Vine Copula

1
Vine - Copula

• Presenter Andrew Wey
• Advisor Jong-Min Kim

2
Outline

• Copulas
• Vines
• Example of Vine-Copula
• UNICORN
• Principle Component Analysis
• Conclusion
• Acknowledgements

3
Copula Overview

• First introduced in 1959 by Abe Sklar.
• Has since played an important role in areas of
  probability and statistics especially in
  dependence studies.
• Most easily viewed as connecting two univariate
  marginal distributions to their joint distribution

4
Copula Definition

• A copula is a function C from I2 to I (where I is
  the interval 0,1) that possesses the following
  properties
• For all u, v in I, C(0,v) C(u,0) 0, and
  C(u,1) u and C(1,v) v.
• For every u1, u2, v1, v2 in I such that u1 u2
  and v1 v2, C(u2,v2) C(u2,v1) C(u1,v2)
  C(u1,v1) 0.

5
Sklars Theorem

• Let H be a bivariate distribution function with
  marginal distribution functions F and G. Then
  there exists a copula C such that H(x,y)
  C(F(x),G(y)). Conversely, for any univariate
  distribution functions F and G and any copula C,
  the function H defined above is a bivariate
  distribution function with marginal distributions
  F and G. Furthermore, if F and G are continuous,
  C is unique.

6
Dependence Properties and Measures

• Properties or measures that are invariant under
  strictly increasing transformations of random
  variables
• Positive (and negative) quadrant dependence is an
  example of a particular dependence property.
• Concordance large variables of X tend to occur
  with large values of Y. The probability of which
  are as follows with vectors (Xi,Yf) and (Xh,Yg).
• Probability of Concordance
• P(Xi Xh)(Yf Yg) gt 0
• Probability of Discordance
• P(Xi Xh)(Yf Yg) lt 0

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Specific Dependence Measures

• Kendalls tau - probability of concordance minus
  the probability of discordance of the two vectors
  (X1,Y1) and (X2,Y2).
• Spearmans rho three times the probability of
  concordance minus the probability of discordance
  of the two vectors (X1,Y1) and (X2,Y3).
• Most families of copulas only have a range of
  dependence which can be represented.

Rob and Big Discordant?
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Fréchet Bounds for Copula

• A bivariate distribution H with marginals F and G
  must satisfy the following inequality
• max(F(x) G(y) -1 , 0) H(x,y) min(F(x),G(y))
• Applying the Fréchet bounds to copulas we get the
  following inequality for copulas
• max(u v -1, 0)) C(u,v) min(u,v)

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C(u,v) max(u v 1,0)
C(u,v) min(u,v)
C(u,v) uv
10
Vines!

• Layered acyclical trees.
• Provides a graphical representation of the
  conditional specifications being made on a joint
  distribution.
• The multivariate distribution is represented by
  the product of the marginals and edges of the
  vine.
• We are only interested in a subset of vines
  called regular vines which can be furthered
  seperated
• D-vine
• Canonical vine

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Formal Definition of Vines

• V (T1,,Tm).
• T1 is a tree with nodes N1 1,,n and a set of
  edges denoted E1.
• For i 2,,m, Ti is a tree with nodes Ni which
  forms a subset of N1 U E1 U E2 U U Ei and edge
  set Ei
• A Vine V is a regular vine on n elements if
• m n
• Ti is a connected tree with edge set Ei and node
  set Ni Ei 1, with Ni n (i 1) for i
  1,,n, where Ni is the cardinality of the set
  Ni.
• The proximity condition holds for i 2,,n 1,
  if a a1,a2 and b b1,b2 are two nodes in
  Ni connected by an edge (a1,a2,b1,b2 e Ni 1)
  then a n b 1.

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D-Vine

• If each node in the base tree has a degree of at
  most two, then the vine is a d-vine.

12
23
34
Base Tree
1
2
3
4
132
243
12
23
34
Second Tree
1423
132
243
Third Tree
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D-Vine Joint Distribution
14
Canonical Vine
2
12

• If each tree has an unique node of degree n i,
  then the vine is a canonical vine.

13
1
3
Base Tree
14
4
13
231
12
Second Tree
241
14
3412
231
241
Third Tree
15
Canonical Vine Joint Distribution
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Analysis of Variation within Vine-Copula

• Two hypothetical situations in which data was
  originally simulated (using UNICORN) from a
  correlation matrix to form a baseline data set.
• Then used the baseline data set to determine
  correlations for a d-vine and canonical vine.
  Then simulated data sets using the new d-vine and
  canonical vine.
• Basic statistical inference and principle
  component analysis was used in the analysis of
  variation within the d-vine and canonical vine.

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Simulation using UNICORN
Determination of Correlations
Statistical Analysis
D-vine Specification
D-vine Data Set
Correlation Matrix
Baseline Data Set
Analysis
C-vine Specification
C-vine Data Set
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UNICORN

• Uncertainty Analysis with Correlations
• Fairly intuitive and easy to use interface
• Solely used for simulation of data sets in our
  example
• Has interesting analysis tools that can be used
  in analysis in particular, a cobweb plot

Cobweb Plot Example
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Quick Overview of Principle Component Analysis

• Typically used as method for dimensional
  reduction
• Given a correlation/covariance matrix
• Find the eigenvectors
• Find the eigenvalues
• The components of PCA are the eigenvectors ranked
  by largest to smallest corresponding eigenvalues

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First Scenario Light Bulb Lifetimes

• Four Variables
• Light 1
• Light 2
• Light 3
• Light 4
• C-Vine Organization
• Base tree pole Light 1
• Second tree pole Light 2 given Light 1

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UNICORN Example
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Light Bulb Lifetimes Basic Statistics
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Light Bulb Lifetimes PCA Results
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Light Bulb Lifetimes PCA Biplots
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Second Scenario Spending Analysis

• Four Variables
• Clothes
• Food
• Housing
• Entertainment
• D-vine Organization Clothes Food Housing
  Entertainment
• C-vine Organization
• Base tree pole Housing
• Second tree pole Clothes given Housing

26
Spending Analysis Basic Statistics
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Spending Analysis PCA Results
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Spending Analysis PCA Biplots
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Conclusion

• Copulas Useful in dependence studies
• Vines- Provide useful graphical interpretations
  of the conditional specifications being made on a
  multivariate distribution
• Examples resulted in mixed results in comparing
  d-vines with canonical vines

30
Acknowledgements

• Thanks to Jong-Min Kim who has advised me through
  this project.
• Special thanks to Peh Ng and Engin Sungur for
  providing invaluable help and motivation
  throughout my UMM career.

31
References

• Roger B. Nelsen, An Introduction to Copulas, 2006
• Tim Bedford and Roger Cooke, Vines- A New
  Graphical Model for Dependent Random Variables,
  2002
• Dorota Kurowicka and Roger Cooke, Uncertainty
  Analysis with High Dimensional Dependence
  Modelling, 2006
• Roger B. Nelsen, Copulas, Characterization,
  Correlation, and Counterexamples, 200
• Brian Everitt and Graham Dunn, Applied
  Multivariate Data Analysis Second Edition, 2001
• K. Aas, C. Czado, A. Frigesse, H. Bakken,
  Pair-Copula Constructions of Multiple Dependence,
  2006