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 $\ell_1$-penalized estimator of spectral precision matrix based on local Whittle likelihood maximization. We develop fast $\textit{pathwise coordinate descent}$ 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.

翻译：谱精度矩阵，即谱密度矩阵的逆，是多元时间序列频域分析中的核心研究对象。估计谱精度矩阵是计算平稳时间序列部分相干性和图模型选择的关键步骤。当多元时间序列的维数处于中等至较大规模时，传统的谱密度矩阵估计量（如平均周期图）往往呈现严重病态，因此需要采用涉及复变量优化的恰当正则化策略。本文提出复图Lasso（CGLASSO），一种基于局部Whittle似然最大化的谱精度矩阵$\ell_1$惩罚估计量。我们开发了快速的路径坐标下降算法，用于在大维时间序列数据集上实现CGLASSO。该算法的核心依赖于复矩阵与实矩阵之间的环同构，该同构可将一系列复变量优化问题映射为类似实变量优化问题。这一发现可能具有独立价值，可更广泛地应用于复值数据的高维统计分析。我们同时给出了所提估计量的完整非渐近理论，表明只要潜在谱精度矩阵具有适当的稀疏性，在高维框架下即可实现一致估计。我们通过模拟数据集比较了CGLASSO与竞品方案的性能，并利用真实fMRI数据集构建了脑区间的偏相干网络。