Description Usage Arguments Details
Compute the best positive approximant for use in the STCOS model, under several prespecified covariance structures.
1 2 3  cov_approx_randwalk(Delta, S)
cov_approx_blockdiag(Delta, S)

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 finelevel process over T = N / n time points. 
Let \bm{Σ} be an N \times N symmetric and positivedefinite 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 positivedefinite 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 finelevel support having n areas, using T time steps. Therefore N = n T represents the dimension of covariance for the finelevel 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.
cov_approx_randwalk
assumes \bm{Σ} is based on the
autocovariance function of a random walk
\bm{Y}_{t+1} = \bm{Y}_{t} + \bm{ε}_t, \quad \bm{ε}_t \sim \textrm{N}(\bm{0}, \bm{Δ}).
so that
\bm{Γ}(s,t) = \min(s,t) \bm{Δ}.
cov_approx_blockdiag
assumes \bm{Σ} is based on
\bm{Y}_{t+1} = \bm{Y}_{t} + \bm{ε}_t, \quad \bm{ε}_t \sim \textrm{N}(\bm{0}, \bm{Δ}).
which are independent across t, so that
\bm{Γ}(s,t) = I(s = t) \bm{Δ},
The block structure is used to reduce the computational burden, as N may be large.
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