(Fast) Stationary Kalman Filter

A simple and fast C++ implementation of the Kalman Filter for stationary data with time-invariant system matrices and missing data.

SKF(X, A, C, Q, R, F_0, P_0, loglik = FALSE)

Arguments

X: numeric data matrix ($T \times n$).
A: transition matrix ($rp \times rp$).
C: observation matrix ($n \times rp$).
Q: state covariance ($rp \times rp$).
R: observation covariance ($n \times n$).
F_0: initial state vector ($rp \times 1$).
P_0: initial state covariance ($rp \times rp$).
loglik: logical. Compute log-likelihood?

Value

Predicted and filtered state vectors and covariances.

F: $T \times rp$ filtered state vectors.
P: $rp \times rp \times T$ filtered state covariances.
F_pred: $T \times rp$ predicted state vectors.
P_pred: $rp \times rp \times T$ predicted state covariances.
loglik: value of the log likelihood.

Details

The underlying state space model is:

$$\textbf{x}_t = \textbf{C} \textbf{F}_t + \textbf{e}_t \sim N(\textbf{0}, \textbf{R})$$ $$\textbf{F}_t = \textbf{A F}_{t-1} + \textbf{u}_t \sim N(\textbf{0}, \textbf{Q})$$

where $x_t$ is X[t, ]. The filter then first performs a time update (prediction)

$$\textbf{F}_t = \textbf{A F}_{t-1}$$ $$\textbf{P}_t = \textbf{A P}_{t-1}\textbf{A}' + \textbf{Q}$$

where $P_t = Cov(F_t)$. This is followed by the measurement update (filtering)

$$\textbf{K}_t = \textbf{P}_t \textbf{C}' (\textbf{C P}_t \textbf{C}' + \textbf{R})^{-1}$$ $$\textbf{F}_t = \textbf{F}_t + \textbf{K}_t (\textbf{x}_t - \textbf{C F}_t)$$ $$\textbf{P}_t = \textbf{P}_t - \textbf{K}_t\textbf{C P}_t$$

If a row of the data is all missing the measurement update is skipped i.e. the prediction becomes the filtered value. The log-likelihood is computed as $$1/2 \sum_t \log(|St|)-e_t'S_te_t-n\log(2\pi)$$ where $S_t = (C P_t C' + R)^{-1}$ and $e_t = x_t - C F_t$ is the prediction error.

For further details see any textbook on time series such as Shumway & Stoffer (2017), which provide an analogous R implementation in astsa::Kfilter0. For another fast (C-based) implementation that also allows time-varying system matrices and non-stationary data see FKF::fkf.

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.

Hamilton, J. D. (1994). Time Series Analysis. Princeton university press.

Examples