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.
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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.
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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
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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
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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.
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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.
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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.
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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. |