Rosenblatt Transformation¶
(1)¶
is to use the Rosenblatt transformation
which is a diffeomorphism from
into the standard space
, where distributions are normal, with zero mean, unit variance and unit correlation matrix (which is equivalent in that normal case to independent components).
(2)¶
The cumulative distribution function of the conditional variable
is defined by:
Rosenblatt transformation: Let in
be a continuous random vector defined by its marginal cumulative
distribution functions
and its copula
. The
Rosenblatt transformation
of
is
defined by:
(3)¶
where both transformations , and
are given by:
(4)¶
(5)¶
API:
See the available Rosenblatt transformations.
References:
Ditlevsen and H.O. Madsen, 2004, “Structural reliability methods,” Department of mechanical engineering technical university of Denmark - Maritime engineering, internet publication.
Goyet, 1998,”Sécurité probabiliste des structures - Fiabilité d’un élément de structure,” Collège de Polytechnique.
Der Kiureghian, P.L. Liu, 1986,”Structural Reliability Under Incomplete Probabilistic Information”, Journal of Engineering Mechanics, vol 112, no. 1, p85-104.
H.O. Madsen, Krenk, S., Lind, N. C., 1986, “Methods of Structural Safety,” Prentice Hall.
Rosenblatt, “Remarks on a Multivariat Transformation”, The Annals of Mathematical Statistics, Vol. 23, No 3, pp. 470-472.