Description Usage Arguments Details Value Author(s) Examples
calibrates the clipping height b
of the rLS-filter in a time-invariant, linear, Gaussian state space model
1 2 |
Z |
observation matrix in the (ti-l-G-SSM); see below |
S |
prediction error covariance matrix (of the classical Kalman filter) in the (ti-l-G-SSM); see below |
V |
observation error covariance matrix in the (ti-l-G-SSM); see below |
r |
SO/IO-contamination radius |
b |
given clipping height |
eff |
efficiency w.r.t. classical Kalman filter in the ideal model |
rlow |
lower bound for SO/IO-contamination radius |
rup |
upper bound for SO/IO-contamination radius |
repl |
number of replicates used for a LLN-approximation of the expectations needed in this calibration |
upto |
an upper bound to |
IO |
logical of length 1: Is it rLS.IO ( |
seed |
if not missing: argument to |
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.
The clipping height b
given Z, V, and prediction error covariance S_{t|t-1}
(of the classical Kalman filter) is either calibrated to a given efficiency eff
in the
ideal model or to given (SO/IO)-radius r
about the ideal model.
If this radius is unknown, to a given radius interval [rlow
,rup
],
{0 <= \code{rlow} < \code{rup} <= 1 }
a least favorable radius r0
is distinguished giving a radius minimax procedure.
The expectations needed for this calibration are calculated by a LLN
approximation with repl
replicates;
The hierarchie is done as follows: when argument eff
is present, this is used,
otherwise, if argument r
is given, this is used; and still otherwise,
the corresponding radius interval is used.
If b
is given, rLScalibrate
only determines the corresponding
efficiency loss eff
and corresponding radius r
for which b
would be SO/IO optimal.
a list containing components clipping height b
,
efficiency loss eff
and least favorable/resp. given radius r
for the rLS filter
Peter Ruckdeschel Peter.Ruckdeschel@itwm.fraunhofer.de,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | require(robKalman)
##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
### limiting prediction error covariance
SS <- limitS(S = SS0, F = F0, Q = Q0, Z = Z0, V = V0i)
### calibration b
# by efficiency in the ideal model
# efficiency = 0.9
(B1 <- rLScalibrateB(eff = 0.9, S = SS, Z = Z0, V = V0i))
# by contamination radius
# r = 0.1
(B2 <- rLScalibrateB(r = 0.1, S = SS, Z = Z0, V = V0i))
# by contamination radius interval
# rlow = 0.05, rup= 0.5
(B3 <- rLScalibrateB(rlow = 0.05, rup= 0.5, S = SS, Z = Z0, V = V0i))
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