kalmanUnivariate {sparseDFM}

Univariate filtering (sequential processing) for fast KFS

Description

Univariate treatment (sequential processing) of the multivariate Kalman filter and smoother equations for fast implementation. Refer to Koopman and Durbin (2000).

Usage

kalmanUnivariate(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 univariate filtering 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

Koopman, S. J., & Durbin, J. (2000). Fast filtering and smoothing for multivariate state space models. Journal of Time Series Analysis, 21(3), 281-296.

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