Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes
Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes
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DOI：
10.1093/biomet/asab046
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发表时间：
2019-10
期刊：
影响因子：
2.7
通讯作者：
Javier Zapata;Sang-Yun Oh;Alexander Petersen
中科院分区：
文献类型：
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作者：
Javier Zapata;Sang-Yun Oh;Alexander Petersen
多元函数数据的协方差结构可能非常复杂，特别是如果多元维度很大，这使得将标准多元数据的统计方法扩展到函数数据设置具有挑战性。例如，高斯图形模型最近已被扩展到设置多元函数数据，通过应用多元方法截断基展开的系数。然而，与多变量数据相比，一个关键的困难是协方差算子是紧凑的，因此不可逆。在本文中的方法解决了一般问题的协方差建模的多元函数数据，特别是功能高斯图形模型。作为第一步，提出了一个新的概念的多元函数数据的协方差算子的可分性，称为部分可分性，导致一个新的Karhunen-Loève型扩展这样的数据。接下来，部分可分性结构被证明是特别有用的，以提供一个定义良好的功能高斯图形模型，可以确定与一系列的有限维图形模型，每个相同的固定尺寸。这促使一个简单而有效的估计程序，通过联合图形套索的应用。通过模拟和分析运动任务期间的功能性大脑连接来评估图形模型估计方法的经验性能。
The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extensions of statistical methods for standard multivariate data to the functional data setting challenging. For example, Gaussian graphical models have recently been extended to the setting of multivariate functional data by applying multivariate methods to the coefficients of truncated basis expansions. However, a key difficulty compared to multivariate data is that the covariance operator is compact, and thus not invertible. The methodology in this paper addresses the general problem of covariance modelling for multivariate functional data, and functional Gaussian graphical models in particular. As a first step, a new notion of separability for the covariance operator of multivariate functional data is proposed, termed partial separability, leading to a novel Karhunen–Loève-type expansion for such data. Next, the partial separability structure is shown to be particularly useful in order to provide a well-defined functional Gaussian graphical model that can be identified with a sequence of finite-dimensional graphical models, each of identical fixed dimension. This motivates a simple and efficient estimation procedure through application of the joint graphical lasso. Empirical performance of the method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task.