Description Usage Arguments Details Value Note Author(s) References Examples
These functions are used internally by package robKalman
1 2 3 | EuclideanNorm(x)
Huberize(x, b, norm=EuclideanNorm, ...)
limitS(S, F, Z, Q, V, tol = 10^-4, itmax = 1000)#
|
x |
a numeric vector |
b |
clipping bound for |
norm |
a function with a numeric vector |
... |
additional arguments to function in argument |
S |
initial state covariance (see below) |
Z |
observation matrix (see below) |
V |
observation error covariance (see below) |
F |
innovation transition matrix (see below) |
Q |
innovation covariance (see below) |
tol |
a tolerance bound for determining when the sequence S_{t|t-1} has stabilized |
itmax |
a maximal number of iterations for |
We work in the setup of the time-invariant, linear, Gaussian state space model (ti-l-G-SSM) with p dimensional states x_t and q dimensional observations y_t, with initial condition
x_0 ~ N_p(a,S),
state equation
x_t = F x_{t-1} + v_t, v_t ~ N_p(0,Q), t>=1,
observation equation
y_t = Z x_t + e_t, e_t ~ N_q(0,V), t>=1,
and where all random variable x_0, v_t, e_t are independent.
In this setup, in most cases (confer, e.g., Anderson and Moore (Appendix)) the prediction error covariances of the classical Kalman filter converge.
EuclideanNorm(x)
returns the Euclidean norm of x
,
Huberize(x,b,norm)
huberizes x
to length b
measured in norm norm
,
limitS(S, F, Z, Q, V)
returns the limiting prediction error covariance
of the classical Kalman Filter, in the time-invariant
state space model (S,F,Z,Q,V).
limitS
does no dimension checking!
Peter Ruckdeschel Peter.Ruckdeschel@itwm.fraunhofer.de,
Anderson, B.D.O. and More, J.B. (1979): Optimal filtering.
Information and System Sciences Series. Prentice Hall.
Ruckdeschel, P. (2001) Ans\"atze zur Robustifizierung des
Kalman Filters. Bayreuther Mathematische Schriften, Vol. 64.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | require(robKalman)
x <- matrix(1:4)
EuclideanNorm(x)
Huberize(x,b=5) ## does clipping
Huberize(x,b=11) ## does no clipping
##Hyper parameter of a lin. time-inv. Gaussian SSM
SS0 <- matrix(0, 2, 2)
F0 <- matrix(c(.7, 0.5, 0.2, 0), 2, 2)
Q0 <- matrix(c(2, 0.5, 0.5, 1), 2, 2)
Z0 <- matrix(c(1, -0.5), 1, 2)
V0i <- 1
SS <- limitS(S = SS0, F = F0, Q = Q0, Z = Z0, V = V0i)
|
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