Bitlis Eren University Journal of Science and Technology 00 (2017) 000–111
Available online at www.dergipark.ulakbim.gov.tr/beuscitech/
Journal of Science and Technology
E-ISSN 2146-7706
Modelling Temperature Measurement Data by Using Copula
Functions
Ayşe METİN KARAKAŞ
Bitlis Eren University, Department of Statistics, TR-13000, Bitlis Turkey
ARTICLE INFO
Article history:
Received 00 December 0000
Received in revised form 00 January 0000
Accepted 00 February 0000
Keywords:
Copula function, Archimedean copula
function, Kendall tau, Spearman rho,
Temperature, Akaike information criteria
ABSTRACT
In this study, methods of copula estimation are used and the temperature measurement data of
the four regions located at the same positions in the range of 01.01.2008 - 30.04.2009 was
modeled with copula functions. For dependence structures of the data sets, it is calculated Kendall
Tau and Spearman Rho values which are nonparametric. Based on this method, parameters of
copula are obtained. A clear advantage of the copula-based model is that it allows for maximumlikelihood estimation using all available data. The main aim of the method is to find the
parameters that make the likelihood functions get its maximum value. With the help of the
maximum-likelihood estimation method, for copula families, it is obtained likelihood values. These
values, Akaike information criteria (AIC) are used to determine which copula supplies the
suitability for the data set.
© 2017. Turkish Journal Park Academic. All rights reserved.
2. Material and Method
1. Introduction
Copulas are functions that link the marginal distributions to
their joint distribution. The notion of copulas is well
understand, it is now known that their empirical estimation is
stronger. In bivariate status, copulas can be used to description
nonparametric measures of dependence for random variables.
Asymmetric models of dependence are suited like those that
exceed correlation and linear association, and then copulas
move a specific role in developing additional status and
measures. Copulas are helpful extensions and generalizations of
approaches for modeling joint distribution and dependence that
have seem in the literature. Succinctly defined copulas are
functions that appropriate multivariate distributions to their
one-dimensional margins. Copulas are considered in different
applications. Especially, these functions are used in the extreme
value theory. While theoretical properties of these objects are
now fairly understood, inferences for copula models are not
extent.
* Corresponding author.
Tel.: 05388394839
E-mail address:[email protected]
2.1. Copula Theory
2
The copula is defined as a C : [0,1]  [0,1] that ensures the
limiting conditions
C  u , 0  C  0, u   0 and

C u,1  C 1, u   u, u  0,1 .
4
(u1, u2 , v1, v2 )  [0,1] , such that, u1  u2 , v1  v2


 
 
 

C u2 , v2  C u2 , v1  C u1 , v2  C u1, v1  0
Ultimately, for twice differentiable and 2-increasing property
can be replaced by the condition
Bitlis Eren University Journal of Science and Technology 00 (2017) 000–111
c (u , v ) 
where
2
 C (u , v )
uv
0
-
uniform random U1 , U 2 , ..., U n variables, the joint distribution
function C is defined



C u1, u2 , , un ,   P U1  u1,U 2  u2 , U n  un .
2.1.1. Sklar Theorem
X
and Y
be random variables with continuous
distribution functions FX
 
and FY , with
 
FX X
FY Y are uniformly distributed on the interval
there is a copula such that for all x, y  R ,
and
[0,1]. Then,
 
function  t  lnt


,   1;


 1
C (u , v )  exp [(  ln u )  (  ln v ) ]
 X , Y  is the joint distribution function for
pair FX  X  , FY Y  provided FX and FY continuous.
[1,2,3,5,6,8,9,10,11,12,13]
This Archimedean copula is defines with the help of generator

t
1
function  (t ) 
,   1,  / 0

 
C (u , v )  (u

v

 1).
(6)
Where  is the copula parameter restricted to (0,  ). This
copula is also asymmetric, but with more weight in the left tail
[10].
This Archimedean copula is defines with the help of generator
 t
e
1
function;  t   ln
,  R / 0 ;

e
1
The normal copula;
1
C (u1, u2 ;  )  G ( (u1 ), (u2 );  )
1(u1) 1(u2 )
1



2 12


2 (1   )

(3)

 
 ( s 2  2 st  t 2 

 dsdt
2
 2(1   ) 
C u , v 

 ln  1 
 

1
where is the cdf of standard distribution, and is the standard
bivariate normal distribution with correlation parameter
limited to the interval (-1,1). [1,12]
2.1.3. Archimedean Copula
and provides:

 (1)  0,  (0)  .

'
For all t  (0,1) ,  (t )  0 , 
e u  1e v  1 
e  1 

(7)

where  is the copula parameter restricted to 0,  [10].
2.1.7. Joe Copula
define a function  : [0,1]  [0,  ] which is continuous
is decreasing, for all
t  (0,1)    t   0 ,  is convex.
 has an inverse 
(5)
2.1.6. Frank Copula
2.1.2. Gaussian Copula
1

2.1.5. Clayton Copula
(2)
The copula C for
Let
(4)
This Archimedean copula is defines with the help of generator

FXY ( X , Y )  C ( F ( x ), F ( y )) .
X
Y
the
[ (u )   (v)].
where  is the copula parameter restricted to (1, ] .This
copula is asymmetric, with more weight in the right tail. Beside
this, it is extreme value copula [10].
Here  is dependence parameter. [1,2,3,5,6,7,8,11,12]
Let
( 1)
2.1.4. Gumbel Copula
c(u, v) is the copula density. In the following, for

C (u , v )  
(1)
This Archimedean copula is defines with the help of generator
function
 
C u , v 
1  1 u





 1 v



 ( 1 u

 1/
 1  v 

   
: 0,   0,1 , which has the same
( 1)
(0)  1 and 
properties out of 
Archimedean Copula is defined by
( 1)
( )  0.
The
 
Where  is the copula parameter restricted to 1,  . This
(8)
Bitlis Eren University Journal of Science and Technology 00 (2017) 000–111
copula family is similar to the Gumbel. The right tail positive
dependence is stronger more than Gumbel [10].
2.2. Dependence Structure of Copulas
and Cn  C as n   .Here the null hypothesis H 0  C  
of independence of X and Y , the distribution of  n is normal
with zero
In this section, we explore ways in which copulas can be used in
the study of dependence or association between random
variables. There are varieties of ways to discuss and to measure
dependence. Dependence properties and measures of
association are interrelated, and so there are many places
where we could begin this study.
2.2.1. Spearman’s Rho
Similar to approach of Pearson correlation coefficient, to
mean
approximate   0.05 ,
n 
n  1 n  z /2  1, 96 [5].
Another measure of dependence is Kendall Tau. This measure
based on ranks given by
P  Qn
n  n

n
 2
 
4
n ( n  1)
( X i  X j )(Yi  Y j )  0 and
n
n 1
 Ri Si  3
.
n ( n  1)( n  1) i 1
n 1
(10)
12  C (u , v ) dudv  3
[0,1]2
these
are
disconcordant
say ( Ri  R j )( Si  S j )  0 .  n is function of copula Cn . As
n   , Cn  C
W
(11)
Also,  n is asymptotically unbiased estimator of
  12  uvdC ( u , v )  3 
[0,1]2
(14)
( X i  X j )(Yi  Y j )  0 . If ( X i  X j )(Yi  Y j )  0 ; we can
write. This coefficient that stated expediently in the form
12
Pn  1
respectively. Here, ( X i , Yi ), ( X j , Y j ) pairs are concordant
where
n 
H0
where Pn and Qn number of concordant and discordant pairs
(9)
1 n
n 1 1 n
R   Ri 
  Si
n i 1
2
n i 1
1 ( n  1) ,thus for
variance
2.2.2. Kendall Tau
compute the correlation between the pairs ( Ri , Si ) of ranks
have been used. Thus, Spearman’s Rho
n
 ( Ri  R )( Si  S )
i 1
 [ 1,1]
n
2 n
2
 ( R  R )  ( Si  S )
i 1 i
i 1
and


1 n
1
 Iij  # j : X j  X i , Y j  Yi ,
n j 1
n
n  4
n
n 1
W
n3
n 1

(15)
4  C (u , v ) dC (u , v )  1
[0,1]2
(12)
written.  n is asymptotically unbiased estimator of  and  n
is normal with zero mean and variance 2(2n  5) {9n( n  1)} .
where the second equality is obtain [5]. This statement
extended Quesada- Molina (1992)
12 n Ri Si
n 1
12  uvdCn (u , v )  3 
3 
n

i

1
n
n 1 n 1
n 1
[0,1]2
(13)
Here the null hypothesis H 0  C   of independence of X
and Y ,
thus
for
H0
9n( n  1) 2(2 n  5)  n  1.96
approximate   0.05 ,
[5].
2.3. Copula Estimation Method
2.3.1. Maximum Likelihood Method (MLE)
Maximum likelihood method is the most used for copula. The
aim of the method is basic to find the parameters that make the
likelihood functions get its maximum value. It is given
Bitlis Eren University Journal of Science and Technology 00 (2017) 000–111
2.3.3. Akaike Information Criteria
f ( x1 , x2 , ..., xn ) 
n
c ( F1 ( x1 ), F2 ( x2 ), ..., Fn ( xn ))  f j ( x j )
j 1
(16)
For the series, to model dependence structure, other selection
criteria are Akaike’s information criterion (AIC) .This;
AIC  2 log L  2k / n
n
c ( F1 ( x1 ), F2 ( x2 ), ..., Fn ( xn )) 

 c ( F1 ( x1 ), F2 ( x2 ), ..., Fn ( xn ))
F1 ( x1 ), F2 ( x2 ), ..., Fn ( xn )

3.1. Data Sets
(17)
Accordingly, the maximum likelihood estimator is
ˆMLE  max l ( )

Here, k is the number of estimated parameter for each model, n
size of sample.
3. Application
T
Let x1t , x2t , ..., xnt
is the sample data matrix; the
t1
likelihood functions can be given
T
l ( )   ln(c ( F1 ( x1t ), F2 ( x2t ), ..., Fn ( xnt ))
t 1
.
T
n
   ln f j ( x jt )
t 1 j 1
(18)
In this study, dependence structure between at the same time
temperature measurements located in similar coordinate
İstanbul, Rome, Baku and Tokyo modeled by copula functions.
The dependence structure between the selected regions is
being done estimated nonparametric method based on using
Kendall tau methods. With the help of this method copulas
family suitable selected data is determined, accordance with
parameters for this family are calculated. In the study, because
of pairwise comparisons  4  correlation analysis table were
 2
 
2.3.2. Inference for marginal (IFM):
obtained. Kendal Tau values are given in the table.1 below.
This method is used to overcome the drawbacks of full
maximum likelihood function. The aim of copula theory is
separate between the univariate margins and the dependence
structure. From equation (18)
T
l ( )   ln(c ( F1 ( x1t , 1 ), F2 ( x2t ,  2 ), ..., Fn ( xnt ,  n ),  )
t 1
(19)
T
n
   ln f j ( x jt ,  j )
t 1 j 1
Table 1. For four areas Kendall Tau (
İstanbul
1
0,675
0,741
0,708
İstanbul
Roma
Bakü
Tokyo
  (1 ,  2 , ...,  n ) and  is vector the
8
Frequency
10
parameters of copula. Accordingly, the fundamental idea of
inference for margins is that it is forecasts the parameters for
marginal distributions and copula separately in two stages.
 ) rank correlation
Roma
0,675
1
0,655
0,636
Bakü
0,741
0,655
1
0,707
Tokyo
0,708
0,636
0,707
1
12
write. In this equation (19) the vector of the parameters for the
univariate marginal
(22)
6
4
2

Estimate the parameters
j
from marginal
0
-2
distributions,

Estimation of the vector of the copula parameters
0
1
2
3
4
5
6
7
İstanbul lowest tempeature
(20)
8
,
used the ˆ  (ˆ1 , ˆ2 , ..., ˆn ) ;
ˆ IFM 
(21)
T
arg max  ln(c ( F1 ( x1t , ˆ1 ), F2 ( x2t , ˆ2 ), ..., Fn ( xnt , ˆn );  )
 t 1
7
6
5
Frequency
T
ˆ j  arg max  ln f j ( x jt ;  j )
t t 1
-1
4
3
2
1
0
-6
-5
-4
-3
-2
-1
0
1
Bakü lowest temperature
2
3
4
5
Bitlis Eren University Journal of Science and Technology 00 (2017) 000–111
10
Table 3. For Clayton family is the parameter
Frequency
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
11
12
İstanbul-Roma
İstanbul-Bakü
İstanbul-Tokyo
Roma-Bakü
Roma-Tokyo
Bakü-Tokyo
Logl
-88,5239
-50,5097
-69,3936
-94,1912
-64,4112
-96,792
𝜃
4,15384
5,72008
4,84931
3,79710
3,49450
4,825939
AIC
177,0519
101,0235
138,7913
188,3865
128,8265
193,5881
Roma lowest temperature
3.4. Kendal Tau and Frank Family Estimation
14
The relationship between Kendall Tau and Frank family;
12
Frequency
10
4

 K ( )  1   1 D1   


8
6
4
2
0
-12
-10
-8
-6
-4
-2
0
2
4
6
8
Tokyo lowest temperature
Table 4.For Frank family is the parameter
Figure 1. Frequency of Istanbul, Baku, Rome and Tokyo lowest
temperature
3.2. Kendal
Estimation
Tau
and
Gumbel
Hougaard
Family
The relationship between Kendall Tau and Gumbel Hougaard
family;
 K ( ) =
Here D is debye function. From this equation given above for
the low temperature Frank family estimations are given in the
following table.
İstanbul-Roma
İstanbul-Bakü
İstanbul-Tokyo
Roma-Bakü
Roma-Tokyo
Bakü-Tokyo
𝜃
10,35253
13,57225
11,78704
9,610621
8,976924
11,73902
Logl
-2313,32
-3005,15
-2635,73
-2165,4
-2034,5
-2631,74
AIC
4626,644
6010,344
5271,464
4330,804
4069,004
5263,484
3.5. Kendal Tau and Joe Family Estimation
 1

The relationship between Kendall Tau and Joe family;
From this equation given above for the low temperature
Gumbel Hougaard family estimations are given in the following
table.
Table 2. For Gumbel Hougaard family is the parameter
İstanbul-Roma
İstanbul-Bakü
İstanbul-Tokyo
Roma-Bakü
𝜃
3,076923
3,861004
3,424658
2,898551
Logl
902,8711
1312,783
1074,603
775,5424
AIC
-1805,74
-2625,56
-2149,2
-1551,08
Roma-Tokyo
Bakü-Tokyo
2,747253
3,412969
723,4774
1063,465
-1446,95
-2126,93
3.3. Kendal Tau and Clayton Family Estimation
 K ( )  1 
4

 
DJ 
Here D is debye function. From this equation given above for
the low temperature Joe family estimations are given in the
following table.
Table 5. For Joe family is the parameter
İstanbul-Roma
İstanbul-Bakü
İstanbul-Tokyo
Roma-Bakü
Roma-Tokyo
Bakü-Tokyo
𝜃
4,958317
6,507019
5,644045
4,607484
4,310508
5,620964
Logl
-83,2145
-95,1592
-105,571
-33,1887
-26,0271
-58,2653
AIC
166,4331
190,3225
211,461
60,38152
52,05832
116,5347
The relationship between Kendall Tau and Clayton family;
 K ( ) =

4. Results and Discussion
 2
from this equation given above for the low temperature Clayton
family estimations are given in the following table.
The studies have focused on copula forecasting methods and
using the daily minimum temperatures of four different areas
are made in applications. In application it was established to
investigate the relationship between the lowest temperatures of
Bitlis Eren University Journal of Science and Technology 00 (2017) 000–111
the four cities in the world in the time interval 01.01.2008 30.04.2009 by using a dataset with 486 units of the
temperatures of the four cities on modeling the concept of
copulas. In practice, it has been making predictions by choosing
different copulas families with non-parametric copulas
estimation method. This study has been shown that the Gumbel
Hougaard, Clayton, Frank and Joe copulas are statistically better
than the other families for our data set. Since the Gumbel
Hougaard and the Clayton families are the Archimedean copula
classes, they provide easiness in the calculation. Gumbel,
Clayton and Gaussian families are more useful in modeling the
structure of the dependency for low temperature
measurements among the regions. Consequently, the following
equalities can be written for each value of the dependency
parameter.
References

[1] Cherubini U, Luciano E (2001). Value-at-Risk Trade-off and
Capital Allocation with Copulas. Economic Notes 30, 235–256.
[2] Frees EW, Valdez EA (1998). Understanding relationships
using copulas. North American Actuarial Journal 2, 1-25.
[3] Genest C, MacKay V (1986). The joy of copulas: bivariate
distributions with uniform marginal. The American Statistician
40, 280-283.
[4] Genest C, Rivest LP (1993). Statistical inference procedures
for bivariate Archimedean copulas. Journal of the American
Statistical Association 88(423), 1034-1043.
[5] Genest, C, Favre AC (2007). Everything You Always Wanted
to Know About Copula Modelling but Were Afraid to Ask.
Journal of Hydrologic Engineering 12, 347-368.
[6] Genest C, Gendron M, Boudeau-Brien M (2009). The Advent
of Copulas in Finance The European Journal of Finance 15, 609618.
[7] Malevergne Y, Sornette D (2003). Testing the Gaussian
Copula Hypothesis for Financial Assets Dependences.
Quantitative Finance 3 (4), 231-250.
[8] Metin A, Çalık S, (2012). Copula Function and Application
with Economic Data. Turkish Journal of Science and Technology
7(2), 199-204.
[9] Naifar N (2010). Modeling dependence structure with
Archimedean copulas and applications to the iTraxx CDS index.
Journal of Computational and Applied Mathematics 235, 24592466.
[10] Nelsen, RB (1999). An Introduction to Copulas. First
edition. Published by the Springer Verlag, New York, ISBN 9780-387-28678-5.
[11] Sklar, A (1959). Fonctions de Repartition a n Dimensions et
Leurs Marges. Publications de I’lnstitut de Statistique de
I’University de Paris 8, 229-231.
[12] Schweitzer B, Wolff EF (1981). On nonparametric
measures of dependence for random variables. Annals of
Statistics 9, 879-885.
[13] Quesade-Molina, J (1992). A generalization of an identity of
Hoeffding and some applications. J.Ital.Stat.Soc. 3, 405-4.