Description Usage Arguments Details Value Author(s) References See Also
These functions are used internally by package robKalman for the ACM filter
1 2 3 4 | .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, ...)
|
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 |
rob1 |
used to pass on information recursively; here: |
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: |
flag |
character, if "weights" (default), use ψ(t)/t to calculate the weights; if "deriv", use ψ'(t) |
y |
observation |
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 |
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
.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}.
Peter Ruckdeschel Peter.Ruckdeschel@itwm.fraunhofer.de,
Bernhard Spangl bernhard.spangl@boku.ac.at,
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.
internalKalman
, internalrLS
, recFilter
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