(Fast) Fixed-Interval Smoother (Kalman Smoother)

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).

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.

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.

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.

Examples