# Routines for the simulation of AO-contaminated state space models

### Description

For testing purposes, with these routines, AO-contaminated observations from a multivariate time-invariant, linear, Gaussian state space model may be generated

### Usage

1 2 3 | ```
rcvmvnorm(runs, mi, Si, mc, Sc, r)
simulateState(a, S, F, Qi, mc=0, Qc=Qi, runs = 1, tt, r=0)
simulateObs(X, Z, Vi, mc=0, Vc=Vi, runs = 1, r=0)
``` |

### Arguments

`runs` |
number of runs to be generated |

`mi` |
mean of the ideal multivariate normal distribution |

`mc` |
mean of the contaminating multivariate normal distribution |

`Si` |
covariance of the ideal multivariate normal distribution |

`Sc` |
covariance of the contaminating multivariate normal distribution |

`r` |
convex contamination radius/probability |

`a` |
mean of the initial state |

`S` |
initial state covariance (see below) |

`F` |
innovation transition matrix (see below) |

`Qi` |
ideal innovation covariance (see below) |

`Qc` |
contaminating innovation covariance (see below) |

`tt` |
length of the simulated series of states/observations |

`Z` |
observation matrix (see below) |

`Vi` |
ideal observation error covariance (see below) |

`mc` |
contaminating observation error mean (see below) |

`Vc` |
contaminating observation error covariance (see below) |

`X` |
series of states on basis of which the observations are simulated |

### 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,*

**ideal observation equation**

*y_t = Z x_t + e_{t;id}, e_{t;id} ~ N_q(0,V_i), t>= 1,*

**realistic observation equation**

*y_t = Z x_t + e_{t;re}, e_{t;re} ~ (1-r) N_q(0,V_i) + r N_q(m_c,V_c), t>= 1,*

and where all random variable *x_0*, *v_t*, *e_{t;id}*
[respectively, *e_{t;re}*] are independent.

For notation, let us formulate the classical Kalman filter in this context:

**(0) ininitial step**

*x_{0|0} = a*

*\code{ }* with error covariance

*S_{0|0} = Cov(x_0-x_{0|0}) = S*

**(1) prediction step**

*x_{t|t-1} = F x_{t-1|t-1}, t>=1*

*\code{ }* with error covariance

*S_{t|t-1} = Cov(x_t-x_{t|t-1}) = F S_{t-1|t-1} F' + Q*

**(2) correction step**

*x_{t|t} = x_{t|t-1} + K_t (y_t - Z x_{t|t-1}), t>=1*

*\code{ }* for Kalman Gain

*K_t = S_{t|t-1} Z' (Z S_{t|t-1} Z' + V )^-*

*\code{ }* with error covariance

*S_{t|t} = Cov(x_t-x_{t|t}) = S_{t|t-1} - K_t Z S_{t|t-1}*

### Value

`rcvmvnorm(mi, Si, mc, Sc, r)`

returns a (pseudo) random variable drawn from

*(1-r){\cal N}_q(m_i,S_i)+r {\cal N}_q(m_c,S_c)*

`simulateState`

simulates a series of *t=*`tt`

states plus one initial state from the (ti-l-G-SSM) given by the Hyper parameters
— yielding a matrix *p x (t+1)*

`simulateObs`

, on bases of the series of states `X`

(initial state included) simulates a series of observations of length
`tt`

according to the Hyper parameters — giving a matrix *q x t*

### Author(s)

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

### See Also

`internalrLS`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
require(robKalman)
a0 <- c(1, 0)
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)
TT <- 100
Z0 <- matrix(c(1, -0.5), 1, 2)
V0i <- 1
m0c <- -30
V0c <- 0.1
ract <- 0.1
X <- simulateState( a = a0, S = SS0, F = F0, Qi = Q0, tt = TT)
Y <- simulateObs(X = X, Z = Z0, Vi = V0i, mc = m0c, Vc = V0c, r = ract)
``` |