Contramonotonic systems in the analysis of the structure of multivariate distributions
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.
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ABSTRACT: The book presents a concise yet mathematically complete treatment of modern utility theories that covers nonprobabilistic preference theory, the von NeumannMorgenstern expectedutility theory and its extensions, and the joint axiomatization of utility and subjective probability.Operational Research Quarterly (19701977) 01/1971; 22(3):308309.
Page 1
SIMULATION OF BEHAVIOR AND INTELLIGENCE
Contramonotonic 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
Email: 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
ndimensional 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 socalled Equilibrium State in the sense of Nash [6].
Approaching the analysis of the structure of a measurement density function in
ndimensional 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 onedimensional” 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 onedimensional projections. Indeed, an
equilibrium state identified by means of the given vector index is parameterized by socalled
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 socalled contramonotonic
system; actually, the first part on multiparameter contramonotonic 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 soughtafter 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. Contramonotonic systems over a family of parameters
Here a monotonic system represents first a oneparameter 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 oneparameter 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 contramonotonic 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 contramonotonicity 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 socalled decoupled multiparameter family of
functions p . The family is said to be decoupled 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 decoupled multiparameter 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 multimodal 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 contramonotonicity 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 contramonotonicity 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 contramonotonic family of functions.
To find the central clusters of a multivariate distribution in ndimensional space we
invoke the notion of a multiparameter contramonotonic 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 onedimensional 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 socalled Nashoptimal
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Equilibrium State there is a simple technique for finding solutions at least in decoupled
family of contramonotonic 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 decoupled family of contramonotonic 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 decoupled
family of contramonotonic 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. Contramonotonic systems over a family of segments
A multiparameter family of contramonotonic 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
Page 8
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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 settheoretic 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 oneparameter family of contramonotonic 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 contramonotonic 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 multiparameter 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
contramonotonicity 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 decoupled family of contramonotonic 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 multimodal empirical distribution in a multidimensional space.
We first investigate the case of a onedimensional (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 contramonotonicity 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 contramonotonicity 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
Page 10
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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 decoupled family of contramonotonic 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 zerovaluedness 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 decoupled family of contramonotonic 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 decoupled family of contramonotonic 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 decoupled
families of contramonotonic functions
The ensuing discussion rests heavily on the contramonotonicity 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 contramonotonicity 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 contramonotonic 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 contramonotonic 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 contramonotonic 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 =
.
Page 13
12/12
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 contramonotonicity 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 contramonotonicity 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 .
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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 DecisionMaking, 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.