SKF function

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

Returns

Predicted and filtered state vectors and covariances. - F: $T x rp$ filtered state vectors.

• P: $rp x rp x T$ filtered state covariances.

• F_pred: $T x rp$ predicted state vectors.

• P_pred: $rp x rp x T$ predicted state covariances.

• loglik: value of the log likelihood.

Details

The underlying state space model is:

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

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

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

where $P(t) = Cov(F(t))$. This is followed by the measurement update (filtering)

$$\textbf{K}_t = \textbf{P}_t \textbf{C}' (\textbf{C P}_t \textbf{C}' + \textbf{R})^{-1}K(t) = P(t) C' inv(C P(t) C' + R)$$ $$\textbf{F}_t = \textbf{F}_t + \textbf{K}_t (\textbf{x}_t - \textbf{C F}_t)F(t) = F(t) + K(t) (x(t) - C F(t))$$ $$\textbf{P}_t = \textbf{P}_t - \textbf{K}_t\textbf{C P}_tP(t) = P(t) - K(t) C P(t)$$

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

$$1/2 \sum_t \log(|St|)-e_t'S_te_t-n\log(2\pi)1/2 sum_t[log(det(S(t))) - e(t)' S(t) e(t) - n log(2 pi)]$$

where $S(t) = inv(C P(t) C' + R)$ 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.

Examples

# See ?SKFS

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.

See Also

FIS SKFS