internalKalman: Internal functions of package robKalman for the classical...
In robKalman: Robust Kalman Filtering

Description Usage Arguments Details Value Author(s) See Also

These functions are used internally by package robKalman

.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)