Statistical analysis of extreme events in a nonstationary context via a Bayesian framework. Case study with peak-over-threshold data

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HAL Id: hal-00452224

https://hal.archives-ouvertes.fr/hal-00452224

Submitted on 1 Feb 2010

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Statistical analysis of extreme events in a nonstationary context via a Bayesian framework. Case study with

peak-over-threshold data

Benjamin Renard, M. Lang, P. Bois

To cite this version:

Benjamin Renard, M. Lang, P. Bois. Statistical analysis of extreme events in a nonstationary context via a Bayesian framework. Case study with peak-over-threshold data. Stochastic Environmental Re- search and Risk Assessment, Springer Verlag (Germany), 2006, 21 (2), p. 97 - p. 112. �10.1007/s00477- 006-0047-4�. �hal-00452224�

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Author-produced version of the article published in Stochastic environmental research and risk assessment, 2006, 21 (2), 97-112.

The original publication is available at http://www.springerlink.com/

doi : 10.1007/s00477-006-0047-4

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Statistical analysis of extreme events in a non- stationary context via a Bayesian framework. Case study with peak-over-threshold data.

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1. Introduction

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2. An overview of the Bayesian approach

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3. Models for extreme values analysis

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/ : / +%10&7 ,&5! ! 9! ,& 0& !6!1785$ ,7 ≤ <= 10%17#!50+05, 0%&7 ,1! $!

+%""%:0&6'

!+%1! 5$,&6! +%""%: 0&2!#!&2!& 6,//,207 1098 0%&7 + !15$,&6! ₌ +%""%: 0&2!#!&2!& 6,//,207 1098 0%&7

=

= −

778/0&60&2!#!&2!&5! 9! :!!& $! 5$,&6! #%0& τ^{,&2 $!} ^#10^%120^{7 10}^{98 0}^%&0^{7 $!&'}

L L

L ₌ L₌ ₌ L L

0 0

π

− − − π τ

=

×

+&%#10%1;&%:"!26! ,9%8 $! !-07 !&5! %+,5$,&6! 07,<,0",9"! $! 7,/! 7#!50+05, 0%&7 +%1?8,& 0"!7:0 $7,/! #1%9,90"0 49!+%1! ,&2,+ !15$,&6! 5,&9! 5$%7!&'

/0& =

= ∀ = − *

+, 1!&2 07,778/!2 0& $! #,1,/! !17 < "! _/> 9! :% 0/! #%0& 7 ,&2 ≤ <= 10%17#!50+05, 0%&7,1! $!&'

+%""%: 0&2!#!&2!& 6,//,207 1098 0%&7 +%""%: 0&2!#!&2!& 6,//,207 1098 0%&7

=

= −

=

$! #10%1207 1098 0%&5,& $879! :10 !&'

L L

L ₌ L₌ ₌ L L

0 0

π

− − −

=

×

D 7#1!<0%87"4 !-#",0&!2 0& $! </ 5,7! !?8, 0%& * 5,& 9! 87!2 % 5%/#"! ! $! #10%1 7#!50+05, 0%&7

",7 $! ,##1%,5$ #1%#%7!2 94 %"!7,&2 ,:& DDF :,7!-#%7!2 $!1! 9!5,87!

1!60%&,"%1!/#0105,";&%:"!26! 07%+ !& !-#1!77!2 0& !1/7%+?8,& 0"!7<,"8!7 1, $!1 $,&

#,1,/! !17<,"8!7 %:!<!1 ," !1&, 0<! #10%1207 1098 0%&75,& 9! 87!2 :0 $%8 ,220 0%&,"

20++058" 0!7:0 $0& $! #1!7!& !2+1,/!:%1;

(8)

5. Computations on posterior distributions

$! /! $%272!75109!20&7!5 0%&7 ,&2(,""%: ,4!7 $!%1!/ %9! 87!2 %5%/#8 ! $!

#%7 !10%1207 1098 0%&12 ; 3 %:!<!1 $! 0& !61,"%+ $! 2!&%/0&, %1%+!?8, 0%& 07 6!&!1,""40& 1,5 ,9"! 12 ; 307 $87%&"4;&%:&8# %,5%&7 ,& %+#1%#%1 0%&,"0 4'

L L

1 ∝ 1 π )

%1!%<!1 $! $06$ 20/!&70%&,"0 4 %+12 ; 3 /,;!7 $07 +8&5 0%& 20++058" % $,&2"! 0&

#1,5 05! !- 7!5 0%&7#1!7!& 7%/! /! $%27 %%<!15%/! $0721,:9,5; !,2!170& !1!7 !2 0&+81 $!1!-#"%1, 0%&%+#%7 !10%1207 1098 0%&7 $,& $%7! 2!75109!29!"%: 5,&1!+!1 % ,&&!1

DDF

5.1. Parameters estimation via marginal densities

$!12 ; 3 ? /,160&,"072!+0&!294'

L L

? = ? ? =

1θ = 1θ θ θ θ ₋ θ ₊ θ

&5! ,6,0& $070& !61, 0%& 07%+ !&0& 1,5 ,9"! % ,&,"4 05,"!-#1!770%&%+ $! #%7 !10%1 /,160&,"07 $!& ,<,0",9"! & ," !1&, 0<! 07 % 70/8", ! , 7,/#"! +1%/ $! I%0& #%7 !10%1 207 1098 0%&12 ; 3 :%/,0&/! $%275,&9! 87!2+%1 $07#81#%7!' $! ! 1%#%"07 ,7 0&67 ,"6%10 $/ ! 1%#%"07 ,&2 ",/ D(DA ! 1%#%"07 ! ," D. A ,7 0&67 D ) ,&2 $!

610224 0997 7,/#"0&6 0 !1 ,&2 ,&&!1 DD $! ", !1 07 5%&5!# 8,""4 !,70!1 % 8&2!17 ,&2 ,&2 07 $87#1!7!& !2 0& 2! ,0"0& &&!- ," $%86$ $! +%1/!1/! $%2 /,49!

/%1! !++050!& 0&7%/! 70 8, 0%&7

0<!& , 7,/#"! %+@ <!5 %17 ₌ _@ = θ θ₌ ₌ _@ , 7,/#"! +1%/ $! ? /,160&,"07 70/#"4 %9 ,0&!2 94 5%&702!10&6 θ_? ₌ _@ ^{7 0}^{/, 0}%& 5,& $!& 9! /,2! 94 /!,&7 %+ $07 7,/#"!' $! /!,& %1 $! /!20,& 60<!7 , #%0& !7 0/, ! ,&2 , 5%&+02!&5!

0& !1<,"5,&9! 5,"58", !2945%/#8 0&6"%:!1,&28##!1!/#0105,"?8,& 0"!7 % 05! $, $!

<,"8!7 θ_? ₌ _@ ,1! &% 0&2!#!&2!& ,",16! 7,/#"! 70=! 07 $871!?801!2 %,<%02,&490,7 28! %,8 %5%11!", 0%& 7!&70 0<0 4,&,"47075%8"29! &!5!77,14 %2! !1/0&! ,510 05,"7,/#"! 70=! ,9%<! :$05$!7 0/, !77 ,47 ,9"!

5.2. Detection of changes

J ! ,##"0!2 $! /! $%2%"%642!75109!294 !11!,8" , ," )))9 9,7!2%& ,4!7+,5 %17 E,77 ,&2 ,+ !14 DD. % 2! !5 5$,&6!7 0& !&<01%&/!& ,"7!10!7 ! 87 2!+0&! $!

/,160&,"207 1098 0%&%+ ,7+%""%:'

L L

1 < = 1 <

70/#"! 5,"58", 0%&7$%:7 $, 12 ;<3 07 $! 2!&%/0&, %1%+ $! ,4!7 $!%1!/ ,##"0!2 %,#,1 058",1/%2!"<'

L L L

1 < = 1 < π <

77 , !2 #1!<0%87"4 $07?8,& 0 45,& 6!&!1,""4&% 9! !-#"050 "45%/#8 !2 $09 DD.

#1%#%7!2 , 70/#"! /! $%2 +%1 !7 0/, 0&6 $! /,160&,"207 1098 0%& %+ 9,7!2 %& 610224 0997 7,/#"0&6 2! ,0"!2 2!7510# 0%& %+ $07 ,##1%,5$ 07 /,2! 0& &&!- !

{

^< ^<

}

^$

Ω = = Ω ∪ Ω :$!1! $! !-#%&!& $ 2!&% !7 $! 5%/#"!/!& %+,7! 9! ,

(9)

#,1 0 0%& %+ $! 5%""!5 0%& %+ /%2!"7 5%&702!1!2 7 ,& !-,/#"!

{

^<⁾^!-# ^<^!-# ^<^!-#^<⁾¹ ^< ¹ ^< ¹

} {

^<⁾^!-# ^<⁾¹

} {

^<^!-#^< ¹ ^<^!-# ^< ¹

}

Ω = = ∪ 5%8"2 9!

,#,1 0 0%& +%1 2, ,/%2!"7 %1! 6!&!1,""4 $! #,1 0 0%&_{Ω ∪ Ω}^$ 5,& ,;! $! +%1/

G7 , 0%&,14 <7 &%& 7 , 0%&,14 /%2!"7H %1 G #,1,/! !17 <7 #,1,/! !17 /%2!"7H 778/! $, O 07 , #10%1 #1%9,90"0 4 207 1098 0%& %&Ω $! #%7 !10%1 #1%9,90"0 4 %+, /%2!"<?5,& $!& 9!

5%/#8 !2 :0 $ $! ,4!7 $!%1!/'

L L

L

? ?

?

1 < <

1 <

1 < <

π π

=

= (

$! ,4!7 +,5 %1 9! :!!& /%2!"7< ,&2<?07 $!& 2!+0&!2 94' L

L

L L

?

? ? ?

1 <

1 < <

+ 1 < < 1 <

π

= π = .

5%/#%70 ! ,4!7 +,5 %1 9! :!!& Ω ! Ω^$ 5,& ,"7% 9! 5%/#8 !2' L

$ L

$ $

< <

1 < <

+ 1 < <

π π

∈Ω ∈Ω

Ω Ω

∈Ω ∈Ω

= F

$! ,4!7 +,5 %1 #1%<02!7 , /!,781!/!& %+ $! #!1 0&!&5! %+, $4#% $!707 5%/#,1!2 % ,&% $!1 5,& $87 9! 5%/#,1!2 % 5",7705," "0;!"0$%%2 1, 0%7 %:!<!1 /%2!" "0;!"0$%%2 07

&% 5%/#8 !2 94 1!#",50&6 $! #,1,/! !17 94 $!01 !7 0/, !7 98 94 0& !61, 0&6 %& $!

#10%1 207 1098 0%& %1! 6!&!1,""4 $07 ,##1%,5$ 07 &% , 5",7705," !7 #1%5!281! $! ,4!7 +,5 %1 <,"8! 2!#!&27 %& #10%1 7#!50+05, 0%&7 0& ,220 0%& % 7,/#"0&6<,10,&5! $87 0 5,&&%

9! 0& !1#1! !2 ,7 , 107; 0& !1/7 %+"0/0 !11%1 #!15!& ,6! :$!& 1!7,/#"0&6 70=! !&27 % 0&+0&0 4 98 1, $!1 0& !1/7 %+ , +,"7! 2!5070%& 5%7 7!! !11!,8" ))) +%1 +81 $!1

!-#",&, 0%&7 ",7705,""4 $! :% !11%1 4#!7 ,1! ,778/!2 % $,<! $! 7,/! 5%7 $! ,4!7 +,5 %1 07 $!& 5%/#,1!2 % :0 $ 61!, !1 5!1 ,0& 4 +%1 $06$ %1 &!,1 =!1% <,"8!7 E,77 ,&2

,+ !14 DD.

5.3. Frequency analysis.

!5 0%& . #1%<02!7 , :,4 %+70/8", 0&6, 7,/#"! +%1 $!12; 3 207 1098 0%& :$05$ 1!", !7 % !7 0/, 0%& 8&5!1 ,0& 4 !5 0%& . 2!75109!7 $! 5%/#8 0&6%+#%7 !10%1 #1%9,90"0 0!7 %+ $!

/%2!"7',7 , 5%""!5 0%& %+/%2!"7 5,&&% 9! 5%&702!1!2 % ,""4 !-$,87 0<! , #,1 %+ $!

/%2!""0&6 8&5!1 ,0& 4 07 $87 5%& 1%""!2 $!7! :% ;0&27 %+8&5!1 ,0& 4 5,& 9! ,;!& 0& % ,55%8& 0& +1!?8!&54 ,&,"4707 94 $! +%""%:0&6#1%5!281!'

!#!, @ 0/!7'

,&2%/ 5$%05! %+ , /%2!"<^A :0 $ #1%9,90"0 0!712</; 3: 12<=; 3 ,&2%/ 5$%05! %+ , #,1,/! !1<!5 %1 +1%/ $! 7,/#"! %+12 <^A3

%/#8 , 0%& %+ 123 $!1 ?8,& 0"! 0& /%2!"<^A:0 $ #,1,/! !1 , 0/!

$! ?8,& 0"!75%""!5 0%& %9 ,0&!2 5,& 9! 5%&702!1!2 ,7, 7,/#"! %+ $! ?8,& 0"! 207 1098 0%&

, 0/! 0&5"820&6 !7 0/, 0%& ,&2 /%2!""0&6 8&5!1 ,0& 0!7 5,& $879! 87!2 % #1%<02! ,

#%0& !7 0/, ! , 5%&+02!&5! 0& !1<," %1 !<!& , 207 1098 0%& !7 0/, ! +%1 123 + 2, ,

(10)

7 , 0%&,104 07 ?8!7 0%&,9"! $! +8&5 0%& ₁ #1%<02!7 , 2!7510# 0%& %+ $! #%7709"! 5$,&6!0& !1/7%+?8,& 0"!210+ :0$ 0/!

6. A particular case: the exponential distribution

# % $07 #%0& $! /! $%27 87!2 0& $07 #,#!1$,<! 9!!& 2!75109!2 ,7 6!&!1,""4 ,7

#%7709"! $!+1,/!:%1;07 $877 0"",##"05,9"!:0$% $!1/%2!"7 !<!1 $!"!77 207 1098 0%&7 9!"%&60&6 % $!!-#%&!& 0,"+,/0"4 5,& 9! 1!, !2 0& ,70/#"!1:,4 :0$ ,&,"4 05,"1!78"7 9!0&6 ,<,0",9"! :0$%8 870&6 /! $%27 & $! &!- 7!5 0%&7 #%7 !10%1 /,160&,"

207 1098 0%&7 ,&2 $! /,160&,"207 1098 0%& %+%97!1<, 0%&7 ,1! 5%/#8 !2 +%1 !-#%&!& 0,"

/%2!"7 "!,20&6 % 7897 ,& 0,"70/#"0+05, 0%&7 % 05! $, ,1!& ,&2 !1&0!1 )) ,"7%

7855!!2!2 0& 2!10<0&6 , +1!!/! $%2 0& $!5,7!%+,7 , 0%&,14 6!&!1,"0=!2 ,1! % 207 1098 0%&

6.1. Stationary model

$!"0;!"0$%%2+%1<₎^!-#/%2!"07'

L L !-#

1 1 % σ

λ λ

=

∏

= − −

>0!:!2 ,7 , +8&5 0%& %+λ ^$0^{7 "}⁰^;!"⁰^{$%%2 0}^{7 #1%#%1}⁰%&," % ,& 0&<!17! 6,//, #2+

B02 ; 3'

{ }⁾

L !-#

B0 = ₊ − _>

Γ *

$07#,1 058",10407;&%:&,7 $!5%&I86,54#1%#!1 4 $87 0+O2λ³ ⁰^778##%7!2^%+^%"^"^%:

,&0&<!17!6,//,207 1098 0%&:0$#,1,/! !17 ,&2 $!#%7 !10%1207 1098 0%&%+λ⁰^7'

L L

!-#

1 λ π λ 1 λ

σ λ λ

λ

+ +

∝

+ −

= −

Γ

=

D

# % #1%#%1 0%&,"04 λ ^#%7^!10^%120⁷¹⁰⁹⁸⁰^%&0⁷⁷⁰^"",&0&<!17!6,//,207 1098 0%& :0$ 8#2, !2#,1,/! !17'

σ

′ = +

′ = + − )

$!/,160&,"207 1098 0%&%+%97!1<, 0%&707+0&,""470/#"4%9 ,0&!2940& !61, 0&62λ^3'

!-#

L )

!-#

1 < λ λ

λ λ λ ^′+

′

=

= − ′

Γ Γ ′

= Γ ′

(11)

6.2. Step change model

@%""%:0&6 $! ,9"! &% , 0%&7 $!<^!-# "0;!"0$%%207'

L !-# % !-# %

1

τ τ τ

τ

σ σ

λ λ λ λ

−

= = +

− −

= − −

@%1 +0-!2 τ ^{$! "}⁰^;!"⁰^{$%%2 0}^{7 #1%#%1}⁰%&," % $! #1%285 %+ :% 0&<!17! 6,//, 207 1098 0%&7 778/0&6 0&2!#!&2!&5! 9! :!!& $! 5$,&6! #%0& ,&2λ ^<,"^8!7^{$! #10}^%1 207 1098 0%&07'

L L

B0 B0

π = λ λ π τ

:$!1!O2τ³ ⁰^7,&420^751!^!20⁷¹⁰⁹⁸⁰%&%& &5!,6,0& ,70/#"!5,"58", 0%&7$%:7

$, +%1,+0-!2τ ^$!#%7^!10^%120⁷¹⁰⁹⁸⁰^%&0⁷⁷⁰^""#1%#%1 0%&," %,#1%285 %+0&<!17!6,//, 2!&700!7'

L L L L

L L

1 B0 B0 1

B0 B0

λ λ π τ

λ λ τ

∝

′ ′ ′ ′

=Γ Γ

=

(

:0$'

%

τ

τ τ

σ σ

=

= +

′ = +

′ = + −

.

$!/,160&,"#%7 !10%1207 1098 0%&5,&201!5 "49!5%/#8 !2940& !61, 0%&+1%/ $!I%0&

#%7 !10%1207 1098 0%&'

(12)

L

!-#

L 1

1

τ

τ λ λ τ λ λ

π τ

λ λ λ τ λ τ

π τ λ λ

λ λ λ τ λ τ

π τ λ

′ ′

− ′ +

= ′

− ′ +

= ′

∝

′ ′

Γ Γ

∝ ′ ′

∝

′ ′

∝ Γ −

′

∝ Γ ′

∝ ′ !-#

λ

− ′

F

@0&,""4 $!/,160&,"207 1098 0%&%+%97!1<, 0%&707' L !-#

1 <

τ

λ λ τ λ λ τ

− π τ

′ ′

=

′ ′

Γ Γ

= Γ Γ ′ ′

6.3. Trend model

$!"0;!"0$%%25,&9!:10!&,7'

) )

L !-# %

1 σ

λ ₌ λ λ ₌ λ

= − −

+ +

∏

^*

@%1+0-!2λ/ $07"0;!"0$%%2 <0!:!2 ,7,λ! +8&5 0%&07#1%#%1 0%&," % ,&0&<!17!6,//,

#2+ &+%1 8&, !"4 &%5%&I86,54#1%#!1 4,##!,17+%1 $!λ/#,1,/! !1 &4#10%1207 1098 0%&

5,& $879!5$%7!& +%1λ/ +%10&7 ,&5!,&%1/,"207 1098 0%& :0$#,1,/! !17 ,&2 $!

I%0& #10%1207 1098 0%&07 $!&'

) L L

B0 @

π = λ λ D

"!,20&6 % $!+%""%:0&6#%7 !10%1207 1098 0%&'

) )

)

L

!-#

1 1

% π

σ λ

λ π λ λ λ

λ λ

+ +

=

∝

− −

= − + −

Γ + +

=

∏

⁾

$! +8&5 0%& 5,& 9! !-#"050"4 0& !61, !2 :0$ 1!7#!5 %λ! $8760<0&6 $! /,160&,"

#%7 !10%1207 1098 0%&%+λ/'

(13)

) )

L

!-#

1

%

λ λ λ λ

λ λ

σ π λ λ

+ =

=

∝

Γ + −

= −

Γ + − +

+

=

∏

$!/,160&,"#%7 !10%1207 1098 0%&%+λ! ,&2 $!/,160&,"207 1098 0%&%+%97!1<, 0%&75,&

9!5%/#8 !294 &8/!105,"0& !61, 0%&'

) )

!-#

) )

L L 1 1 <

λ λ λ λ

λ λ λ λ λ λ

∝

=

7. Application

$! ,4!70,&/! $%2:,7,##"0!2 %,7 1!,/ +"%: 7!10!7+1%/ $! 1P/!10<!1, 85!&

0%07 D(;/ %8 $ ,7 %+@1,&5! ,0"47 1!,/ +"%:7:!1!,<,0",9"!9! :!!& $!4!,17 D) ,&2 )) J ! !- 1,5 !2 , 7!10!7%+0&2!#!&2!& #!,;7,9%<! , / 7 $1!7$%"2

"!,20&6 %,7,/#"!%+D <,"8!7 <!1 $1!7$%"2<,"8!7,&20& !1,110<,"77!10!7:!1!7 820!2 +0681! $!+%1/!1:,77 820!2:0$ $!70-/%2!"7<₎^!-# <^!-# <^!-#<₎¹ < ¹ ,&2< ¹

$!", !1:0$<₎^!-# <^!-#,&2<^!-#

7.1. Prior specification

$!+017 7 !#5%&707 !2%+7#!50+40&6,20751! !#10%1207 1098 0%&%& $!/%2!"75%""!5 0%&

:!5%&702!1!2 &%12!1 %,<%02,&490,70& $!$4#% $!707:!:07$!2 %!<,"8, ! 7 , 0%&,14

&%&7 , 0%&,14 ,&2 -#%&!& 0," !&!1,"0=!2 ,1! % $!+%""%:0&6 #1%9,90"00!7:!1!

5$%7!&'

!-#

) )

!-# !-#

) .

1

1 1

< <

< < < <

π π

π π π π

= =

= = = =

+%1 $!%<!1 $1!7$%"27!10!7 ,&2

!-#

)

!-# !-#

) .

<

< <

π

π π

=

= =

+%1 $!0& !1,110<,"77!10!7 & $07/,&&!1!,5$$4#% $!707$,2,&!?8,"#10%1#1%9,90"04

$!7!5%&27 !#5%&707 !2%+!<,"8, 0&6%81#10%1;&%:"!26!%&?8,& 0"!7 ,7!-#",0&!20&

7!5 0%& ( J ! ,##"0!2 $! /! $%2%"%64 #1%#%7!2 94 ,"Q, ,&2 182$%//! DD ,&2 M,<!""!! ," DDD ;&%:&,7 $! 2@ +"%: 281, 0%&+1!?8!&54 /%2!"7 7,+017 7 !# :!

!7 0/, !2 $! 0&7 ,& ,&!%87 2075$,16! ?8,& 0"! :0$ #1%9,90"04 ) D 94 870&6 $! 18#!20- +%1/8", @! ," D*) D* '

(14)

) * ) D ) D=C *)

:$!1!C 07 $!:, !17$!2,1!, _{! #}07 $!2,0"41,0&+,""?8,& 0"!:0$#1%9,90"04) D ,&2 07,1!60%&,"0&2!- $! %5%7!281, 0%&D @! ," D*) D* :$05$5$,1,5 !10=!7

$!+"%%2281, 0%& :,7 $!&5%/#8 !2'

R

) D

"%6D = −) FD )+ "%6C + (

:$!1! 07 $!/!,&,&&8,"1,0&+,"",&2 $!/!,&,&&8," !/#!1, 81! "" $!#,1,/! !17

&!!2!2 +%1 $!5,"58", 0%& %+ !#,&2D ,1!,<,0",9"!%& /,#7 $! 2@ /! $%2%"%645,&

$!& 9!87!2 % 2!285!+"%%2 ?8,& 0"!7:0$ % $!1281, 0%&7%1+1!?8!&50!7 945%/#,107%&

:0$ $1!!1!+!1!&5!2, ,7! 7 1!#1!7!& , 0<!%+ $! $1!!/,0&61%8#7%+$421%"%605,"+"%%2 1!60/!7 0& @1,&5! $! 1!+!1!&5! 2, , 7! 780,9"! +%1 $! 1P/! 10<!1$,2 ,"1!,24 9!!&

5$%7!&94 182$%//! DD.

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(16)

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Acknowledgements

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Bibliography

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