Covariance-Approximation: Best Approximation to Covariance Structure

Description Usage Arguments Details

Description

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

Usage

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Arguments

Delta

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

S

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

Details

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.