Innovation step in the Kalman filter
Arguments
- z
Vector of measured values at time t + 1
- 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.
- H
Matrix relating the values in x to the values in y.
- R
Cholesky decomposition of the measurement noise covariance matrix.#'
Value
Named list of the innovation ("y"), the Cholesky decomposition
of its covariance matrix ("S"), and the Kalman gain ("K")
Details
In the innovation step, we compute how much error we expect in the measurement if the movement equation is correct in its prediction. The expected "innovation" is defined as:
$$\mathbf{y}_i = \mathbf{z}_i - H \mathbf{x}_i$$ where \(\mathbf{z}\) is the measurement, \(H\) is the measurement matrix, and \(\mathbf{x}\) is the prediction obtained through the prediction step. We also compute the covariance matrix for this innovation, which is defined as:
$$\Sigma_i = H P_i H^T + R$$ where \(R\) contains the (assumed) measurement covariances. It is interesting to note that the value of \(\Sigma\) is always symmetric for each value of \(H\). Whether its Cholesky decomposition exists, however, will depend on both \(H\) and \(P\).
In the final step, we compute the Kalman gain using both the covariance of the prediction and the total covariance of the measurement process through the following equation:
$$K_i = P_i H^T \Sigma_i^{-1}$$ As one may notice, the Kalman gain is a measure of how much we can trust the prediction, specifically by putting the prediction covariance over the total variance. It is thus related to a measure of reliability.