Auxiliary routine for rICfil: solves - if possible - by explicit integration and Newton-Algorithm the following equation: E [|AX|^2 \min{1,b/|AX|}]=1, E [|AX|^2 \min{1,b^2/|AX|^2}]=(1+e)p for X ~ N_p (0,unit(p))
Auxiliary routine for rICfil: solves - if possible - by explicit integration and Newton-Algorithm the following equations: (separate clipping in 1 dimension of normal scores X=X1+X2, X1,X2 indep.) E [A (X1 \min{1,b/|AX1|} +X2) (X1+X2) ]=1, E [A^2 (X1 \min{1,b/|AX1|} +X2)^2]=(1+e) /(S1+S2) for
Auxiliary routine for rICfil Calibrates the robust IC's for a given State Space model to a given relative efficiency loss in terms of the MSE in the ideal model. The state-space model is assumed to be in the following form: y_t = H x_t + v_t x_t = F x_t-1 + w_t x_0 ~ (mu,Sig), v_t ~ (0,Q), w_t ~
Auxiliary routine for rLSfil Calibrates the robust LS- Filter for a given State Space model to a given relative efficiency loss in terms of the MSE in the ideal model. The state-space model is assumed to be in the following form: y_t = H x_t + v_t x_t = F x_t-1 + w_t x_0 ~ (mu,Sig), v_t ~ (0,Q),
Produces T i.i.d. Variates from an eps-contamination Model P= (1-eps) N(mid,Cid) + eps K with K=N(mcont,Ccont) if DirNorm ==0 with K=dirac(mcont) if DirNorm == -1 with K=dirac( +/- mcont) if DirNorm == 1
Calculates a filtered time series for a state space model (uni- or multivariate) with time variable system matrices using the Kalman filter. Furthermore, gkalfilter gives the value of the log likelihood function.
Calculates covariance matrices for the smoothed series of a state space model (uni- or multivariate) with one lag. The quantlet gkallag needs a pre-run of gkalfilter. The state space model has the form (for the notation, see Harvey 1989): State equation alpha_t = c_t + T_t alpha_t-1 + e^s_t M
Calculates the innovations v_t and the standardized v^s_t residuals for a state space form that is estimated with the Kalman filter. As input, the output from the Kalman filter is needed. See the help to gkalfilter or the tutorial for a thorough discussion of the model.
calculates a smoothed time series for a state space model (uni- or multivariate) using the Kalman smoother. The quantlet gkalsmoother needs a pre-run of gkalfilter.
Auxiliary routine for rICfil: - if possible - generates for Scores Lambda~N(0,FI) (FI:: Fisher-Info) a Hampel-Krasker-IC psi to efficiency loss e, i.e. E psi Lambda' = unit(p) E psi=0 (1) E |psi|^2= (1+e) tr (FI^{-1}) (2) and psi= A Lambda w_b w_b=min(1,b/|A Lambda|) for di
Auxiliary routine for rICfil: - if possible - generates for Lambda=Lambda1+Lambda2, Lambda1~N(0,S1), Lambda2~N(0,S2) indep a Hampel-Krasker-IC psi to efficiency loss e, i.e. E psi Lambda' = EM, E psi=0 (1) E |psi|^2= (1+e) tr ((S1+S2)^{-1}) and psi= A (Lambda1 w_b + Lambda2) w_b=min(1,b/|
Auxiliary routine for rICfil: - if possible - it solves for Lambda~N(0,FI), where FI represents the Fisher-Info, following equations: A^{-1} =E [ Lambda Lambda' w_b ] (1) w_b=min(1,b/|A Lambda|) using a fixed-point-algorithm
Auxiliary routine for rICfil: It solves - if possible - for Lambda1~N(0,S1) and Lambda2~N(0,S2) following equations independently: A^{-1} =E [ Lambda1 Lambda1' w_b ] + E [ Lambda2 Lambda2' ] (1) w_b=min(1,b/|A Lambda1|) using a fixed-point-algorithm
Simulates observations and states of a given state-space-model - just as kemitor by Petr Franek (quantlib times) - but this time also the states are returned. The state-space model is assumed to be in the following form: y_t = H x_t + ErrY_t x_t = F x_t-1 + ErrX_t x_0 =
Calculates a filtered time serie (uni- or multivariate) using the Kalman filter equations. The state-space model is assumed to be in the following form: y_t = H x_t + v_t x_t = F x_t-1 + w_t x_0 ~ (mu,Sig), v_t ~ (0,Q), w_t ~ (0,R) All parameters are assumed known.
Auxiliary routine for rICfil: calculates for dimension p=2 diag(E[ YY' u min(b/|aIhY|,u) ]) and diag(E[ YY' min(b/|aIhY|,u)^2 ]) for u square root of a Chi^2_2-variable, and Y~ufo(S_2) indep of u by using a polar representation of Lambda:= I^{1/2} Y u, u = | I^{-1/2} Lambda |, Y=I^{-1/
Auxiliary routine for rICfil: calculates for dimension p=2 (E[ YY' u min(b/|aIhY|,u) ]) and (E[ YY' min(b/|aIhY|,u)^2 ]) for u square root of a Chi^2_2-variable, and Y~ufo(S_2) indep of u by using a polar representation of Lambda:= I^{1/2} Y u, u = | I^{-1/2} Lambda |, Y=I^{-1/2} Lambd
Calculates a filtered time serie (uni- or multivariate) using a robust, recursive Filter based on LS-optimality, the rLS-filter. Additionally to the Kalman-Filter one needs to specify the degree of robustness one wants to achieve; this is done either by specifying a clipping height or by specifying
Auxiliary routine for rlsfil: solves E [ |X-MYw_b(MY)|^2]=(1+e)E [ |X-MY|^2] - if possible - by MC-integration for X ~ N_n(0,Sigt), v ~ N_m(0,Q) indep. M = Sigt H'(Q+HSigt H')^{-1} Y = HX+v, w_b(x)=min(1,b/|x|)
Auxiliary routine for rlsfil: solves E [ |X-MYw_b(MY)|^2]=(1+e)E [ |X-MY|^2] - if possible - by numerical integration for X ~ N(0,Sigt), v ~ N(0,Q) indep. M=Sigt H'(Q+HSigt H')^{-1} Y=HX+v, w_b(x)=min(1,b/|x|)
Calculates a filtered time serie (uni- or multivariate) using a robust, recursive Filter based on LS-optimality, the rLS-filter. additionally to the Kalman-Filter one needs to specify the degree of robustness one wants to achieve; this is done either by specifying a clipping height or by specifying
Auxiliary routine for rICfil: calculates for dimension p>(=)2 diag(E[ YY' u min(b/|aIhY|,u) ]) and diag(E[ YY' min(b/|aIhY|,u)^2 ]) for u square root of a Chi^2_p-variable, and Y~ufo(S_2) indep of u by using a polar representation of Lambda:= I^{1/2} Y u, u = | I^{-1/2} Lambda |, Y=I^{
Auxiliary routine for rICfil: calculates for dimension p>(=)2 (E[ YY' u min(b/|aIhY|,u) ]) and (E[ YY' min(b/|aIhY|,u)^2 ]) for u square root of a Chi^2_p-variable, and Y~ufo(S_2) indep of u by using a polar representation of Lambda:= I^{1/2} Y u, u = | I^{-1/2} Lambda |, Y=I^{-1/2} La