internalACM: Internal functions of package robKalman for the ACM filter

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

Description

These functions are used internally by package robKalman for the ACM filter

Usage

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.getcorrCovACM(S1, K,  Z, W=diag(nrow(Z)))
.ACMinitstep(a, S, ...) 
.ACMpredstep(x0, S0, F, Q, i, rob0, s0, ...)
.ACMcorrstep(y, x1, S1, Z, V, i, rob1, dum, psi, apsi, bpsi, cpsi, flag, ...)

Arguments

a

mean of the initial state

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)

i

the time instance

K

Kalman gain K_t

W

weight matrix

dum

dummy variable for compatibility with ... argument of calling function

s0

scale of nominal Gaussian component of additive noise

S1

prediction error covariance S_{t|t-1} of the ACM filter

S0

filter error covariance S_{t-1|t-1} of the ACM filter

rob0

not used here; included for compatibility reasons; set to NULL

rob1

used to pass on information recursively; here: st time-dependent scale parameter

psi

influence function to be used (default: Hampel's ψ function, which is the only one available at the moment)

a,b,c

tuning constants for Hampel's ψ-function, (default: a=b=2.5, c=5.0)

flag

character, if "weights" (default), use ψ(t)/t to calculate the weights; if "deriv", use ψ'(t)

y

observation y_t

x0

(ACM)- filtered state x_{t-1|t-1}

x1

(ACM)- predicted state x_{t|t-1}

...

not used here; for compatibility with signatures of other "step"-functions

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.

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}

FURTHER DETAILS TO BE FILLED

Value

.getcorrCovACM determines filter error covariance S_{t-1|t-1} of the ACM filter.
.ACMinitstep calculates x_{0|0}.
.ACMpredstep calculates the ACM-x_{t|t-1}.
.ACMcorrstep calculates the ACM-x_{t|t}.

Author(s)

Peter Ruckdeschel Peter.Ruckdeschel@itwm.fraunhofer.de,
Bernhard Spangl bernhard.spangl@boku.ac.at,

References

Martin, R.D. (1979): Approximate Conditional-mean Type Smoothers and Interpolators.
Martin, R.D. (1981): Robust Methods for Time Series.
Martin, R.D. and Thomson, D.J. (1982): Robust-resistent Spectrum Estimation.

See Also

internalKalman, internalrLS, recFilter


robKalman documentation built on May 31, 2017, 2:17 a.m.