Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model representing the conditional relationships between the functional variables. However, in many applications of multivariate functional data, the graph is known and existing functional GGM methods cannot preserve a given graphical constraint. In this manuscript, we demonstrate how to conduct multivariate functional analysis that exactly conforms to a given inter-variable graph. We first show the equivalence between partially separable functional GGM and graphical Gaussian processes (GP), proposed originally for constructing optimal covariance functions for multivariate spatial data that retain the conditional independence relations in a given graphical model. The theoretical connection help design a new algorithm that leverages Dempster's covariance selection to calculate the maximum likelihood estimate of the covariance function for multivariate functional data under graphical constraints. We also show that the finite term truncation of functional GGM basis expansion used in practice is equivalent to a low-rank graphical GP, which is known to oversmooth marginal distributions. To remedy this, we extend our algorithm to better preserve marginal distributions while still respecting the graph and retaining computational scalability. The insights obtained from the new results presented in this manuscript will help practitioners better understand the relationship between these graphical models and in deciding on the appropriate method for their specific multivariate data analysis task. The benefits of the proposed algorithms are illustrated using empirical experiments and an application to functional modeling of neuroimaging data using the connectivity graph among regions of the brain.
Multivariate Functional Data Modeling with Time-varying Clustering
We consider the situation where multivariate functional data has been collected over time at each of a set of sites. Our illustrative setting is bivariate, monitoring ozone and PM $_{10}$ levels as a function of time over the course of a year at a set of monitoring sites. The data we work with is from 24 monitoring sites in Mexico City which record hourly ozone and PM $_{10}$ levels. We use the data for the year 2017. Hence, we have 48 functions to work with. Our objective is to implement model-based clustering of the functions across the sites. Using our example, such clustering can be consid…
Principal points and elliptical distributions from the multivariate setting to the functional case
随机向量$\mathbf{X}$的$k$主点被定义为使$\mathbf{X}$与集合中最近点之间的期望平方距离最小化的点的集合。Flury（1990，1993）、Tarpey（1995）和Tarpey，Li and Flury（1995）对它们进行了深入研究。对于它们的处理，检查通常限于椭圆分布族。在本文中，我们将以前的结果推广到函数椭圆分布的情况，即，当处理可分希尔伯特空间${\cal H}$上的随机元时。高斯过程的主点在Tarpey和Kinateder（2003）中定义。在本文中，我们推广的概念，主点，自洽点和椭圆分布，使他们适合在这个功能框架。结果连接自洽性和特征向量的协方差算子重新获得在这个新的设置以及一个明确的公式的$k=2$的情况下，以便包括椭圆分布的随机元在${\cal H}$。
Global Depths for Irregularly Observed Multivariate Functional Data
Two frameworks for multivariate functional depth based on multivariate depths are introduced in this paper. The first framework is multivariate functional integrated depth, and the second framework involves multivariate functional extremal depth, which is an extension of the extremal depth for univariate functional data. In each framework, global and local multivariate functional depths are proposed. The properties of population multivariate functional depths and consistency of finite sample depths to their population versions are established. In addition, finite sample depths under irregularl…
Sparse Functional Boxplots for Multivariate Curves
This paper introduces the sparse functional boxplot and the intensity sparse functional boxplot as practical exploratory tools. Besides being available for complete functional data, they can be used in sparse univariate and multivariate functional data. The sparse functional boxplot, based on the functional boxplot, displays sparseness proportions within the 50\% central region. The intensity sparse functional boxplot indicates the relative intensity of fitted sparse point patterns in the central region. The two-stage functional boxplot, which derives from the functional boxplot to detect outl…
Multivariate Functional Principal Component Analysis for Mixed-Type mHealth Data: An Application to Mood Disorders
Texas A&M University（德克萨斯农工大学）；University of South Carolina（南卡罗来纳大学）；National Institute of Mental Health（美国国家精神卫生研究所）；Johns Hopkins Bloomberg School of Public Health（约翰霍普金斯大学布隆伯格公共卫生学院）
Derivation of an EM algorithm for constrained and unconstrained multivariate autoregressive state-space (MARSS) models
本报告提出了一个期望最大化（EM）算法的约束多元自回归高斯状态空间（MARSS）模型的最大似然参数值的估计。MARSS模型可以写成：x（t）=Bx（t-1）+u+w（t），y（t）=Zx（t）+a+v（t），其中w（t）和v（t）分别是具有方差-协方差阵Q和R的多元正态误差项。MARSS模型是一类动态线性模型和向量自回归状态空间模型。Shumway和Stoffer在1982年提出了这类模型的无约束期望最大化算法，此后许多研究人员提出了特定类型的约束MARSS模型的期望最大化算法。在这份报告中，我提出了一个通用的期望最大化算法的约束MARSS模型，其中的约束是在参数矩阵（B，u，Q，Z，a，R）内的元素。约束采用vec（M）=f+Dm的形式，其中M是参数矩阵，f是固定值的列向量，D是乘数矩阵，m是估计值的列向量。这允许各种各样的约束参数矩阵形式。该演示文稿适用于时变MARSS模型，其中时间变化通过每个参数的固定（意味着未估计）f（t）和D（t）矩阵进入。该算法允许y和部分确定性系统中的缺失值，其中0出现在Q或R的对角线上。在Comprehensive R Archive Network（CRAN）的MARSS R包中提供了使用该算法估计MARSS模型的开源代码。
Graphical Principal Component Analysis of Multivariate Functional Time Series
School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学统计学院，应用统计中心）
