Description Usage Arguments Details Value Author(s) See Also
These functions are used internally by package robKalman
1 2 3 4 5 6 7 | .getDelta(S1, Z, V)
.getKG(S1, Z, Delta)
.getcorrCov(S1, K, Z)
.getpredCov(S0, F, Q)
.cKinitstep(a, S, ...)
.cKpredstep(x0, S0, F, Q, ...)
.cKcorrstep(y, x1, S1, Z, V, ...)
|
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) |
Delta |
Covariance of Delta y_t |
K |
Kalman gain K_t |
S1 |
prediction error covariance S_{t|t-1} of the classical Kalman filter |
S0 |
filter error covariance S_{t-1|t-1} of the classical Kalman filter |
y |
observation y_t |
x0 |
(classical Kalman)- filtered state x_{t-1|t-1} |
x1 |
(classical 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}
.getDelta
calculates the covariance of Delta y_t
for S1 =
S_{t|t-1}, Z
, V
as above
.getKG
calculates the Kalman Gain for S1 =
S_{t|t-1}, Z
, V
as above
.getcorrCov
calculates S_{t|t} for S1 =
S_{t|t-1}, K =
K_t and Z
as above
.getpredCov
calculates S_{t|t-1} for S0 =
S_{t-1|t-1}, F
, and Q
as above
.cKinitstep
calculates x_{0|0} for a
, S
as above
The return value is a list with components
x0
(the filtered value)
S0
(the filter error covariance)
.cKpredstep
calculates x_{t|t-1} for x0 =
x_{t-1|t-1}, S0 =
S_{t-1|t-1}, and F
, Q
The return value is a list with components
x1
(the predicted values),
S1
(the prediction error covariance),
Ind
(the indicators of clipped runs)
.cKcorrstep
calculates x_{t|t} for x1 =
x_{t|t-1}, y =
y_{t},
S1 =
S_{t|t-1}, and Z
, V
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,
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