Covariance-Approximation: Best Approximation to Covariance Structure

Description Usage Arguments Details


Compute the best positive approximant for use in the STCOS model, under several prespecified covariance structures.





Covariance (n \times n) for observations within a time point for the process whose variance we wish to approximate.


Design matrix (N \times r) of basis functions evaluated on the fine-level process over T = N / n time points.


Let \bm{Σ} be an N \times N symmetric and positive-definite covariance matrix and \bm{S} be an N \times r matrix with rank r. The objective is to compute a matrix \bm{K} which minimizes the Frobenius norm

\Vert \bm{Σ} - \bm{S} \bm{C} \bm{S}^\top {\Vert}_\textrm{F},

over symmetric positive-definite matrices \bm{C}. The solution is given by

\bm{K} = (\bm{S}^\top \bm{S})^{-1} \bm{S}^\top \bm{Σ} \bm{S} (\bm{S}^\top \bm{S})^{-1}.

In the STCOS model, \bm{S} represents the design matrix from a basis function computed from a fine-level support having n areas, using T time steps. Therefore N = n T represents the dimension of covariance for the fine-level support.

We provide functions to handle some possible structures for target covariance matrices of the form

\bm{Σ} = ≤ft( \begin{array}{ccc} \bm{Γ}(1,1) & \cdots & \bm{Γ}(1,T) \\ \vdots & \ddots & \vdots \\ \bm{Γ}(T,1) & \cdots & \bm{Γ}(T,T) \end{array} \right),

where each \bm{Γ}(s,t) is an n \times n matrix.

The block structure is used to reduce the computational burden, as N may be large.

stcos documentation built on July 1, 2020, 10:42 p.m.