Regularized Estimation of Sparse Spectral Precision Matrices
arxiv(2024)
摘要
Spectral precision matrix, the inverse of a spectral density matrix, is an
object of central interest in frequency-domain analysis of multivariate time
series. Estimation of spectral precision matrix is a key step in calculating
partial coherency and graphical model selection of stationary time series. When
the dimension of a multivariate time series is moderate to large, traditional
estimators of spectral density matrices such as averaged periodograms tend to
be severely ill-conditioned, and one needs to resort to suitable regularization
strategies involving optimization over complex variables.
In this work, we propose complex graphical Lasso (CGLASSO), an
ℓ_1-penalized estimator of spectral precision matrix based on local
Whittle likelihood maximization. We develop fast pathwise coordinate
descent algorithms for implementing CGLASSO on large dimensional time series
data sets. At its core, our algorithmic development relies on a ring
isomorphism between complex and real matrices that helps map a number of
optimization problems over complex variables to similar optimization problems
over real variables. This finding may be of independent interest and more
broadly applicable for high-dimensional statistical analysis with
complex-valued data. We also present a complete non-asymptotic theory of our
proposed estimator which shows that consistent estimation is possible in
high-dimensional regime as long as the underlying spectral precision matrix is
suitably sparse. We compare the performance of CGLASSO with competing
alternatives on simulated data sets, and use it to construct partial coherence
network among brain regions from a real fMRI data set.
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