D:\FINAL_TEX\...\Narine.DVI


Mathematical Problems of Computer Science 42, 63{72, 2014.

Detection of H eter ogeneity in T hr ee-Dimensional Data

Sequences: Algor ithm and Applications

E vg u e n i A . H a r o u t u n ia n , Ir in a A . S a fa r ya n ,

A r a m R . N a z a r ya n a n d N a r in e S . H a r u t yu n ya n

Institute for Informatics and Automation Problems of NAS RA

e-mail: evhar@ipia.sci.am, irinasafaryan@yandex.ru, aram.nazaryan@gmail.com,

narineharutyunyan57@gmail.com

Abstract

We present a nonparametric algorithm which allows reducing the investigations of
changes of the joint distribution of chronologically ordered multidimensional random
sequence to the investigations of some one-dimensional conditional distributions. The
algorithm is implemented with the statistical software package R. The action of the
program is demonstrated on applications. The ¯rst one concerns the retrospective
analysis of the changes in the concentration of chemical components of ground water
preceding major seismic events. The second refers to the de¯nition of cut-points in
two-dimensional life time data sets of the imatinib-treated chronic myeloid leukemia
patients.

Keywords: Change-point problem, Rank score test,Threshold copula, Cut-point
selection method.

1 . In t r o d u c t io n

Mo n it o r in g o f s o m e c o m p le x s ys t e m le a d s t o a ve c t o r o f o b s e r va t io n s c o n s is t in g o f in p u t va r i-
a b le s ( p r e d ic t o r s ) , o u t p u t va r ia b le s ( r e s p o n s e s ) , a n d a ls o c o n c o m it a n t ( c a t e g o r iz in g ) va r i-
a b le s , p o s s ib ly in ° u e n c in g s ig n i¯ c a n t ly o n t h e jo in t d is t r ib u t io n o f t h e in p u t a n d o u t p u t
va r ia b le s . E a c h o f t h e s e va r ia b le s c a n b e d is c r e t e , c o n t in u o u s o r o f n o n -n u m e r ic a l t yp e .

Cla s s ī c a t io n o f o b s e r va t io n s t o s t a t is t ic a lly h o m o g e n e o u s a n d s ig n ī c a n t ly d is t in c t
g r o u p s is n e c e s s a r y fo r fo r e c a s t in g a n d t a kin g a d e qu a t e c o n t r o l a c t io n s . S u c h t a s k a r is e s
in p r o b le m s o f m e d ic a l a n d t e c h n ic a l d ia g n o s t ic s in a n a lyz in g a n d fo r e c a s t in g c a t a s t r o p h ic
e ve n t s in n a t u r e , a n d a ls o in a c t u a r ia l a n d ¯ n a n c ia l m a t h e m a t ic s .

Th e r e la t ive ly s m a ll o r m o d e r a t e d e p e n d e n c e is o f in t e r e s t in s e is m o lo g ic a l a p p lic a t io n s .
N e ls e n [1 ] a n d Ma r i e t a l [2 ] h a ve d is c u s s e d t h e fa m ilie s o f b iva r ia t e d is t r ib u t io n s wit h a
s m a ll o r m o d e r a t e d e p e n d e n c e . It is a s s u m e d t h a t t h e d a t a , o b t a in e d fr o m d i®e r e n t p la c e s o f
o b s e r va t io n , a r e we a kly d e p e n d e n t . Th is d e p e n d e n c e is d e t e r m in e d o n ly b y t h e fa c t t h a t t h e
va r ia b le s b e lo n g t o t h e s a m e s ys t e m , i.e ., t o t h e s a m e g e o g r a p h ic r e g io n a n d t h e r e fo r e , t h e y
a r e e xp o s e d t o t h e s a m e e n vir o n m e n t a l c o n d it io n s . Ch a n g e o f t h e s t r u c t u r e o f d e p e n d e n c e
is c o n n e c t e d wit h t h e fa c t t h a t t h e wh o le s ys t e m is p r e p a r in g t o m o ve fr o m o n e s t a t e t o
a n o t h e r , a n d s u c h a c h a n g e is in s o m e wa y a p r e c u r s o r t o t h is t r a n s it io n .

6 3



6 4 Detection of Heterogeneity in Three-Dimensional Data Sequences: Algorithm and Applications

In m e d ic a l a p p lic a t io n s , s u c h d e p e n d e n c e is o b s e r va b le in t h e r e s e a r c h o f t wo -d im e n s io n a l
fu n c t io n s o f s u r viva l.

In t h is p a p e r we s o lve t h e p r o b le m o f c la s s ī c a t io n fo r t h e c a s e o f ve c t o r s ( Xn; Yn; Zn ) ,
n = 1 ; N, wh e r e X a n d Y a r e c o n t in u o u s , a n d t h e c o n c o m it a n t va r ia b le Z c a n b e a r b it r a r y.

Th e c a s e is o f p a r t ic u la r in t e r e s t fo r p r a c t ic e wh e r e t h e s h a r e d va r ia b le Z is a s e qu e n c e
o f o r d in a l n u m b e r s o f o b s e r va t io n s r a n ke d c h r o n o lo g ic a lly, o r a c c o r d in g t o s o m e o t h e r c o n -
c o m it a n t va r ia b le . If t h e fa c t o r t h a t in ° u e n c e s t h e c h a n g e s o f d e p e n d e n c e is t h e t im e , t h e n
t h e d e ¯ n it io n o f t h e m o m e n t o f c h a n g e s is a m u lt id im e n s io n a l ve r s io n o f t h e fa m o u s p r o b le m
a b o u t \ d is o r d e r " ( c h a n g e -p o in t d e t e c t io n p r o b le m ) . S t a g in g , b ib lio g r a p h y a n d s t a t e o f t h e
a r t a r e p r e s e n t e d in t h e b o o k o f B o r o vko v [3 ]. N o n p a r a m e t r ic m e t h o d s o f d e t e c t io n a r e
p r e s e n t e d in t h e b o o k o f B r o d s ky a n d D a r kh o vs ky [4 ]. Th e o r e t ic a l s u b s t a n t ia t io n o f n o n -
p a r a m e t r ic a lg o r it h m s b a s e d o n r a n k s t a t is t ic s fo r d e t e c t in g c h a n g e -p o in t in o n e -d im e n s io n a l
c a s e wa s o b t a in e d b y S a fa r ya n [5 ].

Fo r t h e t wo -d im e n s io n a l c a s e t h e fo llo win g wa s s t a t e d . L e t ( Xn; Yn ) , n = 1 ; N b e a
c h r o n o lo g ic a lly o r d e r e d t wo -d im e n s io n a l r a n d o m s e qu e n c e , s t a t is t ic a l p r o p e r t ie s o f wh ic h
c h a n g e in s o m e u n kn o wn m o m e n t ( c h a n g e -p o in t ) . A s in t h e o n e -d im e n s io n a l c a s e , we a s s u m e
t h a t t h e r e e xis t s a n in d e x ¸ 2 [¢ ; 1 ¡ ¢ ]; 0 < ¢ < 1 =2 , wh ic h d e t e r m in e s t h e in d e x o f
o b s e r va t io n n¸ = [¸N] s u c h , t h a t t h e o b s e r va t io n ( Xn; Yn ) h a s a t wo -d im e n s io n a l d is t r ib u t io n
fu n c t io n F (n) ( x; y ) wh ic h c a n b e wr it t e n a s :

F (n) ( x; y ) = F1 ( x; y ) Ifn · n¸g + F2 ( x; y ) Ifn > n¸g; n = 1 ; N: ( 1 )
wh e r e F1 ( x; y ) 6= F2 ( x; y ) a n d I ( A) is t h e in d ic a t o r o f t h e e ve n t A.

S in c e we a r e in t e r e s t e d in t h e c h a n g e o f d e p e n d e n c e , t h e s a m e r e la t io n c a n b e wr it t e n
wit h t h e c o p u la s

C(n) ( u; v ) = C1 ( u; v ) Ifn · n¸g + C2 ( u; v ) Ifn > n¸g; n = 1 ; N : ( 2 )
R e c a ll t h a t t h e c o p u la o f t wo r a n d o m va r ia b le s ( R V s ) X a n d Y wit h a jo in t d is t r ib u t io n
fu n c t io n F ( x; y ) is a fu n c t io n C ( u; v ) , d e ¯ n e d b y t h e r e la t io n

C ( FX ( x ) ; FY ( y ) ) = F ( x; y ) ;

o r
C ( u; v ) = F ( F ¡1 ( u) ; G¡1 ( v ) ) ;

wh e r e FX ( x ) a n d FY ( Y ) a r e m a r g in a l d is t r ib u t io n fu n c t io n s a n d F
¡1 a n d G¡1 a r e qu a s i

in ve r s e fu n c t io n s d e ¯ n e d a s F ¡1 ( u ) = in ffx : F ( x ) > ug. If t h e m a r g in a l d is t r ib u t io n s a r e
c o n t in u o u s , t h e n t h is r e p r e s e n t a t io n is u n iqu e [1 ]. E xp e d ie n c y o f a p p lic a t io n o f c o p u la s in
t h e d is c r e t e c a s e is d is c u s s e d in t h e a r t ic le o f B la g o ve s c h e n s ky [6 ], wh e r e t h e b a s ic e le m e n t s
o f t h e t h e o r y o f c o p u la s a r e a ls o p r e s e n t e d . Th e m a xim u m like lih o o d e s t im a t o r fo r t h e c o p u la
fu n c t io n c h a n g e -p o in t a r e o b t a in e d b y D ia s a n d E m b r e c h t s in [7 ]. In t h e a r t ic le b y B r o d s ky
et al [8 ], a n e s t im a t e o f t h e c h a n g e -p o in t o f 7 -d im e n s io n a l c o p u la is o b t a in e d o n t h e b a s is o f
m u lt iva r ia t e m o d ī c a t io n o f t h e K o lm o g o r o v-S m ir n o v s t a t is t ic . U n fo r t u n a t e ly, t h is s t a t is t ic
is n o t ve r y c o n ve n ie n t fo r p r a c t ic a l c a lc u la t io n s , a n d a ls o e ± c ie n c y o f a K o lm o g o r o v-S m ir n o v
s t a t is t ic wit h r e s p e c t t o t h e s t a t is t ic o f r a n k s c o r e e ve n in o n e -d im e n s io n a l c a s e is e qu a l t o
z e r o .

In t h is p a p e r a h e u r is t ic a lg o r it h m is p r o p o s e d t h a t a llo ws t o r e d u c e t h e in ve s t ig a t io n
o f c h a n g e s in t h e jo in t d is t r ib u t io n o f m u lt iva r ia t e r a n d o m s e qu e n c e , o r d e r e d c h r o n o lo g i-
c a lly, o r a c c o r d in g t o s o m e o t h e r c a t e g o r ic a l va r ia b le , t o t h e e xa m in a t io n o f c h a n g e s in t h e
c o r r e s p o n d in g o n e -d im e n s io n a l s e qu a n c e o f c o n d it io n a l d is t r ib u t io n s wit h t h e a p p lic a t io n o f
a p p r o p r ia t e ly s e le c t e d r a n k s c o r e s s t a t is t ic s . Th e a lg o r it h m wa s ¯ r s t in t r o d u c e d in [9 ].



E. Haroutunian, I. Safaryan, A. Nazaryan and N. Harutyunyan 6 5

N o w we fo r m a liz e t h e n o t io n o f we a k d e p e n d e n c e in t e r m s o f t h e c o p u la . L e t X a n d Y b e
R V s wit h c o n t in u o u s m a r g in a l d is t r ib u t io n fu n c t io n s F ( x ) a n d G( y ) , t h e jo in t d is t r ib u t io n
fu n c t io n F ( x; y ) a n d t h e c o r r e s p o n d in g c o p u la C ( u; v ) : E a c h c o p u la is a s u r fa c e in t h e u n it
c u b e , s o e a c h d is t a n c e b e t we e n s u r fa c e s z0 = uv a n d z1 = C ( u; v ) s h o u ld yie ld a m e a s u r e
o f d e p e n d e n c e b e t we e n X a n d Y . Th e m o s t fa m o u s o f t h e m a r e t h e P e a r s o n c o r r e la t io n
c o e ± c ie n t r a n d t h e S p e a r m a n r a n k c o r r e la t io n c o e ± c ie n t ½:

r ( X; Y ) =
1

D ( X ) D ( Y )

Z 1
0

Z 1
0

( C ( u; v ) ¡ uv ) dF ¡1 ( u) dG¡1 ( v ) ; ( 3 )

wh e r e D s t a n d s fo r a s t a n d a r d d e via t io n a n d

½( X; Y ) = 1 2
Z 1
0

Z 1
0

( C ( u; v ) ¡ uv ) dudv: ( 4 )

Th e s m a lle r is S p e a r m a n ½( X; Y ) , t h e we a ke r is t h e d e p e n d e n c e [3 ]. B e lo w s o m e e xa m p le s
o f we a k d e p e n d e n c e o f c o p u la s a r e g ive n , wh ic h a r e r e le va n t fo r t h e c o n s id e r e d a p p lic a t io n s .

Case 1. Two fa m ilie s o f o n e -p a r a m e t e r c o p u la s d e s c r ib in g t h e r e la t ive ly we a k d e p e n d e n c e
b e t we e n t h e r a n d o m va r ia b le s X a n d Y a r e p r e s e n t e d b y N e ls e n [1 ]. Th is is t h e Fa r lie -
Gu m b e l-Mo r g e n s t e r n ( FGM) c o p u la :

Cµ ( u; v ) = uv + µuv ( 1 ¡ u ) ( 1 ¡ v ) ; µ 2 [¡ 1 ; 1 ];

a n d t h e A li-Mic h a il-H a q c o p u la ( A MH ) :

Cµ ( u; v ) =
uv

1 + µuv ( 1 ¡ u ) ( 1 ¡ v ) ; µ 2 [¡ 1 ; 1 ]:

Co e ± c ie n t s ( 3 ) a n d ( 4 ) fo r c o p u la s FMG a n d A MH c h a n g e wit h in t h e lim it s [¡ 1 =3 ; 1 =3 ].
Case 2. Th e s e c o n d va r ia n t r e la t e s t o t h e fa c t t h a t t h e p r e d ic t o r c a n a ls o b e a g r o u p in g

va r ia b le , i.e ., c o n t a in o n e o r m o r e c u t -p o in t s . In t h is c a s e , t h e h ig h c o r r e la t io n c o e ± c ie n t
b e t we e n t h e p r e d ic t o r a n d t h e r e s p o n s e c a n n o t b e in t e r p r e t e d a s a s ig n o f d e p e n d e n c e o f o n e
t o t h e o t h e r . Th is d e p e n d e n c e in t h e m o n o g r a p h o f t h e B la g o ve s c h e n s ky [1 0 ] is n a m e d fa ls e
o r s p u r io u s a n d s o m e e xa m p le s o f wh y s u c h a s p u r io u s d e p e n d e n c e m a y a r is e a r e g ive n . In
[1 1 ] t h e fa ls e d e p e n d e n c e is d e ¯ n e d a s a t h r e s h o ld d e p e n d e n c e . H e r e we r e m in d t h e d e ¯ n it io n
o f h o m o g e n e it y o f a n R V wit h r e s p e c t t o a n o t h e r , wh ic h is e qu iva le n t t o t h e d e ¯ n it io n o f
in d e p e n d e n c e .

W e c a ll a n R V Y h o m o g e n e o u s wit h r e s p e c t t o R V X if fo r a ll p a ir s ( x; y ) o n t h e p la n e
t h e fo llo win g c o n d it io n a l p r o b a b ilit ie s a r e e qu a l:

P r ( Y · y=X · x) = P r ( Y · y=X > x ) : ( 5 )

If t h e r e e xis t s a u n iqu e va lu e o f x = ¹ s u c h t h a t fo r a ll y 2 R,

P r ( Y · y=X · x ) = P r ( Y · y=X · ¹) ; f or x · ¹; ( 6 )

P r ( Y · y=X > x ) = P r ( Y · y=X > ¹) ; for x > ¹; ( 7 )

2 . Th e R e s c a lin g P r o c e d u r e a n d S t e p s o f t h e A lg o r it h m

2 .1 Co p u la s o f W e a k D e p e n d e n c e



6 6 Detection of Heterogeneity in Three-Dimensional Data Sequences: Algorithm and Applications

a n d
P r ( Y · y=X > x ) 6= P r ( Y · y=X > ¹) ; ( 8 )

t h e n t h e s t a t is t ic a l d e p e n d e n c e b e t we e n X a n d Y is c a lle d one-thr eshold a n d t h e va lu e ¹
is c a lle d a thr eshold.

A c t u a lly o n e -t h r e s h o ld d e p e n d e n c e is a n e xa m p le o f d e p e n d e n c e in s in g le p o in t ¹. In a
s im ila r wa y M- t h r e s h o ld d e p e n d e n c e wit h M > 1 c a n b e d e ¯ n e d .

Case 3. D e p e n d e n c e a r is in g in t h e t im e m e a n s t h a t a d e p e n d e n c e b e t we e n X a n d Y
o c c u r s a t s o m e u n kn o wn m o m e n t o f t im e , i. e ., in r e la t io n ( 2 ) C1 ( u; v ) = uv: A p r a c t ic a l
e xa m p le o f d e p e n d e n c e a r is in g in t h e t im e g ive n b y S t a kh e e v [1 2 ] is d e d ic a t e d t o e a r t h qu ke
g e o c h e m ic a l p r e c u r s o r s .

W e s o lve t h e a b o ve s t a t e d p r o b le m a b o u t t h e d e t e r m in a t io n o f c h a n g e m o m e n t o f c o p u la
fu n c t io n wit h o u t ¯ xin g in a d va n c e s o m e kin d o f we a k d e p e n d e n c e , s in c e if X a n d Y a r e
we a kly d e p e n d e n t like in Ca s e s 1 o r 3 , t h e n t h e y ve r ify ( 8 ) . It is a s s u m e d t h a t t h e r e is s o m e
u n o b s e r ve d r a n d o m va r ia b le Z1, wh ic h c h a n g e s t h e c o p u la fu n c t io n in s o m e t h r e s h o ld p o in t ,
t h e n a s it is s h o wn in [1 3 ], X is h e t e r o g e n e o u s wit h r e s p e c t t o Y , a n d vic e ve r s a . Ta kin g
o n e o f t h e va r ia b le s fo r t h e b a s e , we ¯ n d t h e t h r e s h o ld va lu e , wh ic h is c h a n g e m o m e n t o f t h e
c o p u la fu n c t io n . Th e a p p r o a c h t o c u t -p o in t d e t e c t io n u s in g t h e c h a n g e -p o in t id e n t i¯ c a t io n
t e c h n iqu e s is in t r o d u c e d in [1 4 ].

Step1. Selection of a base var iable.
L e t RXi = # ( Xn : Xn · Xi; n = 1 ; N ) a n d RYi = # ( Yn : Yn · Yi; n = 1 ; N ) ; i = 1 ; N

b e r a n ks o f s e qu e n c e s fXngNn=1 a n d fYngNn=1.
W e d e ¯ n e t wo s e qu e n c e s o f r ank scor e statistic a s fo llo ws :

W iN ( n) = n= ( N ¡ n) ( T iJ ( n) ¡ A( J ) ) ; n = 1 ; N; ( 9 )

wh e r e

T iJ ( n) = 1 =n
nX

i=1

J ( Ri= ( N + 1 ) ) ; ( 1 0 )

a n d

A( J ) =
Z 1
0

( J ( u ) du ) ; ( 1 1 )

wh e r e Ri is RXi in t h e ¯ r s t c a s e o r is RYi in t h e s e c o n d c a s e a n d J ( u ) is a s c o r e fu n c t io n
[1 5 ]. Th e change points by time a r e

n̂i = a r g m in
[¢ N]·n·[(1¡¢ )N ]

W iN ( n) ; 0 < ¢ < 1 = 2 ; i = 1 ; 2 : ( 1 2 )

W e d e ¯ n e t h e s t a n d a r d iz e d s t a t is t ic

W ¤iN =
r

N ( 1 ¡ n
N

)
n

N
S ( J ) W iN ( n) ;

wh e r e S ( J ) =
R 1
0 J

2 ( u) du ¡ (
R 1
0 J ( u ) du )

2. If fo r b o t h va r ia b le s W ¤iN · z®, wh e r e z® is t h e
qu a n t ile o f t h e le ve l ® fo r s t a n d a r d n o r m a l d is t r ib u t io n , is d e t e c t e d a n d we c h o o s e t h e b a s e

2 .2 P r o b le m Fo r m u la t io n fo r S o m e H e t e r o g e n e it y Mo d e ls

2 .3 S t e p s o f A lg o r it h m



E. Haroutunian, I. Safaryan, A. Nazaryan and N. Harutyunyan 6 7

va r ia b le fo r wh ic h t h e e xt r e m u m is o b s e r ve d e a r lie r . Fo r o t h e r c a s e s , t h e c h o ic e o f b a s e
va r ia b le is a r b it r a r y.

Step 2. E stimation of the moment homogeneity violation
In wh a t fo llo ws we d e n o t e t h e b a s e va r ia b le s b y X, r a n ks o f va r ia b le Y r ear r anged in

o r d e r o f in c r e a s in g o f t h e b a s e va r ia b le b y RXYn ; n = 1 ; N , a n d r a n k s c o r e s t a t is t ic c a lc u la t e d
wit h r e a r r a n g e d r a n ks b y W XN ( n) ; n = 1 ; N. Th e m o m e n t n¹ o f t h e vio la t io n o f h o m o g e n e it y
o f r e s p o n s e va r ia b le r e la t ive t o t h e b a s e va r ia b le is d e ¯ n e d b y ( 1 2 ) , wh e r e W iN ( n) is r e p la c e d
b y W XN ( n) .

Th e fo u n d e d m o m e n t o f c h a n g e is a c e r t a in r a n k o f b a s e va r ia b le , a n d c o r r e s p o n d in g t o
t h is r a n k t h e va lu e o f va r ia b le is t h e cut-point. In d e x o f t h e m o m e n t in c h r o n o lo g ic a lly
o r d e r e d s e qu e n c e c o r r e s p o n d in g t o c u t -p o in t is t h e m o m e n t o f c h a n g e o f t h e c o p u la . U n -
fo r t u n a t e ly, in t h e r e a l d a t a qu it e a lo t o f s u c h c o in c id in g va lu e s a r e e n c o u n t e r e d . D i®e r e n t
r a n ks c o r r e s p o n d t o t h e m s in c e in a c c o r d a n c e wit h o u r r a n kin g s ys t e m , t h e s m a lle r r a n k
r e c e ive s a c h r o n o lo g ic a lly e a r lie r o b s e r va t io n .

Step 3. Cor r ection of the moment of change of the copula visually using
gr aphical r epr esentations

Fo r g r a u n d wa t e r s it is p r e s e n t e d o n Fig . 2 . b y s c a t t e r p lo t s . Fo r s u r viva l t im e d a t a t h e
c o p u la d e n s it y h is t o g r a m s a r e o b t a in e d b u t n o t in c lu d e d in t h e p a p e r fo r r e s t r ic t io n o f t h e
vo lu m e o f t h e t e xt .

Th e c o n c e p t o f s e is m o g r a p h ic g e o c h e m ic a l a n o m a ly is n o t c le a r ly d e ¯ n e d . U s u a lly, t h e s e -
qu e n c e o f o b s e r va t io n s b e t we e n t h e t wo e a r t h qu a ke s c o n t a in s t wo s p e c i¯ c p o in t s : t h e s t a r t
o f t h e a c c u m u la t io n o f c h a n g e s a n d r e le a s e t o t h e le ve l o f qu a s i-p e r m a n e n t ( \ g e o c h e m ic a l
qu ie s c e n c y" ) . Th u s , t h e o b s e r va t io n s in t h e p e r io d p r io r t o t h e e a r t h qu a ke c a n b e c o n s id e r e d
a s a c h r o n o lo g ic a lly o r d e r e d r a n d o m s e qu e n c e wit h t wo u n kn o wn p o in t s o f t h e \ d is o r d e r " .
P r o c e s s in g o f t h e d a t a o b t a in e d o n a n u m b e r o f s t a t io n s o f t h e h yd r o g e o c h e m ic a l o b s e r va t io n
n e t wo r k o f t h e N a t io n a l S e r vic e fo r S e is m ic P r o t e c t io n ( N S S P ) o f A r m e n ia , u s in g n o n p a r a -
m e t r ic a lg o r it h m s , d e s ig n e d t o c h a n g e s in t h e s t a t is t ic a l p r o p e r t ie s o f t h e r a n d o m s e qu e n c e s ,
c o n ¯ r m e d t h is m o d e l o f a n o m a lie s . Th e p r e s e n t e xa m p le is r e la t e d t o t h e d e t e r m in a t io n o f
t h e t im e o f c h a n g in g t h e s t r u c t u r e o f t wo -d im e n s io n a l r e la t io n s h ip s fo r t h e t h r e e o b s e r va t io n
s t a t io n s o n t h e e ve o f t h e h e liu m c o n t e n t o f t h e S p it a k e a r t h qu a ke . N o t e t h a t t h e c o r r e la t io n
a n a lys is p r e s e n t e d in t h e b o o k o f P e t r o s ia n [1 6 ] d id n o t d e t e r m in e a s ig n i¯ c a n t s t a t is t ic a l
d e p e n d e n c e o n h e liu m ( He ) b e t we e n t h e o b s e r va t io n s o f t wo s t a t io n s .Th e a c t u a l d a t a in t h e
e xa m p le is t h e c o n t e n t o f h e liu m in t h e g r o u n d wa t e r o n t h e e ve o f m a jo r s e is m ic e ve n t s .
D e ¯ n it io n o f c h a n g e m o m e n t s o f t h e s e t wo s e qu e n c e s wa s c a r r ie d o u t in a kn o wn m a n n e r b y
m e a n s o f t h e W ilc o xo n s t a t is t ic .

If we c h o o s e ¢ = 0 :1 , t h e n t h e le ft 3 0 va lu e s o f t h e W ilc o xo n s t a t is t ic will b e c u t t o b o t h
s id e s a n d t h e m in im u m va lu e o u t s id e t h e c r it ic a l b o r d e r -1 .9 6 c o m e s t o t h e p o in t 1 5 0 . Th u s ,
d a t a fr o m t h e m o n it o r in g s t a t io n A r a r a t , h a vin g a r a n k 1 5 0 , c o r r e s p o n d s t o t h e t im e o f
t h e c h a n g e o f h o m o g e n e it y o f h e liu m in s t a t io n K a ja r a n wit h r e s p e c t t o t h e s t a t io n A r a r a t .
Th e o r e t ic a lly, t h is m e a n s s o m e fo r m o f we a k d e p e n d e n c e ( c a s e 1 -3 ) b e t we e n t h e va lu e s o f
t h e in d ic a t o r s o f t h e s e t wo s t a t io n s . Th e r a n k 1 5 0 c o r r e s p o n d s t o t h e va lu e o f c u t -p o in t
¹ = 2 8 4 a n d t im e 0 6 .0 2 .8 8 ( 2 / 6 / 8 8 ) . W e p r e s e n t a t wo -d im e n s io n a l s c a t t e r p lo t u n t il t h e

3 . A n a lys is o f R e a l D a t a

3 .1 Co m p a r is o n o f V a r ia t io n s o f t h e Co m p o n e n t s o f Gr o u n d wa t e r



6 8 Detection of Heterogeneity in Three-Dimensional Data Sequences: Algorithm and Applications

t im e m o m e n t 0 6 .0 2 .8 8 b y o n e c o lo r , a n d a ft e r it wit h a n o t h e r .

(a)

(b)

(c)



E. Haroutunian, I. Safaryan, A. Nazaryan and N. Harutyunyan 6 9

(d)
Fig. 1. (a)data of helium at the station Kajaran are on the vertical axis, the horizontal axis

is for the number of observations over time. (b) Wilcoxon statistic over time, (c) data for station
Kajaran ordered according to increasing of values of the index of helium on Ararat station are
on the vertical axis, the horizontal axis is for Rank values of the helium in station Ararat. (d)
Wilcoxon statistic for Kajaran by Ararat.

Fig. 2. Two-dimentional data of helium classi¯ed over time (Kajaran -Ararat).

W e s e e t h a t t h e t wo -d im e n s io n a l o b s e r va t io n s a r e s u ± c ie n t ly we ll s e p a r a t e d in t im e .
H o we ve r , t h e fo llo win g r a n ks : 1 4 4 ( 2 9 .0 9 .8 7 ) , 1 4 5 ( 2 9 .1 1 .8 7 ) , 1 4 6 ( 0 1 .1 2 .8 7 ) , 1 4 7 ( 1 1 .1 2 .8 7 ) ,
1 4 9 ( 0 7 .0 1 .8 8 ) 1 5 0 ( 0 6 .0 2 .8 8 ) , 1 5 1 ( 2 8 .0 4 .8 8 ) , 1 5 2 ( 3 0 .0 4 .8 8 ) c o r r e s p o n d t o t h e va lu e 2 8 4 . W e
c o n s t r u c t e d t wo -d im e n s io n a l s c a t t e r p lo t s fo r a ll o f m o m e n t s c o r r e s p o n d in g t o t h e c u t -p o in t
¹ = 2 8 4 a n d m a d e s u r e t h a t t h e b e s t d ivis io n b y t im e c o r r e s p o n d e d t o 0 2 .0 6 .8 8 .



7 0 Detection of Heterogeneity in Three-Dimensional Data Sequences: Algorithm and Applications

R e c e n t ly a lo t o f wo r ks h a ve b e e n d e vo t e d t o t h e s t u d y o f t wo -d im e n s io n a l s u r viva l fu n c t io n s
wh ic h in c lu d e a ls o a p p lic a t io n s d e s c r ib in g a lg o r it h m s a n d p r o g r a m s in R c o d e s s u c h a s in
[1 7 ,1 8 ]. Th e s a m e a lg o r it h m wa s a p p lie d t o t h e a n a lys is o f t wo -d im e n s io n a l life t im e d a t a o f
e xp e c t a n c y o f p a t ie n t s wit h m ye lo id le u ke m ia . Th e s e qu e n c e fXngNn=1 is t h e life t im e d a t a
d e n o t in g t h e d a t e o f d ia g n o s is , a n d t h e s e qu e n c e fYngNn=1 d e n o t e s t h e life t im e fr o m t h e p o in t
o f t h e r a p e u t ic t r e a t m e n t . Ca t e g o r iz in g va r ia b le Z is t h e a g e o f p a t ie n t , t h e s a m p le s iz e is
4 1 3 p a t ie n t s . In t h is e xa m p le , t h e va r ia b le s a r e s t r o n g ly d e p e n d e n t , h o we ve r ,e ve n in t h is
c a s e , t h e p r o p o s e d a lg o r it h m wo r ks . Th e c u t -p o in t is fo u n d : 6 0 ye a r s o f a g e , a ft e r wh ic h
t h e c o p u la fu n c t io n is c h a n g e d . Fo r t wo -d im e n s io n a l h is t o g r a m o f t h e d e n s it y c o p u la s u c h
s e p a r a t io n is vis ib le e ve n t o t h e e ye . Th e m a in c o n c lu s io n is t h a t a ft e r 6 0 ye a r s , t h e life t im e
is o n ly s lig h t ly d e p e n d e n t o n t h e t h e r a p e u t ic t r e a t m e n t .

4 . Co n c lu s io n

Th e p r e s e n t e d a lg o r it h m s h o ws t h e p r o s p e c t s o f a p p lic a t io n o f t h r e s h o ld c o p u la m e t h -
o d s a n d m ixe d s a m p lin g t o d e t e r m in a t io n o f a n o m a lie s in m u lt id im e n s io n a l h yd r o g e o c h e m -
ic a l d a t a o c c u r r in g p r io r t o e a r t h qu a ke s a n d fo r s p a t ia lly c o r r e la t e d s u r viva l d a t a . Fu r t h e r
t h e o r e t ic a l e la b o r a t io n a n d im p le m e n t a t io n o f p r o g r a m s fo r t h e ir r e a liz a t io n a r e a d m is s a b le .
It is d e s ir a b le t o ¯ t t in g c o p u la s C1 ( u; v ) a n d C2 ( u; v ) fo r u s in g t h e g o o d n e s s -o f-̄ t t e s t s .

A c kn o wle d g e m e n t

Th is wo r k wa s s u p p o r t e d in p a r t b y S CS o f ME S o f R A u n d e r Th e m a t ic P r o g r a m N o S CS
1 3 { 1 A 2 9 5 .

Refer ences

[1 ] R . V . N e ls e n , An Introduction to Copulas, S p r in g e r , N e w Y o r k, 2 0 0 6 .

[2 ] D . D . Ma r i, S . K o t z , Correlation and D ependence, L o n d o n Im p e r ia l Co lle g e P r e s s .
2 0 0 4 .

[3 ] A . A . B o r o vko v, M athematical Statistics, ( in R u s s ia n ) , M., " N a u ka " , 2 0 0 7 .

[4 ] B . E . B r o d s ky, B .S . D a r kh o vs ky, Nonparametric M ethods in Change-P oint P roblems.
K lu ve r , D o r d r e c h t , 1 9 9 3 .

[5 ] I. A . S a fa r ya n , Nonparametric algorithms for monitoring of time series, P h D Th e s is
( in R u s s ia n ) , Y e r e va n , 1 9 9 8 .

[6 ] U . N . B la g o ve s c h e n s ky, \ Th e m a in e le m e n t s o f t h e t h e o r y o f c o p u la s " , ( in R u s s ia n ) ,
Applied E conometrics, N2, p p . 1 1 3 -1 3 1 , 2 0 1 2 .

[7 ] A . D ia s a n d P . E m b r e c h t s , \ Ch a n g e -p o in t a n a lys is in ¯ n a n c e a n d in s u r a n c e " , In New
R isk M esures in Investment and R egulation, W illey, New York,p p . 2 0 0 3 .

[8 ] B . E . B r o d s ky, G.I. P e n ika s a n d I.A . S a fa r ya n , \ D e t e c t in g s t r u c t u r a l c h a n g e s in t h e
c o p u la m o d e ls " , ( in R u s s ia n ) , Applied E conometrics, 4(16), p p . 3 -1 6 , 2 0 0 9 .

3 .2 A n a lys is o f S u r viva l Fu n c t io n o f P a t ie n t s wit h Mye lo id L e u ke m ia



E. Haroutunian, I. Safaryan, A. Nazaryan and N. Harutyunyan 7 1

[9 ] E . A . H a r o u t u n ia n , I.A . S a fa r ya n , H .M. P e t r o s ya n a n d A . R . Ge vo r kia n , \ On id e n t i¯ -
c a t io n o f a n o m a lie s in m u lt id im e n s io n a l h yd r o g e o c h e m ic a l d a t a a s e a r t h qu a ke p r e c u r -
s o r s " , M athematical P roblems of Computer Science, vo l. 4 0 , p . 7 6 , 2 0 1 3 .

[1 0 ] U . N . B la g o ve s c h e n s ky, Secrets of the correlations, ( in R u s s ia n ) , M ., " N a u c h n a ya
kn ig a " , 2 0 0 8 .

[1 1 ] E . A . H a r o u t u n ia n a n d I. A . S a fa r ya n , \ Co p u la s o f t wo -d im e n s io n a l t h r e s h o ld m o d e ls " ,
M athematical P roblems of Computer Science, vo l. 3 1 , p p . 4 0 -4 8 , 2 0 0 8 .

[1 2 ] U .I. S t a h e e v, \ Ge o c h e m ic a l p r e c u r s o r s o f e a r t h qu a ke s " , ( in R u s s ia n ) , R ussian Chemi-
cal J ournal, vo l. X L IX , n o . 4 , p p . 1 1 0 -1 1 9 , 2 0 0 5 .

[1 3 ] E . A . H a r o u t u n ia n a n d I. A . S a fa r ya n , \ On e s t im a t io n o f t h r e s h o ld p a r a m e t e r in
t h r e e -d im e n s io n a l c o p u la s m o d e l" , Abstracts of International Conference on Computer
Science and Information Technologies, Yerevan, p p . 1 2 9 -1 3 1 . 2 0 1 1 .

[1 4 ] B . L a u s e n a n d M. S c h u m a c h e r , \ Ma xim a lly s e le c t e d r a n k s t a t is t ic s " . B iometrics, vo l.
4 8 , p p . 7 3 -8 5 , 1 9 9 2 .

[1 5 ] E . A . H a r o u t u n ia n a n d I.A . S a fa r ya n , \ N o n p a r a m e t r ic c o n s is t e n t e s t im a t io n o f r a n d o m
s e qu e n c e c h a n g e m o m e n t " , ( in R u s s ia n ) , M athematical P roblems of Computer Science,
vo l. 1 7 , p p . 7 6 -8 5 , 1 9 9 7 .

[1 6 ] H . M. P e t r o s ya n , P recursors and P rognosis of E arthquakes on the Territory of the
R epublic of Armenia, ( in R u s s ia n ) , Y e r e va n , 2 0 0 9

[1 7 ] J. P a ik a n d Zh . Y in g , \ A c o m p o s it e like lih o o d a p p r o a c h fo r s p a t ia lly c o r r e la t e d s u r viva l
d a t a " , Computational Statistics and D ata Analysis, vo l. 5 6 , p p . 2 0 9 -2 1 6 , 2 0 1 2 .

[1 8 ] P . F. S u a n d Ch .-I. L i, Y u . S h yr , \ S a m p le s iz e d e t e r m in a t io n fo r p a ir e d r ig h t -c e n s o r e d
d a t a b a s e d o n t h e d i®e r e n c e o f K a p la n -Me ie r e s t im a t e s " , Computational Statistics and
D ata Analysis, vo l. 7 4 , p p . 3 9 -5 1 , 2 0 1 4 .

Submitted 22.08.2014, accepted 10.11.2014.

ºé³ã³÷ ïíÛ³ÉÝ»ñÇ Ñ³çáñ¹³Ï³ÝáõÃÛáõÝÝ»ñÇ ³Ýѳٳë»éáõÃÛ³Ý
ѳÛïݳµ»ñáõÙ. ³É·áñÇÃÙ ¨ ÏÇñ³éáõÃÛáõÝÝ»ñ

º. гñáõÃÛáõÝÛ³Ý, Æ. ê³ý³ñÛ³Ý, ². ܳ½³ñÛ³Ý ¨ Ü. гñáõÃÛáõÝÛ³Ý

²Ù÷á÷áõÙ

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å³ï³Ñ³Ï³Ý ѳçáñ¹³Ï³ÝáõÃÛ³Ý Ñ³Ù³ï»Õ µ³ßËÙ³Ý Ñ»ï³½áïáõÙÁ ѳݷ»óÝ»É áñáß
Ùdzã³÷ å³ÛÙ³Ý³Ï³Ý µ³ßËáõÙÝ»ñÇ Ñ»ï³½áïÙ³ÝÁ: ²É·áñÇÃÙÁ Çñ³Ï³Ý³óí³Í ¿
Íñ³·ñÙ³Ý íÇ׳ϳ·ñ³Ï³Ý R É»½íÇ ÙÇçáóáí: Ìñ³·ñÇ ·áñÍ»ÉÇáõÃÛáõÝÁ óáõÛó ¿ ïñí»É
»ñÏáõ ÏÇñ³éáõÃÛáõÝÝ»ñÇ íñ³` ë»ÛëÙÇÏ ¨ µÅßϳϳÝ:



7 2 Detection of Heterogeneity in Three-Dimensional Data Sequences: Algorithm and Applications

Îáíàðóæåíèå íåîäíîðîäíîñòè â òðåõðàçìåðíûõ
ïîñëåäîâàòåëüíîñòÿõ äàííûõ:

àëãîðèòì è ïðèëîæåíèÿ

Å. Àðóòþíÿí, È. Ñàôàðÿí, À. Íàçàðÿí è Í. Àðóòþíÿí

Àííîòàöèÿ

Ìû ïðåäñòàâëÿåì íåïàðàìåòðè÷åñêèé àëãîðèòì, ïîçâîëÿþùèé ñâîäèòü
èññëåäîâàíèå èçìåíåíèé ñîâìåñòíîãî ðàñïðåäåëåíèÿ ìíîãîìåðíîé ñëó÷àéíîé
ïîñëåäîâàòåëüíîñòè ê èññëåäîâàíèþ íåêîòîðûõ îäíîìåðíûõ óñëîâíûõ ðàñïðå-
äåëåíèé. Àëãîðèòì ðåàëèçîâàí â ñðåäå ñòàòèñòè÷åñêîãî ÿçûêà R. Äåéñòâèå
ïðîãðàììû ïîêàçàíî íà äâóõ ïðèëîæåíèÿõ: ñåéñìîëîãè÷åñêîì è ìåäèöèíñêîì.