PROCEEDINGS of the Union of Scientists - Ruse

Book 5

Mathematics, Informatics and Physics

Volume 12, 2015

RUSE

PPRROOCCEEEEDDIINNGGSS OOFF TTHHEE UUNNIIOONN OOFF SSCCIIEENNTTIISSTTSS -- RRUUSSEE

EEDDIITTOORRIIAALL BBOOAARRDD

EEddiittoorr iinn CChhiieeff

Prof. Zlatojivka Zdravkova, PhD

MMaannaaggiinngg EEddiittoorr

Assoc. Prof. Tsetska Rashkova, PhD

MMeemmbbeerrss

Assoc. Prof. Petar Rashkov, PhD Prof. Margarita Teodosieva, PhD

Assoc. Prof. Nadezhda Nancheva, PhD

PPrriinntt DDeessiiggnn

Assist. Prof. Victoria Rashkova, PhD

UUnniioonn ooff SScciieennttiissttss -- RRuussee

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PROCEEDINGS of the Union of Scientists – Ruse

ISSN 1314-3077

ISSN 1314-3077

PR O C E E D I N G S O F THE UN I O N O F SCI E N TIS TS – RU S E V O L . 12 / 2015

3

Proceedings

of the Union of Scientists – Ruse

Contains five books:

1. Technical Sciences

2. Medicine and Ecology

3. Agrarian and Veterinary Medical Sciences

4. Social Sciences

5. Mathematics, Informatics and Physics

MATHEMATICS , INFORMATICS AND PHYSICS

PR O C E E D I N G S O F TH E UN I O N O F SCI E N TI S TS – RU S E V OL . 12 / 2015 4

BOARD OF DIRECTORS OF THE US - RUSE

1. Prof. Hristo Beloev, DSc – Chairman

2. Assoc. Prof. Vladimir Hvarchilkov – Vice-Chairman

3. Assoc. Prof. Teodor Iliev – Secretary in Chief

SCIENTIFIC SECTIONS WITH US - RUSE

1. Assoc. Prof. Aleksandar Ivanov – Chairman of “Machine-building Sciences and Technologies" scientific section

2. Prof. Ognjan Alipiev – Chairman of "Agricultural Machinery and Technologies" scientific section

3. Assoc. Prof. Ivan Evtimov– Chairman of "Transport" scientific section

4. Assoc. Prof. Teodor Iliev – Chairman of "Electrical Engineering, Electronics and Automation" scientific section

5. Assist. Prof. Diana Marinova – Chairman of "Agrarian Sciences" scientific section

6. Svilen Dosev, MD – Chairman of "Medicine and Dentistry" scientific section

7. Assoc. Prof. Vladimir Hvarchilkov – Chairman of "Veterinary Medical Sciences" scientific section

8. Assist. Prof. Anton Nedjalkov – Chairman of "Economics and Law" scientific section

9. Assoc. Prof. Tsetska Rashkova – Chairman of "Mathematics, Informatics and Physics" scientific section

10. Assoc. Prof. Ljubomir Zlatev – Chairman of "History" scientific section

11. Assoc. Prof. Rusi Rusev – Chairman of "Philology" scientific section

12. Prof. Penka Angelova, DSc– Chairman of "European Studies" scientific section

13. Prof. Antoaneta Momchilova - Chairman of "Physical Education, Sport and Kinesiterapy" section

CONTROL PANEL OF US - RUSE

1. Assoc. Prof. Jordanka Velcheva

2. Assoc. Prof. Nikolai Kotsev

3. Assist. Prof. Ivanka Dimitrova

EDITOR IN CHIEF OF PROCEEDINGS OF US - RUSE

Prof. Zlatojivka Zdravkova

MATHEMATICS , INFORMATICS AND PHYSICS

PR O C E E D I N G S O F TH E UN I O N O F SCI E N TI S TS – RU S E V OL . 12 / 2015 5

The Ruse Branch of the Union of Scientists in Bulgaria was founded in 1956. Its first Chairman was Prof. Stoyan Petrov. He was followed by Prof. Trifon Georgiev, Prof. Kolyo Vasilev, Prof. Georgi Popov, Prof. Mityo Kanev, Assoc. Prof. Boris Borisov, Prof. Emil Marinov, Prof. Hristo Beloev. The individual members number nearly 300 recognized scientists from Ruse, organized in 13 scientific sections. There are several collective members too – organizations and companies from Ruse, known for their success in the field of science and higher education, or their applied research activities. The activities of the Union of Scientists – Ruse are numerous: scientific, educational and other humanitarian events directly related to hot issues in the development of Ruse region, including its infrastructure, environment, history and future development; commitment to the development of the scientific organizations in Ruse, the professional development and growth of the scientists and the protection of their individual rights.

The Union of Scientists – Ruse (US – Ruse) organizes publishing of scientific and popular informative literature, and since 1998 – the “Proceedings of the Union of Scientists- Ruse".

BOOK 5

"MATHEMATICS, INFORMATICS AND

PHYSICS"

VOLUME 12

CONTENTS

Mathematics

Neli Keranova, Nako Nachev .................................................... 7 Simple components of semisimple group algebras of finite P-groups with minimal commutants

Evelina Veleva .........................................................................15 Marginal densities of the wishart distribution corresponding to cycle graphs

Ivan Georgiev, Juri Kandilarov .................................................23 Immersed interface finite element method for diffusion problem with localized terms

Veselina Evtimova ..................................................................33 Exploring the possibilities for A timely provision of service to patients at an emergency medical aid centre

Tsetska Rashkova ..................................................................38 Teaching group theory via transformations

Stefka Karakoleva, Ivan Georgiev, Slavi Georgiev, Pavel Zlatarov .........................................................................48 Results from computer mathematics education for motivated students at Ruse University

Informatics

Valentin Velikov, Mariya Petrova .............................................58 Subsystem for graphical user interfaces creating

Victoria Rashkova ....................................................................66 Data protection with digital signature

Desislava Baeva ......................................................................75 Translating a SQL application data to semantic Web

Kamelia Shoylekova ................................................................80 Information system “Kaneff centre”

Rumen Rusev ..........................................................................85 Software system for digital analysis of fingernail imprints in forensic medicine

Metodi Dimitrov ........................................................................90 Daily life applications of the modular self reconfigurable robots

Galina Atanasova ....................................................................94 The critical thinking essence and its relationship with algorithm thinking development

Galina Atanasova ....................................................................99 Critical thinking skills improvement via algorithmic problems

Georgi Dimitrov, Galina Panayotova ...................................... 106 Aspects of Website optimization

MATHEMATICS , INFORMATICS AND PHYSICS

PR O C E E D I N G S O F TH E UN I O N O F SCI E N TI S TS – RU S E V OL . 12 / 2015 6

BOOK 5

"MATHEMATICS, INFORMATICS AND

PHYSICS"

VOLUME 12

Physics

Galina Krumova ..................................................................... 114 An approach to description of monopole excitations in nuclei

Nikolay Angelov ..................................................................... 120 Influence of speed and frequency of process laser marking of products of structural steel

Nikolay Angelov ..................................................................... 125 Determination of working intervals of power density and frequency for laser marking on samples from steel HS18-0-1 Applications

Valerij Dzhurov ...................................................................... 131 Radiolocation parameter determination of blasting materials

MATHEM ATICS

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web: suruse.uni-ruse.bg

MARGINAL DENSITIES OF THE WISHART DISTRIBUTION CORRESPONDING TO CYCLE GRAPHS1

Evelina Veleva

Angel Kanchev University of Ruse

Abstract: The aim of the paper is to suggest a technique for the calculation of marginal densities of

the Wishart distribution, corresponding to cycle graphs. Each non-decomposable graph consists of at least one cycle with 4 or more edges. Using the results for decomposable graphs, it is shown how by Monte Carlo method for numerical integration a marginal density, corresponding to any cyclic graph can be computed at a given point.

Keywords: Wishart distribution, non-decomposable graph, marginal density, covariance matrix, graphical Gaussian models

INTRODUCTION

The distribution of the sample covariance matrix for a sample from a multivariate normal distribution is derived in 1928 by Wishart ([11]) and is known as the Wishart distri-

bution. It is a multivariate generalization of the2 ( chi-square) distribution and is present

in almost all textbooks on multivariate statistical analysis. Wishart distribution is subject to research and generalization since its introduction in 1928 until now. The study of this dis-tribution is relevant and up to date as evidenced by the publication of more new materials on the subject (see e.g. [1], [6], [7]). A p p random matrix with Wishart distribution

W ( , )p n , where 1p n and is a positive definite p p matrix, has probability den-

sity of the form

1( 1) / 2 (W )/2

, , / 2 / 2

1(W) (detW)

2 /2 (det )

n p tr

p n np n

p

f en

(1)

for every real p p positive definite matrix W , where ( )p is the multivariate Gamma

function defined as ( 1) / 4

1[ (1 )/2]

pp p

p jj

and det( ) , ( )tr denote the

determinant and the trace of a matrix. When the observed multivariate normal distributed

random variables are mutually independent, as well as its inverse 1 are diagonal ma-

trices. We shall denote the elements of the matrix 1 by ,i j . The equality , 0i j means

that the corresponding pair of variables iX and jX are conditionally independent given all

other 2p variables. The presence of conditional independence between pairs of factors

given the others underlies graphical Gaussian models ([5]). In some scientific fields, such as genetics, majority of pairs of factors are conditionally independent given the rest. The

elements of the sample covariance matrix for which the elements of the matrix 1 are

non-zero form a set of sufficient statistics for the estimation of the covariance matrix (see [2]). When modelling covariance matrices it is sufficient to include in the model only

those covariance coefficients for which , 0i j . The respective marginal density of the

1This paper contains results of the work on project No 2015 - FPHHC – 03, financed by „Scientific Re-

search” Fund of Ruse University.

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Wishart distribution which will be used for the estimation of the parameters of the model

can be obtained by integrating the density (1) with respect to those ,i jw for which , 0i j .

By integrating the density (1), some marginal densities of the Wishart distribution are derived in [8] in explicit form. The bounds of the integration however, cannot always be

exactly obtained. Let ,( )i jWW be a p p positive definite random matrix with an arbi-

trary distribution. Let M be a subset of the set ,{ ,1 }i jW i j p of non-diagonal elements

of W. The marginal density CMf , obtained after integration of the density function of W

with respect to the variables from M , can be represented by a graph MG with the set of

vertices {1, , }V p . For every element ,i jW of the set ,{ ,1 }\i jW i j p M we draw in the

graph an undirected edge between the vertices „ i ” and „ j ”. For instance, the graph on

Figure 1 presents the joint (marginal) density of the random variables from the set

1,2 2,3 3,4 4,5 5,6 6,7 7,8 1,8{ , , , , , , , }W W W W W W W W . For a graph with 8p vertices, the number of all

possible edges is 2

8C 28. In this case the set M consists of the rest 20 variables with

respect to which the density function of W is integrated.

Fig.1. Cycle graph Fig.2. Decomposable graph

The next Proposition is proved in [9].

Proposition 1. The bounds of the integration of the density function of W with re-spect to all the elements of M can be exactly obtained if and only if the corresponding

graph MG is decomposable.

We shall recall some basic concepts from graph theory (see [5]). Let ( , )G V E be a

graph with a finite set of vertices V and a set of undirected edges E . G is called a com-

plete graph if every pair of distinct vertices is connected by an edge. A subset of vertices

U V defines an induced subgraph UG of G which contains all the vertices U and any

edges in E that connect vertices in U . A clique is a complete subgraph that is maximal, that is, it is not a subgraph of any other complete subgraph.

Definition 1. A graph G is decomposable if and only if the set of cliques of G can be

ordered as ( 1C , 2C ,..., kC ) so that for each 2, ,i k if 1

1

i i

i

j

j

S C C

then i lS C for

some l i .

For a decomposable graph MG , if , 0i j for every element ,i jW of M , the marginal

density CMf of the Wishart distribution has a compact form, derived in [9] and given below

by Proposition 2. For an arbitrary p p matrix A and a subset {1,2, , }p we shall

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denote by A[ ] the submatrix of A , composed of the rows and columns with numbers

from . By we denote the number of elements of a set .

Proposition 2. Let ,( )i jWW has Wishart distribution W ( , )p n and M be a sub-

set of the set ,{ ,1 }i jW i j p of non-diagonal elements of W. Let , 0i j for every ele-

ment ,i jW of M . Let the graph MG be decomposable with k cliques 1C , 2C ,..., kC , or-

dered according to Definition 1 and iV be the set of vertices of iC , 1, ,i k . Then the joint

density CMf of the elements of the set ,{ ,1 }\C

i jM W i j p M can be written in the

form

0, , [ ]1

0

0, , [ ]2

(W [ ])

(W )

(W [ ])

i i

C

i i

k

iV n VikM

iU n Ui

f V

f

f U

, (2)

where 0 ,W ( )i jw is a p p symmetric matrix such that , 0i jw for ,i jW M ; , , (W)p nf is

the Wishart density function given by (1) and 1 1( )i i iU V V V , 2, ,i k .

Obtaining of marginal densities corresponding to non-decomposable graphs is dis-cussed in [8]. According to Proposition 1, they cannot be expressed in explicit form. For a

non-decomposable graph MG the marginal density CMf can be calculated numerically for

every set of fixed values for the random variables from the set ,{ ,1 }\C

i jM W i j p M .

These values can be written in the form of a matrix 0W , where the elements corresponding

to the elements of M are zeros. For the calculation of 0(W )CMf in [10], the using of Monte

Carlo method for numerical integration is suggested. Let 1M and 2M be sets such that

1) 1 2 ,{ ,1 }i jM M M W i j p and the densities

1CM

f and 2CM

f correspond to

decomposable graphs;

2) the density 2CM L

f

, where 1\L M M , corresponds to a decomposable graph.

The choice of 1M and 2M is not unique and is always possible. Then 0(W )CMf can

be written as

10

0 0

0

(W ,x)(W ) (W ,x) x

(W ,x)

C

C

M

MD

ff g d

g

, (3)

where D is the domain of definition of 1CM

f , the integration is with respect to all variables

from the set L and 0(W ,x)g is a probability density function on D . In [10], 0(W ,x)g is pro-

posed to be the conditional probability density function

0(W ,x)g 2

0/(W ,x)CL M

f 2

2

0

0

(W ,x)

(W )

C

C

M L

M

f

f

, (4)

derived by Proposition 2 under the condition that , 0i j for every element ,i jW of 2M . By

(3), 0(W )CMf is presented as the expectation of the random variable

10

0

(W , )

(W , )

CMf

g

ξ

ξ, (5)

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where ξ is a random vector with density function 0(W ,x)g . Hence 0(W )CMf can be esti-

mated by the mean value of a large number of realizations of (5). The main difficulty there-fore is the generation of realizations of a random vector with a given probability density

function 0(W ,x)g . Each non-decomposable graph consists of at least one cycle with 4 or

more edges. The aim of this paper is to be applied the proposed approach for the calcula-tion of marginal densities of the Wishart distribution, corresponding to cycle graphs with any number of edges, greater than 3.

MARGINAL DENSITIES CORRESPONDING TO CYCLE GRAPHS

Let W be a random matrix with Wishart distribution W ( , )p n . Let us consider the

joint density of the variables from the set 1,2 2,3 3,4 1, 1, 1,1 ,{ , , , , , , , , }C

p p p p pM W W W W W W W ,

presented by the continuous edges on the graph of Figure 2. We have to choose the sets

1M and 2M , 1 2 ,{ ,1 }i jM M M W i j p , such that the densities

1CM

f and 2CM

f cor-

respond to decomposable graphs. The density 2CM L

f

, where 1\L M M , also have to cor-

respond to a decomposable graph. This conditions will be satisfied if we choose

1,3 1,4 1,5 1, 1{ , , , , }pL W W W W so that 1

CM is presented by all edges on the graph of Figure 2

and 2 1,\{ }C C

pM M W . Let 1,2 2,3 1, 1, 1,1 ,, , , , , ,p p p p py y y y y y

be fixed values for the random

variables from the set CM . We suppose that , 0i jy for i j and 2

, , ,i i j j i jy y y for i j ,

otherwise these values cannot be elements of a positive definite matrix. Let us denote by

0W the matrix 0

0 ,W ( )i jw such that

0 , , ,,

0

C C

i j i j j ii j

y if W M or W Mw

otherwise

.

The variables 1,3 1,4 1, 1, , , px x x

correspond to the elements of the set L with respect to

which we have to integrate the density 1CM

f . Let Wx be the matrix ,W ( )x

x i jw such that

, , ,

0,,

i j i j j ix

i ji j

x if W L or W Lw

w otherwise

.

The graph 1MG , shown on Figure 2, is decomposable with 2p cliques iC ,

1, , 2i p with sets of vertices {1, 1, 2}iV i i , 1, , 2i p . Assuming , 0i j for

, 1i jW M , according to Proposition 2, the density 1CM

f has the form

1

2

3, ,Σ[ ]

12

2, ,Σ[ ]

2

(W [ ])

(W )

(W [ ])

i

C

i

p

n V x i

ix pM

n U x i

i

f V

f

f U

,

where {1, 1}iU i , 2, , 2i p .

The probability density function 0(W ) (W ,x)xg g will be computed under the condi-

tion that , 0i j for every element ,i jW of 2M . The graph 2MG is decomposable with 1p

cliques with sets of vertices { , 1}iT i i , 1, , 1i p respectively. Therefore, according to

Proposition 2

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2

1

2, ,Σ[ ] 0

10 1

1, ,Σ[{ }] 0

2

(W [ ])

(W )

(W [{ }])

i

C

p

n T i

ipM

n i

i

f T

f

f i

.

The graph, corresponding to the marginal density 2CM L

f

consists of all edges on Fig-

ure 2 except the edge connecting the vertices 1 and p . This graph is decomposable with

2p cliques with sets of vertices iV , 1, , 3i p and 1pT respectively. Consequently

1

2

3

2, ,Σ[ ] 1 3, ,Σ[ ]

13

1, ,Σ[{ 1}] 2, ,Σ[ ]

2

(W [ ]) (W [ ])

(W )

(W [{ 1}]) (W [ ])

p i

C

i

p

n T x p n V x i

ix pM L

n p x n U x i

i

f T f V

f

f p f U

,

hence

2

2

2

3 2

3, ,Σ[ ] 1, ,Σ[{ }]

1 2

/ 3 20

2, ,Σ[ ] 2, ,Σ[ ]

2 1

(W [ ]) (W [{ }])(W )

(W ) (W )(W )

(W [ ]) (W [ ])

iC

C

C

i i

p p

n V x i n i xxM L i i

x xL M p p

Mn U x i n T x i

i i

f V f if

g ff

f U f T

. (6)

Each realization of the random variable (5) will have the form

21

2

2

2, ,Σ[ ] 03, ,Σ[ ] 21

2

2, ,Σ[ ] 21, ,Σ[{ }] 0

2

(W [ ])(W ) (W [ ])

(W ) (W [ ])(W [{ }])

iCp

p

p

n T ix n V x pM i

p

x n U x pn i

i

f Tf f V

g f Uf i

. (7)

For the calculation of (7) we must have a realization 1,3 1,4 1, 1{ , , , }px x x x of a random

vector 1,3 1,4 1, 1{ , , , }p ξ with density function (6). Let us consider the transformation

2 2

1,3 1,2 2,3 1,3 1,1 2,2 1,2 2,2 3,3 2,3

2,2

1( )( )x y y z y y y y y y

y

, (8)

2 2

1,4 1,3 3,4 1,4 1,1 3,3 1,3 3,3 4,4 3,4

3,3

1( )( )x x y z y y x y y y

y

, (9)

2 2

1, 1 1, 2 2, 1 1, 1 1,1 2, 2 1, 2 2, 2 1, 1 2, 1

2, 2

1( )( )p p p p p p p p p p p p p p

p p

x x y z y y x y y yy

. (10)

The inverse transformation is

1,3 2,2 1,2 2,3

1,3 2 2

1,1 2,2 1,2 2,2 3,3 2,3( )( )

x y y yz

y y y y y y

,

1,4 3,3 1,3 3,4

1,4 2 2

1,1 3,3 1,3 3,3 4,4 3,4( )( )

x y x yz

y y x y y y

,

1, 1 2, 2 1, 2 2, 1

1, 1 2 2

1,1 2, 2 1, 2 2, 2 1, 1 2, 1( )( )

p p p p p p

p

p p p p p p p p p

x y x yz

y y x y y y

.

It is easy to see that all elements above the main diagonal in the Jacobian J ,

1,3 1,4 1, 1

1,3 1,4 1, 1

, , ,

, , ,

p

p

z z z

x x x

J are equal to zero. Consequently

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1, 11,3 1,4

1,3 1,4 1, 1

detp

p

zz z

x x x

J

2,2 3,3

2 2 2 2

1,1 2,2 1,2 2,2 3,3 2,3 1,1 3,3 1,3 3,3 4,4 3,4( )( ) ( )( )

y y

y y y y y y y y x y y y

2, 2

2 2

1,1 2, 2 1, 2 2, 2 1, 1 2, 1( )( )

p p

p p p p p p p p p

y

y y x y y y

2

21/2 1/2

2 3

1 2

det(W [{ }])

det(W [ ]) det(W [ ])

p

x

i

p p

x i x i

i i

i

T U

.

Theorem 1. Let 1,3 1,4 1, 1, , , pz z z

be realizations of independent and identically distrib-

uted random variables 1,3 1,4 1, 1, , , p

with density function

42 2

1

2(1 )

2 1

2 2

n

n

zn

, ( 1,1)z . (11)

Then the variables 1,3 1,4 1, 1, , , px x x

defined by equalities (8) – (10) are realizations of

random variables 1,3 1,4 1, 1, , , p

with joint density function (6).

Proof. The joint density function of the random variables 1,3 1,4 1, 1, , , p

is

1,3 1,4 1, 1( , , , )pf z z z

3

4 4 42 2 22 2 21,3 1,4 1, 1

1

2(1 ) (1 ) (1 )

2 1

2 2

p

n n n

p

n

z z zn

,

for 1, ( 1,1)iz , 3, , 1i p . Using (8), it is easy to check that

1,1 1,2 1,3

2 2 2

1,2 2,2 2,3 1,1 2,2 1,2 2,2 3,3 2,3 1,3

2,2

1,3 2,3 3,3

1det ( )( )(1 )

y y x

y y y y y y y y y zy

x y y

.

Consequently 2,2 12

1,3

1 2

det(W [ ])1

det(W [ ])det(W [ ])

x

x x

y Vz

T T .

Analogously, for 2, , 3i p

1,1 1, 1 1, 2

2 2 2

1, 1 1, 1 1, 2 1,1 1, 1 1, 1 1, 1 2, 2 1, 2 1, 2

1, 1

1, 2 1, 2 2, 2

1det ( )( )(1 )

i i

i i i i i i i i i i i i i i i

i i

i i i i i

y x x

x y y y y x y y y zy

x y y

and hence for 2, , 3i p

1, 12

1, 2

1

det(W [ ])1

det(W [ ])det(W [ ])

i i x i

i

x i x i

y Vz

U T

.

The joint density of the random variables 1,3 1,4 1, 1, , , p gets the form

3

1,3 1,4 1, 1

1

2( , , , ) det

2 1

2 2

p

p

n

f x x xn

J

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21

4

22, 2 32,2 1 3,3 2

1 2 2 3 3 2

det(W [ ])det(W [ ]) det(W [ ]).

det(W [ ])det(W [ ]) det(W [ ])det(W [ ]) det(W [ ])det(W [ ])

n

p p x px x

x x x x x p x p

y Vy V y V

T T U T U T

2 4

3 2 32 2

0

2 1

3 32 32 2

0

1 2

1 det(W [{ }]) det(W [ ])2

.2 1

det(W [ ]) det(W [ ])2 2

n np p p

x i

i i

n np p

i x i

i i

n i V

n

T U

.

With respect to the variables 1,3 1,4 1, 1, , , px x x

the density function (W )xg , given by (6),

can be written in the form

(W )xg

43 2

1

33 2

2

det(W [ ])

det(W [ ])

np

x i

i

np

x i

i

V

C

U

,

if the matrices W [ ]x iV are positive definite. Here C is a function of the values

1,2 2,3 1, 1, 1,1 ,, , , , , ,p p p p py y y y y y but does not depend on

1,3 1,4 1, 1, , , px x x . This completes the

proof. The distribution with density function (11) is known as Pearson distribution of the

second type (see [4]) and also as power semicircle distribution (see [3]). Random variables with this distribution can be easily generated (see [4], p. 481) using the quotient of the dif-ference and the sum of two Gamma distributed random variables.

For every 3p realizations 1,3 1,4 1, 1, , , pz z z

of a random variable with density function

(11) by formulas (8) – (10) we get a realization 1,3 1,4 1, 1{ , , , }px x x x of a random vector

1,3 1,4 1, 1{ , , , }p ξ with density function (6).

If 1x , …, Nx are N realizations of ξ then according to (7)

0(W )CMf 2

2

2

2, ,Σ[ ] 03, ,Σ[ ] 21

21 2, ,Σ[ ] 2

1, ,Σ[{ }] 0

2

(W [ ]) (W [ ])1

(W [ ])(W [{ }])

ip i

p i

p

n T i Nn V pi

pi n U p

n i

i

f T f V

N f Uf i

x

x

.

REFERENCES [1] S. A. Andersson, T. Klein. On Riesz and Wishart distributions associated with

decomposable undirected graphs. Journal of Multivariate Analysis, Volume 101, Issue 4, April 2010, 789-810.

[2] T. W. Anderson, An Introduction to Multivariate Statistical Analysis. John Wiley & Sons, N. Y., 3ed., 2003.

[3] O. Arizmendi, V. Perez-Abreu. On the non-classical infinite divisibility of power semicircle distributions. Commun. Stoch. Anal. 4 (2010), 161-178, in honor of Gopinath Killianpur.

[4] L. Devroye. Non-Uniform Random Variate Generation. New York, Springer-Verlag, 1986.

[5] S. L. Lauritzen. Graphical Models. Clarendon Press, Oxford, 1996.

MATHEM ATICS

P R O C E E D I N G S O F T H E U N I O N O F S C I E N T I S T S – R U S E V O L . 12 / 2015

22

[6] G. Letac, H. Massam. Wishart distributions for decomposable graphs. Ann. Statist. 35 (2007), 1278-1323.

[7] H.F. Strydom, N.A.S. Crowther. Maximum likelihood estimation of parameter structures for the Wishart distribution using constraints. J Stat Plan Infer, 143 (4), 2013, 783–794.

[8] E. Veleva. Some marginal densities of the Wishart distribution. Math. and Education in Math., 41 (2012), 273-278.

[9] Е. Veleva. Marginal densities of the Wishart distribution. Pliska Stud. Math. Bulgar., 22 (2013), 225-236.

[10] Е. Veleva. Wishart distribution on non-decomposable graphs. Proc. of Annual Conf. Rousse Univ., Rousse, Bulgaria, 2013, vol. 52, book 6.1, 39 - 43. (in Bulgarian).

[11] J. Wishart. The generalized product moment distribution in samples from a normal multivariate population. Biometrika, 20 (1928), 32–52.

CONTACT ADDRESS Assoc. Prof. Evelina Veleva, PhD Department of Applied Mathematics and Statistics Angel Kanchev University of Ruse 8, Studentska Str., 7017 Ruse, Bulgaria Phone: (++359 82) 888 606 E-mail: eveleva@uni-ruse.bg

МАРГИНАЛНИ ПЛЪТНОСТИ НА РАЗПРЕДЕЛЕНИЕТО НА УИШАРТ, СЪОТВЕТСТВАЩИ НА ЦИКЛИЧНИ ГРАФИ

Евелина Велева

Русенски университет “Ангел Кънчев”

Резюме: Целта на статията е да предложи техника за изчисляване на маргиналните

плътности на разпределението на Уишарт, съответстващи на циклични графи. Всеки неразло-жим граф се състои от поне един цикъл с 4 или повече дъги. Използвайки резултатите за разло-жими графи е показано как с помощта на метода Монте Карло за числено интегриране, марги-нална плътност, съответстваща на произволен цикличен граф, може да бъде пресметната в зададена точка.

Ключови думи: разпределение на Уишарт, неразложим граф, маргиналнa плътност, кова-риационна матрица, графични Гаусови модели.

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