internalrLS: Internal functions of package robKalman for the rLS filter

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


This function is used internally by package robKalman for the rLS filter


.rLScorrstep(y, x1, S1, Z, V, i,  rob1=NULL, b, norm=EuclideanNorm, ...)



observation matrix (see below)


observation error covariance (see below)


clipping height b for the rLS filter


the time instance


a function with a numeric vector x as first argument, returning a norm of x - not necessarily, but defaulting to, Euclidean norm; used by rLS filter to determine "too" large corrections


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


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


observation y_t


(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,


Ruckdeschel, P. (2001) Ans\"atze zur Robustifizierung des Kalman Filters. Bayreuther Mathematische Schriften, Vol. 64.

See Also


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