Prediction step in the Kalman filter

Usage

kf_predict(
  x0,
  P0,
  F,
  W,
  u = matrix(0, nrow = length(x0), ncol = 1),
  B = matrix(0, nrow = length(u), ncol = length(u))
)

Arguments

x0: Vector of values of the movement equation at time t.
P0: Cholesky decomposition of the covariance of x0 at time t.
F: Transition matrix, relating values of X at time t to those at t.
W: Cholesky decomposition of the process noise covariance matrix.
u: Vector of values for the external variables at time t. By default an empty vector.
B: Matrix that connects the external variables in u to the values in x. By default an empty matrix.

Value

Named list of predicted values for $\mathbf{x}$ ("x") and $P$ ("P") at the next time point.

Details

In this step, we use the movement equation to predict the new values of x based on the initial condition defined by the location $\mathbf{x}$ and the covariance $P$ at time $t_{i - 1}$, using the equation:

$$\mathbf{x}_t = F_i \mathbf{x}_{i - 1} + B_i \mathbf{u}_i + \mathbf{\epsilon}_i$$ where $F$ is the movement transition matrix, $B$ scales the external influences $\mathbf{u}$, and $\mathbf{\epsilon}$ defines the error, and where all terms are defined at the time point $t_i$.

The covariance $P$ is also updated based on this equation, representing the certainty around the prediction $\mathbf{x}_i$. This covariance is updated through the following equation:

$$P_i = F_i P_{i - 1} F_i^T + W_i$$ where $W$ represents the covariance matrix of $\epsilon$.

For stability purposes, we use Cholesky decompositions of the covariance matrices rather than the actual covariance matrices. This needs to be taken into account when making your own model.

Prediction step in the Kalman filter

Usage

Arguments

Value

Details

See also