Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Library: kalman

abinfonewton Auxiliary routine for rICfil: solves - if possible - by explicit integration and Newton-Algorithm 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))
absepnewton Auxiliary routine for rICfil: solves - if possible - by explicit integration and Newton-Algorithm (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 X=X1+X2, X1 ~ N(0,S1), X2 ~ N(0,S2) indep1

betrnormE Auxiliary routine for ricfil: calculates the E [ |X| (|x|<t) ], X an n-dim standard normal variate
betrnormF Auxiliary routine for ricfil: calculates the cdf of |X|, X an n-dim standard normal variate
betrnormV Auxiliary routine for ricfil: calculates E [ |X|^2 (|x|<t) ], X an n-dim standard normal variate
calibrIC 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 ~ (0,R)

All parameters are assumed known.

calibrLS 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), w_t ~ (0,R)

All parameters are assumed known.

epscontnorm 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

ew2inn Auxiliary routine for rICfil: calculates E[ min(t^2,u^2) ] for u square root of a Chi^2_p-variable, (recursively in dimension p)
ewinn Auxiliary routine for rICfil: calculates E[ u min(t,u) ] for u square root of a Chi^2_p-variable, (recursively in dimension p)
ICerz 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 dim p==1 a Newton-Algo is used for both a and b, for dim p>=2 for A a fixed-point-algorithm and for b a "careful" bisection method is used. Integration for A and p==2 is done by a Romberg-procedure. Integration for A and p>2 is done by a MC-procedure.

ICerzsep 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/|A Lambda1|)

For dim p==1 a Newton-Algo is used for both a and b, for dim p>=2 for A a fixed-point-algorithm and for b a "careful" bisection method is used. Integration for A and p==2 is done by a Romberg-procedure. Integration for A and p>2 is done by a MC-procedure.

itera Auxiliary routine for rICfil:

- if possible - solves for Lambda~N(0,FI) (FI:: Fisher-Info)

A^{-1} =E [ Lambda Lambda' w_b ] (1)

w_b=min(1,b/|A Lambda|)

using a fixed-point-algorithm

iteras Auxiliary routine for rICfil:

- if possible - solves for Lambda1~N(0,S1),Lambda2~N(0,S2) indep.

A^{-1} =E [ Lambda1 Lambda1' w_b ] + E [ Lambda2 Lambda2' ] (1)

w_b=min(1,b/|A Lambda1|)

using a fixed-point-algorithm

kalmanmain sets defaults for library kalman
kalmantest Tests the quantlets of the kalman library.
kemitor2 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 = mu

kfilter2 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.

nmomnorm Auxiliary routine for ricfil: calculates the n-th moment of a standard normal variate truncated at t, i.e. E [X^n (X<t)] for X~N(0,1)
numint2 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^{-1/2} Lambda /u

the integrals are evaluated stepwise, first conditioning on Y and calculated "analytically" using Ewinn, Ew2inn and then the outer integration is done by a Romberg-Procedure along the directions Y, parametrized by a sin-cos-representation.

numint2m 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} Lambda /u

the integrals are evaluated stepwise, first conditioning on Y and calculated "analytically" using Ewinn, Ew2inn and then the outer integration is done by a Romberg-Procedure along the directions Y, parametrized by a sin-cos-representation.

rICfil 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 a relative loss w.r.t. the classical Kalman Filter in the ideal model in terms of MSE. 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.

rlsbnorm, 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|)

rlsbnorm1 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|)

rlsfil 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 a relative loss w.r.t. the classical Kalman Filter in the ideal model in terms of MSE. 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.

stointp 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^{-1/2} Lambda /u

The integrals are evaluated stepwise, first conditioning on Y and calculated "analytically" using Ewinn, Ew2inn and then the outer integration is done by MC-Integration along the directions Y, parametrized by a sin-cos-representation.

stointpm 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} Lambda /u.

The integrals are evaluated stepwise, first conditioning on Y and calculated "analytically" using Ewinn, Ew2inn and then the outer integration is done by MC-Integration along the directions Y, parametrized by a sin-cos-representation.


Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

(C) MD*TECH Method and Data Technologies, 21.9.2000