Lecture on Copulas Part 1 - George Washington …dorpjr/EMSE280/Copula/Lecture...EMSE 280 - Copula...
Transcript of Lecture on Copulas Part 1 - George Washington …dorpjr/EMSE280/Copula/Lecture...EMSE 280 - Copula...
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr
Lecture on Copulas: Part 1______________________________________
J. René van Dorp1
Faculty Web-Page: www.seas.gwu.edu/~dorpjr
March 29-th, 2010
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 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 1
OUTLINE______________________________________
1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 2
1. INTRODUCTION...CDF Theorem
______________________________________
Theorem: Let be a continuous random variable with distribution funnction\JÐ † Ñ ] \ ] œ JÐ\Ñ. Let be a transformation of such that . The distribution of] Ò!ß "Ó is uniform on .
Proof: For a uniform random variable on we haveY Ò!ß "Ó
T <ÐY Ÿ ?Ñ œ ?ß a? − Ò!ß "Ó
Hence, we need to show that T<Ð] Ÿ CÑ œ Cß aC − Ò!ß "ÓÞ Since we have thatJÐBÑ œ T<Ð\ Ÿ BÑ − Ò!ß "Ó B ] œ JÐ\Ñ for all values of it follows that hassupport .Ò!ß "Ó
T <Ð] Ÿ CÑ œ T<ÒJ Ð\Ñ Ÿ CÓ œ T<ÖJ ÒJ Ð\ÑÓ Ÿ J ÐCÑ×
œ T<Ò\ Ÿ J ÐCÑÓ œ J ÒJ ÐCÑÓ œ C
" "
" " Þ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 3
1. INTRODUCTION...Bivariate Normal PDF
______________________________________
• Probability density function of a bivariate normal distribution:
\ œ µ QZ RÐ ß Ñß œ \\
" "
# #. D Mean Vector ,. .
.
Covariance Matrix D 5
5œ
G9@Ð\ ß\ Ñ
G9@Ð\ ß\ Ñ #
" " #
" ###
0ÐBß CÑ œ /B: Ð Ð "
# l l 1 D
. D .B Ñ B Ñw "
• Independence in case of the bivariate normal distribution implies (and viceversa):
D D5 5
5 5œ œ
! "Î !
! ! "Î # #
" "# ## #
",
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 4
1. INTRODUCTION...BVN PDF - Independence
______________________________________-3
.0
-2.0
-1.1
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-0.1
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1.3
2.0
2.8
. œ œ ß œ !Þ!ß T <Ð] Ÿ !l\ Ÿ !Ñ ¸ !Þ&!
! !! , 1
1D 3
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 5
1. INTRODUCTION...BVN PDF - Dependence
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. œ œ ß ¸ !Þ%)$ß T <Ð] Ÿ !l\ Ÿ !Ñ ¸ !Þ(& !! , 1
1D 33
3
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 6
1. INTRODUCTION...Sklar's (1959) Theorem
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Sklar’s Theorem (1959). Given a joint CDF for random variablesJÐB ßá ß B Ñ" 8
\ ßá ß\ J Ð † Ñßá J Ð † ÑÞ J ÐB ßá ß B Ñ" 8 \ \ " 8 with marginal CDFs Then can" A
be written as a function of its marginals:
JÐB ßá ß B Ñ œ GÖJ ÐB Ñßá J ÐB Ñ×" 8 \ " \ 8 , " 8
where ( , . . . , ) is a joint distribution function with uniform marginals.G ? ? Ò!ß "Ó" 8
Moreover, if each is continuous, then ( , . . . , ) is unique, and if eachJ ÐB Ñ G ? ?\ 3 " 83
J ÐB Ñ\ 33 is discrete, then C is unique on
V+8ÒJ Ð † ÑÓ ‚á ‚ V+8ÒJ Ð † ÑÓ\ \" 8
where is the rangeV+8ÒJ Ð † ÑÓ J Ð † ÑÞ\ \8 8
For and ( , . . . , ) continuous and differentiable case one has:J Ð † Ñ G ? ?\ " 8"
0ÐB ßá ß B Ñ œ 0 ÐB Ñ ‚á ‚ 0 ÐB Ñ ‚ -ÖJ ÐB Ñßá J ÐB Ñ×" 8 \ " \ 8 \ " \ 8" 8 " 8
where is copula pdf or -Ð? ßá ß ? Ñ" 8 dependence function.
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 7
1. INTRODUCTION...Sklar's (1959)- The Bivariate Case
______________________________________
• , : such that \ ] \ µ KÐ † Ñß ] µ LÐ † Ñw w w wContinuous random variables
• , H( ): KÐ † Ñ † Cumulative distribution functions - cdf's.
• The mapping is called the\ Ä \ œ KÐ\ Ñ Ê Ów w \ µ Ò!ß "Uniformprobability integral transformation e.g. Nelsen (1999).
• Any bivariate joint distribution of can be transformed to a bivariateÐ\ ß ] Ñw w
copula { } - Sklar (1959).Ð\ß ] Ñ œ KÐ\ ÑßLÐ] Ñw w
• Thus, a bivariate copula is a bivariate distribution with uniform marginals.
• As such, many authors studied copulae indirectly.
• Gaussian and Student-t Copulae (of this construct) were studied explicitlyÞ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 8
1. INTRODUCTION...BVN Normal Copula
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BVN - Independence BVN Copula - IndependenceÐ\ ß ] Ñ Ð\ß ] Ñw w
\ µ Y Ò!ß "Óß ] µ Y Ò!ß "Óß Ð\ß ] Ñ µ -Ð? ß ? Ñ œ "ß aÐ? ß ? Ñ − Ò!ß "Ó" # " ##
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 9
1. INTRODUCTION...BVN Normal Copula
______________________________________
0
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BVN - Dependence BVN Copula - DependenceÐ\ ß ] Ñ Ð\ß ] Ñw w
\ µ Y Ò!ß "Óß ] µ Y Ò!ß "Óß Ð\ß ] Ñ µ -Ð? ß ? Ñ Ð? ß ? Ñ − Ò!ß "Ó" # " ##,
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 10
1. INTRODUCTION...Summary
______________________________________
• The dependence relationship between two random variables is \ ß]w w obscured
by the marginal densities of and \ ] Þw w
• One can think of the copula density as the densities that filters or extracts the
marginal information from the joint distribution of and .\ ]w w
• To describe, study and measure statistical dependence between random \ ß]w w
variables one may study the copula densities.
• Two random vectors and ) share the same dependenceÐ\ ß ] Ñ Ð\ ß ]" " # #
relationship when their copula densities are the same.
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 11
1. INTRODUCTION...Inverse CDF Theorem
______________________________________• Vice versa, to build a joint distribution between two random variables
\ µ KÐ † Ñ ] µ LÐ † Ñ Ò!ß "Ó' and ' , one may construct first the copula on and#
utilize the inverse transformation and ( ).K Ð † Ñ L †" "
Theorem: Let be a continuous random variable with distribution funnction\JÐ † Ñ ] Y µ Ò!ß "Ó ] œ J ÐYÑ ]. Let be a transformation of such that . Then"
also has distribution function .JÐ † Ñ
Proof: For a uniform random variable on we haveY Ò!ß "Ó
T <ÐY Ÿ ?Ñ œ ?ß a? − Ò!ß "Ó
Hence, we need to show that T<Ð] Ÿ CÑ œ JÐCÑÞ
T <Ð] Ÿ CÑ œ T<ÒJ ÐYÑ Ÿ CÓ œ T<ÖJ ÒJ ÐYÑÓ Ÿ J ÐCÑ×
œ T<ÒY Ÿ J ÐCÑÓ œ J ÐCÑ
" "
Þ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 12
1. INTRODUCTION...Risk Management?
______________________________________Example 1: Let be the value of Stock . Let .W 3 V œ A W ß A œ "ß A !3 3 3 3 3
3œ" 3œ"
8 8 "5% Value-at-Risk" of a Portfolio is defined as follows:
T<ÐV Z EVÑ œ !Þ!&
Gaussian Copulas have been used to model dependence between ÐW ßá ß W Ñ" 8
Example 2: Four satellite are needed in orbit to make three dimensional
observations of the earth magnetosphere. The orbit each selected so that each
islocated at the corner point of a predetermined tethahedron, when crossing
regions of interest within the magnetosphere.
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 13
1. INTRODUCTION...Risk Management?
______________________________________Lifetime of the indivdual satelites are random but dependent. Let be the\%ßW
lifetime of the system if four satellites are put into orbit and if a fifth\&ßW
redundant satelite if put into orbit. Copulas can be used to study the difference
between and . Is reliability improvement of sending a fifth satelite into\ \%ßW &ßW
orbit worth its cost when taking dependence into account?
\ œ Q38Ð\ ß\ ß\ ß\ Ñ%ßW " # $ %
\ œ Q+B Q38Ö\ ß\ ß\ ß\ ×ßQ38Ö\ ß\ ß\ ß\ ×ß&ßW " # $ % " # $ &Q38Ö\ ß\ ß\ ß\ ×ß Q38Ö\ ß\ ß\ ß\ ×ß" # % & " $ % &
Q38Ö\ ß\ ß\ ß\ ×# $ % &
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 14
1. INTRODUCTION...Risk Management?
______________________________________Example 3: Consider the following project network representating the constuction
of a ship. The activity completion times are dependent random variables. Copulas
can be used to construct the dependence between these random variables and
evaluate the project completion time distribution.
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EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 15
OUTLINE______________________________________
1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 16
2. COPULA CONSTRUCTION...Archimedean Copulas
______________________________________
• Genest and Mackay (1986) used for copulaan algabraic methodconstruction.
• , with - : :À Ð!ß "Ó Ä Ò!ß∞Ñ Ð"Ñ œ ! a convex decreasing function The generatorfunction.
• They possess :joint cdf and probability density function (pdf)
GÖBß Cl Ð † Ñ× œ Ð"ÑÖ ÐBÑ ÐCÑ× ÐBÑ ÐCÑ Ÿ !
!:
: : : : : "
elsewhere
-ÖBß Cl Ð † Ñ× œ Ð#ÑÖGÐBß CÑ× ÐBÑ ÐCÑ
Ò ÖGÐBß CÑ×Ó:
: : :
:
'' ' '' $
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 17
OUTLINE______________________________________
1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 18
3. ARCHIMEDEAN EXAMPLES...Clayton Copula
______________________________________
• Generator Function:
: : αÐ>Ñ œ > "ß Ð=Ñ œ Ð" =Ñ ß " "Îα α 0Þ
• Cumulative Distribution Function:
GÐBß Cl Ñ œ Þα ÐBÑ ÐCÑ " ß ! "Î
α αα
α
• Probability Density Function:
-ÐBß Cl Ñ œ ß ÞÐ" Ñ ÐBCÑ
ÐBCÑ B C ÐBCÑα α
αα α α α
α
" "#α
α 0
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 19
3. ARCHIMEDEAN EXAMPLES...Clayton Copula
______________________________________
T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ !Þ(&ß ¸ "Þ*"&α
0 10
1
0
10
1
0.00
1.00
2.00
3.00
4.00
5.00
Joint Probability Density Function Contour Plot PDF
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 20
3. ARCHIMEDEAN EXAMPLES...Gumbel Copula
______________________________________
• Generator Function:
: : αÐ>Ñ œ Ð >Ñ ß Ð=Ñ œ /B:Ð = Ñß "ln α α" "Î Þ
• Cumulative Distribution Function:
GÐBß Cl Ñ œ Þα IB: Ð BÑ Ð CÑ ß " ln lnα αα"Î
α
• Probability Density Function:
-ÐBß Cl Ñ œ IB: Ð BÑ Ð CÑ ‚Ð BÑ Ð CÑ
B C
Ð BÑ Ð CÑ Ð "Ñ Ð BÑ Ð CÑ ß "Þ
α
α α
ln lnln ln
ln ln ln ln
α αα α α
α α α αα α
"Î " "
#Î # "Î #
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 21
3. ARCHIMEDEAN EXAMPLES...Gumbel Copula
______________________________________
T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ !Þ(&ß ¸ "Þ**(α
0
10
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5.00
0 10
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Joint Probability Density Function Contour Plot PDF
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 22
3. ARCHIMEDEAN EXAMPLES...Frank Copula
______________________________________
• Generator Function:
: : α α ‘Ð>Ñ œ ß Ð=Ñ œ Ò" / Ð/ "ÑÓß − ÏÖ!×/ "
/ "ln ln
α
αα
>" " =
• Cumulative Distribution Function:
"
αln " ß − ÏÖ!×
Ð/ "ÑÐ/ "Ñ
/ "
B B
α α
αα ‘
• Probability Density Function:
-ÐBß Cl Ñ œ " ‚/ / Ð/ "ÑÐ/ "Ñ
/ " / "Ð/ "ÑÐ/ "Ñ Ð/ "ÑÐ/ "Ñ
/ " / "" " ß − ÏÖ!×
αα
α ‘
B C B B
"
B B B B
"
α α α α
α α
α α α α
α α
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 23
3. ARCHIMEDEAN EXAMPLES...Frank Copula
______________________________________
T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ !Þ(&ß ¸ %Þ)(&α
0
10
1
0.00
1.00
2.00
3.00
4.00
5.00
0 10
1
Joint Probability Density Function Contour Plot PDF
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 24
OUTLINE______________________________________ 1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 25
4. COPULA CONSTRUCTION...Genealized Diagonal Band
______________________________________
• Cooke and Waij (1986) used for copula constructiona geometric method
0
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x
y
(0, 1−θ)
(1, θ)
DB2
DB1
DB3
A B
y(0, θ−1)
(1, 2−θ)
(0, 1)
(1, 0)
(0, 0)
(1, 1)
0
10
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3.00
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x
y
(0, 1−θ)
(1, θ)
DB2
DB1
DB3
A B
y(0, θ−1)
(1, 2−θ)
(0, 1)
(1, 0)
(0, 0)
(1, 1)
A: Gray area support of a copula comprisedHFÐ Ñ)
of sub-areas B: Example of a copula.HF ß 3 œ "ß #ß $à HFÐ!Þ&Ñ3
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 26
4. COPULA CONSTRUCTION...Diagonal Band Copula
______________________________________
• Diagonal Band (DB) copula possess pdf:
GÖBß Cl × œ Ð$Ñ"ÎÐ" Ñ ÐBß CÑ − HF ∪HF"ÎÖ#Ð" Ñ× ÐBß CÑ − HF!
)))
" $
#
elsewhere
• , Analagous to Archimedean copula Bojarski (2001) generalized copula via HFÐ Ñ) agenerator function | .0Ð † Ñ)
• Generator function | is a with support .0Ð † Ñ Ò Ó) )symmetric pdf "ß " )
• Lewandowski (2005) showed that Bojarski's (2001) GDB Copulae areequivalent to Fergusons (1995) family of copulae with joint pdf:
-ÐBß CÑ œ Ö1ÐlB Cl 1Ð" l" B ClÑ× Ð%Ñ"
#, 1Ð † Ñ Ò!ß "Ópdf on
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 27
4. COPULA CONSTRUCTION...Generalized DB Copula
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• For sampling efficiency of generator would be desirable.inverse cdf 0Ð † l Ñ)
• Consider Van Dorp and Kotz's (2003) symmetric Two-Sided (TS) pdf's À
0 Dl:Ð † l Ñ× œ ‚"
#
:ÐD "l Ñ " D Ÿ !ß:Ð" Dl Ñ ! D "ß
{ , for , for G
GG Ð&Ñ
that too uses the generating pdf concept. Pdf has support :ÐDÑ :ÐDÑ Ò!ß "ÓÞ
• The associated with inverse cdf (or quantile function) Ð%Ñ
J ?l:Ð † l Ñ× œ! ? Ÿ ß
? "ß"
"#
"#
{, for
, for G
G
GT Ð#?l Ñ "
" T Ð# #?l ÑÐ'Ñ
"
"
where is the quantile function of T Ð † l Ñ" < :Ð † l ÑÞG
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 28
4. CONSTRUCTION...GDB Copula with TS Gen. PDF
______________________________________
• Bivariate pdf is constructed where and 1ÐBß CÑ ß \ µ YÒ!ß "Ó the conditionalpdf 1ÐClBÑ has the following form À
1ÖClBß × œ Ð(Ñ:Ð † l Ñ l:Ð † l Ñ×G G0ÖB C B " Ÿ C Ÿ B ", ,
• From , and it follows that:\ µ YÒ!ß "Ó Ð(Ñ TS framework pdf Ð%Ñ
1ÖBß Cl œ Ð)Ñßß
:Ð † l Ñ× ‚"
#
:Ð" l Ñ:Ð" l Ñ
GGG B C " B C Ÿ !ß
B C ! B C "ß
• From , a bivariate pdf | is constucted on the unit squareÐ)Ñ -ÐBß C Ñ:Ð † l ÑGÒ!ß "Ó 1ÖBß Cl# of outsideby folding back the probability masses :Ð † l Ñ×Gthe unit square onto it,Ò!ß "Ó# using "folding" lines and .C œ " C œ !
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 29
4. CONSTRUCTION...GDB Copula with TS Gen. PDF
______________________________________
010
1
0.00
1.00
2.00x
y
1
1
x – y = – 1
x – y = 1
B
x – y = 0
(0,0)
(1,1)
(1,0)
(0,1)
C
2A
4A3A
1A
x + y = 1
x + y = 0
x + y = 2
x
y 1A3A
2A4A
A: pdf B: Areas 1ÐBß CÑ Ð)Ñà E ß 3 œ "ßáß%à3
C: pdf with on .-ÖBß Cl × Ð"!Ñ :ÐDÑ œ #D Ò!ß "Ó:Ð † l ÑG
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 30
4. CONSTRUCTION...GDB Copula with TS Gen. PDF
______________________________________
• Relationship between and -ÖBß Cl × 1ÖBß Cl:Ð † l Ñ :Ð † l Ñ×G G in 8Ð Ñ À
-ÖBß Cl × œ
1ÖBß Cl 1ÖBß Cl ß1ÖBß Cl 1ÖBß # Cl ß
:Ð † l Ñ Ð*Ñ
:Ð † l Ñ× :Ð † l Ñ×:Ð † l Ñ× :Ð † l Ñ×
G
G GG G ! B C Ÿ "
" B C Ÿ #,.
• Combining with Ð*Ñ Ð)Ñ now yields À
-ÖBß Cl × œ
"
#‚
" ß" ß
ß
:Ð † l Ñ
:Ð l Ñ :Ð" l Ñ:Ð l Ñ :Ð" l Ñ:Ð l Ñ :Ð" l Ñ:Ð l Ñ
G
G GG GG GG
B C B C ÐBß CÑ − EB C B C ÐBß CÑ − E
B C " B C ÐBß CÑ − EB C "
"
#
$
,,,
:Ð" l ÑB C ÐBß CÑ − E ÞG ß %
Ð"!Ñ
• Note in Ð"!Ñ -ÐCß B œ -ÐBß CÑ \ µ YÒ!ß "Ó Ê ] µ YÒ!ß "ÓÑ . Hence,
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 31
4. CONSTRUCTION...Joint CDF
______________________________________
• Pdf of GDB copula with TS pdf with generating pdf:ÐDlGÑ À
-ÖBß Cl × œ
"
#‚
" ß" ß
ß
:Ð † l Ñ
:Ð l Ñ :Ð" l Ñ:Ð l Ñ :Ð" l Ñ:Ð l Ñ :Ð" l Ñ:Ð l Ñ
G
G GG GG GG
B C B C ÐBß CÑ − EB C B C ÐBß CÑ − E
B C " B C ÐBß CÑ − EB C "
"
#
$
,,,
:Ð" l ÑB C ÐBß CÑ − E ÞG ß %
• Cdf of GDB copula with TS gen. pdf :ÐDlGÑ and cdf TÐDl ÑG follows as:
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
Ð""Ñ
ÐBß CÑ − E Þ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 32
OUTLINE______________________________________
1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 33
5. GDB EXAMPLES WITH TS GEN. PDF...Triangular PDF
______________________________________
• Substitution of generating pdf :ÐDÑ œ #D with support in yieldsÒ!ß "Ó Ð"!Ñ
-ÐBß CÑ œ # ‚" ß " ßß ß C ÐBß CÑ − E B ÐBß CÑ − E
B ÐBß CÑ − E C ÐBß CÑ − E Þ" #
$ %
, ,,
010
1
0.00
1.00
2.00
A B
1.0
0.5
1.0 0.5 0.02.0 1.5
1.5
2.01.00.5 1.50.0
0.5
1.0
1.5
x
y1A
3A
2A4A
1A
3A
2A
4A
A: Copula density ; B: Density contour plot-ÖBß C× Þ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 34
5. GDB EXAMPLES WITH TS GEN. PDF...Triangular PDF
______________________________________• Substitution of pdf :ÐDÑ œ #D Ð""Ñ in and yields:generating cdf TÐDÑ œ D#
GÖBß C× œ ‚"
$
B $BC 'BC ß
C $B C 'BCß
C $C $CÐB "Ñ $B $B "ß
B $B $BÐC "Ñ $C $C "ß
$ #
$ #
$ # # #
$ # # #
ÐBß CÑ − E
ÐBß CÑ − E
ÐBß CÑ − E
ÐBß CÑ − E
"
#
$
%
,,,
Þ
0
10
1
0.00
0.20
0.40
0.60
0.80
1.00
x
y
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
z
Gen
erat
ing
PD
F of
TS
Fra
emew
ork
Graph of joint triangular copula cdf GÐBß CÑ given aboveÞ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 35
5. GDB EXAMPLES WITH TS GEN. PDF...Slope PDF
______________________________________
:ÐDl Ñ œ #Ð Dß ! Ÿ Ÿ #Þα α# "Ñα α
In figure below: 0.583T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ ¸ ß œ "Þ&Þα
010
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
A B
A: Slope generating pdf; B: GDB Copula with TS Gen. PDF in AÞ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 36
5. GDB EXAMPLES WITH TS GEN. PDF...Power PDF
______________________________________
:ÐDl8Ñ œ 8D ß 8 !Þ8"
In figure below: 0.750, T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ œ 8 œ $Þ
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
Pow
er P
DF
C D
C: Power generating pdf; D: GDB Copula with TS Gen. PDF in AÞ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 37
5. GDB EXAMPLES WITH TS GEN. PDF...Ogive PDF
______________________________________
:ÐDl7Ñ œ Ö# 7 " D 7D ×ß7 !7 #
$7 %( ) . 7 7"
In figure below: 0.750, T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ ¸ 7 œ %Þ*"'Þ
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
Ogi
ve P
DF
E F
E: Ogive generating pdf; F: GDB Copula with TS Gen. PDF in AÞ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 38
5. GDB EXAMPLES WITH TS GEN. PDF...Uniform PDFÒ ß "Ó)
______________________________________
:ÐDl Ñ œ ß Ÿ D Ÿ "ß ! Ÿ Ÿ "ß"
" ) ) )
)
In figure below: 0.750, T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ ¸ œ !Þ&Þ)
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
Uni
form
PD
F
G H
G: Uniform gen pdf; H: Ò ß "Ó Þ) GDB Copula with TS Gen. PDF in AÞ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 39
5. GDB EXAMPLES WITH TS GEN. PDF...Beta PDF
______________________________________
:ÐDl+ß ,Ñ œ B Ð" BÑÐ+ ,Ñ
Ð+Ñ Ð,Ñ
>
> >+" ,", ,+ !ß , !
In figure below: 0. 0, T<Ð] Ÿ !Þ&l\ Ÿ !Þ&Ñ ¸ & + œ , œ &Þ
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
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 40
OUTLINE______________________________________
1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 41
7. SAMPLING PROCEDURE...Archimedean Copula
______________________________________
• - : À Ð!ß "Ó Ä Ò!ß∞Ñ The archimedean copula generator function.
• Let be the cdf of a random variable such thatKÐ † Ñ ^
!
∞=> "KÐ>Ñ/ .> œ Ð=ÑÞ:
In other words, is the Laplace transform of the cdf :"Ð=Ñ KÐ † Ñ
• Clayton Copula: : α" "Î "Ð=Ñ œ Ð" =Ñ ß ß ^ µ K+77+Ð ß "Ñαα0
• Gumbel Copula: : α #" "Î "Ð=Ñ œ /B:Ð = Ñß "ß ^ µ W>+,6/Ð ß "ß ß !Ñßαα
.# œ Ð Ñ cos 1α
α
#
• Frank Copula: : α α ‘" " =Ð=Ñ œ Ò" / Ð/ "ÑÓß − ÏÖ!×ln α , T<Ð^ œ 5Ñ œ Ò Ð" ÑÓ ß œ " / Þ) α5
5" ln ) )
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 42
7. SAMPLING PROCEDURE...Archimedean Copula
______________________________________
Algorithm (Marshall and Olkin, 1988):
Step 1: Sample from a uniform random variable on ,? Y Ò!ß "Ó
Step 2: Sample from a uniform random variable on ,@ Z Ò!ß "Ó
Step 3: Sample from ,D KÐ † l Ñα
Step 4: Evaluate ,? œ • ln?D
Step 5: Evaluate ,@ œ • ln@D
Step 6: B œ ? Ñß:"Ð •
Step 7: .C œ @ Ñ:"Ð •
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 43
OUTLINE______________________________________
1. INTRODUCTION
2. COPULA CONSTRUCTION - ARCHIMEDEAN
3. ARCHIMEDEAN EXAMPLES
4. COPULA CONSTRUCTION - GENERALIZED DIAGONAL BAND
5. GENERALIZED DIAGONAL BAND EXAMPLES
6. SAMPLING PROCEDURE - ARCHIMEDEAN COPULA
7. SAMPLING PROCEDURE - GDB COPULA
8. SELECTED REFERENCES
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 44
7. SAMPLING PROCEDURE...GDB Copula
______________________________________
x
y
1
y = 1
1. Sample x in [0,1]
2. Sample z in [-1,1]
3. y = z + x
4. If y < 0 then y = −y
5. If y > 1 Then y =1−( y−1)
ALGORITHM:
z
x
y = x
y
y
y = −1
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 45
7. SAMPLING PROCEDURE...GDB Copula
______________________________________
• random variable with ^ symmetric Two-Sided (TS) pdf À
0 Dl:Ð † l Ñ× œ ‚"
#
:ÐD "l Ñ " D Ÿ !ß:Ð" Dl Ñ ! D "ß
{ , for , for G
GG Ð&Ñ
where is a generating pdf with support :ÐDÑ Ò!ß "ÓÞ
Step 1: Sample from a uniform random variable on .B \ Ò!ß "Ó
Step 2: Sample from a uniform random variable on .? Y Ò!ß "Ó
Step 3: If then else ? Ÿ D œ T Ð#?Ñ " D œ "T Ð# #"#
" " ?Ñ
Step 4: C œ D B
Step 5: If then C ! C œ C
Step 6: If then C " C œ " ÐC "Ñ
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 46
7. SAMPLING PROCEDURE...GDB Copula
______________________________________
• For the generating densities herein we have for arbitrary quantile level; − Ð!ß "Ñ:
T
ß :ÐDl Ñß Á "ß
; ß :ÐDl8Ñß
; ß :ÐDl7Ñß
Ð" Ñ; ß
"
Ð# Ñ Ð# Ñ %Ð "Ñ;#Ð "Ñ
"Î8
#Ð7"Ñ #Ð7"Ñ7 7 7
#$7%
#ÎÐ7#Ñ
Ð;l Ñ œ<
α α αα
#
α α
) ) :ÐDl Ñß)
• One could favor the power pdf and uniform pdf's due to least number ofoperations.
EMSE 280 - Copula Lecture: Part 1 J.R. van Dorp; www.seas.gwu.edu/~dorpjr - Page 47
8. COPULAE...Selected References
______________________________________
Cooke, R.M. and Waij, R. 1986 Monte carlo sampling for generalized knowledge dependence, withÐ ÑÞ
application to human reliability. , 6 (3), pp. 335-343.Risk AnalysisGenest, C. and Mackay, J. (1986). The joy of copulas, bivariate distributions with uniform marginals. The
American Statistician, 40 (4), pp. 280-283.Ferguson, T.F. (1995). A class of symmetric bivariate uniform distributions. , 36 (1), pp.Statistical Papers
31-40.S Kotz and J R van Dorp (2009), Generalized Diagonal Band Copulas with Two-Sided GeneratingÞ Þ Þ
Densities, , published online before print November 25,Decision AnalysisDOI:10.1287/deca.1090.0162.
Marshall, A W. and I. Olkin (1988) Families of multivariate distributions. JÞ ournal of the American StatisticalAssociation, 83, 834–841.
McNeil, A., R. Frey, and P. Embrechts (2005). .Quantitative Risk Management: Concepts,Techniques and ToolsPrinceton University Press.
Nelsen, R.B. (1999). . Springer, New York.An Introduction to CopulasSklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. 8, Publ. Inst. Statist. Univ. Paris,
pp. 229-231.Van Dorp, J.R and Kotz, S. (2003). Generalizations of two sided power distributions and their
convolution. , 32 (9), pp. 1703 - 1723.Communications in Statistics: Theory and Methods