, utilizing derivatives of optimal transport maps over time. The resulting time dynamics of optimal transport maps are illustrated with time-varying distribution data that include yearly income distributions, the evolution of mortality over calendar years, and data on age-dependent height distributions of children from the longitudinal Z\"urich growth study.