kalmanMultivariate {sparseDFM}

Classic Multivariate KFS Equations

Description

Implementation of the classic multivariate Kalman filter and smoother equations of Shumway and Stoffer (1982).

Usage

kalmanMultivariate(X, a0_0, P0_0, A, Lambda, Sig_e, Sig_u)

Arguments

X

n x p, numeric matrix of (stationary) time series

a0_0

k x 1, initial state mean vector

P0_0

k x k, initial state covariance matrix

A

k x k, state transition matrix

Lambda

p x k, measurement matrix

Sig_e

p x p, measurement equation residuals covariance matrix (diagonal)

Sig_u

k x k, state equation residuals covariance matrix

Details

For full details of the classic multivariate KFS approach, please refer to Mosley et al. (2023). Note that n is the number of observations, p is the number of time series, and k is the number of states.

Value

logl log-likelihood of the innovations from the Kalman filter

at_t k \times n, filtered state mean vectors

Pt_t k \times k \times n, filtered state covariance matrices

at_n k \times n, smoothed state mean vectors

Pt_n k \times k \times n, smoothed state covariance matrices

Pt_tlag_n k \times k \times n, smoothed state covariance with lag

References

Mosley, L., Chan, TS., & Gibberd, A. (2023). sparseDFM: An R Package to Estimate Dynamic Factor Models with Sparse Loadings.

Shumway, R. H., & Stoffer, D. S. (1982). An approach to time series smoothing and forecasting using the EM algorithm. Journal of time series analysis, 3(4), 253-264.