=Paper=
{{Paper
|id=Vol-2386/paper13
|storemode=property
|title=Multi-Fuzzy Sets as Aggregation Subjective and Objective Fuzziness
|pdfUrl=https://ceur-ws.org/Vol-2386/paper13.pdf
|volume=Vol-2386
|authors=Yuri M. Minaev,Oksana Yu. Filimonova,Julia I. Minaeva
|dblpUrl=https://dblp.org/rec/conf/momlet/MinaevFM19
}}
==Multi-Fuzzy Sets as Aggregation Subjective and Objective Fuzziness==
https://ceur-ws.org/Vol-2386/paper13.pdf
Multi-Fuzzy Sets as Aggregation Subjective and
Objective Fuzziness
Yuri M. Minaev1[0000-0002-1168-1927], Oksana Yu. Filimonova2[0000-0001-6630-3344],
Julia I. Minaeva 2[0000-0002-2367-1507]
1 National Aviation University, Kyiv 03058,
2 Kyiv National University of Construction and Architecture, 03037, Kyiv
min_14@ukr.net, filimonova1209@ukr.net,
jumin@bigmir.net
Abstract. The non-factor fuzziness is considered as one of the forms of uncer-
tainty modeling by detecting hidden knowledge contained in the original data
array (cloud of data) and fuzzy set. It is shown that the objective component of
the fuzziness consists of 2 parts: 1-subset of ordered pairs, calculated on the basis
of tensor decomposition structured as a 2D or 3D tensor of the initial data array;
2 - the immersion of the interval of possible values of the fuzzy set into a special
matrix (Toeplitz), followed by a tensor decomposition. The results obtained: it is
shown that in the general case analysis of uncertainty must be performed using a
multi-fuzzy set; with restrictions on the formation of the membership function it
is suggested to use a subset of ordered pairs on the basis of objective fuzziness;
the developed algorithm for reducing the multi-fuzzy set to FS-1 type. The fol-
lowing examples illustrate the effectiveness of the proposed methodology.
Keywords: multi-fuzzy set, a subset of ordered sequences, tensor decomposi-
tion, special matrix, objective fuzziness.
1 Introduction
The theory of fuzzy sets (TFS) as a method and apparatus for solving problems in un-
certain conditions has shown its high efficiency. However, at present, its application
has encountered a number of difficulties due, in particular, to the following factors:
the emergence of fundamentally new problems associated with the complication of
technological processes, which limits the possibility of assigning a heuristic mem-
bership function (MF); the need to work with data of super large volumes and high
measurements, which require consideration of their life cycle;
philosophical interpretation of the phenomenon of fuzziness, which declares in the
composition of this phenomenon the objective and subjective components, involves
taking into account its influence on the structure and type of fuzzy set (FS), because
the standard FS is constructed only on the basis of the subjective component - FS,
that is, the standard type of FS is a special case;
the initial data set (IDS), on the basis of which the universal set (US) is calculated,
is the bearer of hidden knowledge, but at present, research on the influence of the
structure of IDS on the form and structure of FS has not been carried out;
IDS containing fuzziness (or inaccuracy), allows to distinguish several types of for-
mal (objective) fuzziness in the form of subsets of ordered pairs (SOP).
Let's remind that FS A in X is SOP:
A x, A ( x) | x X , A [0,1] (1)
where A is the function of membership or degree of membership [1] (also the degree
of compatibility or truth) x in A , which reproduce X in the space of membership M
[2]. When M contains only two points 0 and 1, A is not fuzzy and A is identical to
the characteristic function of the classical set.
In this regard, it is advisable to turn to the theory of non-factors [3], because the first
non-factors were identified and studied within the framework of the problems of TFS,
specifically fuzzy mathematics [4]. Four non-factors are considered in the theory of
non-factors: inaccuracy, insufficiency, ambiguity and fuzziness due to the fact that their
formal representation is associated with the use of a range of values. Note that in the
appendix to different categories of data and knowledge, non-factors are treated differ-
ently, for example, the fuzziness of data, fuzziness of conclusions, the fuzziness of the
problem statement - these are completely different fuzziness. At the same time, one of
the principal differences lies precisely in the fact that the considered fuzziness has dif-
ferent proportions of the objective and subjective components.
In this context, only data fuzziness is considered. Separately, pay attention to inac-
curacy, in the work an example of modeling of inaccuracy is given by a subset of or-
dered sequences. It is shown in [5] that inaccuracy is a universal factor inherent in all
real parameters. In [6], it has been shown that inaccuracies and insufficiency are asso-
ciated with the use of the range of values, which simplifies the consideration of their
meaningful differences; they do not have the "external resemblance", but they are in-
terrelated, moving to each other at different levels of modeling.
2 Review of the Literature
2.1 Main statements
M. Black [7] first applied the many-valued Lukasiewicz logic to lists as multiplicative
of objects and called such sets indeterminate (vague), L. Zade based on the logic of
Lukasevich constructed a complete algebraic system [8]. This was the first publication
in which the term fuzzy set was used and the main idea of fuzziness was formulated.
Attempts to give a philosophical interpretation of the actual positions of TFS made in
the works [9, 10], in the works [11, 12] considered the problem of fuzziness as a scien-
tific concept. Let's pay attention that interest in the conceptual foundations of TFS, due
to the expansion of TFS to new areas, new tasks, solve which using standard concepts
is difficult.
Here are the main provisions of work [11]. Fuzziness is a concept that characterizes
the non-continuity of the transition from the absence of manifestations of the full dis-
covery of the quality of objects, properties and relations of the real world, which is
reflected in the cognitive and intellectual activity of the individual. The fuzziness in the
real world is manifested in 2 forms - audio and visual, blur (fuzziness) of the sound
signal, fuzziness (blurring) of the image, although the forms of the emergence of the
virtual (conceivable abstract) fuzziness are much larger.
The methodological basis for considering the phenomenon of fuzziness and the con-
cept that it expresses is the formalization of intuitive notions of fuzziness. For this pur-
pose, an analysis of the forms of the appearance of fuzzy bones is used to identify the
components that make up its contents and the synthesis of the system of interconnec-
tions of the fuzziness of the object's property in the process of cognitive activity of the
subject. For the formalization of intuitional notions of fuzziness, the method of deter-
mination through abstraction is used; the method of an equation is used in the process
of understanding the ontological grounds of the phenomenon of fuzziness and the ap-
pearance of the relationship between its objective and subjective aspects.
One of the most important conclusions of the work [11] is that the dialectical nature
of the relationship between the subjective and objective aspects of fuzziness is deter-
mined, this important conclusion is confirmed in the works of Nobel laureates D.
Kahneman and A. Tversky [13], which showed that the use of common sense (the main
tool for the formation and assessment of subjective fuzziness) is not rational in all cases,
the person rational supposed abnormalities of rational behavior. The second important
conclusion from the work [11] is that the phenomenon of fuzzy, having objective and
subjective aspects, differing in depth of reflection of the essence of processes, must be
studied in a complex manner, while simultaneously considering both types.
Since fuzziness is a characteristic, first of all, the cognitive process, then the solution
to this issue is the most effective method from the establishment of consistency patterns
to the construction of a formal system based on them. In work [11], this approach is
realized by revealing intuitive notions of fuzziness, establishing their specific content
and further formalization in order to identify their logical structure, and in the further
clarification of the relationship between the formal and substantive aspects of the con-
cept of fuzziness. If the fuzziness is understood as a property, then, as noted by Cana-
dian researchers A.M. Norwich and I.B. Turksen, it is also necessary to distinguish be-
tween objective and subjective fuzziness.
The theory of non-factors proposed by A.S. Narinyany, devotes great attention to the
analysis of non-precision, which is a multidimensional characteristic of fuzziness, be-
ing one of the most common forms of its expression. The content of the concept of
fuzzy contains a consistent series of lower-level abstractions, which in turn, in each
case, contain abstractions of an even lower level, both explicit and intuitive; the com-
mon thing for them is that they characterize the power of the real objects, phenomena,
relations and concepts.
Touching TFS, we note that in the general case in the formation of FS (process of
fuzzification) objective fuzziness is to calculate the interval US, subjective - in the
choice (heuristic) MF, while it should be borne in mind that the basis of the methodo-
logical approach to the analysis of the relationship objective and subjective fuzziness
is the rationale for choosing one of two aspects of the concept of fuzziness as defining
in relation to another. In this context authors, objective inaccuracy is additionally de-
termined by the artificial blurring of the US with the help of the immersion procedure
in a special matrix.
In [14], it is shown that the problem of uncertainty (which, unfortunately, reduces
only to one of the non-factors - fuzziness) inevitably arises in the simulation of complex
systems, in which a person plays an active role. In order to obtain significant conclu-
sions about the behavior of a complex system, it is necessary, in principle, not only to
abandon the high standards of accuracy and severity which are characteristic of rela-
tively simple systems, but to consider alternatives, not only varying with different
standard FS, but also changing the approach to computation SOP.
Objective fuzziness is a description of the objects, properties and relations of the real
world, not dependent on consciousness; subjective fuzziness, in turn, characterizes the
process of knowledge, the human way of thinking and knowledge of objective fuzziness
measures reflection of the internal patterns of the object in its external manifestations.
Subjective fuzziness is defined as a measure of the reflection of the phenomena of the
external world into the inner world of the individual, which allows for the personality
of the human psyche to be taken into account. At least at the level of sensation and
perception, subjective fuzziness has an objective basis, which, in turn, gives grounds to
believe that objective obscurity is decisive in relation to subjective.
Subjective and objective types of fuzziness also differ in depth of reflection of the
essence of processes and in relation to these types of basic categories of dialectics. Re-
flecting the contradictory process of cognition, objective and subjective fuzziness,
while in a dialectical contradiction, adequately describe the phenomenon of fuzziness
in its totality, indicating their dialectical unity. The study of various interpretations of
fuzziness, their analysis and generalization showed that no interpretation completely
does not cover the content of the concept under study and does not fully reflect its
essence. Therefore, there are prerequisites for considering the fuzziness as a theoretical
concept, in which any preliminary interpretation will act as a separate case. From a
methodological point of view, the interpretation of fuzziness as a theoretical concept
has the essential advantage that such an understanding of fuzziness does not allow to
reduce any one interpretation of this concept into a general rank and, at the same time,
deny any of them due to its incompleteness.
2.2 List of main symbols and abbreviations
In table 1 is presented main abbreviations, that are used in the article.
Table 1. List of main symbols and abbreviations
Notations Definitions
X Matrix (or tensor)
x vector
X(:,j), X(i,:) j-th column, i-row of matrix
Frobenius norm
F
Kronecker product
FS Fuzzy set
MF Membership functions
US Universal set
TFS Theory of fuzzy set
IDS Initial data set
SOP Set of ordered pairs
KP Kronecker product
PCA Principal component analysis
MFS Multi-fuzzy set
SVD Singular value decomposition
3 Problem Statement
The main object of TFS - FS x x / x , x [0,1], x X {x} , has the property
that for its presence there is sufficient minimum number of statistical characteristics of
IDS: (xmin, xmean, xmax) X, obtained on the basis of IDS, and an expert assessment of
the type of FS. At the same time, a number of structural characteristics that are endowed
with IDS can be obtained based on the discovery of hidden knowledge contained in the
Data Mining (IDS), is not used. The presence of additional characteristics in conditions
of uncertainty can affect the decision-making - to correct the heuristic adopted FS, or
in the most unfavorable case - the impossibility of appointing an FS - to offer as an FS
equivalent subset of ordered pairs formed on the basis of IDS, or else.
In TFS, the notion of the nearest (to fuzzy x ) crisp set is considered, the closeness
is understood as the proximity of the norm, in particular, Euclid, Frobenius, or else-
2
where, that is x x0 min , where x ( x x )1n , x X , x [0,1];
F
, {0;1}, x X . Note that the F-norm of the matrix A with the
n
x 0 x x0 x0
0
1
dimension mn is defined as A trace A A , the sets x x and
1/2 n
H
F x
1
x x must be represented in the matrix form, in particular, in the form of the
n
0
x0 1
KP- x x respectively.
T T
and x x0
The concept of the nearest object in the work is expanded by forming the tasks of
determining the subsets of ordered pairs: a) nearest to the given FS; b) nearest to the
structured IDS. Let's pay attention to the following. The nearest crisp set approximates
the fuzzy, subset of ordered pairs nearest to the standard fuzzy can be used for compar-
ison or estimation or to be more convenient for further processing, since it always has
the same form and obtained on the basis of formal transformations, which excludes the
participation of an expert. Accordingly, the following tasks are formulated.
An object in uncertainty is represented by an IDS (or array), in the general case, a
data cloud, for example, an array of data obtained when measuring a certain value
with devices with limited accuracy in conditions of interference. To work with this
object, its representation is used in the form of some statement such as "approximately
equal to ...", "close to ...", etc., which, in turn, can be presented to FS x x / x ,
x 0,1 , x X x , where x 0,1 is MF, which is appointed by the expert
heuristically, as a rule, from the composition of the means of a specific application
package, which is used in the work.
Need to show:
the existence, in addition to the standard FS x( j ) x( j ) / x( j ) , x( j ) [0,1] ,
x X of a collection of SOP that is endowed with properties of FS (having one
( j)
of the components as a weight function) are analyzed on the basis of formal proce-
dures and represent the object no less effectively than FS; it is important that the
specified SOP are obtained by detecting hidden knowledge (intellectual analysis)
that are found both in the components of the FS and in the IDS;
the possibility of using the SOP as an FS, if there are restrictions in the designation
of FS;
the expediency of submitting an object in conditions of the uncertainty of a multi of
fuzzy sets containing FS and SOP that characterize this object;
the possibility of using methods of fuzzy mathematics, developed for type FS-1
type, on a multi-fuzzy set (MFS);
real examples of the solution of the above tasks.
4 Materials and Methods
Definition 1. Subjective fuzziness is a subset of ordered pairs xi xi , one of its
components of which xi
0,1 plays the role of weight function and obtained on the
basis of mental activity of a person (in particular, common sense); objective fuzziness
xi 0,1 obtained on the basis of formal calculations.
Formation of a subset of ordered pairs based on the blurring of the vector of input
data [15]. One of the tasks that were posed by the authors was to use the logical and
semantic proximity of tasks that arise in the processing of noisy (blurring) images and
procedures of blurring (fuzzification) input values in the formation of FS. As it is
known, the restoration of blurry and tired images, for example, magnetic resonance
tomography (MRT) is the main problem in digital images. Some elements of the blind
deconvolution method used to restore a crisp (sharp) image from blurring and noisy
MRT images can be used to solve problems, in particular, to make decisions under
uncertainty, using TFS.
Recall that blind deconvolution is a method of restoring a sharp (crisp) version of a
blurred image when the source of blurring is unknown. In some applications, particu-
larly in the physical imaging problems, the blurring process is not known and needs to
be restored with the image. A similar situation occurs when using TFS - the process of
fuzzification x xi / xi , xi [0,1] , fuzzifier(x)→x Ux of the initial data array or
a separate component of this mathematical object - a scalar, variable, vector, etc., is not
coded and entirely depends on the position (knowledge, experience and other qualities)
of an expert. Without denying the rationality of this approach, we duplicate the expert
by performing formal fuzzification of the input by immersing the input vector into a
special matrix and, having obtained a new matrix, we execute a singular decomposition
over it, define a rank-1 matrix (row and column) which, according to the Young-Ekkart
theorems [16] we shall consider as a subset of ordered pairs possessing properties of
informally formed FS.
It is known [17] that if one imagines a degradation (image, signal) in the form of a
linear fashion, i.e. g (x, y) = f (x, y) h (x, y) + (x, y), where - a symbol of Krone-
ker's product, f (x, y) is the initial image, h (x, y) is the function of scattering points are
unknown functions, then the restoration is reduced to the problem of blind deconvolu-
tion [18]. The expression for g (x, y) is also considered in the vector-matrix form:
g H f , where the matrix [H] can be constructed from the function h of the
discrete point distribution and has the structure of the so-called special matrix.
These can be the Toeplitz matrix, Hankel matrix or a combination of them: for ex-
ample, a block matrix in Toeplitz form with Hankel-shaped blocks or the like. General
view - the method of forming a Toeplitz matrix from a vector (in this case US) - is
shown in Fig.1. This matrix-vector form can be useful in analyzing the problem of
fuzziness, which is the basis for the formation of a SOP with the properties of the FS.
Fig. 1. Formation of the Toeplitz matrix from the vector [t0, t1, ..., tn-2, tn-1] as the process of US
fuzziness - a method of the formal (objective) formation of SOP as an analogue of the MF
It is shown [19] that in most applications where using a blurred image B restore its
sharp and clear version X, the most common model of the blurring process is given as
B = A [X] + N, where A is the blurring process, be it linear or nonlinear process, and
N denotes some additive noisy process. For digital imaging X and B are discrete vectors
or arrays, and as a result, the linear blur A corresponds to a matrix-matrix operator. In
the case when the blur operator A is invariant with respect to the displacement, the
corresponding blurring matrix will be well structured and may be, for example,
Toeplitz, circular or Hankel. The algebra of calculations with structured matrices is
considered in the paper [20]; in this paper we consider the cut singular decomposition
of the matrices because it allows the obtaining of the SOP, and the first component of
which is the US, and the second - weight function, that is, SOP structurally and seman-
tically coincides with FS, although calculated on the basis of formal relations.
Tensor models. A tensor is a d-linear form or d-dimensional array: A a i1i 2...id
.
Tensor has: dimensionality (order) d - number of indices (measurements, axes, direc-
tions), size n1. . .nd - the number of readings per axis [21]. An example of the sepa-
ration of variables is singular decomposition A a va ua . The canonical tensor
a 1.r
decomposition also appears in terms of the Kronecker's product (KP) of the matrices.
Case d 3 differs substantially from the case d = 2: the approximation of a tensor with
a decrease in rank is a complex multidimensional optimization task. In [22], the effi-
ciency of using tensor approximations for calculations is shown, and the d-dimensional
tensor can be considered as a vector-column with d-dimensional indexation. An ex-
treme case is the transformation of a vector of size N = 2d into a d-tensor of measure-
ments 22 ... 2.
When maintaining the total number of elements of the tensor, the effectiveness of
its use (representation of the tensor, the possibility of performing algebra operations,
the formation of FS-2 or n-types, etc.) can be significantly increased by increasing the
number of measurements and reducing the number of readings for each dimension. One
of the specific characteristics of a tensor is its quantization-the introduction of addi-
tional (virtual) measurements. While maintaining the total number of elements of the
tensor, the effectiveness of its presentation can be substantially increased by increasing
the number of measurements and reducing the number of readings for each dimension.
The main object of the TFS - SOP is to some extent universal, but in a number of
cases it can be used in other. forms [23], in particular, when performing operations of
fuzzy mathematics and fuzzy logic than is customary in the TFS. FS-1 type
a a / a , a U , a [0,1] , regardless of the form of MF and the method of its for-
mation, can be represented by means of a KP in the form of a 2D tensor:
. The FS-tensor T with the help of the singular decomposition
T
a Ta a a a
procedure can be compared to the SOP:
p
a a / p a , a U , p a [0,1] (2)
which has the property when the SOP , p a is nearest (in the sense of the F-norm) to
p
the FS a , a a min .
F
Note that in the TFS, in addition to the most common types of FS-1 type, designed
for the simulation of the simplest types of uncertainty, in which the values of the MF
from the interval [0,1], proposed higher-level FS, allow for, in particular, the consider-
ation of multidimensional uncertainties. Conceptually, these types of FS are defined in
[24]. For example, the type FS-2 type is defined as an FS of a universal set X in which
the FN is fuzzy subsets of the interval [0,1], expanded with the FS-n type, where n = 2,
3, ... is the FS, in which MF is FS -n-1 type. The tensor model of the FS-2 type,
, a U , a , [0,1], in particular, has the form of a block ten-
a a / a a / a a
a
sor a T a , where Ta a a a .
The basis of matrix (tensor) decompositions. Since the appearance of matrix decom-
positions, later extended to tensors, they were treated as a decomposition on a basis and
weighted coefficients [25], which are the bases of all decompositions, are most clearly
visible in the methodology of Principal component analysis (PCA) and singular value
decomposition (SVD) . But if one considers that one of the interpretations of the FS
and MF is that the FS is a subset of ordered pairs, one of whose components is a uni-
versal set (basis in the notation of the PCA), the second is the heuristic significance of
each element of the US in truth, in particular some assertions (weighted coefficients in
the notation of the PCA, we can see the identity in the definition and semantics of the
FS and the matrix (tensor) decomposition. SVD is a decomposition of a real (complex)
matrix in order to bring it to a canonical form.
In the writings of authors it is shown that SVD can be used as a procedure for de-
termining subsets of ordered pairs similar to FS. This allows us to expand the possibil-
ities of singular expansion - from solving standard tasks - approaching the least-squares
method, compression of images, etc. - to tasks in conditions of uncertainty. An example,
when the matrix 33 is represented by a subset of ordered pairs (Tabl.2) in such way:
(abs(u(:,1)*s(1,1,),abs(v(:,1)))
the KP of a component of which allows a new matrix (initial approximation) with an
accuracy of not less than 1%, is given below. Note that the absolute values are left and
the right singular vectors have the meaning of weight functions, in addition,
sum (v (:, 1)2)=sum (u (:, 1)2 ) = 1.
Table 2. Example SVD and its properties.
initial matrix singular decompositions: [u s v] = svd(x)
x= u= s= v=
3.82 3.71 3.28 -0.56 0.06 -0.83 11.16 0 0 -0.61 -0.38 0.69
4.79 4.63 3.41 -0.67 -0.62 0.41 0 0.85 0 -0.60 -0.35 -0.72
3.12 3.02 3.40 -0.49 0.78 0.39 0 0 0.01 -0.52 0.86 0.01
norm(x,'fro')= x1=kron(u(:,1)*s(1,1),v(:,1)T) norm(x1,'fro')= 11 .16
= 11.20 norm(x1,'fro')= norm(x,'fro') ,
x1x, sum(v(:,1)2= sum(u(:,1)2=1
In [26], the SVD model is proposed for the N-th order of tensors.
Theorem 26 (N-order SVD). Each complex (I1 × I2 × • • • × IN) -the tensor A an
be written as a product A = S ×1U(1) ×2U(2) • • • ×N U(N), in which:
1 2 n
1. U ( n ) U (In ) U 2(In )...U (In ) - unitary (In × In) - matrix,
2. S is a complex (I1 × I2 × • • × IN), the tensor for which the sub-tensors obtained by
fixation of the n-th index in α have the properties of ordering: for all possible values
of n.
Decomposition of the 3rd order tensor in Fig. 2. A comparison of matrix and tensor
theorems shows a clear analogy between two cases, in particular, the left and right sin-
gular vectors of the matrix are generalized as n-modal singular vectors
J J 2 J3
Fig. 2. Representation of the 3-rd order tensor A C 1 as a product of the 3-rd order
J1 J 2 J 3 J I J I J I
tensor B C and matrix U
(1)
R 1 1 , U (2) R 2 2 , U (3) R 3 3 .
Links between high-order (HOSVD) and matrix SVD. Let HOSVD for A be given
H
by Theorem 2. Then there A ( n ) U ( n) ( n) V ( n) is SVD for A ( n ) , where the diago-
R I n I n and the column wise extended orthonormal matrix
(n)
nal matrix
(n) I n 1I n 2 I N I1I 2 I n 1I n
V C are defined in accordance
(n) diag 1(n) , 2(n) ,... I(n) ,
(3)
n
,
T
V ( n) H S( n) U ( n 1) U ( n 2) ...U ( N 1) U (1) U (2) ... U ( n 1) (4)
~
in which S ( n ) - a normalized version S( n ) with rows, varied according to the scale to
~
one-dimensional length S( n ) ( n ) S ( n ) .
Multi-fuzzy sets as a way of storing, analyzing and processing data in conditions of
uncertainty. Given that in the general case, as a model of uncertainty (fuzziness), has
an objective and subjective component, it should be recognized that the MFS will be
more adequate to the model than the application of the FS. In addition, MFS is a unique
way of representing, analyzing and processing large data, a subset of ordered se-
quences, the most natural form of representation of uncertainty. The theory of multiple
fuzzy sets in terms of multidimensional membership functions is an extension of a se-
ries of related theories: fuzzy sets, L-fuzzy sets, intuitionistic fuzzy sets, which are dis-
cussed in [27], from which the basic definitions are taken.
Definition 2. Let X be a non-empty set and {Li: iP} be the family of complete
lattices. Multi FS A in X is a set of ordered sequences:
A { x, 1 ( x), 2 ( x),..., i ( x),... : x X } , where i LiX , for iP.
We recall that a complete lattice is a partially ordered set in which any non-empty
subset has an exact upper and lower bound, usually called the union and intersection of
the elements of a subset.
Remark 1. If the sequences of membership functions have only k-terms (the end-of-
number of terms), k is called dimensional A . If Li = [0, 1] (for i = 1, 2, ..., k), then the
set of all the multi-fuzzy sets in X of dimension k are denoted by MkFS (X). The func-
tion of multi membership (plural membership) μ A is a function of X in Ik such that for
all x in X, μA (x) = μ1(x), μ2 (x), ..., μk (x) . For simplicity, we denote the multi-FS
A { x, 1 ( x), 2 ( x),..., k ( x) : x X } as = (μ1, μ2, ..., k). In this work, Li = [0, 1]
(for i = 1, 2, ..., k), and some properties of the multi-fuzzy sets of dimension k are given.
Definition 3. Let k be a positive integer and let it go , in MkFS (X) such, that
=( μ1, ..., μk ) = { x, μ1 (x), μ2 (x),..., μk (x): x X} and = ( 1,...,k ) ={ x, 1 (x),
2(x), ..., k (x): x X}, then we have the following relations and operations:
1. if and only if i i to all i=1,2,…,k;
2. = if and only if i =i to all i=1,2,…,k;
3. = = (1 1,…, k k)={ x, max (1(x),1(x)),…, max(k(x),k(x)): xX};
4. = (1 1,…, k k)={ x, min (1(x),1(x)),…, min(k(x),k(x)): xX};
5. + = = (1+1,…, k+k)={x,1(x)+1(x) - 1(x) 1(x),…,k(x)+k(x)-k(x)
1kx): x X };
Remark 2. Let μ = (μ1, μ2) be the fuzzy set of measure 2 and let μ1 (x), μ2 (x) be the
gradation of the membership and non- membership of the quantity x in μ, respectively.
If μ1(x) + μ2(x) ≤ 1, then μ is an intuitionist FS. Consequently, every intuitionist FS in
X is a multi-fuzzy set in the X dimensionality of 2, and each intuitionist fuzzy operation
is a multi-fuzzy mapping on the multi fuzzy sets. But multi-FS does not necessarily
have to be an intuitionistic fuzzy set, for example, multi-FS μ = {(x, μ1 (x), μ2 (x)):
μ1(x)= .9, μ2 (x) = .8, x X} is not an intuitionistic FS.
Arithmetic on multi-fuzzy sets:
A+B={(x, A+B(x), A+B(x))|xX}, where A+B(x)=A(x)+B(x)-A(x) B(x),
A+B(x)= A(x)B(x),
AB={(x, AB(x), AB(x))|xX}, where AB(x)=A(x)B(x) , AB(x)= A(x) +
B(x) - A(x)B(x).
We note that between the subjective and the objective fuzziness, which are charac-
terized, above all, by the proximity of the F-norm and the value of the defuzzified value
of the FS, there are certain interconnections, and there are cases when these values are
close or coincide.
5 Experiment
Assume that an object is an inaccurately measured set (array) of values, which is struc-
tured as a matrix of 88
A=
[7.91 8.03 7.77 8.25 4.71 5.60 7.70 5.49;
6.96 6.41 8.74 7.42 5.82 4.36 7.09 4.83;
5.05 5.22 6.14 3.82 3.39 6.48 5.77 8.25;
4.74 7.22 8.28 3.07 8.93 7.56 6.41 3.09;
5.05 6.28 4.04 8.36 6.50 6.18 7.77 7.61;
6.20 5.67 8.88 4.19 5.54 6.84 3.36 8.83;
7.36 7.17 4.63 4.79 6.09 4.25 6.62 8.94;
4.86 6.73 4.51 6.97 5.00 5.28 3.30 7.73]
The object obtained by the above-mentioned method (under uncertainty) is forma-
lized: a) in the form of an FS with a heuristic designated MF using standard library MF;
b) SOP №1, which is formed on the basis of immersion of the interval US in a special
matrix; c) SOP №2, which is formed on the basis of IDS (Fig. 3). Recall that in the
general case we have 3 channels of formalization of uncertainty:
standard (for TFS) - FS a a / A , A [0,1], a A is formed heuristic, MF A
is selected by an expert from the existing set of MF, which is placed in the working
package of applications, US A = {a} is formed on the basis of IDS; additionally
formed 2D tensor TA: a a / A T A a ( A )T ;
the formation of an objectively blurred object, the method of immersion of the vector
(the interval of values of US) into a special matrix (Toeplitz, Hankel, etc.)
T Ai toeplitz (a ) , the singular decomposition of which allows calculating the SOP
a (1) a / A(1) , A(1) [0,1], a A(1) A ;
structuring of IDS A in the form of a 2D tensor (a matrix of dimension mm),
T A2 reshape( A, m, m) , a singular decomposition of which allows the calculation
(2)
of the SOP a a / A(2) , A(2) [0,1], a A(2) A .
Fig. 3. Formation of multi-fuzzy sets as the structuring of 3 matrices: Kronecker product com-
ponent FS, IDS and fuzziness US fuzzy set.
Difficulties in the formation of MFS from three objects: FS a , SOP a (1) and a (2)
consist in the fact that they are constructed on different US
A(1) A, A(2) A, A(1) A(2) A , therefore the following steps of the algorithm are
as follows: 1-st way - consider 2D tensors as the frontal slices of the 3D tensor with
the following the high-order tensor decomposition, that is, the implementation of the
procedure [U1, U2, U3, S] = svd3 (rank, ), a fragment of the algorithm and an exam-
ple showing its performance is given below:
1. .An unstructured array of initial data is specified: X={xi}, i=1,n;
2. .Formation FS: x {x / x } , x[0,1], x X={xj}, j=1,m; m – number of -levels
in FS x , m2 n ;
3. Formation of 3D tensor x based on frontal slices: {x(:,:,1), x(:,:,2),x(:,:,3)} -fig.4:
x(:,:,1)= x(x)T, where - symbol of a tensor product; x(:,:,2)=reshape(X, m, m) –
structuring of IDS in the form of a 2D matrix; x(:,:,3)= toeplitz([x1,…,xm] – fuzzi-
ness US;
4. High-order singular decomposition of the 3D tensor: [U1,U2,U3,S]=svd3(rank, X)
[28]; reconstruction of a complete tensor:
F = tmul( tmul( tmul( S, U1, 1), U2, 2), U3, 3).
Fig. 4. Representation of the object in conditions of uncertainty in the form of a 3D tensor
5. Formation of a subset of ordered sequences – MFS:
m=max(abs(U2(:,1)).* abs(U3(:,1)))
sort([abs(U1(:,1)*S(1))*m,abs(U2(:,1))/max(abs(U2(:,1))),
abs(U3(:,1))/max(abs(U3(:,1)))]),
where m is the norm factor, introduced for compatibility of the results in the 2D-tensor.
In the MFS notation we have:
x=abs(U1(:,1)*S(1))*m,
(1) =abs(U2(:,1))/max(abs(U2(:,1))),
(2)=abs(U3(:,1))/max(abs(U3(:,1))),
Δ
that is x {x/ μ (1) ,μ (2) } .
Due to the fact that the solution under uncertainty conditions is most often adopted
on the basis of the standard FS-1 type {x / }, x X, [0,1], it is expedient to bring
Δ
the MFS x {x/ μ (1) , μ (2) } to form {x / }. This can be realized in two ways: the
first is formation of FS on the basis of singular vectors of the 3D tensor (fig.5).
Fig. 5. Formation of FS on the basis of singular vectors of the 3D tensor.
The second method is to immerse the set of frontal slices x (:,:, 1), x (:,:, 2) and x
(:,:, 3) in the Toeplitz matrix: 1 = toeplitz ([x (:,: , 1), x (:,:, 2), x (:,:, 3)]) we have
1 = [x (:,:, 1) x (:,:, 2) x (:,:, 3); x (:,:, 2) x (:,:, 1) x (:,:, 2); x (:,:, 2) x (:,:, 1)]. The
singular decomposing of matrix 1 allows one to obtain a SOP x {x / } which
represents the integral characteristic of the object under uncertainty conditions.
Example. An object in conditions of uncertainty (inaccuracy of measurements, the
effect of interference, etc.) is described in the form of a statement more than 3 and
not more than 9 and represents one of the realizations of the expression X = 3 + rand
(9) * 6; for its formalization by an expert, the proposed FS x = (x / x) with a triangular
MF of the form: trimf = trimf (x, [3.00 6.00 9.00]), xX = [3.06 3.76 4.47 5.17 5.88
6.59 7.29 8.00 8.70], defuzzified value of FS, calculated by the center of gravity (CG)
method and the F-norm are given below: def ( x ) = 5.98, x F =norm ( x , 'fro') = 18.54.
In addition to heuristic FS, this object can be characterized by a further number of sub-
sets of ordered pairs, endowed with properties of FS, which is accepted as a standard,
the results of calculations are given in Table 3.
Table 3. Comparative characteristics FS and SOP
FS x Subsets of ordered pairs
Structuring and de- Immersion of US Truncated Integral model
composition IDS into a special models col- of uncertainty
(Toeplitz) matrix umns 1 and 3
1 2 3 4 5
3.17 0 5.64 0.84 5.38 0.80 3.17 0 5.13 0.66
3.87 0.25 6.04 0.86 5.46 0.82 4.56 0.50 5.27 0.68
4.56 0.50 6.07 0.87 5.46 0.82 5.96 1.00 5.50 0.74
5.26 0.75 6.32 0.88 5.70 0.85 7.35 0.50 5.59 0.77
5.96 1.00 6.72 0.90 5.70 0.85 8.74 0 5.67 0.78
6.65 0.75 6.80 0.90 6.10 0.91 5.38 0.80 6.00 0.81
7.35 0.50 6.94 0.91 6.10 0.91 5.46 0.82 6.18 0.86
8.04 0.25 7.35 0.94 6.69 1.00 5.70 0.85 6.29 0.88
8.74 0 7.69 1.00 6.69 1.00 6.10 0.91 6.40 0.94
6.69 1.00
|| x ||F ...def ( x)|| x ||F ...def ( x) || x ||F ...def ( x) || x ||F ...def ( x) || x ||F ...def ( x)
18.74 5.96 20.13 6.65 18.01 5.96 14.08 5.96 17.55 5.83
13.30 5.90
6 Results and Discussion
Thus, in conditions of uncertainty, under which in this case we will understand the
situation when a semantically uniquely identified object (for example, the result of
measuring a certain value in the conditions of interference), the known initial data array,
which allows structuring and determine the interval of values, which is accepted as a
US for the formation of FS, an object in conditions of uncertainty can be represented
by the totality of FS and SOP.
In this case, the standard FS x {x/x }, x [0,1], x X intended heuristic;
SOP1 x1 {x/ x1 }, x1 [0,1], x X1 is formed on the basis of formal procedures
for simulating the fuzziness process by immersion of the US interval into a special
matrix (Toeplitz) and subsequent singular decomposition;
SOP2 x2 {x/ x 2 }, x 2 [0,1], x X 2 is formed on the basis of structuring the IDS
in the form of a matrix and the subsequent singular decomposition.
All SOP and FS have the properties: proximity to F-norm, defuzzified values-
def ( x) def ( x1 ) def ( x2 ) takes place inclusion - X1X, X2 X. It should be noted
separately that the integral model of uncertainty containing the of the Kronecker prod-
uct components of FS, IDS, the matrix of the Toeplitz model of fuzziness, which is
considered as a block vector, immersed in a special matrix, has the value || x ||F and
def ( x ) that is practically the same as the FS, but the length of the interval US , on
which the SOPs are defined, is much shorter than the corresponding magnitude for FS
(1.27 versus 5.5). Failure to take into account this factor can have a significant effect
on decision making as a result of the use of fuzzy mathematics. FS by its nature gives
guaranteed partial overvaluation, it can be effectively used as a control parameter in
decision-making.
Operations of fuzzy mathematics in the system of subsets of ordered pairs calculated
on the basis of the singular decomposition of the Toeplitz matrices formed by immers-
ing the real component of the FS into a special matrix.
Let 2 fuzzy number (or FS) be given: a {a / a }, a 0,1, aX; b {b / b } ;
b 0,1, bX; the principle of fuzzy expansion the result of a mathematical (arith-
metic) operation on them allows you to represent in the form c a *f b {c / c }
where *f{+, -, *, /}, c = a * b, c min( a , b ), c [0,1], c X .
Suppose that the FS a and b are represented in the form of FS with a triangular or
trapezoidal MF, that is, trimf (x, [a, b, c]) or trapmf (x, [a,b,c,d]), the choice of these
FS is due to the fact that they explicitly contain the interval of values for trimf x=Iac,
for trapmf x = Iad. If the number of α-levels is n, then the FS as the SOP has the form
of matrices with the dimension 2n, the 1-st vector-column (a1a2 ...an )T or (b1b2 ...bn )T
used to form the Toeplitz matrix simulating the process of blurring (fuzzines) the inter-
val.
a1
a1
a
2 a2
a . , (5)
.
.
a
n a
n
a1(T ) (T )
a1 a1 a2 ... an a1
a (T ) (T )
a
2 a
2 a1 ... an 1 2 a2
. .
a toeplitz (a) Ta svd (T ) a . (6)
a (n)
. . .
. .
.
a a (T )
an n n 1 ... a1 an a(T )
n
~
The procedure for b is performed similarly. The singular decomposition of the
~ (n)
Toeplitz matrixes Ta and Tb allows us to calculate the SOP a ( n ) and b , which are
~
structurally and functionally similar to a and b , because || a (n)
||F || a ||F ,
~ (n) ~
~
def a ( n ) def a , || b ||F || b || , def b ( n) def b .
The result of the operation a *f b is nearest to the F-norm and the defuzzification
~ ( n) ~ ( n)
value of the result a *f b .
Table 4 shows the results of an arithmetic operation, which is performed on a FS
(with expertly assigned MF) - a subjectivistic fuzziness, and over a SOP calculated on
the basis of formal interval blurring models by immersion of interval (US) into the
Toeplitz matrix.
Table 4. Fuzzy arithmetic in the SOP system, formed on the basis of immersion
the interval of possible values of FS into a special (Toeplitz) matrix
Standart FS Subsets of ordered pairs, calculated on the ba-
x =[4:4/8:8] ; a=trimf(x,[4 6 8 ]); sis of blurring (fuzziness) of the universal set
x1=[7:4/8:11] ;b=trapmf(x1,[7 8 10 11]) by immersion into the Toeplitz matrix
a b c=a+b an bn cn =an +bn
4.00 0 7.00 0 11.00 0 5.61 0.85 8.63 0.90 14.24 0.85
4.50 0.25 7.50 0.50 12.00 0.25 5.64 0.85 8.65 0.90 14.29 0.85
5.00 0.50 8.00 1.00 13.00 0.50 5.70 0.86 8.71 0.91 14.40 0.86
5.50 0.75 8.50 1.00 14.00 0.75 5.78 0.87 8.79 0.91 14.56 0.87
6.00 1.00 9.00 1.00 15.00 1.00 5.89 0.89 8.90 0.93 14.78 0.89
6.50 0.75 9.50 1.00 16.00 0.75 6.02 0.91 9.03 0.94 15.06 0.91
7.00 0.50 10.00 17.00 0.50 6.19 0.94 9.20 0.96 15.39 0.94
7.50 0.25 1.00 18.00 0.25 6.39 0.97 9.39 0.98 15.78 0.97
8.00 0 10.50 19.00 0 6.62 1.00 9.61 1.00 16.22 1.00
0.50
11.00 0
F-norms and defuzzification values
18.49 6.00 27.13 9.00 45.69 15.00 18.18 6.00 27.13 9.00 45.03 15.01
Remark 3. Specificity of the Toeplitz matrix is such that it must be formed on the
basis of a vector of US with a dimension of 1m, m = 2n, whereas the FS is based on a
US measure of 1n. The SOP with the dimension of 2m, which is calculated as a result
of the singular decomposition of the Toeplitz matrix, has the same content in two adja-
cent lines. In order to be able to compare on the basis of calculated F-norms and de-
fuzzified values of FS and SOP, measuring 2n and 2m, respectively, SOP are cut to
a dimension of 2n, using only odd lines.
7 Conclusions
1.The complication of tasks in conditions of uncertainty, due, above all, the complica-
tion of technological processes requiring automation, the need to take into account the
data of super-large volumes and measurements, which are represented by rank-1 ten-
sors, leads to the need to take into account the situation when the MF, cannot be funda-
mentally formed or have insurmountable difficulties in their formation. Another reason
that determines the limited use of TFS for solving problems under uncertainty is that
under real conditions, the object is described not only by the standard MF, which is
designated heuristically, but also by a number of SOP, caused by the discovery of latent
knowledge that takes into account the structural features of the initial data array and the
value interval - US, on which the MF and SOP are formed.
2. The technology of blurring of the interval on which the FS is formed is proposed
as a procedure for immersing this interval (vector) into a special matrix (Toeplitz), the
singular decomposition of which allows one to calculate the SOP, which is one of the
characteristics of uncertainty. It is shown that this SOP is the nearest (in the sense of
the F-norm) to the heuristic FS intended for this range and can be to some extent a
verification test for the expediency of using the adopted MF.
3.The necessity (in a number of cases) of the simultaneous consideration of the sub-
jective component of the fuzziness in the form of the KP of the components of the FS
(2D tensor mm), structured IDS and the blurred interval of FS (objective fuzziness),
characterizing the uncertainty, by creating a block of the Toeplitz Matrix, the singular
decomposition of which allows for a generalized SOP, which integrally characterizes
the uncertainty. Over SOP, obtained by means of singular decompositions (2D and 3D
tensors), it is possible to apply without limitation all processing methods developed for
standard FS.
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