Explanation of Variability through Optimal Transport

1715753Tabak研究者开发了一种方法来解释由已知和未知的外部因素引起的数据变异性。这在所有数据密集型领域都有应用，包括公共卫生、经济学和环境研究。它为个性化医疗提供了一种天然的工具，因为它既可以从数据中过滤每个病人的特征（如年龄和性别）的影响，又可以将预测具体化，以解释每个病人的具体情况。它还允许分析在不同地点和时间用不同仪器采集的气象数据，以及根据已知和未知外部因素的非结构化时间序列。研究生和博士后参与项目工作。合并数据集和去除可变性的外部因素的问题在最优传输数学理论的扩展中找到了一个非常自然的公式：这两个问题都涉及规范化（即传输）原始数据，这样就不会留下指向来源数据集或混淆变量值的信号。另一方面，运输需要优化，最大限度地减少与其他来源相关的可变性的去除。本项目扩展了最优运输理论来处理这些问题。扩展包括诸如寻找只有有限数量的样本已知的概率密度连续体的重心等问题。研究生和博士后参与项目工作。

1715753Tabak The investigator develops a methodology for the explanation of variability in data attributable to known and unknown external factors. This has applications in all data-intensive fields, including public health, economics, and environmental studies. It provides a natural tool for personalized medicine, as it can both filter from data the influence of each patient's characteristics, such as age and sex, and particularize the predictions to account for the specifics of each patient. It also permits the analysis of meteorological data taken at different locations and times and with different instruments, and of unstructured time series depending on known and unknown external factors. Graduate and postdoctoral students participate in the work of the project. The problems of amalgamating datasets and removing external factors of variability find a very natural formulation in an extension of the mathematical theory of optimal transport: both problems concern normalizing (i.e. transporting) the original data so that no signal is left that points to the dataset of provenance or the values of the confounding variables. On the other hand, the transport needs to be optimized, minimizing the removal of variability associated to other sources. This project extends the theory of optimal transport to handle these problems. The extensions include problems such as finding the barycenter of a continuum of probability densities of which only a finite amount of samples are known. Graduate and postdoctoral students participate in the work of the project.