# Calibration of clipping height b

### Description

calibrates the clipping height `b`

of the rLS-filter in a time-invariant, linear, Gaussian state space model

### Usage

1 2 |

### Arguments

`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 |

### Details

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.

### Value

a list containing components clipping height `b`

,
efficiency loss `eff`

and least favorable/resp. given radius `r`

for the rLS filter

### Author(s)

Peter Ruckdeschel Peter.Ruckdeschel@itwm.fraunhofer.de,

### Examples

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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