Updating step in the Kalman filter

Updating step in the Kalman filter

Usage

kf_update(x, P, y, H, K)

Arguments

x

Vector of predicted values of the movement equation at time t + 1.

P

Cholesky decomposition of the estimation covariance matrix at time t + 1.

y

Vector of innovations at time t + 1

H

Measurement matrix connecting measurement to movement.

K

Kalman gain

Value

List containing the smoothed value of \(x\) ("x") together with the Cholesky decomposition of its covariance ("P")

Details

In the update step, we use the measurement innovation \(\mathbf{y}_i\) and the predicted value \(\mathbf{x}_i\) and combine both in a guess of what the latent state of the system should be. For this, we use the following equation:

$$\hat{\mathbf{x}}_i = \mathbf{x}_i + K \mathbf{y}$$ where \(\mathbf{x}\) is the prediction coming from the prediction step, \(\mathbf{y}\) is the innovation derived in the innovation step, \(K\) is the Kalman gain, and \(\hat{\mathbf{x}}\) is the latent state of the system.

We also compute the covariance of this update:

$$\hat{P}_i = (I - K H) P_i$$ where I is the identity matrix, \(H\) is the measurement matrix, \(K\) is the Kalman gain, \(P_i\) is the covariance of the prediction derived in the prediction step, and \(\hat{P}_i\) is the estimated certainty around the guess \(\hat{\mathbf{x}}_i\).

Note that the values \(\hat{\mathbf{x}}_i\) and \(\hat{P}_i\) are used as initial conditions for the next time step \(t_{i + 1}\), therefore serving as input in the prediction step and starting the cycle all over again.

See also

kalman_filter, kf_predict, kf_innovation