simulateSScont: Routines for the simulation of AO-contaminated state space...
In robKalman: Robust Kalman Filtering

Description Usage Arguments Details Value Author(s) See Also Examples

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

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)

`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

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}

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

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

internalrLS

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)