THESIS
2023
1 online resource (xiv, 102 pages) : color illustrations
Abstract
Quantifying the relationships between components of a complex system is a valuable
yet challenging task, especially when the number of variables is large. The
Gaussian graphical model (GGM) incorporates an undirected graph that represents
the conditional dependence between variables. It has a wide variety of
applications, such as biological networks, social networks, and stock networks.
The precision matrix, also known as the inverse covariance matrix or concentration
matrix, encodes the partial correlation between pairs of variables given the
others. Nonzero entries in the precision matrix correspond to edges in the graph,
making GGM a potent tool for analysis with interpretability.
Challenges in GGM include the feasibility of methods and high computational
costs in the high-dimensiona...[
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Quantifying the relationships between components of a complex system is a valuable
yet challenging task, especially when the number of variables is large. The
Gaussian graphical model (GGM) incorporates an undirected graph that represents
the conditional dependence between variables. It has a wide variety of
applications, such as biological networks, social networks, and stock networks.
The precision matrix, also known as the inverse covariance matrix or concentration
matrix, encodes the partial correlation between pairs of variables given the
others. Nonzero entries in the precision matrix correspond to edges in the graph,
making GGM a potent tool for analysis with interpretability.
Challenges in GGM include the feasibility of methods and high computational
costs in the high-dimensional settings, the estimation accuracy of large precision
matrix, inference on edge existence, and joint estimation of multiple graphs. To
address these challenges, we propose the f̠lexible and accurate G̠aussian graphical
model (FLAG), which utilizes the random effects model for pairwise conditional
regression, making it feasible in the high-dimensional settings and free
from sparsity assumptions on the precision matrix. The critical advantage of FLAG is its capability to perform statistical inference on each entry in the
precision matrix, quantifying the uncertainty associated with each edge. The
minorize-maximization (MM) algorithm and parameter-expanded expectation-maximization
(PX-EM) algorithm are designed to estimate the variance components
efficiently.
To handle multiple pairs, the algorithm is further accelerated by low-rank update,
which significantly reduces the computational cost by eliminating repeated
eigen-decompositions of large matrices. The capability of element-wise statistical
inference makes FLAG method easily extended to joint estimation of multiple
graphs with little additional computational cost. The FLAG method with meta-analysis
applies hypothesis tests to determine whether the same pair in different
groups share the same partial correlation and adjusts the shared one using meta-analysis.
The FLAG with covariate adjusted (FLAG-CA) method regards the
extra information that differs across groups as fixed effects and estimates the
variance components based on an extended model using the revised MM and
EM algorithms.
The proposed method FLAG and its extensions are evaluated using simulation
data in the block-magnified precision matrix, graph with hub structure, and
multiple graph settings, as well as the real data applications, including gene
expression in the human brain, term association in the university websites, and
stock prices in the U.S. financial market.
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