Lecture on Copulas Part 2dorpjr/EMSE280/Copula... · EMSE 280 - Copula Lecture: Part 2 J.R. van Dorp; dorpjr - Page 1 OUTLINE _____ 1. ORDINAL MEASURES OF ASSOCIATION: OVERVIEW

EMSE 280 - Copula Lecture: Part 2 J.R. van Dorp; www.seas.gwu.edu/~dorpjr

Lecture on Copulas: Part 2______________________________________

J. René van Dorp1

Faculty Web-Page: www.seas.gwu.edu/~dorpjr

November 16, 2011

1 Department of Engineering Management and Systems Egineering, School of Engineering andApplied Science, The George Washington University, 1776 G Street, N.W. Suite 101, WashingtonßD.C. 20052. E-mail: [email protected].

EMSE 280 - Copula Lecture: Part 2 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 1

OUTLINE______________________________________

1. ORDINAL MEASURES OF ASSOCIATION: OVERVIEW

2. ORDINAL MEASURES OF ASSOCIATION: ARCHIMEDEAN

3. ORDINAL MEASURES OF ASSOCIATION: GDB

4. COPULA PARAMETER ELICITATION: ARCHIMEDEAN

5. COPULA PARAMETER ELICITATION: GDB

6. COPULA SELECTION: AN ENTROPY APROACH

7. A VALUE OF INFORMATION EXAMPLE


1. ORDINAL MEASURES OF ASSOCIATION...Overview

______________________________________

• Positive (negative) dependence between and \ µ KÐ † Ñ ] µ LÐ † Ñw w

when of one go with of the other.large values large (small) values

• In case of positive (negative) dependence, and are said to be\ ]w w

concordant (disconcordant).

• Classical measures for :the degree of positive or negative dependence

Blomquist's (1950) , Kendall's (1938) and Spearman's (1904) ." 7 3=

• All three measures attain .values ranging from to " "

• All three are Hence,ordinally invariantÞ

3 3= =w w w wÐ\ ß] Ñ œ Ð\ß] Ñß Ð\ß] Ñ œ ÖKÐ\ ÑßLÐ] Ñ×where , etc.

• Recall, is a copula.\ ] µ YÒ!ß "Óand and thus the joint pdf of Ð\ß ] Ñ


1. ORD. MEASURES OF ASSOCIATION...Blomquist's "

______________________________________

• Excellent review of classical measures , and is given by Kruskal (1958)." 7 3=

"Ð\ß] Ñ œ %GÐ ß Ñ "" "

# #, where is copula cdfGÐ † ß † Ñ

X1 < 0.5

X1 > 0.5

Y1 < 0.5

Y1 > 0.5

1

-1Y1 < 0.5

Y1 > 0.5 1

-1EMV = β

A

Blomqvist’s β


1. ORD. MEASURES OF ASSOCIATION...Kendalls's 7

______________________________________

• Let -Ð † ß † Ñ GÐ † ß † Ñ \ ß ] Ñ µ GÐ † ß † Ñ, be the copula pdf and cdf , and let Ð 3 3

3 œ "ß # be two independent bivariate samples from the copula.

7Ð\ß] Ñ œ % GÐBß CÑ-ÐBß CÑ.B.C " ! !

" "

.

X1 < X2

X1 > X2

Y1 < Y2

Y1 > Y2

1

-1Y1 < Y2

Y1 > Y2 1

-1EMV = τ

B

Kendall’s τ


1. ORD. MEASURES OF ASSOCIATION...Spearman's 3=

______________________________________

• Let -Ð † ß † Ñ GÐ † ß † Ñ \ ß ] Ñ µ GÐ † ß † Ñ 3 œ "ß #ß $, be the copula pdf , let Ð 3 3

be three independent bivariate samples from the copula.

3=! !

" "

Ð\ß] Ñ œ "# BC-ÐBß CÑ.B.C $ Þ

X1 < X2

X1 > X2

Y1 < Y3

Y1 > Y3

1

-1Y1 < Y3

Y1 > Y3 1

-1EMV = ρs

C

Spearman’s ρs


1. ORD. MEAS OF ASSOC...ÞTail Dependence

______________________________________

• Lower and upper tail dependence measures are in vogue, particularly inproblem contexts dealing with modeling the joint occurrence of extremeevents, such as insurance and modeling of default risk in finance.

• Recent to the copula approach may be credited to burst of attention theGaussian copula which has been widely adopted by the "financial quants" .

• Embrechts (2008) even refers to this attention as "the copula craze".

• Unfortunately, some (see, e.g., Gaussian copula for theSalmon, 2009) blamed 2008 financial crash lack of, in part due to lower and upper tail dependence.

• These measures too are , although theyordinal measures of associationfocus primarily on modeling positive dependence and not negativedependence.



______________________________________

• \ µ KÐ † Ñ ] µ LÐ † Ñ' , ' , :Lower tail dependence -P

-P" "œ T<Ö] Ÿ L ÐBÑl\ Ÿ K ÐBÑ×

B Æ !

œ T<Ð] Ÿ Bl\ Ÿ BÑ œ ßB Æ ! B Æ !

GÐBß BÑ

B

lim

lim lim

' '

• \ µ KÐ † Ñ ] µ LÐ † Ñ' , ' , Upper tail dependence -Y :

-Y" "œ T<Ö] L ÐBÑl\ K ÐBÑ×

B Å "

œ T<Ð] Bl\ BÑ œ ÞB Å " B Å "

" #B GÐBß BÑ

" B

lim

lim lim

' '

• Clayton and Gumbel copulae exhibit lower and upper tail dependence,respectively.



______________________________________

• For GDB copula cdf we have ,GÖBß Cl ×:Ð † l ÑG - -P Yœ œ ! similar tothe Gaussian copulae (Embrechts et. al, 2002).

• Blomquist's Kendall's and Spearman's are more applicable in contexts" 7 3ß W

dealing with the modeling of joint events in general, not extremes per se.

• Blomquist's Kendall's and Spearman's pertain to full copula support" 7 3ß W

and not just to their asymptotic extreme values.

• Clayton and Gumbel copulae couldCaution to those who believe that the serve as the panacea as an alternative to the Gaussian Copula.

• Heteroscedastic behavior of financial processes their dependencesuggestscannot be modelled using a copula with ,a constant correlation over timeregardless of a copula displaying tail dependence or not.


OUTLINE______________________________________









2. ORD. MEAS OF ASSOC...ÞArchimedean Copula

______________________________________

• Population expressions for , and are" 7 3 - -ß ß= P Y :

"

7

3

-

-

Ð\ß ] Ñ œ %GÐ ß Ñ "

Ð\ß ] Ñ œ % GÐBß CÑ-ÐBß CÑ.B.C "

Ð\ß ] Ñ œ "# BC-ÐBß CÑ.B.C $

" "# #

! !

" "

= ! !

" "

P

Y

,,

,

œB Æ !

œB Å "

lim

lim

GÐBßBÑB

"#BGÐBßBÑ"B

• Archimedean copula with generator function pdf :Ð † Ñ (e.g Genest andMackay, 1986):

" α

7 α

3 α

-

-

Ö\ß ] l × œ % "

Ö\ß ] l × œ

Ö\ß ] l ×

: : :" " "# #

Ð>ÑÐ>Ñ

Ö# ÐBÑ×B

Ö Ð Ñ Ð Ñ× ,,

: No expression available.% .> "

!

" ::

: :

w

"

=

P

Y

œB Æ !

œ

lim

limB Å "

"#B"B: :"Ö# ÐBÑ×



______________________________________

• Clayton Copula: : αÐ>Ñ œ > "ß − Ð!ß∞Ñα

"

7

3- -

Ð\ß ] Ñ œ − Ò!ß "Ó

Ð\ß ] Ñ œ − Ò!ß "Ó

Ð\ß ] Ñœ # ß œ !

% # " "αα

""Î

,,

: No expression available,

αα

α

#

=

P Y

α ¸ "Þ*"& Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ(&

0.00

0.25

0.50

0.75

1.00

0 5 10 15 20α

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρ

β

τ

1.915

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X>x| α )P

r(Y>x

|X>x

, α )

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X<x| α )

Pr(Y

<x|X

<x, α

)



______________________________________

• Gumbel Copula: : αÐ>Ñ œ > "ß − Ð!ß∞Ñα

"

7

3- -

Ð\ß ] Ñ œ % ‚ − Ò!ß "Ó

Ð\ß ] Ñ œ − Ò!ß "Ó

Ð\ß ] Ñœ !ß œ # #

Ð"Î#Ñ "Ð# Ñ

"

"Îα ,,

: No expression available,

αα

=

P Yα

α ¸ "Þ**( Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ(&

0.00

0.25

0.50

0.75

1.00

1 6 11 16

α

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρ

β

τ

1.997

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X>x| α )P

r(Y>x

|X>x

, α )

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X<x| α )

Pr(Y

<x|X

<x, α

)



______________________________________

• Frank Copula: : α ‘Ð>Ñ œ ß − ÏÖ!×ln / "/ "

α

α

>

"

7

3-

Ð\ß ] Ñ œ − Ò "ß "Ó

Ð\ß ] Ñ œ − Ò "ß "Ó

Ð\ß ] Ñ

%α ln

" "

" Ò" HÐ ÑÓ ß HÐ Ñ œ .>

Ð/ "Ñ/ "

% " >! / "

# #

>

α

α ,

,: No expression available,

α α

αα α

=

P Yœ !ß œ !Þ-

α ¸ %Þ)(& Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ(&

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

-20 -15 -10 -5 0 5 10 15 20α

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρ

β

τ

4.875

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Pr(X>x| α )

Pr(Y

>x|X

>x, α

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Pr(X<x| α )

Pr(Y

<x|X

<x, α

)


OUTLINE______________________________________









3. ORD. MEAS OF ASSOC...ÞGDB Copula with TS Gen. PDF

______________________________________

• Population expressions for , and ar" 7 3=

"

7

3

-

-

Ð\ß ] Ñ œ %GÐ ß Ñ "

Ð\ß ] Ñ œ % GÐBß CÑ-ÐBß CÑ.B.C "

Ð\ß ] Ñ œ "# BC-ÐBß CÑ.B.C $

" "# #

! !

" "

= ! !

" "

P

Y

,,

,

œB Æ !

œB Å "

lim

lim

GÐBßBÑB

"#BGÐBßBÑ"B

• We have for GDB copula with TS pdf with generating pdf and:Ð † l ÑG^ µ TÐ † l Ñß T Ð † l ÑG G the generating cdf:

" G G

7 G G

3 G G G- -

Ö\ß ] l:Ð † l Ñ× œ #IÒ^l Ó "

Ö\ß ] l:Ð † l Ñ× œ l

Ö\ß ] l:Ð † l Ñ× œ œ !ß

,,

.#IÒ^ Ó = .= % = = .= "#

! !

" " # T Ð l Ñ T Ð l Ñ # #G G

=

P Y

% l Ó 'IÒ^ l Ó "IÒ^$ #

œ !



______________________________________

• Power pdf : :ÐDl8Ñ œ 8D ß 8 !ß8"

"

7

3

Ö\ ] × œ

Ö\ ] × œ

Ö\ ] × œ

, ,

,

l

l

l

:Ð † l8Ñ ß − Ò "ß "Óß

:Ð † l8Ñ ß − Ò "ß "Óß

:Ð † l8Ñ ß

8"8"8" 8"8# Ð8"ÑÐ8#ÑÐ#8"Ñ

=Ð8"ÑÐ8'ÑÐ8#ÑÐ8$Ñ − Ò "ß "Ó.

- -P Yœ !ß œ !

8 œ $ Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ(&

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X>x|n)

Pr(Y

>x|X

>x,n

) 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X<x|n)

Pr(Y

<x|X

<x,n

)

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0 5 10 15 20

n

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρβτ3.00



______________________________________

• Ogive pdf: :ÐDl7Ñ œ Ö# 7 " D 7D ×ß7 !ß7#$7%

7 7"( )

"

7

Ö\ ] × œ

Ö\ ] × œ

,

,

l

l

:Ð † l7Ñ ß − Ò!ß "Óß

:Ð † l7Ñ

7Ð7"ÑÐ$7)ÑÐ7 ÑÐ7 ÑÐ$7%Ñ3 47Ð7"ÑÐ"'#7 #'%$7 ")"$#7 ''"!)7 "%!!$#7&)))!Ñ

Ð

' & % $

7$ÑÐ7%ÑÐ7'ÑÐ#7&ÑÐ$7%Ñ Ð$7)ÑÐ$7"!Ñ# ß − Ò!ß "Óß

:Ð † l7Ñ ß − Ò!ß "Ó3=7Ð7"ÑÐ$7 (!7 %#%7($'ÑÐ7%ÑÐ7&ÑÐ7'ÑÐ7)ÑÐ$7%ÑÖ\ ] × œ, l

$ #

.-P œ !ß-Y œ !

7 œ %Þ*"' Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ(&

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Pr(X>x|m)

Pr(Y

>x|X

>x,m

) 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X<x|m)

Pr(Y

<x|X

<x,m

)

0.00

0.25

0.50

0.75

1.00

0 5 10 15 20

m

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρ

β

τ

4.916



______________________________________

• U pdfÒ ß "Ó À) :ÐDl Ñ œ ß Ÿ D Ÿ "ß ! Ÿ Ÿ ""

") ) )

)ß

" ) )7 ) )

3 )

Ö\ ] × œÖ\ ] × œ

Ö\ ] × œ

, , ,

ll Ð #

l Ð" Ñ

:Ð † l Ñ ß − Ò!ß "Óß:Ð † l Ñ ÑÎ$ß − Ò!ß "Óß

:Ð † l Ñ ß − Ò!ß "Ó

)

) ) )=# .

- -P Yœ !ß œ !

) œ !Þ& Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ(&

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X>x| θ )P

r(Y>x

|X>x

, θ )

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Pr(x<x| θ )

Pr(Y

<x|X

<x, θ

)

0.00

0.25

0.50

0.75

1.00

0 0.25 0.5 0.75 1

θ

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρ

β

τ

0.5



______________________________________

• Slope pdf : :ÐDl Ñ œ #Ð Dß ! Ÿ Ÿ #ßα α# "Ñα α

" α α

7 α

3 α α

Ö\ ] × œ

Ö\ ] × œ

Ö\ ] œ

, ,

,

l

l

l

:Ð † l Ñ ß − Ò ß Óß

:Ð † l Ñ ß − Ò ß Óß

:Ð † l Ñ ß − Ò ß Ó

" " " "$ $ $ $% % %"& "& "&

=# #& &

%"&

# #& &

α

.- -P Yœ !ß œ !

α œ "Þ(& Ê T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ T<Ð] !Þ&l\ !Þ&Ñ œ !Þ'#&

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pr(X<x| α )

Pr(Y

<x|X

<x, α

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Pr(X>x| α )

Pr(Y

>x|X

>x, α

)

-0.50

-0.25

0.00

0.25

0.50

0 0.5 1 1.5 2

α

Ord

inal

Mea

sure

s of

Ass

ocia

tion

ρβτ1.75


3. ORD. MEAS OF ASSOC...ÞReflection Property

______________________________________

• Let be pdf , ;ÐDl Ñ ^ œ " ^ ^ µ :ÐDl Ñ ÊG Gw ;ÐDl Ñ œ :Ð" Dl ÑG G .

• -ÖBß Cl; × œ -ÖBß Cl ×ÐDl Ñ :Ð" Dl ÑG G obtained via a right angle rotation.

0

1

0

1

0.00

1.00

2.00

x

y

Graph of rotated copula .using:Ð" Dl Ñ œ #Ð" DÑG



______________________________________

• We have for GDB copula with TS gen. pdf and ::Ð † l Ñ ^ µ :Ð † l ÑG G

" G G

7 G

3 G G G

Ö\ß ] l:Ð † l Ñ× œ #IÒ^l Ó "

Ö\ß ] l:Ð † l Ñ× œ

Ö\ß ] l:Ð † l Ñ× œ

,,

.#IÒ^ Ó = .= % = = .= "#

! !

" " # T Ð l Ñ T Ð l Ñ # #G G

= % l Ó 'IÒ^ l Ó "IÒ^$ #

• Let be pdf , ;ÐDl Ñ ^ œ "^ ^ µ :ÐDl ÑG Gw Ê

" " "7 7 73 3 3

Ö\ß ] l × œ Ö\ß ] l × œ Ö\ß ] l ×Ö\ß ] l × œ Ö\ß ] l × œ Ö\ß ] l ×Ö\ß ] l × œ Ö\ß ] l × œ

;ÐDl Ñ :Ð" Dl Ñ :ÐDl Ñ;ÐDl Ñ :Ð" Dl Ñ :ÐDl Ñ;ÐDl Ñ :Ð" Dl Ñ

G G GG G GG G

,,

= = =Ö\ß ] l ×:ÐDl ÑG .

• on :ÐDl Ñ ÊG symmetric Ò!ß "Ó Ê :Ð" Dl ÑG Gœ :ÐDl Ñ ß ß ´ !" 7 3=



______________________________________

:ÐDl+Ñ œ B Ð" BÑÐ#+Ñ

Ð+Ñ Ð+Ñ

>

> >+" +",+ ! Ê " 7 3ß ß ´ ! a+ != ,

0

10

1

0.00

1.00

2.00

3.00

4.00

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

z

Slo

pe P

DF

I J

G: Beta generating pdf; H: GDB Copula with TS Gen. PDF in AÞ


OUTLINE______________________________________









4. COPULA PARAM. ELICITATION...Procedure

______________________________________

• \ µ KÐ † Ñß ] µ LÐ † Ñ ÖKÐ\ ÑßLÐ] Ñ× œ Ð\ß ] Ñ µ GÖBß Cl ×' ' , ' ' :Ð † l ÑG

• Elicit: T<Ð Ÿ l Ÿ Ñ œ T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ] C \ B' ' ' '!Þ& !Þ& 1.

• This falls within elicitation procedure the conditional fractile estimationmethod for eliciting degree of dependence - Clemen and Reilly (1999).

• We have for Blomquist's "

" 1Ö\ß ] × œ # " œ # "T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ

• Hence, elicitation of 1 is equivalent to an indirect elicitation of Blomquist's" which has a more straightforward interpretation as and 7 3=Þ


4. COPULA PARAM. ELICITATION...Archimedean Copulas

______________________________________

• General expression for Archimean Copulas with generator : αÐ † Ñ| :

1 α αÐ\ß ] l Ñ œ T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ #GÐ ß l Ñ" "

# #

œ # ‚ : : :"Ö Ð Ñ Ð Ñ×" "

# #

• For different Archimean Copulas with generator herein : αÐ † Ñ| À

1 αÐ\ß ] l Ñ œÐ "Ñ

Ð"Î#Ñ

"

2α α

α

+1 "Î

#

" Ð/ "Ñ/ "

− Ò!Þ&ß "Ó

− Ò!Þ&ß "Ó

− Ò!Þ&ß "Ó

, Clayton Copula,, Gumbel Copula,

, Frank Copul

"Î

Î# #

α

α

αln a,

• Having elicited one solves for using a numerical produres, except in case1 αof Gumbel copula.


4. COPULA PARAM. ELICITATION...Archimedean Copulas

______________________________________

• Assume that an expert has assessed a value À

1 GÖ\ ] ×, l œ !Þ(&:Ð † l Ñ ÊT<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ

Clayton Copula,Gumbel Copula,Frank Copula,

ααα

¸ "Þ*"&¸ "Þ**(¸ %Þ)(&

,,.

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Clayton 0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Gumbel 0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

FrankArchimedean Archimedean Archimedean


OUTLINE______________________________________









5. COPULA PARAM. ELICITATION...GDB Copulas

______________________________________

GÖBß Cl × œ

ß

ß

ß:Ð † l Ñ

B T ÐDl Ñ.D

C TÐDl Ñ×.D

B T ÐDl Ñ.D

C

G

G

G

G

"# "

"# "

"#

BC

B"

BC

BC#

C"

BC$

" C

"

B

"

ÐBß CÑ − E

ÐBß CÑ − E

ÐBß CÑ − E

,

,

,

TÐDl Ñ.D"#BC"

BC%

"G ß ÐBß CÑ − E Þ

Ê

"

#œ T<Ð] Ÿ !Þ&ß\ Ÿ !Þ&Ñ œ1 GÖ ß l × œ

" "

# #:Ð † l Ñ Ö" TÐDl Ñ×.D

"

#G G

!

"

• With ^ µ IÒ^ Ó œ ßT ÐDl ÑßG l Ö" TÐDl Ñ×.DG G!

" thus we have:

1 GÖ\ß] l:Ð † l Ñ× œ IÒ^l ÓG Þ Ð"#Ñ

• Thus, having elicited one solves for using .1 < the method of moments



______________________________________

• For different GDB Copulas with generating pdf's herein:Ð † l † Ñ

1 G

α

Ö\ ] × œ, l:Ð † l Ñ

Ð − Ò ß Ó :ÐDl Ñ

8ÎÐ8 "Ñ − Ò!ß "Ó :ÐDl8Ñ

− Ò!Þ&ß "Ó :ÐDl7

# ÑÎ'α " #

$ $

Ð7#Ñ$7% Ð7%ÑÐ7$Ñ

$7'

, , Slope pdf,, , Power pdf,

,#

Ñß

Ð "ÑÎ# − Ò!Þ&ß "Ó :ÐDl Ñ Ò ß "Ó

Ogive pdf,, , U pdf.) ) )

• Having elicited one solves for and in closed form for GDB copula1 α )ß 8with slope, power or U generating pdf.Ò ß "Ó)

• Having elicited one solves for using a numerical procedure in case of1 7GDB copula with Ogive generating pdf.



______________________________________

• Assume that an expert has assessed a value À

1 GÖ\ ] ×, l œ !Þ(&:Ð † l Ñ ÊT<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ

GDB - Power Gen. Density,GDB - Ogive Gen. Density,GDB - ,

8 œ $7 ¸ %Þ*"'

œ !Þ&

,,

.U Gen. DensityÒ ß "Ó) )

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Power(n) Ogive(m) U[θ,1]GDB GDB GDB


OUTLINE______________________________________









6. COPULA SELECTION...How to pick one?

______________________________________

• Assume that an expert has assessed a value

1 GÖ\ ] ×, l œ !Þ(&:Ð † l Ñ T <Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ .

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Power(n) Ogive(m) U[θ,1]

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Clayton 0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Gumbel 0

10

1

0.00

1.00

2.00

3.00

4.00

5.00


GDB GDB GDB



______________________________________

• All six copulas above match the constraint:

1 GÖ\ ] ×, l !Þ(&:Ð † l Ñ œ T<Ö] !Þ&l\ !Þ&× œ

• Lower tail dependence Pick Clayton CopulaÊ

• Upper tail dependence Pick Gumbel CopulaÊ

• If neither is required, pick one that is uniform as possible?

ClaytonGumbelFrankPower

À ¸ "Þ*"& Ê œ !Þ&ß ¸ !Þ%)*

À ¸ "Þ**( Ê ¸ !Þ''"'ß œ !Þ&ß œ !Þ&

À ¸ %Þ)(& Ê ¸ !Þ'#%%ß œ !Þ&ß ¸ !Þ%%)

Ð8Ñ À 8 œ $ Ê

α " 7

α " 7

α " 7

3

3

3

3

=

=

=

=

¸ !Þ''#&ß

Ð8Ñ œ !Þ'!!!ß

Ð7Ñ œ !Þ'!&*ß

Ð Ñ œ !Þ'#&!ß

" 7

" 7

) ) " 7

œ !Þ&ß œ !Þ%"%ß

Ð7Ñ À 7 ¸ %Þ*"' Ê œ !Þ&ß œ !Þ%"&ß

HFÐ Ñ À œ !Þ& Ê œ !Þ&ß œ !Þ%"(ß

Ogive 3

3 )=

=


6. COPULA SELECTION...Smallest ?3=

______________________________________

• Can we select the one with the smallest rank correlation in general?

• Pdf :Ð † l Ñ Ò!ß "Ó Ê IÒ^l Ó œ :Ð † l ÑG < 1 G symmetric on Ö\ ] × œ Þ, l "#

• Pdf :Ð † l Ñ Ò!ß "Ó Ê ´ !ß ´ !ß ´ !ÞG 3 " 7 symmetric on =

• When elicited 1 GÖ\ ] × œ œ, it seems tol:Ð † l Ñ T <Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ "#

intuitive to select copula with independent uniform marginals.

• Hence, answer is: No.


6. COPULA SELECTION...Max. Entropy?

______________________________________

• Select copula that between it and copula withminimizes distanceindependent uniform marginals.

• Kullback-Liebler distance the relative information measures of onecandidate pdf with respect to pdf given by0ÐBß CÑ 1ÐBß CÑ

MÐ0 l1Ñ œ 0ÐBß CÑ Ö0ÐBß CÑÎ1ÐBß CÑ×.B.C ln .

• Setting and yields:0ÐBß CÑ œ -ÖBß C× 1ÐBß CÑ œ ?ÐBß CÑ œ "ß

MÐ-l?Ñ œ -ÐBß CÑ Ö-ÐBß CÑ×.B.C Ð%!Ñ W-

ln ,

• The quantity is known as of the pdf I œ MÐ-l?Ñ -ÐBß CÑÞthe entropy

• Bedford and Meeuwissen (1997) constructed maximum entropy copulaegiven a correlation constraint that are .minimally informative


6. COPULA SELECTION...Max. Entropy

______________________________________

• Bedford and Meeuwissen's (1997) maximum entropy copulae given acorrelation constraint, do not possess closed form pdf and cdf.

• Select amongst a set of GDB copula that matches a specified constraint thatone that is (or has maximum entropy).minimally informative

• Utilizing over a 100 by 100 grid over we havenumerical integration Ò!ß "Ó ß#

MÖ-ÐBß CÑl:Ð † l Ñ× œ!Þ#"$' :ÐDl8Ñß 8 œ $ß!Þ#### :ÐDl7Ñß7 œ %Þ*"'!Þ$%!! :ÐDl Ñß œ !Þ& Ò ß "Ó

<) ) )

, Power pdf,, , Ogive pdf, . U pdf

• Summarizing, given set by the constraint 1 œ !Þ(&T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ ,the relative information approach above would suggest to use the GDBcopula with the power( ) generating pdf with 8 Ð#'Ñ 8 œ $.


6. COPULA SELECTION...Max. Entropy

______________________________________

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Max Entropy Gumbel

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Max Entropy Clayton

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Max Entropy Frank

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Max Entropy TSP

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Max Entropy Ogive

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Max Entropy Diagonal Band

Archimedean Archimedean Archimedean

GDB GDB GDB

X-Axis: Y-Axis:Rank Correlation Relative Information Entropyœ



______________________________________

• Assume that an expert has assessed a value

1 GÖ\ ] ×, l œ !Þ(&:Ð † l Ñ T <Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ .

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Power(n) Ogive(m) U[θ,1]

0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Clayton 0

10

1

0.00

1.00

2.00

3.00

4.00

5.00

Gumbel 0

10

1

0.00

1.00

2.00

3.00

4.00

5.00


GDB GDB GDB


OUTLINE______________________________________









7. A VALUE OF INFORMATION EXAMPLE...Description

______________________________________

• The farmer needs to (with a total worthprotecting his/her crop of orangesof $50,000) against freezing weather with the objective of .minimizing losses

• Temperature X (in Fahrenheit) that night.µ YÒ#%ß $%Ó

• (below freezing) Farmer looses entire crop X $# Ê without protection.

• Two protection alternatives: or with $ or $ ,Burners Sprinklers "!ß !!! $ß !!!respectively, in mobilization cost.

• Effectiveness of both is uncertain. Farmer assesses all-in loss F ÐWÑ to varybetween $25000 ($28,000) and $35,000 ($33,000) with a most likely+ œ , œvalue of $27,000 ($29,000) .7 œ if it freezes

• Assume and to be F W Ê IÒFÓ œtriangular distributed $29,000ßIÒWÓ œ $30,000.


7. A VALUE OF INFORMATION EXAMPLE...Decision Tree

______________________________________

50

0

Burners

Sprinklers

Do Nothing

Pr(T < 32)

10

3

E[B] = 29

E[S] = 30

Pr(T > 32)

Pr(T < 32)

Pr(T > 32)

Pr(T < 32)

Pr(T > 32)

25.20

24.60

40

24.60


7. A VALUE OF INFORMATION EXAMPLE...Dependence on X

______________________________________

• Effectiveness sprinkler insular layer of freezing option is based on an water on the oranges.

• Effectiveness burner gas usage. option is based on

• The farmer assesses a % chance ( % chance) that the burning loss *! '! F(sprinkler loss ) is ( when the temperature W , = Ñ Xabove it median value !Þ& !Þ&

is . Hence, we have:below its median value #*J

T<ÐF , l X #*Ñ œ !Þ*ß T <ÐW = l X #*Ñ œ !Þ'ß!Þ& !Þ&

where $28,675 and $29,838, ¸ = ¸ Þ!Þ& !Þ&

• Model dependence between ( ) and using a GDB copula with a powerF W X(slope) generating density with ( )8 œ "Î"" œ !Þ% Þα


7. A VALUE OF INFORMATION EXAMPLE...EVPI Freezing

______________________________________

• To reduce losses further, the farmer considers consulting either a clairvoyantExpert A a clairvoyant Expert B on or on "Freezing" the temperature .X

50

Burners

Sprinklers

Do nothing

E[B] = 29

E[S] = 30 Pr(T < 32)

0

Burners

Sprinklers

Do nothing

10

3Pr(T > 32)

0

29

23.20

EVPI = 24.6 – 23.2 = 1.4


7. A VALUE OF INFORMATION EXAMPLE...EVPI Temperature X

______________________________________

• EVPI on the temperature from Expert B is since itX more complicatedrequires evaluation of and IÒFl>Ó IÒWl>ÓÞ

• K > ß IÒFl>Ó = œ #&!!iven we evaluate using realizations using the steps:

Step 1: Recall, B œ Ð X µ Y8309<7Ò#%ß $%ÓÑ>#%$%#%

Step 2: from GDBSample quantile levels C3ß 3 œ "ßá ß = Ð\ß ] Ñcopula with power generating density for .Ð8Ñ Fß 8 œ "Î""

Step : $ IÒFl>Ó œ L ÐC Ñß"=3œ"

="

3• F µ Ð à à Ñ L Ð † ÑTriang $25,000 $27,000 $35,000 , is the inverse cdf or"

quantile function of $28,000 $29,000 $33,000 .FÞ W µ X<3+81Ð à à ÑEvaluation of is analogous.IÒWl>Ó



______________________________________

• .X œ ÐX lX $#Ñ µ Ò#%ß $#Ó X µ Ò#%ß $%Ów U since U

50

Burners

Sprinklers

Do nothing

E[B|T’=t]

E[S|T’=t]Pr(T < 32)

0

Burners

Sprinklers

Do nothing

10

5Pr(T > 32)

0

T’ T’=t<32

EVPI = 24.6 – 22.96 = 1.64

28.70

22.96



______________________________________

$25

$27

$29

$31

$33

$35

24.0 26.0 28.0 30.0 32.0

Temperature (in F)

Loss

(in

$100

0's)

E[B|t] E[S|t]

29]}|[{][ ' == tBEEBE T

30]}|[{][ ' == tSEESE T

SB

7.28]}'|[],'|[({' =TSETBEMinET



______________________________________

• EVPI "freezing" $1, , EVPI "freezing" $1,64¸ %!! ¸ !

• Summarizing, the farmer is willing to pay $240 dollars more for perfectinformation on the temperature .X

• Optimal decision switches to Sprinkler option when providesExpert A "Freezing" information.

• Optimal decision switches to Sprinkler option when providesExpert B "temperature " information, where .> #' > $#

• When provides "temperature " information, where theExpert B > #% > #'ßoptimal decision remains the Burner option.

QUESTIONS?


7. COPULAS...Selected References

______________________________________

Bojarski, J. . (2001) A new class of band copulas - distributions with uniform marginals.Technical Publication, Institute of Mathematics, Technical University of Zielona Góra.

Samuel Kotz and Johan René van Dorp , (2009) Generalized Diagonal Band Copulaswith Two-Sided Generating Densities, , published online before printDecision AnalysisNovember 25, DOI:10.1287/deca.1090.0162.

Cooke, R.M. and Waij, R. Ð Ñ1986 Þ Monte carlo sampling for generalized knowledgedependence, with application to human reliability. , 6 (3), pp. 335-343.Risk Analysis

Genest, C. and Mackay, J. . (1986) The joy of copulas, bivariate distributions withuniform marginals. 40 (4), pp. 280-283.The American Statistician,

Ferguson, T.F. . A class of symmetric bivariate uniform distributions. ,(1995) Statistical Papers36 (1), pp. 31-40.

Nelsen, R.B. . . Springer, New York.(1999) An Introduction to CopulasSklar, A. . (1959) Fonctions de répartition à n dimensions et leurs marges. Publ. Inst.

Statist. Univ. Paris, 8, pp. 229-231.Van Dorp, J.R and Kotz, S. . (2003) Generalizations of two sided power distributions

and their convolution. , 32 (9), pp. 1703 -Communications in Statistics: Theory and Methods1723.

Documents

Lecture on Copulas Part 2dorpjr/EMSE280/Copula... · EMSE 280 - Copula Lecture: Part 2 J.R. van Dorp; dorpjr - Page 1 OUTLINE _____ 1. ORDINAL MEASURES OF ASSOCIATION: OVERVIEW