Latent deformation models for multivariate functional data and time‐warping separability
Latent deformation models for multivariate functional data and time‐warping separability
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DOI：
10.1111/biom.13851
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发表时间：
2021-07
期刊：
影响因子：
1.9
通讯作者：
Cody Carroll;H. Müller
中科院分区：
文献类型：
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作者：
Cody Carroll;H. Müller
多变量函数数据提出了在单变量函数数据中没有发现的理论和实践并发症。其中之一是多元函数数据的分量函数为正并且受到相互时间弯曲的情况。也就是说，分量过程表现出共同的形状，但除了受试者特定的时间扭曲之外，还受到跨其域的系统相位变化的影响，其中每个受试者都有自己的内部时钟。这激发了一种新的多元函数数据模型，该模型通过利用新的时间扭曲可分性假设将这种相互时间扭曲连接到基于潜在变形的框架。这种可分性假设允许有意义的解释和降维。由此产生的潜在变形模型是非常适合代表常见的功能向量数据。所提出的方法将每个分量的随机幅度因子与多元函数数据向量分量之间基于群体的配准相结合，并包括潜在群体函数，该函数对应于共同的底层轨迹。我们为模型的所有组成部分提出了估计器，从而实现了多变量函数数据和下游分析（如Fréchet回归）的基于数据的表示。当曲线被完全观察到或观察到测量误差时，建立收敛速率。的有用性的模型，解释，和实际方面的说明，在模拟和应用程序的多变量人类生长曲线和多变量环境污染数据。
Multivariate functional data present theoretical and practical complications that are not found in univariate functional data. One of these is a situation where the component functions of multivariate functional data are positive and are subject to mutual time warping. That is, the component processes exhibit a common shape but are subject to systematic phase variation across their domains in addition to subject‐specific time warping, where each subject has its own internal clock. This motivates a novel model for multivariate functional data that connect such mutual time warping to a latent‐deformation‐based framework by exploiting a novel time‐warping separability assumption. This separability assumption allows for meaningful interpretation and dimension reduction. The resulting latent deformation model is shown to be well suited to represent commonly encountered functional vector data. The proposed approach combines a random amplitude factor for each component with population‐based registration across the components of a multivariate functional data vector and includes a latent population function, which corresponds to a common underlying trajectory. We propose estimators for all components of the model, enabling implementation of the proposed data‐based representation for multivariate functional data and downstream analyses such as Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves and multivariate environmental pollution data.