Sparse Graphical Models for Multivariate Time series

本研究的目的是在谱域中开发稀疏图形模型，以帮助可视化作为多变量时间序列记录的神经生理信号之间的连通性（相关性）。部分相干，部分相关的频谱域模拟，将被用作功能连通性的度量，它识别驱动任何两个分量序列之间的相关性的频率区域，以调整其他分量的线性效应。在神经科学应用中，一个图的顶点可能代表不同的体素，而两个顶点之间的边缘反映了两个体素上信号之间的直接联系。边缘的缺失由两个信号的零部分相干性表示，检测它的能力是构建有意义图的关键。目前计算部分相干性的方法是先估计谱密度矩阵，然后对其进行反演。只要序列的维度或体素数相对于序列的长度较小，这种经典方法就能很好地工作。在估计高维fMRI时间序列的部分相干性时，存在严重的计算复杂性和统计稳定性问题。稳定性和复杂性总是受到光谱平滑程度和待反转矩阵大小等因素的影响。本研究的目标是通过直接使用惩罚正态似然估计逆谱密度矩阵来完全避免这些问题，类似于最近高斯图形模型稀疏估计的发展导致了快速图形套索方法。它将利用一个未被充分利用的事实，即在每个频率上多元平稳过程的谱密度矩阵实际上是具有复项的相同维数的随机向量的协方差矩阵。谱域方法通常用于分析生物、物理和工程科学领域的多变量时间序列数据。本研究将高斯图形模型的一般概念和技术从标准的多变量数据提升到谱域的多变量时间序列设置，并开发了结构化协方差（精度）矩阵的图形套索方法。提议的工作本质上是跨学科的，可以直接应用于神经科学数据的分析。它对高维数据分析的关注对金融市场、流行病学、环境监测和全球变化等多变量时间序列数据的收集具有直接影响。一名研究生将参与研究项目，研究结果将纳入研究生课程，并在统计领域以外的研究人员可以参加的研讨会和讲习班上提出。

The objective of this research is to develop sparse graphical models in the spectral domain to help visualize connectivity (correlation) among neurophysiological signals recorded as a multivariate time series. Partial coherence, the spectral-domain analogue of the partial correlation, will be used as a measure of functional connectivity which identifies the frequency region that drives the correlation between any two component series adjusted for the linear effects of the others. In the neuroscience applications, the vertices of a graph may represent different voxels while an edge between two vertices reflect a direct connection between the signals at the two voxels. The absence of an edge is indicated by a null partial coherence for the two signals, and the ability to detect it is the key in the construction of a meaningful graph. At present, partial coherence is computed by estimating the spectral density matrix first and then inverting it. This classical approach works well so long as the dimension of the series or the number of voxels is small relative to the length of the series. Serious computational complexity and statistical stability problems arise when estimating the partial coherences for high-dimensional fMRI time series. The stability and complexity are invariably influenced by factors such as the degree of spectral smoothing and size of the matrix to be inverted. The goal of this research is to completely avoid these issues by estimating the inverse spectral density matrix directly using the penalized normal likelihood in analogy with the recent developments in sparse estimation of Gaussian graphical models leading to the fast graphical lasso methodology. It will exploit an under-utilized fact that the spectral density matrix of a multivariate stationary process at each frequency is actually the covariance matrix of a random vector of the same dimension with complex entries.Spectral-domain methodologies are commonly used in the analysis of multivariate time series data arising from biological, physical and engineering sciences. This research elevates the general concepts and techniques of Gaussian graphical models from the standard multivariate data to the multivariate time series setup in the spectral domain, and develops graphical lasso methodology for structured covariance (precision) matrices. The proposed work is interdisciplinary in nature with immediate applications to the analysis of neuroscience data. Its focus on high-dimensional data analysis has immediate impacts on settings where multivariate time series data are collected such as in financial markets, epidemiology, environmental monitoring and global change. A graduate student will be involved in the research project, the results will be incorporated in graduate courses and presented in seminars and workshops accessible to researchers outside the field of statistics.