Abstract

The smoothness of subdiagonals of the Cholesky factor of large covariance matrices is closely related to the degree of nonstationarity of autoregressive models for time series data. Heuristically, one expects for nearly stationary covariance matrix entries in each subdiagonal of the Cholesky factor of its inverse to be approximately the same in the sense that the sum of the absolute values of successive differences is small. Statistically, such smoothness is achieved by regularizing each subdiagonal using fused-type lasso penalties. We rely on the standard Cholesky factor as the new parameter within a regularized normal likelihood setup which guarantees: (a) joint convexity of the likelihood function, (b) strict convexity of the likelihood function restricted to each subdiagonal even when n < p, and (c) positive-definiteness of the estimated covariance matrix. A block coordinate descent algorithm, where each block is a subdiagonal, is proposed, and its convergence is established under mild conditions. Lack of decoupling of the penalized likelihood function into a sum of functions involving individual subdiagonals gives rise to some computational challenges and advantages relative to two recent algorithms for sparse estimation of the Cholesky factor, which decouple row-wise. Simulation results and real data analysis show the scope and good performance of the proposed methodology. Software for our method is freely available in R language. Supplementary materials for this article are available online.

摘要

大协方差矩阵的 Cholesky 因子的次对角线的平滑度与时间序列数据的自回归模型的非平稳性程度密切相关。启发式地，人们期望其逆的 Cholesky 因子的每个次对角线中的几乎平稳的协方差矩阵条目在连续差异的绝对值之和很小的意义上大致相同。从统计上讲，这种平滑度是通过使用融合型套索惩罚对每个次对角线进行正则化来实现的。我们依靠标准 Cholesky 因子作为正则化似然设置中的新参数，它保证：(a) 似然函数的联合凸性，(b) 即使当n < p时，似然函数的严格凸性也仅限于每个次对角线和 (c) 估计的协方差矩阵的正定性。提出了一种块坐标下降算法，其中每个块是一个次对角线，并在温和的条件下建立收敛。缺乏将惩罚似然函数解耦为涉及各个次对角线的函数之和，这导致了一些计算挑战，并且相对于最近两种用于 Cholesky 因子稀疏估计的算法（逐行解耦）具有一些计算挑战和优势。仿真结果和实际数据分析显示了所提出方法的适用范围和良好性能。适用于我们方法的软件可以免费使用 R 语言。本文的补充材料可在线获取。