Methods for imposing dependencies

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Methods for imposing dependencies

• Dorota Kurowicka

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Model
Input
Output
x1
x2
x3
y
x4

xk
xi input factors
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Dependencies
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Measures of dependence
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Rank correlation

• Always exists
• Does not depend on marginal distributions
• Arbitrary continuous invertible one dimensional
  marginal distributions
• can be joined with any rank correlation value
  in the interval -1,1
• Measures monotonic relationships
• Successful expert elicitation procedure have
  been developed for rank
• correlation

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Copula
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Normal copula
A density of the normal copula with correlation
?0.8
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Bivariate dependence
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Copula
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Copula
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Diagonal band copula
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Franks copula
Archimedean copulas
?(0,1?0,?) convex,decreasing with positive
second derivative and such that ?(1)0
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Which copula?

• If there is data available
• If our information consists of rank correlation
  between variables

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Which copula?

• sampling

uniform on (0,1), independent
where
samples form
1
F
VU
v
0
1
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Multivariate extensions
with invertible cdfs
and rank correlation matrix R
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Multivariate copulas
n-dimensional Franks copula
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Markov Trees
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Vines
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Vines
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Vines
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• Joe, Multivariate Models and Dependence
  Concepts, Chapman Hall, 1997.
• Nelsen, An Introduction to Copulas, Springer,
  1999.
• Saltelli, Chan, Scott, Sensitivity Analysis,
  Wiley, 2000.
• Doruet Mari and Kotz , Correlations and
  Dependence, Imperial College Press 2001.

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