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）是多变量时间序列频域分析中的核心研究对象。谱精度矩阵的估计是计算平稳时间序列部分相干性与图模型选择的关键步骤。当多变量时间序列的维度适中或较高时，传统谱密度矩阵估计量（如平均周期图）往往呈现严重病态性，此时需要借助涉及复变量优化的适当正则化策略。本文提出复数图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$\textit{径路坐标下降}$算法，用于在大维度时间序列数据集上实现CGLASSO。算法开发的核心依赖于复数矩阵与实数矩阵之间的环同构，该同构性可将多个复变量优化问题映射为类似的实变量优化问题。这一发现可能具有独立研究价值，且可更广泛地应用于高维复数数据的统计分析。我们同时提出了所提出估计量的完整非渐近理论，表明当潜在谱精度矩阵满足适当稀疏性时，在高维场景下仍可实现一致估计。我们通过模拟数据集比较CGLASSO与竞争方法的性能，并利用其实例构建真实fMRI数据集大脑区域间的部分相干网络。