Modeling Dependence in Uncertainty Analysis

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Modeling Dependence in Uncertainty Analysis

• Roger M. Cooke
• Inst. Appl. Mathematics
• Delft University of Technology
• r.m.cooke_at_its.tudelft.nl
• http//ssor.twi.tudelft.nl/risk/.

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How to Represent Dependence

• Via functional relations
• Via dependence trees
• Via vines

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Functional relations

• XT X1Xn independent standardized
• C ? COV(n x n) C LLT L lower triangular
• Y LX has covariance matrix C.
• (what are the marginals Y1,Yn????)

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Dependence trees

• Marginals are given, how characterize dependence?

bloodpressure
Fat
weight
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Rank Correlation

• Copula bivariate uniform with (rank) correlation
• (i.e. correlation between percentiles)
• Diagonal Band Min Information Elliptical Copula

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r(bldpr,Fat)90 r(bldpr, wgt)80Cobweb
Plots, (min inf copua)natural scale
Percentile scale
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r(bldpr,Fat)90 r(bldpr, wgt)80Cobweb
Plots, (Elliptical copua)natural scale
Percentile scale
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More Dependences

• With n vbls, there are n-choose-2 correlations
• There are n-1 edges in a tree

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Belief Nets (directed acyclic graph)

• Cant simply assign correlations to arcs wont
  be positive definite

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Elliptical Copula

• Partial correlation Conditional correlation
• Conditional correlations dont depend on value of
  conditioning vbls
• Partial correlations on a Regular Vine uniquely
  determine correlation matrix
• Partial correlations on a Regular Vine are
  algebraically independent
• Replace (conditional) independence with
  (conditional) indifference (correlation zero)

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Associate (conditional) rank correlations
r(B,F W,H)
r(B,W)
r(B,H W)
r(W,F)
r(W,H F)
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Regular (D-) Vine Nested set of treesEdges are
joined only if they have common child
r(F,H)
r(W,F)
r(B,W)
Weight
Bldprssr
Fat
Height
r(B,F W)
r(W,H F)
r(B,H W,F)
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Assess conditional rank correlations

• r(B,F W)
• Suppose an individual is chosen with weight
  60kg, suppose for him/her the fat in diet is
  above the median value for 60kg people, what is
  your probability that for him/her the blood
  pressure is also above the median?

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Unconditional CobwebsVine
Tree
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Conditional Cobwebs (top 10 bldprssr)Vine
Tree