kalmanMultivariate function

Classic Multivariate KFS Equations

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

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

Returns

logl log-likelihood of the innovations from the Kalman filter

at_t $k x n$, filtered state mean vectors

Pt_t $k x k x n$, filtered state covariance matrices

at_n $k x n$, smoothed state mean vectors

Pt_n $k x k x n$, smoothed state covariance matrices

Pt_tlag_n $k x k x n$, smoothed state covariance with lag

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.

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.