=Paper=
{{Paper
|id=Vol-2177/paper-06-a003
|storemode=property
|title=
Estimation of Heavy Tail Dependence Based on Copulas
for the Precipitation Analysis
|pdfUrl=https://ceur-ws.org/Vol-2177/paper-06-a003.pdf
|volume=Vol-2177
|authors=Leonid A. Sevastianov,Nikita D. Rassakhan,Eugeny Yu. Shchetinin
}}
==
Estimation of Heavy Tail Dependence Based on Copulas
for the Precipitation Analysis
==
https://ceur-ws.org/Vol-2177/paper-06-a003.pdf
40
UDC 519.246.5
Estimation of Heavy Tail Dependence Based on Copulas for the
Precipitation Analysis
Leonid A. Sevastianov* , Nikita D. Rassakhan† , Eugeny Yu. Shchetinin‡
*
Department of Applied Probability and Informatics,
Peoples’ Friendship University of Russia (RUDN University),
6 Miklukho-Maklaya str., Moscow, 117198, Russia
†
Department of Applied Mathematics
Moscow State Technology University “STANKIN”
3a Vadkovsky Ln., Moscow, 127055, Russian Federation
‡
Financial University under the Government of the Russian Federation
Leningradsky pr. 49, 111123, Moscow, Russian Federation
Email: sevastianov_la@rudn.university, rassahan@gmail.com, riviera-molto@mail.ru
Consideration of tail dependence is a very important part of risk analysis in many applied
sciences that is measured in order to estimate the risk of simultaneous extreme events. Usually
the tail dependence coefficient is the measurement in question. Pearson correlation coefficient
unfortunately is not a suitable measure for estimating dependencies between two quantities
in the context of simultaneous occurrence of extreme events when these events are of interest
for the researcher because it takes extreme events into account with the same weight as it
takes “normal” events although dependence of extreme values may slightly differ.
Present work emphasizes the importance of taking into account tail dependencies in bivariate
statistical analysis using copulas. Due to increasing frequency of environmental cataclysms
the issue of analyzing risks (e.g. economic losses) and their consequences comes to the fore.
Moreover, researchers should take into consideration consequences of their joint occurrence.
Three non-parametric estimators of tail dependence coefficients were compared in order to
estimate correlation between daily cumulative rainfall totals recorded in central European
part of Russia. The majority of existing estimators depends on threshold 𝑘 and thus there is a
trade-off between variance and bias during the calculation of the best value for 𝑘. For balancing
an algorithm is presented that is based on using moving average filter and then searching the
“stable” part of tail dependence coefficient. Estimate of tail dependence coefficient is assumed
to be equal to mean value on the “stable” part.
Key words and phrases: extreme value theory, spatial modelling, extreme precipitation,
spatial structures of statistical dependence, tail dependence coefficient.
Copyright © 2018 for the individual papers by the papers’ authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VIII Conference
“Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Moscow, Russia, 20-Apr-2018, published at http://ceur-ws.org
Sevastianov Leonid A., Rassakhan Nikita D., Shchetinin Eugeny Yu. 41
1. Introduction
One of the most important parts of multivariate extreme value analysis is the study of
extremal dependencies [13]; basically tail dependence coefficient is used for this purpose.
For bivariate vector (𝑋1 , 𝑋2 ) upper tail dependence coefficient has the following form [5]:
𝜆𝑈 = lim 𝑃 (𝐹1 (𝑋1 ) > 𝑡|𝐹2 (𝑋2 ) > 𝑡) , 𝑡 → 1− , (1)
where 𝐹1 , 𝐹2 are distribution functions of random variables 𝑋1 , 𝑋2 respectively, 0 < 𝑡 6 1
is the threshold.
Using the copula function [8] equation (1) can be written in alternative form [15, 18]:
1 − 2𝑡 + 𝐶(𝑡, 𝑡)
𝜆𝑈 = lim . (2)
𝑡→1− 1−𝑡
2. Main section
Tail dependence coefficient estimation methods are essential analysis tools for extremal
structures that are studied in this paper on precipitation data. Onward we will describe
some of them. Foremost such estimators are non-parametric estimators based on
empirical copula 𝐶 (𝑛) (𝑢, 𝑣) concept [4], [16] with 𝐹(𝑛) (·) as empirical distribution
function.
(︁ )︁ (︁ )︁
(1) (1) (𝑛) (𝑛)
Let 𝑋1 , 𝑋2 , . . . , 𝑋1 , 𝑋2 be independent identically distributed copies
of bivariate random vector (𝑋1 , 𝑋2 ). Using their joint distribution function [12] and
equation (2) an estimator for upper tail dependence (1) coefficient can be derived [7]:
(︁ )︁
1−𝐶 ˆ 1 − 𝑘,1 − 𝑘
𝑛 𝑛
ˆ
𝜆 𝑆𝐸𝐶 ˆ
≡𝜆 𝑆𝐸𝐶
(𝑘) = 2 − , 1 6 𝑘 < 𝑛. (3)
𝑘
𝑛
Then in respect that log (1 − 𝑡) ∼ −𝑡, 𝑡 ≈ 0 next estimator can be obtained:
(︁ )︁
log 𝐶ˆ 1 − 𝑘,1 − 𝑘
𝑛 𝑛
ˆ 𝐿𝑂𝐺 ≡ 𝜆
𝜆 ˆ 𝐿𝑂𝐺 (𝑘) = 2 − , 1 6 𝑘 < 𝑛. (4)
𝑘
log (1 − 𝑛 )
where 𝐶 ˆ denotes empirical copula.
Note that both estimators depend on choice of threshold 𝑘 and thereafter 𝑘𝑡ℎ order
statistic [19]. It is very important to choose the right value for 𝑘 which is not an easy
task due to the trade-off between variance and bias.
Another estimator for upper tail dependence coefficient is suggested in works [9, 10]:
⎡ ⎧ √︂ ⎫⎤
𝑛
⎪ log (︁ 1 (𝑖) )︁ 𝑙𝑜𝑔 (︁ 1 (𝑖) )︁ ⎪
⎪ ^
𝐹 1 𝑋1 ^
𝐹2 𝑋2
⎪
ˆ 𝐶𝐹 𝐺 = 2 − 2 exp ⎢ 1
⎢ ∑︁ ⎨ ⎬⎥
𝜆 ⎣𝑛 log 1
⎥. (5)
⎩ log max (︁𝐹^1 (︁𝑋 (𝑖) )︁,𝐹^2 (︁𝑋 (𝑖) )︁)︁2 ⎪
⎪ ⎪⎦
𝑖=1 ⎪ ⎭
1 2
Main advantage of this equation is that 𝜆 ˆ doesn’t depend on 𝑘. However, copula
𝐶(𝑋1 , 𝑋2 ) must be well approximated with extreme-value copulas for correctness of the
estimator.
It follows from equations (3), (4) that estimators depend on choice of threshold 𝑘
which is determined by balancing variance and bias for estimator according to stability
theorem for 𝜆𝑈 [8]. Increasing the value of 𝑘 leads to reduction of bias and increase
in variance; it goes the same the other way around. For big enough data sample size
𝑛 balance between bias and variance is described by the “stable” part of 𝜆𝑈 plot. An
algorithm for finding this “stable” part is presented in paper [6]:
42 ITTMM—2018
1. Empirical estimation is smoothed with moving average filter window size of which
is equal(︁to 𝑏 = 𝑖𝑛𝑡(0.05𝑛). ˆ ˆ
)︁ Sequence 𝜆1 , . . . , 𝜆𝑛−2𝑏 is obtained as a result.
2. Vector 𝜆 ˆ𝑘 , . . . , 𝜆
ˆ 𝑘+𝑚−1 , where 𝑘 = 1, . . . , 𝑛 − 2𝑏 + 𝑚 − 1, 𝑚 = 𝑖𝑛𝑡
(︀√ )︀
𝑛 − 2𝑏
can be made from the sequence 𝜆 ˆ1 , . . . , 𝜆
ˆ 𝑛−2𝑏 by a sequential search.
3. If the current vector satisfies
𝑘+𝑚−1
∑︁
¯𝑖 − 𝜆
|𝜆 ¯ 𝑘 | 6 2𝜎,
𝑖=𝑘+1
ˆ1 , . . . , 𝜆
where 𝜎 is standard deviation of 𝜆 ˆ 𝑛−2𝑏 then the final expression for 𝜆𝑈
takes the form of
𝑚
1 ∑︁ ¯
𝜆𝑈 = 𝜆𝑘+𝑖−1 .
𝑚 𝑖=1
If the condition is not satisfied after sequential searching, then 𝜆𝑈 = 0.
Example of algorithm realisation using R language is presented below:
lambda_sec <−
( 1 / k ∗ rank_sum ( msk_rank , spb_rank , l e n g t h ( df$Msk ) , k ) )
lambda_sec2 <− 2 − ( 1 / k ∗ rank_sum2 ( msk_rank , mhzsk_rank ,
l e n g t h ( df$Msk ) , k ) )
rank_sum <− f u n c t i o n ( rank1 , rank2 , l g t h , k ) { a = 0
f o r ( i in 1 : lgth ){
a = a + i f e l s e ( ( rank1 [ i ] > l g t h − k )
& ( rank2 [ i ] > l g t h −k ) , 1 , 0 ) } r e t u r n ( a ) }
rank_sum2 <− f u n c t i o n ( rank1 , rank2 , l g t h , k ) {
a = 0; f o r ( i in 1 : l g t h ){
a = a + i f e l s e ( ( rank1 [ i ] > l g t h − k )
| ( rank2 [ i ] > l g t h −k ) , 1 , 0 )
} return (a) }
b = t r u n c ( 0 . 0 5 ∗ l e n g t h ( df$Msk ) )
m = t r u n c ( s q r t ( l e n g t h ( df$Msk −2∗b ) ) )
wow <− 0 ; wow2 <−0; sssuum <− 0
f o r ( i i n 1 : ( l e n g t h ( ls_ma_na)−2∗b+m−1)){
rw <− ls_ma_na [ i : ( i+m) ]
f o r ( l i n 1 :m) sssuum <− sssuum + abs ( rw [ l ]−rw [ 1 ] )
i f ( sd ( rw)>= sssuum /m) {wow <− i
wow2 <− mean ( rw )
b r e a k } sssuum <− 0}
In this study the precipitation data of the All-Russian Research Institute of Hydrom-
eteorological Information — the World Data Center of the Russian Federation is used,
which contains data on daily precipitation in 11 cities of the European part of Russia [11].
The data is freely available on the website http://aisori.meteo.ru/ClimateR and is
represented by a set of tables (a separate table for each city); each table contains daily
rainfall value for the period 1966–2016 years.
Implementation of this algorithm is presented in Fig. 1 [2]. Both plots are using
monthly maximum of precipitation in Moscow and Kostroma to evaluate upper tail
dependence coefficient using estimators 𝜆 ˆ 𝑆𝐸𝐶 (left) and 𝜆 ˆ 𝐿𝑂𝐺 (right). Black line
Sevastianov Leonid A., Rassakhan Nikita D., Shchetinin Eugeny Yu. 43
ˆ
corresponds to 𝜆(𝑘); blue smooth line is 𝜆(𝑘) ˆ after applying moving average filter to
it. Pink transparent plateau is the resulting value for 𝜆 ˆ 𝑈 where placement of plateu
ˆ𝑘 , . . . , 𝜆
corresponds to indexes of vector 𝜆 ˆ 𝑘+𝑚−1 from the algorithm above.
Figure 1. Implementation of the algorithm for finding “stable” part of TDC for
𝜆^ 𝑆𝐸𝐶 (left) and 𝜆
^ 𝐿𝑂𝐺 (right). Moscow and Kostorma were used as a pair of
cities from the area under study
All three estimators (3), (4), (5) for upper tail dependence coefficient were calculated
for 55 pairs of 11 cities under study [1, 3]. Furthermore, Pearson’s correlation coefficient
(PCC) was also calculated in order to compare it with estimators. Results for some of
the pairs are represented by Table 1. As we can see, PCC is quite different from all
other estimators in some cases (the Sp. Petersburg — N. Novgorod pair as an example)
but it actually is close enough to at least one of the estimators for the most pairs.
Table 1
Values for estimators of 𝜆𝑈 and Pearson’s correlation coefficient calculated
for some pairs of cities in the European part of Russia
Pair of cities ^ 𝑆𝐸𝐶
𝜆 ^ 𝐿𝑂𝐺
𝜆 ^ 𝐶𝐹 𝐺
𝜆 PCC
Moscow – Kolomna 0.6417793 0.5358919 0.5210492 0.6021248
Kolomna – Ryazan 0.5561837 0.4623016 0.5022938 0.5509277
Pskov – Smolensk 0.54967 0.437089 0.3722651 0.4314
Kostroma – N.Novgorod 0.6346736 0.415878 0.3856968 0.4766541
Bryansk – Mozhaisk 0.6846937 0.1764084 0.433269 0.6021248
St. Petersburg – N.Novgorod 0.1470922 0.0770384 0.2608262 0.3353886
Smolensk – Moscow 0.488586 0.3740484 0.3968146 0.3952423
St. Petersburg – Pskov 0.5338348 0.4111191 0.4124836 0.4867774
N.Novgorod – Tambov 0.2271567 0.1740808 0.3475754 0.3898402
Pskov – Kostroma 0.5107715 0.4133031 0.3863028 0.5022300
44 ITTMM—2018
Finally the attempt to find the correlation between estimators for upper tail depen-
dence and the distance between cities under study was made [14]. Scatterplots for four
values compared in Table 1 were plotted as a result; they are presented in Fig. 2.
Figure 2. Comparison of 4 estimators and their dependence on the distance ℎ
^ 𝑆𝐸𝐶 — upper left, 𝜆
between observation points: 𝜆 ^ 𝐿𝑂𝐺 — upper right, 𝜆
^ 𝐶𝐹 𝐺 —
lower left, PCC — lower right
It is obvious that there must be an inverse relation between the distance ℎ and
dependence estimators. Therefore 𝜆 ˆ 𝑆𝐸𝐶 is a bad estimator for the problem under study.
Three other estimators (𝜆 ˆ 𝐿𝑂𝐺 , 𝜆
ˆ 𝐶𝐹 𝐺 and PCC) show roughly the same with some
correction, which is why they are considered to be more trustworthy. So it is proposed
ˆ 𝐿𝑂𝐺 and 𝜆
to take the average of 𝜆 ˆ 𝐶𝐹 𝐺 or just 𝜆
ˆ 𝐶𝐹 𝐺 as the resulting estimator for 𝜆𝑈 .
3. Conclusions
This paper highlights the importance of taking into account the tail dependence
coefficient in the context of multivariate frequency analysis using copulas. The three
following nonparametric estimators(𝜆ˆ 𝑆𝐸𝐶 , 𝜆
ˆ 𝐿𝑂𝐺 , 𝜆
ˆ 𝐶𝐹 𝐺 ) have been compared. The aim
of this comparison was to choose the best estimator in the context of our application [20].
No estimator works in every case yet some of them show poor performance thus they
Sevastianov Leonid A., Rassakhan Nikita D., Shchetinin Eugeny Yu. 45
need to be excluded. It is therefore important to pursue research in this field to get the
right estimation for 𝜆𝑈 based on values of 𝜆 ˆ 𝑆𝐸𝐶 , 𝜆
ˆ 𝐿𝑂𝐺 and 𝜆
ˆ 𝐶𝐹 𝐺 .
Most non-parametric estimators have to deal with the choice of the number 𝑘 of
order statistics to be considered in the production of an estimate. This is not an easy
task since it requires a trade-off between variance and bias (small values of 𝑘 cause large
variance and large values of 𝑘 increase the bias).
Frahm et al. [6] introduced a simple algorithm to find the optimal threshold 𝑘 in
order to estimate 𝜆𝑈 . Since this very simple algorithm revealed some potential, we
intend to develop this idea further. 𝜆 ˆ 𝐶𝐹 𝐺 is considered to be the best estimator out of
ˆ 𝑆𝐸𝐶 , 𝜆
𝜆 ˆ 𝐿𝑂𝐺 and 𝜆 ˆ 𝐶𝐹 𝐺 , it looks like PCC with some corrections.
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