Inference for High-dimensional Sparse Spectral Precision Matrices

Gaussian graphical models in spectral domain provide a principled framework for identifying conditional dependence structures in stationary high-dimensional time series. Inference for the spectral precision matrix (SPM) at fixed frequency is challenging because estimation requires smoothing across frequencies, while spectral-domain observations, i.e. discrete Fourier transforms, are only asymptotically independent, have non-sparse precision matrices, and exhibit finite-sample biases that invalidate standard i.i.d. precision matrix inference. We propose an inference framework for sparse high-dimensional SPMs. Our method constructs a debiased complex graphical lasso (deCGLASSO) estimator at a specified frequency. Using asymptotic theory for quadratic forms of stationary multivariate time series, we establish asymptotic normality of the debiased estimator. For each matrix entry, we develop an estimator of the asymptotic covariance by aggregating information across neighboring frequencies. The key theoretical contribution is explicit control of the regularization, truncation bias and smoothing bias. We demonstrate the method's empirical performance on simulated data and real fMRI data.

Graphical models

Precision matrix estimation

High-dimensional time series

Spectral domain inference

Debiased estimators

Confidence intervals 

Section on Statistical Learning and Data Science