Description Usage Arguments Details Value Author(s) References See Also
This function is used internally by package robKalman for the rLS filter
1 | .rLScorrstep(y, x1, S1, Z, V, i, rob1=NULL, b, norm=EuclideanNorm, ...)
|
Z |
observation matrix (see below) |
V |
observation error covariance (see below) |
b |
clipping height |
i |
the time instance |
norm |
a function with a numeric vector |
S1 |
prediction error covariance S_{t|t-1} of the classical Kalman filter |
rob1 |
not used here; included for compatibility reasons; set to |
y |
observation y_t |
x1 |
(rLS Kalman)- predicted state x_{t|t-1} |
... |
not used here; for compatibility with signatures of other "step"-functions |
We work in the setup of the linear, Gaussian state space model (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_t x_{t-1} + v_t, v_t ~ N_p(0,Q_t), t>=1,
observation equation
y_t = Z_t x_t + e_t, e_t ~ N_q(0,V_t), 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_t x_{t-1|t-1}, t>=1
\code{ } with error covariance
S_{t|t-1} = Cov(x_t-x_{t|t-1}) = F_t S_{t-1|t-1} F_t' + Q_t
(2) correction step
x_{t|t} = x_{t|t-1} + K_t (y_t - Z_t x_{t|t-1}), t>=1
\code{ } for Kalman Gain
K_t = S_{t|t-1} Z_t' (Z_t S_{t|t-1} Z_t' + V_t )^-
\code{ } with error covariance
S_{t|t} = Cov(x_t-x_{t|t}) = S_{t|t-1} - K_t Z_t S_{t|t-1}
.rLScorrstep(y, x1, S1, Z, V, b)
calculates rLS-x_{t|t} for arguments b
, x1 =
x_{t|t-1} (from rLS-past),
y =
y_{t}, S1 =
S_{t|t-1}, Z
, and V
as above
The return value is a list with components
x0
(the filtered values), K
(the Kalman gain),
S0
(the filter error covariance),
Delta
(the covariance of Delta y_t),
DeltaY
(the observation residuals Delta y_t),
Ind
(the indicators of clipped runs)
Peter Ruckdeschel Peter.Ruckdeschel@itwm.fraunhofer.de,
Ruckdeschel, P. (2001) Ans\"atze zur Robustifizierung des Kalman Filters. Bayreuther Mathematische Schriften, Vol. 64.
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