Covariance Parameter

lsqrt(A::CovarianceParameter)

Computes a square matrix L such that A = L*L'. L need not be a Cholesky factor.

stein(Σ::CovarianceParameter, Φ::AbstractMatrix)

Computes the output of the stein operator

Σ ↦ Φ * Σ * Φ'.

The type of CovarianceParameter is preserved at the output.

stein(Σ::CovarianceParameter, Φ::AbstractMatrix, Q::CovarianceParameter)

Computes the output of the stein operator

Σ ↦ Φ * Σ * Φ' + Q.

Both Σ and Q need to be of the same CovarianceParameter type, e.g. both SymOrHerm or both Cholesky. The type of the CovarianceParameter is preserved at the output.

schur_reduce(Π::CovarianceParameter, C::AbstractMatrix)

Returns the tuple (S, K, Σ) associated with the following (block) Schur reduction:

[C*Π*C' C*Π; Π*C' Π] = [0 0; 0 Σ] + [I; K]*(C*Π*C')*[I; K]'

In terms of Kalman filtering, Π is the predictive covariance, C the measurement matrix, and R the measurement covariance, then S is the marginal measurement covariance, K is the Kalman gain, and Σ is the filtering covariance. The type of the CovarianceParameter is preserved at the output.

schur_reduce(Π::CovarianceParameter, C::AbstractMatrix, R::CovarianceParameter)

Returns the tuple (S, K, Σ) associated with the following (block) Schur reduction:

[CΠC' + R CΠ; ΠC' Π] = [0 0; 0 Σ] + [I; K](CΠC' + R)[I; K]'

In terms of Kalman filtering, Π is the predictive covariance, C the measurement matrix, and R the measurement covariance, then S is the marginal measurement covariance, K is the Kalman gain, and Σ is the filtering covariance. Both Π and R need to be of the same CovarianceParameter type, e.g. both SymOrHerm or both Cholesky. The type of the CovarianceParameter is preserved at the output.