Covariance Approximation {stcos}

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 \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{\Sigma} 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{\Sigma} - \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{\Sigma} \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{\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 \bm{\Gamma}(s,t) is an n \times n matrix.

• cov_approx_randwalk assumes \bm{\Sigma} is based on the autocovariance function of a random walk

  \bm{Y}_{t+1} = \bm{Y}_{t} + \bm{\epsilon}_t, \quad \bm{\epsilon}_t \sim \textrm{N}(\bm{0}, \bm{\Delta}).

  so that

  \bm{\Gamma}(s,t) = \min(s,t) \bm{\Delta}.

• cov_approx_blockdiag assumes \bm{\Sigma} is based on

  \bm{Y}_{t+1} = \bm{Y}_{t} + \bm{\epsilon}_t, \quad \bm{\epsilon}_t \sim \textrm{N}(\bm{0}, \bm{\Delta}).

  which are independent across t, so that

  \bm{\Gamma}(s,t) = I(s = t) \bm{\Delta},

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