Covariance (n×n) for observations within a time
point for the process whose variance we wish to approximate.
S
Design matrix (N×r) of basis functions evaluated on
the fine-level process over T=N/n time points.
Details
Let Σ be an N×N symmetric and positive-definite
covariance matrix and S be an N×r matrix with
rank r. The objective is to compute a matrix K which minimizes
the Frobenius norm
∥Σ−SCS⊤∥F,
over symmetric positive-definite matrices C. The
solution is given by
K=(S⊤S)−1S⊤ΣS(S⊤S)−1.
In the STCOS model, S represents the design matrix from a basis
function computed from a fine-level support having n areas, using
T time steps. Therefore N=nT 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(T,1)⋯⋱⋯Γ(1,T)⋮Γ(T,T)⎠⎞,
where each Γ(s,t) is an n×n matrix.
cov_approx_randwalk assumes Σ is based on the
autocovariance function of a random walk
Yt+1=Yt+ϵt,ϵt∼N(0,Δ).
so that
Γ(s,t)=min(s,t)Δ.
cov_approx_blockdiag assumes Σ is based on
Yt+1=Yt+ϵt,ϵt∼N(0,Δ).
which are independent across t, so that
Γ(s,t)=I(s=t)Δ,
The block structure is used to reduce the computational burden, as N
may be large.