FIS {dfms}

(Fast) Fixed-Interval Smoother (Kalman Smoother)

Description

(Fast) Fixed-Interval Smoother (Kalman Smoother)

Usage

FIS(A, F, F_pred, P, P_pred, F_0 = NULL, P_0 = NULL)

Arguments

A

transition matrix ($rp \times rp$).

F

state estimates ($T \times rp$).

F_pred

state predicted estimates ($T \times rp$).

P

variance estimates ($rp \times rp \times T$).

P_pred

predicted variance estimates ($rp \times rp \times T$).

F_0

initial state vector ($rp \times 1$) or empty (NULL).

P_0

initial state covariance ($rp \times rp$) or empty (NULL).

Details

The Kalman Smoother is given by:

$\textbf{J}_t = \textbf{P}_t \textbf{A} + inv(\textbf{P}^{pred}_{t+1})$

$\textbf{F}^{smooth}_t = \textbf{F}_t + \textbf{J}_t (\textbf{F}^{smooth}_{t+1} - \textbf{F}^{pred}_{t+1})$

$\textbf{P}^{smooth}_t = \textbf{P}_t + \textbf{J}_t (\textbf{P}^{smooth}_{t+1} - \textbf{P}^{pred}_{t+1}) \textbf{J}_t'$

The initial smoothed values for period t = T are set equal to the filtered values. If F_0 and P_0 are supplied, the smoothed initial conditions (t = 0 values) are also calculated and returned. For further details see any textbook on time series such as Shumway & Stoffer (2017), which provide an analogous R implementation in astsa::Ksmooth0.

Value

Smoothed state and covariance estimates, including initial (t = 0) values.

F_smooth

$T \times rp$ smoothed state vectors, equal to the filtered state in period $T$.

P_smooth

$rp \times rp \times T$ smoothed state covariance, equal to the filtered covariance in period $T$.

F_smooth_0

$1 \times rp$ initial smoothed state vectors, based on F_0.

P_smooth_0

$rp \times rp$ initial smoothed state covariance, based on P_0.

References

Shumway, R. H., & Stoffer, D. S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer.

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter.

See Also

SKF SKFS

Examples

# See ?SKFS