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SIMULATION OF BEHAVIOR AND INTELLIGENCE

Contra-monotonic systems in the analysis of the structure

of multivariate distributions

J. E. Mullat

Department of Economics, Tallinn Technical University (1973 – 1980)

Translated from Avtomatica i Telemekhanika, No. 7, pp. 167 – 175, July, 1981.

Original article submitted June 9, 1980.

005 – 1179/81/4207 – 0986 $7.50 1982 Plenum Publishing Corporation1179

We alert the readers’ obligation with respect to copyrighted material.

Correspondence Address:

Byvej 269

2650 Hvidovre

Copenhagen Denmark

Phone (45) 36 49 33 05

E-mail: joseph_mullat@email.dk

Abstract

The problem of distinguishing condensations in multivariate space of measurements based

on a qualitative vector criterion is presented. We find solutions by a special parameterization

of functions, the values of which decrease in all regions of the definition in inverse

proportion to the values of the parameters.

Keywords: monotonic, distributions, equilibrium, Nash, cluster

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1. Introduction

The analysis of the structure of the probability density function of measurements in an

n-dimensional space is a traditional topic of investigation in such applied fields as

experimental design [1], image analysis [2], the analysis of decision making [3], pattern

recognition [4], etc.

At the conceptual level, the structure of a distribution is customarily represented by a set of

data clusters, sometimes called modes [5]. The analysis of such a structure, indirectly if not

explicitly, is usually reduced to an optimization variational problem. i.e., the maximization

of some scalar performance indexes characterizing the identified clusters. Instead of scalar

performance index, in this article we use a vector index, and base the concept of optimality

on the so-called Equilibrium State in the sense of Nash [6].

Approaching the analysis of the structure of a measurement density function in

n-dimensional space, our standpoint is the equilibrium state concept. It is justified by the

fact that, essentially, what happens, is the replacement here of a single multidimensional

problem by many “almost one-dimensional” problems in projections onto the coordinate

axes. On each axis a cluster is delineated in such a way as to “bind” the axes together in a

rigorously defined way. So, exposed to such a “bind” the cluster on a given axis cannot be

“nudged” without in some measure deteriorating itself on the other axes in the sense of

investigated performance index, subject to the condition that these others are fixed.

The superiority of the proposed approach is not restricted to the indicated “technical detail”

of replacing one multidimensional space by one-dimensional projections. Indeed, an

equilibrium state identified by means of the given vector index is parameterized by so-called

thresholds, which satisfy the density levels of the clusters. In certain special cases, at any

rate, an equilibrium state as the solution of a system of equations can be expressed

analytically in the form of threshold functions, whereupon the identified clusters can be fully

scanned in the spectrum of possible density levels.

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The proposed theory for the identification of clusters of the probability density of

measurements in n -dimensional space is set forth in two parts. In the first part (Sec. 2) the

theory is not taken beyond the scope of customary multivariate functions and it concludes

with a system equations, namely the system whose solution in the form of threshold

functions makes it possible to scan the identified clusters. In the second part (Sec. 3) the

theory now rests on a more abundant class of measurable functions specified by the class of

sets represented on the coordinate axes by at most countable set of unions or intersections of

segments. Overall the construction described in this part is so-called contra-monotonic

system; actually, the first part on multiparameter contra-monotonic systems is also discussed

in these terms (special case).

The fundamental result of the second part does not differ, in any way, from the form of the

system of equations in the first part; the essential difference is in the space of admissible

solutions. Whereas in the system of equations of the first part the solution is a numerical

vector, in the second part it is a set of measurable sets containing the sought-after measurable

density clusters. As the solution of the system of equations, the set of measurable sets serves

as a fixed point of special kind mapping of subsets of multidimensional space. This

particular feature is utilized in an iterative solving procedure.

2. Contra-monotonic systems over a family of parameters

Here a monotonic system represents first a one-parameter and then a multiparameter

family of functions defined on real axis. This type of representation is a special case of a

more general monotonic system described in the next section.

We consider a one-parameter family of functions

)h ; x(

p

defined on the real axis, where

h is a parameter. For definiteness, we assume that an individual copy p of the indicated

family is a function integrable with respect to x and differentiable with respect to h. The

family of functions p is said to be contra-monotonic if it obeys the following condition: for

any pair of quantities l and g such that

gl ≤

the inequality

)g; x ( ) l ; x(

pp

≥

holds for any x.

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The specification of a multiparameter family of functions p is reducible to the following

scheme. We replace the one function p by a vector function

n

,...,,

pppp

21

=

, each j -th

component of which is a copy of the function depending now on n parameters

n21

h ,...,h ,h

,

i.e.,

)h,..., h ,

1

h ; x(

njj

2

pp =

. The contra-monotonicity condition for any pair of vectors

nl,...,l ,

1

ll

2

=

and

n g,..., g ,

1

gg

2

=

such that

kk

gl ≤

)n,...,2 , 1

=

k(

is written in the form

of n inequalities:

)g ,...,g ,

1

g; x()l,..., l ,

1

l ; x(

n2jn2j

pp

≥

.

We note that this condition rigorously associates with the family of vector functions a

componentwise partial ordering of the vector parameters.

We give special attention to the case of a so-called de-coupled multi-parameter family of

functions p . The family is said to be de-coupled if the j -th component of a copy of vector

function p does not depend on the j -th component of the vector of parameters h, i.e., on

hj. Therefore, a copy of function p of a de-coupled multi-parameter family is written in the

form

)h,..., h ,

1

h,...,h , x(

n1jj1j

+−

p

)n,...,1j(

=

.

We now return to the original problem of analyzing a multi-modal empirical distribution in

multidimensional space. We first investigate the case of one axis (univariate distribution).

Let

)x ( p

be the probability density function of points in the x axis. For the contra-

monotonic family p we can choose, for example, the functions

h)x( p)h ; x(

=

p

. It is easy

verified that the contra-monotonicity condition is satisfied.

We consider the following variational problem. With respect to an externally specified

threshold

o

u

)1u0(

o≤≤

let it be necessary to maximize the functional

dx]u)h ; x([)h(

h

h

o

∫

−

+

−=

pP

.

It is clear that for small h the quantity

)h(

P

will be small because of the narrow interval of

integration, while for the large h it will be small by the contra-monotonicity condition.

Consequently, the value of

)h(maxhP

will necessarily be attained foe certain finite

nonzero

o

h .

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It is readily noted that if

)x ( p

is a unimodal density function with zero expectation, then

the maximization of the functional

)h(

P

implies the identification of an interval on the axis

corresponding to a concentration of the density

)x( p

. But if

)x ( p

has a more complicated

form, then the maximum of

)h(

P

specifies an interval in which is concentrated the

“essential part”, in some definite sense, of the density function

)x( p

.

Directly from the form of function

)h(

P

we deduce the following necessary condition for

local maximum (the zero equation of the derivative with respect to h:

0)h(

h

=

∂

∂P

or, in the

expanded form, the equation

o

hy

h

h

y

u2dx| )y; x() h ; h()h ; h

−

(

=++

=

+

∂∫p

−

∂

pp

.(1)

The root of the given equation will necessarily contain one at which

)h(

P

attains a global

maximum. We have thus done with the problem: we found the central cluster points of the

density function on one axis in terms of a contra-monotonic family of functions.

To find the central clusters of a multivariate distribution in n-dimensional space we

invoke the notion of a multiparameter contra-monotonic family of functions p . Let the

family of functions p in vector form be written, say, in the form

∑=

1k

th axis. In the stated sense the goodness of the delineated central cluster is evaluated by the

h

jn1j

)x(p)h,..., h ; x(

=

p

,

where

=

n

k,hh

and

)x(pj

is a projection of the multivariate distribution on the axis j -

multivariate (vector) performance index

n

,...,PPP

1

=

, where

dx]u)h ,...,h ; x([)h,..., h ,

1

h(

j

j

h

∫

−

h

jn1jn2j

−=

pP

(2)

and

j u is the component of the corresponding externally specified multidimensional

threshold vector u:

n u,..., u ,

1

uu

2

=

. As in the one-dimensional case, of course, it is

meaningful to use the given functional only distributions

)x(pj

with zero expectation.

Once the goodness of a delineated cluster has been evaluated by the vector index, it must

be decided, based on standard [7] vector optimization principles, what is an acceptable

cluster. In this connection it desirable to indicate simultaneously a procedure for finding an

extremal point in the space of parameters. It turns out that for so-called Nash-optimal

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Equilibrium State there is a simple technique for finding solutions at least in de-coupled

family of contra-monotonic functions p .

En equilibrium situation (Nash point) in the parameter space

n h ,...,hh

1

=

with indices

j

P is defined as a point

*

n

*

2

*

1

*

h,..., h ,hh

=

such that for every j the inequality

)h,...,h ,...,h()h ,...,h ,

j

h ,

1

h ,...,h(

*

n

*

j

*

1j

*

n

*

j1

*

j

*

1j

PP

≤

+−

holds for any value of

j h . In other words, if there are no sensible bases in the sense of index

j

P on the one ( j -th) axis, then the equilibrium situation is shifted with respect to the

parameter

j h , subject to the condition that the quantities

*

kh ,

jk ≠ , are fixed on all other

axes.

Clearly, a necessary condition at a Nash point in the parameter space (as in the one-

dimensional case) is that the partial derivatives tend to zero, i.e., the n equalities

0)h ,...,h(h/

*

n

*

1jj

=∂∂

P

must hold. The sufficient condition comprises the n inequalities

0)h,...,h(h/

*

n

*

1j

2

j

2

≤∂∂

P

.

An essential issue here, however, is the fact that the necessary condition (equalities)

acquires a simpler form for de-coupled family of contra-monotonic functions than in the

general case. Thus, by the decoupling of the family p the partial derivative

jj

h/ ∂ ∂P

is

identically zero, and the system of equations, see (1) by analogy, with respect to the sought-

after point

*

h is reducible to the form

jn1j1j1jjn1j1j1jj

u2)h,... h ,h ,...,h ;h()h ,...h ,h,..., h ;h(

=+−

+−+−

pp

(3)

Now the sufficient condition is satisfied automatically for any solution

*

h of Eqs. (3).

In conclusion we write out the system of equations for two special cases of a de-coupled

family of contra-monotonic functions p .

1. Let

j h

−

jn1j1j1j

)x(p)h,...,h ,h,...,h ; x(

+−

=

s

p

, where

n21

h...hh

+++=

s

. Then the system

of equations (3) is reducible to the form

j

h

jj

h

jj

u2)h(p)h(p

jj

=+−

−−

ss

)n,...,1j(

=

.

2. Let the role of

)h,...,h ,

1

h ,...,h ; x(

n1jj1j

+−

p

be taken by the function

n

1j1j

21

h

n

h

1j

h

1j

h

2

h

1

)x(p...)x(p)x(p...)x(p)x(p

+−

+−

.

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The system of equations (3) for finding a solution, i.e., an equilibrium situation (Nash

point)

*

h , is written

j

h

jjj

h

jjj

u2)h(p / )h ( p)h(p / )h( p

jj

=+−−

)n,...,1j(

=

,

where

n21

h

n

h

2

h

1

)x(p...)x(p)x(p)x( p

=

is the product of univariate density functions.

We conclude this section with an important observation affecting the vector of thresholds

>=<

n21

u,...,u ,uu

. By straightforward reasoning we infer that each component

*

j h of the

equilibrium situation

*

h is a function of thresholds and

*

h can be represented by a vector

function of thresholds in the form

)u ,...,u ,

1

u(hh

n2

*

j

*

j=

. If the solution of the system of

equations (3) can be expressed analytically, then prolific possibilities are afforded for

scanning the equilibrium situations in the parameter space and, accordingly, selecting an

“acceptable” cluster in the spectrum of existing densities of measurements in a

multidimensional space of thresholds. A similar approach can be used when solutions of Eqs.

(3) are sought by numerical methods.

3. Contra-monotonic systems over a family of segments

A multi-parameter family of contra-monotonic functions used for the analysis of

multivariate distributions, unfortunately, has one substantial drawback. Generally speaking,

there is no way to guarantee the identification of homogeneous distribution clusters in

projection onto the j -th axis, because the segment

]h ,

j

h[

j

−

can contain several distinct

modes. On the other hand, it is sometimes desirable to identify modes by merely indicating a

family of segments containing each mode separately. The construction proposed below

enlarges the possibilities for the solution of such a problem by augmenting the contra-

monotonic systems of the proceeding section in natural way.

Thus, on real axis we consider subsets represented by at most countable set of operations

of union, intersection, and difference of segments. The class of all such subsets is denoted by

B , and each representative subset by

BH ∈

(which we call a B set) is distinguished from

like sets by length m (by measure zero). A set L is congruent with G

)LG(

=

if the

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measure of the symmetric difference

LGD is equal to zero

)0LG(

=

Dm

; a set L is

contained in G

)GL(

⊆

with respect to measure m if

0L\G

=

m

. A measure on the real

axis, being an additive function of sets (the length), is determined by taking to the limit the

length of the sets in the set of unions, intersections, and differences of segments forming the

B set. Then set-theoretic operations over B sets will be understood to mean up to measure

zero. By convention, all B sets of measure zero are indistinguishable.

We associate with every B set H a nonnegative function

)H; x(

p

, which is Borel

measurable (or simply measurable) and whose domain of definition is on the real axis.1 In

other words, in contrast with the one-parameter family of contra-monotonic functions of the

preceding section, the parameter h is now generalized, namely, it is extended to the B set

H . As before, we say that a family of measurable functions p is contra-monotonic if it

obeys the following condition: for any pair of sets L and G such that

GL ⊆

the inequality

)G ; x()L ; x(

pp

≥

holds for any x.

The scheme of specification of a multi-parameter family of functions is analogous to the

previous situation. In place of a scalar function p we now specify a vector function

n

,...,,

pppp

21

=

, each j -th component of which is a copy of a function depending at the

outset on n parameters

n

H ,...,H,H

21

(B sets), i.e.,

)H ,...,H,H; x(

n21jj

pp =

. Again, the

contra-monotonicity condition is reducible to the statement that for any pair of vectors

(ordered sets of B sets) of the form

n L,...,LL

1

=

and

n

G ,...,GG

1

=

such that

kk

GL ⊆

)n ,..., 2 , 1

=

k(

, the following n inequalities are satisfied:2

)G ,...,G; x()L,...,L; x(

n1jn1j

pp

≥

.

These inequalities associate a partial ordering of sets of B sets with a family of vector

functions p in a rigorously defined way.

1 A function

; x(

p

H; x(

p

2 Here x is a point on the j -th axis. This is tacitly understood everywhere.

)H

)

is Borel measurable if for any numerical threshold

u

is measurable:

(:x{

p

o

u the set of all x of the real scale

for which

o

>

}u)H; x

o

>

is B set.

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In the case of a de-coupled family of contra-monotonic functions, where the j -th

component of a copy of the vector function p does not depend on the parameter

j

H , or B

set on the j -th axis of definition of the function

j

p , this component

j

p of the vector

function p is written

)H,...,H,H ; x(

n21jj

pp =

.

Following again the order of discussion of Sec. 2, we now consider the original problem of

analyzing the structure of a multi-modal empirical distribution in a multidimensional space.

We first investigate the case of a one-dimensional (univariate) distribution.

Let

)x( p

be the density function of points on the x axis. In the role of the contra-

monotonic family of functions p , we adopt functions of the form

∫

H

)H(F)x ( p)H ; x(

=

p

,

where

=

dx)x( p)H(F

is the probability of a random variable occurring in a B set under

the probability density function

)x ( p

. It is clear that the contra-monotonicity condition is

satisfied.

We consider the following variational problem. Given the externally specified threshold

o

u

)1u0(

o≤≤

, maximize the functional

∫

H

−=

od]u)H ; x([)H(

mpP

.

The integral here is understood in the Lebegue sense with respect to measure m, where m,

as mentioned before, is the length of the B set on the x axis.

Clearly, the quantity

)H(

P

as a function of the length m (measure of set H ) increases

first and then, as

∞→

H

m

, reverts to zero by the contra-monotonicity condition on the

family of functions p. Therefore, the value of

)H(maxHP

will necessary be attained on a

certain B set of finite measure m (see the analogous assertion in Sec. 2).

It is impossible in the same simple way to deduce directly from the form of the functional

)H(

P

any maximum condition comparable with the like condition of the preceding section

(Eq. 1). To do so would require elaborating the notation of a “virtual translation” from a B

set H to a set H~ similar to it in some sense, in such a way as to establish the necessary

maximum condition. These circumstances exclude the case of a univariate distribution from

further consideration. Nonetheless, as will be shown presently, for multivariate distribution

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there are means for finding a B set that will maximize the function

)H(

P

at least in the

case of a de-coupled family of contra-monotonic functions.

As in the preceding section, we evaluate the goodness of an identified central cluster by the

multivariate (vector) performance index

n

,...,,

PPPP

21

=

:

∫

H

−=

j

jn1n21j

d]u)H,...,H; x([)H,...,H,H(

mpP

,

where

j u is the coordinate of the corresponding multidimensional vector of thresholds u,

specified externally:

n u,..., u ,

1

uu

2

=

.

At this point we call attention to the fact that, in contrast with the analogous multivariate

index of Sec. 2, the given functional now has significance for an arbitrary distribution, rather

than only for the centered condition of zero-valuedness of the expectation. We again look for

the required cluster in multidimensional space as an equilibrium situation according to the

vector index

n

,...,,

PPPP

21

=

, We regard a cluster as a set of B sets

*

n

*

2

*

1

*

H ,...,H,HH

=

such that the following inequalities holds for every j :

)H,...,H,...,H()H,...,H,H,H ,...,H(

*

n

*

j

*

1j

*

n

*

j1j

*

j1

*

1j

PP

≤

+−

)n ,...,1j(

=

.

In a de-coupled family of contra-monotonic functions it is feasible (as in the multi-

parameter case; see Eq. (3) ) to find an equilibrium situation. Equilibrium situations are

sought to be a special technique of mappings of B sets onto real axes.

We define the following type of mappings of B sets onto real axes:

}u)H ; x(:x{)H(V

jjjjj

>=

p

,

where

j u is the threshold involved in the expression for the functional

j

P

)n ,..., 2 , 1

=

j(

.

Thus defined, n such mappings are uniquely expressible in the vector form

}u)H; x (:x{)H(V

>=

p

.

Here

n21

H...HHH

×××=

denotes the direct product of sets

j

H . We define a fixed point

of the mapping

)H(V

as a set

*

H for which the equality

)H(VH

**=

holds.

*

nn

*

jj11jj

*

jj11

*

11

HH,..., ,...,HH,,HH,,HH,...,,..., HH

++−−

****

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Theorem 1. For a de-coupled family of contra-monotonic functions p , a fixed point of the

mapping

)H(V

generates an equilibrium situation according to the vector index

n

,...,,

PPPP

21

=

.

The proof of the theorem is simple. Thus, because

j

p is independent of the parameter

j

H ,

the form of the function

)H ,...,H,H,...,H; x(

*

n

*

j1

*

j1

*

1j

+−

p

does not depend on

j

H . Also, the

set

*

n

*

2

*

1

*

H ...HHH

×××=

in projection onto the j -th axis intersects the set

*

j

H consisting

exclusively of all points x for which

j

*

jj

u)H ; x(

>

p

:

}u)H; x(:x{H

j

*

jj

*

j

>=

p

. It is

immediately apparent that any

j

H distinct from

*

j

H the value of the functional

)H ,...,H,H,H ,...,H(

*

n

*

j1j

*

j1

*

1j

+−

P

for immovable sets

*

k

H

) jk(

≠

cannot be anything but

smaller than the quantity

)H,...,H,H,H,...,H(

*

n

*

j1

*

j

*

j1

*

1j

+−

P

.

It is important, therefore, to find the fixed points of the constructed mapping of B sets.

4. Methods of finding equilibrium state for de-coupled

families of contra-monotonic functions

The ensuing discussion rests heavily on the contra-monotonicity property of a function p .

To facilitate comprehension of the formulations and propositions we use the language of

diagrams reflecting the structure of the relations involved in the constructed mappings of B

sets, in particular the symbol → denoting the relation “set

1

X is nested in set

2

X

)XX(

21⊆

“:

21

XX →

.

All diagrams of the relations between B sets are based on the following proposition: the

relation

21

XX →

(as a consequence of the contra-monotonicity condition on p ) implies

that

)X(V)X(V

21←

.

Now let the mapping V be applied to the original space W of axes on which the functions

j

p

)n,...,2 , 1

=

j(

are defined. After the image

)W(V

has been obtained, we again apply the

mapping V with the B set

)W(V

as its inverse image, i.e., we consider the image

)W(V2

,

and so on. In this way we construct a chain of B sets W ,

)W(V

,

)W(V2

,..., which we call

the central series of the contra-monotonic system.

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The following diagram of nestings of B sets of the central series is inferred directly from

the above stated proposition:

↓ ↓ ↓

V)W(V)W(V)W

→←→

↑_____________ ↑________________ ↑_________________

)W(V)W((VW

5432

←←

...

It is evident from the diagram that there exist in the central series two monotonic chains of

B sets: one shrinking and one growing. The monotonically shrinking chain of B sets

comprises the sequence

...)W(V)W(V

42

←←

with even powers of the mapping V . The

monotonically growing chain is the sequence

...)W(V)W(V)W(V

53

→→→

with odd

powers of V .

It is well known [8] that monotonically decreasing (increasing) chains in the class of B

sets always converge in the limit of sets of the same class. For example, the limit of the sets

)W(V

k2

with even powers is the intersection

)W(VL

k2

1k

∞

=

= I

, and the limit of sets

)W(V

1k2 −

with odd powers is the union

)W(VG

1k2

1k

−∞

=

= U

.

Theorem 2. For the central series of a contra-monotonic system the nesting

GL ⊆

of the

limiting B set L of even powers of the mapping

)X(V

in the limiting B set G of odd powers

of the same mapping is always true.

The theorem follows at once from the diagram of nestings of the central series.

We now resume our at the moment interrupted discussion of the problem of finding a fixed

point of a mapping of B sets, such point generating an equilibrium situation according to the

vector index P (Theorem 1). In contra-monotonic systems, as a rule, the strict nesting

GL ⊂

of limiting B sets holds in the statement of Theorem 2. The equality

GL =

would

imply convergence of the central series in the limit to a single set, namely a fixed pint. In

view of the exceptional status of the equality

GL =

, we give a “more refined” procedure,

which automatically in the number of cases of practical importance yields the desired result,

a solution of the equation

)X(VX =

.

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Procedure for Solving the Equation

)X(VX =

. A chain of B sets

,...,H,H

10

is

generated recursively according to the following rule. Let the set

k

H (where

0

H is any B

set of finite measure) be already generated in the chain. We use the mapping

)X(V

to

transform the following B sets:

)}H(V)H(V{V

kk

2

U

,

}H)H(V{V

kk I

,

}H)H(V{V

kk U

,

)}H(V)H(V{V

kk

2

I

,

which we denote, in order, by

2

kkk

2

k

G,L ,G,L

. By the contra-monotonicity of the family of

functions p it turns out that

2

k L is a subset of

k

G and that

k L is a subset of

2

k

G . Picking any

k A based on the condition

kk

2

k

GAL

⊂⊂

, and then

k

B from the analogous condition

2

kkk

GBL

⊂⊂

, we put the set

1k

H+ following

k

H in the constructed series of B sets equal to

kk

BA U

:

kkk

BAH

U

=

. The sets

k A and

k

B can be chosen, for example, according to

mapping rules in the class of B sets, namely,

}u )]G ; x()L; x([:x{A

kkk

>+=

pp

2

2

1

,

}u)] G; x()L; x([

2

1

:x{B

2

kkk

>+=

pp

.

The conditions imposed on

k A and

k

B are satisfied in this case.

Theorem 3. For the series of sets

)H(V

k to contain the limiting set

)H(V

* as

∞→

k

,

which would be a solution of the equation

)X(VX =

, the following two conditions are

sufficient:

a)

kk

G lim

m

∞→

\

0L2

k=

,

b)

2

kk

G lim

m

∞→

\

0Lk=

.

The plan of the proof is quickly grasped in the following nesting diagrams, which are

consequences of the contra-monotonicity property of the functions p , i.e.,

I.

)H(VGL)H(V

kk

2

kk

2

←→←

,

II.

)H(VGL)H(V

k

22

kkk

←→←

.

Diagrams I and II imply the validity of the two chains:

1)

)H(V

k

2

\

)H(V)H(V

k

2

k ⊆

\

2

kk

LG ⊆

\

k

G ,

2)

)H(V

k \

)H(V)H(V

kk

2

⊆

\

k

2

k

LG ⊆

\

2

k

G .

Page 14

13/12

The first chain implies that for the limiting set

*

H of the series

,...,H,H

10

the equality

)H(V

k

2

m

\

0)H(V

*

= holds, i.e.,

)H(V)H(V

*2*⊂

; the second chain implies the opposite

relation:

)H(V)H(V

**2

⊆

. Consequently,

)H(V

* is the solution of the equation

)X(VX =

:

))H(V(V)H(V

**

=

. Of course, the conditions of the theorem are sufficient for

the existence of a solution of the equation

)X(VX =

, and their absence does not in any way

negate some other solving technique, provided that solutions exist in general. The possibility

that solution

*

H of the equation

)X(VX =

do not exist should certainly not be dismissed.

LITERATURE CITED

1. Finney, D.J., 1964, An Introduction to the theory of Experimental Design, Univ.

Chicago Press.

2. Rosenfeld, A., 1969, Picture Processing by Computer, Academic Press, New York.

3. Fishburn, P.C., 1970, Utility Theory for Decision-Making, Wiley, New York.

4. Aizerman, M.A., Braverman, E.M., Rozonoer, L.I., 1970, The Method of Potential

Functions in Machine Training Theory [in Russian], Nauka, Moscow.

5. Zagoruiko, N.G., and Zaslavskaya, T.I., 1968, Pattern Recognition in Social Research [in

Russian], Sib. Otd. Akad. Nauk SSSR, Novosibirsk.

6. Owen, G. Game theory, 1968, Saunders, Philadelphia.

7. Becker, G.M., and McClintock, C.G., 1967, “Behavioral decision theory,” in: Annual

Review of Psychology, Vol. 18, Stanford, Calif.

8. Shilov, G.E., and Gurevich, B.L., 1967, Integral, Measure, and Derivative [in Russian],

Nauka, Moscow.