(Fast) Fixed-Interval Smoother (Kalman Smoother)
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.