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Monte Carlo Analysis
Correlation Between Gaussian Input Variables
Like in the univariate case, there are standard families of probability distributions for random
vectors. In particular, the multivariate normal distribution with mean vector μ = (μ 1 , …, μ k ) T
and variance-covariance matrix C has the density:
This definition implies that each of the marginal distribution is normal and that the complete
joint distribution is determined once the marginal means and the variance-covariance matrix are
specified.
From its definition, we see also that the density of multinormal distribution is constant on
ellipsoids of the form:
Figure 11-8 shows the contour ellipses of a two-dimensional normal distribution. These contour
ellipses are the iso-distance curves from the mean of this normal distribution corresponding to
the metric C −1 .
Tip
As emphasized, the relationships between the input variables are described with the concept
of linear correlation. In fact “linearity” is strongly related to “gaussianity.” The two terms
could be consider as synonyms in this context. It is possible, however, to describe general
multivariate distributions and to represent dependence structure among the input variables.
Understanding these relationships can be done with the notion of a “copula” function. For an
introduction, see for example Bouye, E., Durrleman, V., Nikeghbali, A., Riboulet, G., Roncalli,
T., Copulas for Finance: A Reading Guide and Some Applications, 2000, Working Paper,
Groupe de Recherche Operationnelle, Credit Lyonnais.
Specifying Linear Correlation with .CORREL
You can define a correlation coefficient between Gaussian parameters using the .CORREL
command.
Refer to the .CORREL command in the Eldo Reference Manual.
The following netlist is a toy example which shows the effect of linear correlation between
input variables. We can indirectly visualize the joint distribution of the Gaussian input P1 and
P2.
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Eldo® User's Manual, 15.3