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密度曲线 Fréchet 回归的 Wasserstein F 检验与置信带AI 解读

Data consisting of samples of probability density functions are increasingly prevalent, necessitating the development of methodologies for their analysis that respect the inherent nonlinearities associated with densities. In many applications, density curves appear as functional response objects in a regression model with vector predictors. For such models, inference is key to understand the importance of density-predictor relationships, and the uncertainty associated with…

多个随机密度的 Wasserstein 协方差AI 解读

A common feature of methods for analysing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fréchet mean is typically used to quantify and visualize the average density of the sample. For one-dimensional densities, the Wasserstein metric is popular due to its theoretical appeal and interpretive value as an optimal transport metric, leading to the Wasserstein–Fréchet…

欧氏预测变量下随机对象的 Fréchet 回归AI 解读

Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Fréchet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in $\mathcal{R}^{p}$, achieved by extending the classical concept of a Fréchet mean to the notion of a…

多元函数型数据可解释协方差的 Fréchet 积分与自适应度量选择AI 解读

For multivariate functional data recorded from a sample of subjects on a common domain, one is often interested in the covariance between pairs of the component functions, extending the notion of a covariance matrix for multivariate data to the functional case. A straightforward approach is to integrate the pointwise covariance matrices over the functional time domain. We generalize this approach by defining the Fréchet integral, which depends on the metric chosen for the…

通过 Hilbert 空间变换开展密度函数的函数型数据分析AI 解读

Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints, densities do not live in a vector space and, therefore, commonly used Hilbert space based methods of functional data analysis are not applicable. To address this problem, we introduce a transformation approach, mapping probability densities to a Hilbert space of functions through a…