LARGE COVARIANCE ESTIMATION THROUGH ELLIPTICAL FACTOR MODELS

Abstract

We propose a general Principal Orthogonal complEment Thresholding (POET) framework for large-scale covariance matrix estimation based on the approximate factor model. A set of high level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms is established to better understand how POET works. Such a framework allows us to recover existing results for sub-Gaussian data in a more transparent way that only depends on the concentration properties of the sample covariance matrix. As a new theoretical contribution, for the first time, such a framework allows us to exploit conditional sparsity covariance structure for the heavy-tailed data. In particular, for the elliptical distribution, we propose a robust estimator based on the marginal and spatial Kendall's tau to satisfy these conditions. In addition, we study conditional graphical model under the same framework. The technical tools developed in this paper are of general interest to high dimensional principal component analysis. Thorough numerical results are also provided to back up the developed theory.

Keywords: approximate factor model; conditional graphical model; elliptical distribution; marginal and spatial Kendall’s tau; principal component analysis; sub-Gaussian family.

Fig 1

Conditional sparse covariance matrix estimation. The 12 plots correspond to logarithms (base 2) of the ratios of average errors of the original and the robust POET estimators, measured in different norms. Data were generated from multivariate t-distribution with degree of freedom ν = 4.2 (black dotted), ν = 7 (blue dashed), ν = ∞ (orange solid) with p from 100 to 1000, n = p/2 and m = 3. 100 simulations were conducted for each p.

Fig 2

Comparison of relative Frobenius norms. The plots correspond to average errors of the original POET (for ν =∞, solid) and the robust POET (for ν = 4.2, dotted). In each setting, we compare ${\Vert {\widehat{\sum }}^{\mathcal{T}}-\sum \Vert }_{\sum }$ (red) and ${\Vert {\widehat{\sum }}_u^{\mathcal{T}}-\sum \Vert }_{\sum }$ (blue) in the left panel and compare ${\Vert {\mathbf{L}}^{-1/2}\left(\widehat{\mathbf{\Gamma }}\widehat{\mathbf{\Lambda }}{\widehat{\mathbf{\Gamma }}}^{\prime }-B{B}^{\prime }\right){\mathbf{L}}^{-1/2}\Vert }_F$ (red) and ${\Vert {\mathbf{L}}^{-1/2}B{B}^{\prime }{\mathbf{L}}^{-1/2}\Vert }_F$ (blue) in the right panel.

Fig 3

Conditional graphical model estimation. The plots correspond to log ratios (base 2) of average errors of the original and the robust POET estimators for ${\widehat{\mathbf{\Omega }}}_u$ and $\widehat{\mathbf{\Omega }}$, measured in spectral norms. Data were generated from multivariate t-distribution with degree of freedom ν = 4.2 (black dotted), ν = 7 (blue dashed), ν = ∞ (orange solid) with p from 50 to 500, n = 0.6p and m = 3. 100 simulations were conducted for each p.