Covariance Approximation {stcos}R Documentation

Best Approximation to Covariance Structure

Description

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

Usage

cov_approx_randwalk(Delta, S)

cov_approx_blockdiag(Delta, S)

Arguments

Delta

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

S

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

Details

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

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

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

K=(SS)1SΣS(SS)1. \bm{K} = (\bm{S}^\top \bm{S})^{-1} \bm{S}^\top \bm{\Sigma} \bm{S} (\bm{S}^\top \bm{S})^{-1}.

In the STCOS model, S\bm{S} represents the design matrix from a basis function computed from a fine-level support having nn areas, using TT time steps. Therefore N=nTN = 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

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

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

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


[Package stcos version 0.3.1 Index]