directional dependence using copulasmnstats.morris.umn.edu/molly olson/directional dependence... ·...
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Directional Dependence using Copulas Molly Olson
Faculty Mentor: Engin Sungur
Objectives
Understanding the ideas of directional dependence, copulas, and spatial statistics
Learn how to develop and the developing of new models
Ultimately, apply to data
Directional Dependence
Causal relationships can only be set through experiments
Not “direction of dependence”
Unemployment and education level
Depends on the ‘way’ it’s looked at or introduced Not limited to positive and negative dependence
Unemployment Education Level
Background and basic definitions:
Directional Dependence
To better understand
Background and basic definitions:
http://assets.byways.org/asset_files/000/008/010/JK-view-up-tree.jpg?1258528818
http://www.upwithtrees.org/wp-content/uploads/2013/07/trees-566.jpg
Copulas
( )kXXX ,,, 21 … ( ) ( ) ( )kXXX xFxFxFk
,,, 21 21…
( )kXXX xxxFk
,,, 21,,, 21……
( ) ( ) ( )kXkXX xFUXFUXFUk
=== ,,, 2211 21…
( )kUUU ,,, 21 … ( )kuuuC ,,, 21 …
Background and basic definitions:
Copulas
Multivariate functions with uniform marginals
Eliminate influence of marginals
Used to describe the dependence between R.V.
Background and basic definitions:
Directional Dependence and Copulas
Symmetric. Example: Farlie-Gumbel-Morgenstern’s family
Cθ(u, v) = uv + θuv(1 － u)(1 － v)
If then we say the pair (U,V) is directionally dependent in joint behavior
Background and basic definitions:
Spatial Statistics
2012, Orth
Now
Background and basic definitions:
PVW (! ) = cos(! )V + sin(! )WCor(U,PVW (! ))
P2VW (!) = !sin(!)V + cos(!)WCor(PVW
1 (!),P2VW ("))
P1VW (!) = cos(!)V + sin(!)W
Procedure
Start with U1, U2, U3, U4
Uniform distribution
http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29
Procedure
Histogram of U4
U4
Frequency
0.0 0.2 0.4 0.6 0.8 1.0
05
1015
2025
30
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Independence between U1 and U2
U1
U2
Procedure
cos!1 sin!1 0 0!sin"1 cos"1 0 00 0 1 00 0 0 1
"
#
$$$$$
%
&
'''''
U1
U2
U3
U4
"
#
$$$$$
%
&
'''''
=
P(1,2)+
P(1,2)!
U3
U4
"
#
$$$$$
%
&
''''',
,
1 0 0 00 cos!2 sin!2 00 !sin"2 cos"2 00 0 0 1
"
#
$$$$$
%
&
'''''
P(1,2)+
P(1,2)!
U3
U4
"
#
$$$$$
%
&
'''''
=
P(1,2)+
P(1,2),3!+
P(1,2),3!!
U4
"
#
$$$$$$
%
&
''''''
,
1 0 0 00 1 0 00 0 cos!3 sin!30 0 !sin"3 cos"3
"
#
$$$$$
%
&
'''''
P(1,2)+
P(1,2),3!+
P(1,2),3!!
U4
"
#
$$$$$$
%
&
''''''
=
P(1,2)+
P(1,2),3!+
P[(1,2),3],4!!+
P[(1,2),3],4!!!
"
#
$$$$$$
%
&
''''''
Procedure Independence between U1 and U2
P(1,2)+
!1 =10!,"1 = 25
!,!2 = 85!,"2 =15
!,!3 = 81!,"3 =17
!
P(1,2)+
P(1,2),3!+
P[(1,2),3],4!!+
P[(1,2),3],4!!!
!1 = 81!,"1 =18
!,!2 = 33!,"2 = 73
!
P(1,2),3!+
P(1,2),3!!
U4
P(1,2)+ vs P(1,2)
1 with !1 = 80!,"1 = 20
!
U1
Procedure
Process
Find exact correlations
We know correlations between Ui’s are 0
Result
2012
One angle
45 90 135 180 225 270 315 360Θ
�1
�0.5
0.5
1
ΡDU DV �Θ�
0
15 °
30 °
45 °
60 °75 °90 °105 °
120 °
135 °
150 °
165 °
180 °
195 °
210 °
225 °
240 °255 ° 270 ° 285 °
300 °
315 °
330 °
345 °
�1.0 �0.5 0.0 0.5 1.0
�1.0
�0.5
0.0
0.5
1.0
ΡDU DV �Θ�
Result Theorem 1. Suppose that (Ui,Uj) are independent uniform random variables on the interval (0,1).
Define P1=cos(α)Ui+sin(α)Uj and P2=-sin(β)Ui+cos(β)Uj. Then,
(1)
(2) Cor(α,β)=-Cor(β,α)
(3) If α=β=θ, then (P1,P2) are independent.
Sin(β-α) Cov P(!,")( ) =
1/12 (1/12)sin(! !")(1 /12)sin(! !") 1 /12
"
#$$
%
&''
Cor P(!,")( ) =1 sin(! !")
sin(! !") 1
"
#$$
%
&''
Sin(α-β)
Result Theorem 2. Suppose that (U1,U2,U3,U4) are independent uniform random variables on the interval (0,1). Define
cos!1 sin!1 0 0!sin"1 cos"1 0 00 0 1 00 0 0 1
"
#
$$$$$
%
&
'''''
U1
U2
U3
U4
"
#
$$$$$
%
&
'''''
=
P(1,2)+
P(1,2)!
U3
U4
"
#
$$$$$$
%
&
''''''
1 0 0 00 cos!2 sin!2 00 !sin"2 cos"2 00 0 0 1
"
#
$$$$$
%
&
'''''
P(1,2)+
P(1,2)!
U3
U4
"
#
$$$$$$
%
&
''''''
=
P(1,2)+
P(1,2),3!+
P(1,2),3!!
U4
"
#
$$$$$$
%
&
''''''
1 0 0 00 1 0 00 0 cos!3 sin!30 0 !sin"3 cos"3
"
#
$$$$$
%
&
'''''
P(1,2)+
P(1,2),3!+
P(1,2),3!!
U4
"
#
$$$$$$
%
&
''''''
=
P(1,2)+
P(1,2),3!+
P[(1,2),3],4!!+
P[(1,2),3],4!!!
"
#
$$$$$$
%
&
''''''
Then Cov P(1,2)+ ,P(1,2),3
!+( ) =1/12 (1/12)cos(!2 )sin(!1 !"1)
(1 /12)cos(!2 )sin(!1 !"1) 1 /12
"
#$$
%
&''
Cor P(1,2)+ ,P(1,2),3
!+( ) =1 cos(!2 )sin(!1 !"1)
cos(!2 )sin(!1 !"1) 1
"
#$$
%
&''
Cov P(1,2)+ ,P[(1,2),3],4
!!+( ) =1/12 !(1 /12)cos(!3)sin(!1 !"1)sin("2 )
!(1 /12)cos(!3)sin(!1 !"1)sin("2 ) 1 /12
"
#$$
%
&''
Cor P(1,2)+ ,P[(1,2),3],4
!!+( ) =1 !cos(!3)sin(!1 !"1)sin("2 )
!cos(!3)sin(!1 !"1)sin("2 ) 1
"
#$$
%
&''
Cov P(1,2)+ ,P[(1,2),3],4
!!!( ) =1/12 (1/12)sin(!3)sin("1 !!1)sin(!2 )
(1 /12)sin(!3)sin("1 !!1)sin(!2 ) 1 /12
"
#$$
%
&''
Cor P(1,2)+ ,P[(1,2),3],4
!!!( ) =1 sin(!3)sin("1 !!1)sin(!2 )
sin(!3)sin("1 !!1)sin(!2 ) 1
"
#$$
%
&''
Cov P(1,2),3!+ ,P[(1,2),3],4
!!+( ) =1/12 (1/12)cos(!3)sin(!2 !"2 )
(1 /12)cos(!3)sin(!2 !"2 ) 1 /12
"
#$$
%
&''
Cor P(1,2),3!+ ,P[(1,2),3],4
!!+( ) =1 cos(!3)sin(!2 !"2 )
cos(!3)sin(!2 !"2 ) 1
"
#$$
%
&''
Cov P(1,2),3!+ ,P[(1,2),3],4
!!!( ) =1/12 !(1 /12)sin(!3)sin("2 !!2 )
!(1 /12)sin(!3)sin("2 !!2 ) 1 /12
"
#$$
%
&''
Cor P(1,2),3!+ ,P[(1,2),3],4
!!!( ) =1 !sin(!3)sin("2 !!2 )
!sin(!3)sin("2 !!2 ) 1
"
#$$
%
&''
Cov P[(1,2),3],4!!+ ,P[(1,2),3],4
!!!( ) =1/12 (1/12)sin(!3 !"3)
(1 /12)sin(!3 !"3) 1 /12
"
#$$
%
&''
Cor P[(1,2),3],4!!+ ,P[(1,2),3],4
!!!( ) =1 sin(!3 !"3)
sin(!3 !"3) 1
"
#$$
%
&''
(2)
(1)
(3)
(4)
(5)
(6)
Result Then, a matrix plot of where
P(1,2)+ ,P(1,2),3
!+ ,P[(1,2),3],4!!+ ,P[(1,2),3],4
!!!
P(1,2)+
P(1,2),3!+
P[(1,2),3],4!!+
P[(1,2),3],4!!!
!1 =10!,"1 = 25
!,!2 = 85!,"2 =15
!,!3 = 81!,"3 =17
!
Result
Not uniform anymore
Rank
Simulation
Motion Chart
Future Research
Explore properties up to ‘n’ variables
Find a copula with directional dependence property
Apply these methods to a data set
References [1] Nelsen, R. B., An Introduction to copulas (2nd edn.), New York: Springer, 2006
[3] Sungur, E.A., Orth, J.M., Understanding Directional Dependence through Angular Correlations, 2011
[2] Sungur, E.A., Orth, J.M., Constructing a New Class of Copulas with Directional Dependence Property by Using Oblique Jacobi Transformations, 2012