Extended Kalman filter

Formula: linearize f, h via Jacobians at the current state, then run the linear Kalman recursions on the linearisation.

Usage

EkfF(y, f, h, F, H, Q, R, x0 = NULL, P0 = NULL)

Arguments

y

Observation sequence (scalars).

f

State transition x_t = f(x_{t-1}).

h

Observation map y_t = h(x_t); returns a scalar.

F

d x d Jacobian df/dx at the state.

H

Length-d Jacobian dh/dx at the state.

Q

d x d state noise covariance.

R

Observation noise variance (> 0).

x0

Initial state mean (default zeros).

P0

Initial state covariance (default identity).

Value

List with estimate, state, cov, loglik, n, method.

Details

  predict   x- = f(x),        P- = F P F' + Q
  gain      S  = H P- H' + R, K  = P- H' / S
  update    x  = x- + K (y - h(x-)),  P = P- - K S K'

Scalar observations, d-dimensional state. f and h are the (possibly nonlinear) transition and observation maps; F and H are their Jacobians, supplied either as functions of the state or as constant matrices, in which case the filter degenerates to the plain linear Kalman filter.

References

Schmidt (1966), in Leondes (ed.), Advances in Control Systems 3:293-340; Jazwinski (1970), Stochastic Processes and Filtering Theory, Academic Press.

Examples

set.seed(1)
y <- matrix(cumsum(rnorm(20)), 20, 1)
EkfF(y, f = function(x) x, h = function(x) x, F = matrix(1), H = matrix(1),
     Q = matrix(0.1), R = matrix(1))
#> $estimate
#> [1] 2.602284
#> 
#> $state
#> [1] 2.602284
#> 
#> $cov
#> [1] 0.2701574
#> 
#> $loglik
#> [1] -30.26164
#> 
#> $n
#> [1] 20
#> 
#> $method
#> [1] "Extended Kalman filter"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"