=Paper=
{{Paper
|id=Vol-2917/paper22
|storemode=property
|title=Tensor Models for Data Extraction and Use of Hidden Knowledge in the Environment of Uncertainty Modeled by Fuzzy Sets of 1 and 2 Types
|pdfUrl=https://ceur-ws.org/Vol-2917/paper22.pdf
|volume=Vol-2917
|authors=Yuri Minaev,Oksana Filimonova,Julia Minaeva
|dblpUrl=https://dblp.org/rec/conf/momlet/MinaevFM21
}}
==Tensor Models for Data Extraction and Use of Hidden Knowledge in the Environment of Uncertainty Modeled by Fuzzy Sets of 1 and 2 Types==
https://ceur-ws.org/Vol-2917/paper22.pdf
Tensor Models for Data Extraction and Use of Hidden Knowledge
in the Environment of Uncertainty Modeled by Fuzzy Sets of 1
and 2 Types
Yuri Minaev 1, Oksana Filimonova 2, Julia Minaeva 3
1
Kyiv National University of Technologies and Design, Nemyrovycha-Danchenka Street, 2, Kyiv-01011, Ukraine
2
Kyiv National University of Construction and Architecture, Povitroflotsky Avenue, 31, Kyiv-03680, Ukraine
3
Taras Shevchenko National University of Kyiv, Volodymyrska Street, 64/13, Kyiv-01601, Ukraine
Abstract
We consider the modeling of uncertainty, presented in the form of type-1 fuzzy sets and type-
2 fuzzy sets, 2D and 3D tensors, which allows the use of matrix-tensor algebra (in particular,
Kronecker algebra) to solve decision-making problems under uncertainty along with standard
fuzzy mathematics; it is shown that the tensor decompositions of the formed models allow
obtaining the closest (in the sense of Frobenius norm) subsets of ordered pairs and sequences,
which can be used with limited possibilities of assignment of membership functions or as an
alternative to fuzzy sets in solving fuzzy equations and fuzzy systems. An important type of
hidden knowledge is the ability to obtain the values of matrix (tensor) invariants, presented in
trace form, which significantly affects the quality of decision making.
Tensor models of fuzzy sets make it possible to expand the range of problems to be solved
under conditions of uncertainty, in particular, the use of special matrices (tensors) - Toeplitz,
Hankel, etc. allows to obtain for a given universal set an objective analog of a fuzzy set and to
obtain a comparative assessment of the decision.
Keywords 1
Tensor, fuzzy set, uncertainty, data extraction, hidden knowledge, tensor decomposition,
Kronecker product, matrix (tensor) invariants, Kronecker algebra
1. Introduction
Fuzzy set theory (FST) is now a practically universal apparatus that is used in almost all cases where
there may be uncertainty. The circumstances that, in our opinion, contributed to this phenomenon are
as follows:
the subset of ordered pairs (SOP), which is the main element of the mathematical apparatus of
FST, assumes its flexible modification: depending on the level of uncertainty type-1 FS can extend
to type-2 FS, n-type in general and be a subset of ordered sequences (SOS);
the presence of a component - membership function (MF), which requires virtually no
mathematical constraints (except for convexity) and almost entirely depends on the opinion of the
expert, allows you to adapt the mathematical apparatus to almost any type of real uncertainty
problems.
In [1] it was shown that theoretically SOP can be most rationally used in the analysis of uncertain-
ty in the form of fuzziness (vagueness) and inaccuracy, but real life does not support this thesis. Type-
2 FS was introduced by Zadeh as a continuation of the concept of type-1 FS. Type-2 FS is rational to
MoMLeT+DS 2021: 3rd International Workshop on Modern Machine Learning Technologies and Data Science, June 5, 2021, Lviv-Shatsk,
Ukraine
EMAIL:min_14@ukr.net (Yu. Minaev); filimonova1209@ukr.net (O.Filimonova); jumin@bigmir.net (J. Minaeva)
ORCID: 0000-0002-1168-1927 (Yu.Minaev); 0000-0002-6394-0636 (O.Filimonova); 0000-0002-2367-1507(J. Minaeva)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
use to describe certain types of uncertainty formulated in [2] because the membership function of the
type-2 FS itself is fuzzy and corresponds to the nature and characteristics of particular uncertainty.
Although FST is currently the most common mathematical apparatus for solving uncertainty
problems (this applies to both type-1 FS and type-2 FS), there are virtually no convincing examples of
the effectiveness of type-2 FS compared to type-1 FS, recent studies have shown that in some cases the
standard FST does not allow to solve a number of problems under uncertainty. This is due to the
arithmetic and logical nature of FS, which does not allow the use of FS tensor-matrix analysis directly
and in full. This means, in particular, solutions of large fuzzy equations and systems of fuzzy equations,
where the parameters can be both type-1 FS and type-2 FS.
In [3] shows that fuzzy sets over the last fifty years have laid the foundation for a successful method
of modeling uncertainty and inaccuracy in a way that no other technique has. The use of fuzzy sets in
real computer systems is extremely wide and constantly increasing, which emphasizes the relevance of
research related to the discovery of hidden knowledge, which contains uncertainty and its fuzzy set
models, presented in tensor form. Note that tensors and tensor decompositions are very powerful and
versatile tools that can model a wide variety of inhomogeneous, multi-aspect data. As a result of tensor
decompositions, it is possible to extract useful hidden information from multi-aspect data tensors [4],
including from data under uncertainty.
The object of research is the generalized processing of multidimensional (multi-aspects) and large-
volume data under conditions of uncertainty, which is modeled by fuzzy sets.
The subject of research – tensor models of type-1 and type-2 fuzzy sets, hidden knowledge that can
be extracted, tensor decompositions and the formation of the nearest fuzzy sets, the solution of fuzzy
equations based on the concept of the nearest fuzzy sets.
The purpose of the work is to expand the class of solvable problems under conditions of uncertainty
by extracting hidden knowledge by using tensor models, in particular, the concepts of nearest fuzzy sets
and properties of Kronecker algebra, tensor decompositions
The tasks that need to be solved to achieve the goal of the work are the following:
substantiate the necessity and expediency of representing type-1 FS and type-2 FS by tensor
models, which are based on the use of tensor products of FS components in modeling uncertainty;
show the equivalence of tensor models FS-1 and -2 type with SOP, obtained by singular
decomposition of the tensor model FS;
identify the possibility of 3D tensor representation of uncertainty and identify areas of rational
application of 3D models;
identify hidden knowledge that can be used in tensor modeling of uncertainty;
to develop methods for solving fuzzy equations at the level of matrix equations by using
Kronecker algebra.
2. Problem statement
2.1 List of main symbols and abbreviations
In table 1 are presented the main abbreviations, that are used in the article.
Table 1
Abbreviations
Abbreviation Explanation of the meaning of abbreviations
ЕЕG Electroencephalo graphics
US Universal set
TRS Type Reduced System
TS Time series
T2FS Type-2 fuzzy set
SOS A subset of ordered sequences
SOP A subset of ordered pairs
NKP The nearest Kronecker product
MF Membership function
KP Kronecker product
TP Tensor product
KMIP Iteration procedure Karnik - Mendel
IT2 FS Interval type-2 fuzzy set
ISD an initial set of data
HOSVD High-order singular decomposition
FV Fuzzy variable
FST Fuzzy set theory
FIS fuzzy interval systems
FS Fuzzy set
In table 2 are presented the main nomenclatures that is used in the article.
Table 2
Nomenclature
Symbol Definition
A, A, a, a Tensor, matrix, (column) vector, scalar
The set of real numbers
Outer product
Vec( ) Vectorization operator
Kronecker product
n n-mode product
A n-mode matricization of tensor A
n
A-1 Inverse of A
† Moore-Penrose Pseudoinverse of A
A or A
A 1/ 2
F T
Frobenius norm - trace A A
A(:,i) Spans the entire i th column of A (same for tensors)
A(i,:) Spans the entire i th row of A (same for tensors)
reshape ( ) Rearrange the entries of a given matrix or tensor to a given set of dimensions
a Type-1 fuzzy set:
a a /
a or a a1 ;
a
; a n n 2 , a A,
a 0, 1
1 n
2.2 Main statement
Recently, a number of uncertainty problems have emerged, the solution of which by TFS methods,
in particular, by fuzzy mathematics methods, is either extremely difficult or the result is not
constructive. This is especially true for data processing, which belongs to the category of BIG DATA,
where ultra-high dimensionality is combined with a large amount of data and thus necessitates working
with 3D data, membership functions for which are not tabulated, defining these functions is also
extremely difficult.
Problems related to decision-making based on fuzzy equations and systems of fuzzy equations with
general data (the parameters of the equations can be set in the form of both type-1 FS and type-2 require
the use of new methods and algorithms. Note that modern methods focused on this class of problems,
usually work with fuzzy numbers and contain a large number of assumptions. On the basis of the stated
requirements, the tasks are formulated as follows: representation of type-1 FS in the form of a 2D
x x n n
tensor (matrix) and, accordingly, the tensor product: x T x ; singular
x
decomposition u s v svd T allows you to calculate a subset of ordered pairs of sigmoid-
y y
like shape y x / , y X, 0, 1 such, that:
y
y x y x y
T y , T T , def T def T . (1)
F F
This allows you to implement mathematical operations on fuzzy variables x , y, ..., z at the level of
tensor variables: x T
x , y T y , ..., z T z with the subsequent transformation of the
result into the SOP. A similar algorithm with Kronecker products is implemented for type-2 FS, defined
on US X and presented as
z z z x z
z z / ,z X, 0, 1 , / ; (2)
for a given FS x x /
x , x X , x 0, 1 find a subset of ordered pairs
y y
y y / , y X , 0, 1 , obtained as a result of tensor decompositions, called
the fuzzy nearest set (with respect to FS) x ; in turn the tensor model of type-1 FS x has the
x
appearance X x , nearest is determined by the principle of the nearest Kronecker
product X y
y
2
min; .
F
n2 x n n , by
type-1 FS presented as a tensor product of components x T
implementing the procedure T
x reshape T x , p, q, w , p q w n n ,where T x -
3D tensor (Fig. 1), and high-order singular decomposition which allows you to obtain a subset of
multy multy
order sequences T
x
x / x , x , ... . Note that T
1 2 x
is a multi-fuzzy
set
Figure 1: The sequence of conversion from standard FS x at 2D and 3D tensor models
Formation of 2D tensor model type-2 FS:
x 1 m
1
x1 x x
1 1
x
x X
x
x /
x
Blurring MF . , (3)
0,1
x n xn 1
m
x x
n n
y y
x x y 1 ; ; x y n , (4)
1 1 n n
method of forming a 2D tensor model - double kroneker product of type-2 FS
x y
x n k k
x, T x y , T . (5)
Separately, we pay attention to the possibility of presenting the type-1 FS in the form as
proposed in [5] - x x
x n n or n 1 n .
3. Review of the literature
One of the directions of expanding the use of FS is granular computing, in [6] it was shown that
granular computing is a new computational theory and paradigm that deals with the processing of
information granules, which are defined as a set of information entities grouped together by their
similarity, physical adjacency or indistinguishable ability. In most aspects of human reasoning, these
granules have an uncertain formation, so the concept of detailing fuzzy information (and revealing
hidden information) may be of particular interest for applications where FSs must be converted to crisp
sets to avoid uncertainty.
In [7] a tensor granule formed as a tensor product of FS components is proposed, which allows to
significantly expand the possibilities of FST and, accordingly, to expand the range of solvable problems
under conditions of uncertainty. According to [8], the theory of rough sets is an important approach to
granular calculations.
Tensor models of type-1 and type-2 fuzzy sets (the concept of tensorization) not only significantly
strengthen the arsenal of methods of fuzzy mathematics but also be an additional channel for comparing
the quality of the obtained solutions. The concept of tensorization, as shown in [9], refers to procedures
for generating structured tensors of higher-order from lower-order data formats (vectors, matrices, or
even low-order tensors) or representing very large system parameters in low-order tensor formats.
For any given source data format, the tensor procedure can affect the choice and efficiency of tensor
decomposition in the next step. Records of such a tensor can be obtained using:
a certain permutation, for example, the transformation of the original data into a tensor,
alignment of data blocks or epochs, for example, slices of the third-order tensor are epochs of
multichannel EEG signals, or
increasing the data using, for example, Toeplitz matrices / tensors and G (H)ankel.
Let's pay attention to the last thesis. Procedures for the formation and subsequent deposition of
Toeplitz or Hankel matrices (tensors) formed on universal sets allow us to solve problems under
uncertainty by FST methods under limited conditions of MF assignment, proposed by the authors in
[33]. Recall that the tensor in the general case can be represented by fibers or slices [30, 33].
Note the following. First, the theory of rough sets as a new mathematical tool for the implementation
of procedures (fuzzy) data conclusions is proposed in [10]. In this regard, we note that the vast majority
of works concerning the type-2 FS and their extensions, consider the procedures of fuzzy conclusions,
ie the implementation of fuzzy rules "if A, then B otherwise C", although the number of problems under
uncertainty, where required type-2 FS and their extensions, much larger, especially for fuzzy
mathematics with type-2 FS.
Secondly, as shown in the paper [11]: “Information granules are intuitively attractive constructions
that play a key role in human cognitive activity and decision-making. We perceive complex phenomena
by organizing existing knowledge together with existing experimental evidence and structuring it in the
form of some meaningful, semantically sound entities that are central to all subsequent processes of
world description, environmental reasoning, and decision support.”
According to [12], type-1 FS can directly and effectively model certain types of uncertainty
(according to [1], these are fuzzy and inaccurate), because their MFs are absolutely crisp. On the other
hand, type-2 FS, having fuzzy MF, can model wider classes of uncertainty. The membership functions
of type-1 FS are two-dimensional, while the membership functions of the type-2 FS are three-
dimensional. It is the new third dimension of the type-2 FS that provides additional degrees of freedom,
which allows you to directly model the uncertainties. In type-1 FS membership values are between zero
and one, while the values of fuzzy membership type-2 are considered as the value of type-1 fuzzy
membership, A as a total type-2 FS, is described as follows:
u
A x / x [ f u / u ] / x, J x, : u [ x , A x ] 0,1 , (6)
x x A
X A X J u
x
x
min x , x max x . (7)
A A A A
In [13], the possibility of decomposing an interval type-2 fuzzy logic system into two parallel type-
1 fuzzy systems was considered. This decomposition avoids the problems associated with type
reduction methods, which are usually required in type-2 fuzzy systems. Type-2 fuzzy set (T2 FS) - is a
three-dimensional fuzzy set, in which the primary fuzzy set is characterized by classes of membership,
which are not crisp numbers, and actually fuzzy sets - the so-called. secondary membership functions.
In works [14,15] tensor models of type-1 FS are offered, which allow to apply matrix-tensor methods,
to use tensor-matrix analysis for problems of fuzzy mathematics and if necessary to receive the result
in the form standard for FST. Since type-2 FS is an extension of type-1 FS and this object has an
effective representation in the matrix (tensor) basis, there is a logical desire to expand the application
of tensor methods and models directly for type-2 FS, especially since type-2 FS - 3D measurable object,
i.e. tensor.
Moreover, in [16] it was shown that the use of type-1 fuzzy sets for modeling words is scientifically
incorrect. However, as shown in [30], most likely the reason lies in the fact that insufficient resources
were spent by researchers to develop the actual theory of type-2 FS, as evidenced by the fact that the
proposed operation for type-2 FS is not as effective and understandable as they need to be to satisfy real
application developers, and the lack of real compelling examples of type-2 FS applications
It is known that FS A on the universal set U is characterized by the membership function
: U 0,1 and is recorded as A u u or A u u , when U is discrete
A u U A u U
A
u , u U , u 0,1 . One of the
or continuous respectively, abbreviated record A u /
A A
greatest results of FST is the principle of fuzzy expansion, which allows to fuzzify any mathematical
theory.
Recall the following:
type-1 fuzzy sets are a special case of type-2 fuzzy sets, where for all u ∈ U the set of
degrees of primary membership, namely
u is a singleton (with a maximum value equal to
u
i
i Ju
one);
type-2 FS in the future will be denoted as follows: A
A
x / x, B B x / x ,
xX xX
where
A
x A f u / u , B x B g (u ) / u .
uJu uJu
Type-2 FS has features that not only complicate its use but do not positively affect its prevalence,
especially in the problems of fuzzy mathematics, in particular fuzzy equations and fuzzy equation
systems, because the fuzzy mathematics apparatus designed mainly for type-1 FS and common to type-
2 FS, was unable to solve such problems under uncertainty. This is especially true for the presentation
of type-2 FS in a form suitable for computer implementation, and defuzzification procedures. One of
the ways to solve these and other issues related to the use of FS-2 is to find new forms of representation,
granular form of representation (discussed by the authors earlier) and the geometric approach, which is
considered in [17, 18]. Note that the practical majority of algorithms designed to represent and defuzzify
type-2 FS, developed by J. Mendel.
The authors believe that the uncertainty simulated by the type-2 FS can be more effectively
represented, in particular for the implementation of mathematical operations, by a tensor granule. For
the generalized case of type-2 FS, when the functions of the secondary membership - the third
dimension is of any type, there is a significant computational complexity that has limited their
deployment. The complexity of the calculations in the general case of type-2 FS prevents their
deployment.
Of course, type-2 fuzzy sets exist in three-dimensional environments, this additional dimension
requires the introduction of additional notations. In particular, like type-1 FS, type-2 FS has a domain,
in this case X. The membership level at one point in the domain is a type-1 fuzzy number, known as the
secondary membership function. The domain of the secondary membership function in x, denoted by
Jx, is known as the secondary domain or shared domain. Estimation of membership at point u in the
function of secondary membership in x, denoted x, u , is known as the average level of
A
membership. In works [2, 3] the way of representation of type-2 FS under the name is resulted
Moderate.
It is important to note that the use of type-2 FS requires a preliminary assessment of the quality of
the solution obtained when using type-1 FS. In all cases, the type-2 FS is generated by the type-1 FS,
MF which is called primary. The following questions arise:
if based on IDS for modeling of uncertainty type-1 FS was offered
x x / ( x) , x X ,
x 0,1 , type-2 FS is formed by the erosion of the primary MF
u u
x x, u / , x X , x, u / , x X , u 0,1 ,
u 0,1, the output of the FLS is
a reduced type FS y y /
y , y 0,1 , y X , and defuzzyfied value y def y , then
how the quantities are related y def y and x def x ;
if the quality criteria of FS of all types are not defined, deffuzyfied values, for example, type-2
FS and type-1 FS, if they affect one object of uncertainty, are not considered criteria.
On this basis, we can assume that the task of finding type-1 FS, obtained as a result of the
transformation of some type-2 FS, which have close (or coinciding) defuzzyfied values, is relevant. In
addition, many researchers consider an important problem of accuracy in the application of FST
methods, although accuracy under uncertainty is a conditional concept. The possibility of replacing the
type-2 FS with an equivalent (from the point of view of defaced value) type-1 FS is relevant.
It should be added that this problem is not fundamentally new for the theory and practice of the type-
2 FS. The above cited article [19] proposes ... a new approach to the defuzzification of interval fuzzy
sets of type-2 based on the convolution method, which converts the interval type- 2 FS into an embedded
representative set of type- 1 (RES), the defuzzyfied value of which approaches the corresponding the
value of the type-2 set, it is known that RES as a type-1 set, can be defuzzyfied quite easily.
Available methods of defuzzyfication for discrete type-2 sets, first of all, provide the so-called
comprehensive defuzzyfication. For example, for fuzzy interval type-2 systems (FIS), the
defuzzification stage consists of two parts - the actual type reduction and defuzzification. This type
reduction algorithm was proposed by J. Mendel:
1. All possible built-in type-2 sets must be considered;
2. Minimum average membership found for each built-in set;
3. For each embedded set, the value of the domain of the centroid of type-1 of the embedded set
of type-2 is calculated;
4. For each embedded set, the value of the domain of the centroid of type-1 of the embedded set
of type-2 is calculated (x, z), it is possible that for some values x will be more than one corresponding
value z;
5. For each value of the domain the maximum average estimation is chosen, it creates a subset of
ordered pairs (x, zmax), such that between x and zmax there is an unambiguous correspondence. This
completes the reduction of the type-2 set to (Type Reduced Type) type-1.
The obtained TRS - as a type-1 fuzzy set, is easily defuzzyfied by finding its centroid value. Thus,
the reduction of the type involves the processing of all embedded sets in the type-2 FS, which is what
makes the algorithm to be called "exhaustive defuzzification ". Naturally, there are a lot of built-in sets.
For example, when in the above example type-2 FIS implemented inference using sets that were
sampled on 51 slices on the x and y axes, the number of embedded sets in the aggregate set was
calculated as a value of the order of 2.91063
Although embedded sets are generally easy to process, they create a bottleneck during processing
due to their high dimensionality. As a result, exhaustive defuzzification is an impractical method to use,
the most common method of reducing the type of fuzzy set interval – type-2 is the iterative procedure
Karnik - Mendel (KMIP). The result of reducing the type of the interval of fuzzy sets - type 2 is the
interval - type 1, where the centroid lies between two endpoints. An iterative procedure is an effective
method of finding these endpoints. The center of this set – type-1 (i.e. defuzzyfied value of the set –
type-2) is the center of this interval. Note that this procedure is extended to generalized fuzzy sets - 2
types [19].
In [19] T1 MF between the upper and lower uncertainty bands was found as a representative
embedded set, but the method and concept used are not based entirely on the concept of the influence
of uncertainty on certain data and degrees of affiliation. In [20] is proposed methods of overcoming
difficulties in understanding and interpreting type-2 FS for FLC.
4. Materials and method.
Historically, FST has formed the main object of the theory - a subset of ordered pairs (type-1 FS) as
a procedure for a heuristic blurring of a universal set and its representation as a set of -levels. Further
logic of development of the accepted concept naturally led to type-2 FS, formed as a procedure of
blurring crisp values of membership function type-1 FS. Assuming that the number of -levels in the
representation of uncertainty using FS is large enough (the concept of BIG DATA provides for the use
of FS as one of the possible models), we can show that initially selected by the expert FS (with heuristic
FN) can be simultaneously represented as multi fuzzy set [21] or as type-2 FS in 3D space. The fuzzy
set generalizes FS-1, -2 types, and intuitive sets.
Nearest FS. The ability to represent SOP as 2D and 3D objects requires an assessment of their
proximity. Recall that the problem of finding the nearest (farthest) element, neighbor, etc. is not new to
mathematics. In the last 20 years, it has been replenished with the so-called problem of the nearest
Kronecker product, the solution of which is extremely important for modern mathematics, in particular,
tensor (matrix) analysis. Unfortunately, the use of this powerful device to solve problems under
uncertainty began only in the last 5-7 years [ 22].
Tensor models FS were first proposed in [23], we recall that the main apparatus TFS - a subset of
n2
ordered pairs - is a matrix in , which can be represented as a tensor y (n-number -levels FS).
m p q
In turn, the procedure reshape A, m, p,q allows to represent the initial FS A in space
and get a subset of ordered sequences (triplets), which allows fundamentally from new positions to
implement the analysis of uncertainty in 3D space, in particular, this applies to type-2 FS and so on.
We present the problem of the nearest Kronecker product (NKP) using the paper [24]. It was shown
in [24] that the solution of the NKP problem is associated with the procedure of singular decomposition
of the permutation (vectorized) version of the matrix A. This leads to the problem
B, C R( A) vec( B) vec( C)T , and the fact of minimization is the search for the
F
nearest rank-1 matrix to R A . The nearest rank-1 matrix is a well-known problem of singular
decomposition. In particular, if UT R A V - singular decomposition, the optimum is defined
as:
1 / 2 U :, 1 , vec C 1/ 2
vec B
opt
1
opt 1
V :, 1 (8)
It is important to note that in this case the scaling is arbitrary. Indeed, if B and C is the
opt opt
solution of the NKP problem, and given 0, then B
opt
and 1 / Copt 1 / Copt ,
then and are also optimal. It is accepted that = max abs V :, 1 , it allows to consider B
opt
and
opt
C as SOP, where one of components C 0, 1 , - that gives the chance to apply the TFS device
opt
to the optimum decisions calculated as a result of singular decompositions.
Considering [31, 34] , the ordinary set A nearest to the fuzzy A one is located at the smallest
distance from the given fuzzy set, or in other words has the smallest norm. It is shown that this will be
an ordinary set endowed with the following properties
0, if x 0.5;
A i
A
x 1, if x 0.5;
A i
0, if x 0.5.
A i
In turn, if FS is represented as a tensor model, as shown below,
A a /
a a a1 ; ; a an A = A :, 1 A :, 2 n n (9)
1 n
then the search for the nearest fuzzy set should be implemented by entering 2 prerequisites:
fuzzy set is used to represent uncertainty, all subsequent mathematical procedures are
performed on tensor models, the final result (if necessary) is converted into a subset of ordered pairs,
which is analogous to fuzzy set, always has a sigmoid-like shape and is calculated as a result of
tensor decompositions.
The main advantage of the concept of nearest fuzzy sets (or subsets of ordered sequences) is that:
it is possible to use hidden knowledge, which is contained in the set of initial data (SID) and
accumulated in the FS;
there is an additional channel to obtain information for the formation of MF;
there is a possibility of processing 3D data under conditions of uncertainty and the possibility
of simplified analysis using type-2 FS;
additional possibilities of expanding the classes of solvable problems under conditions of
uncertainty, in particular, the solution of fuzzy equations and systems of fuzzy equations of type-2
FS and multi-fuzzy equations by using the methods of Kronecker algebra.
The specified model can be transformed into the following models:
x
SOP, calculated on the basis of singular decomposition T ;
3D tensor T
x [T x :,:,1 ,...,T x :,:, k ] , presented in the form of frontal slices
0 0 0
T
x :,:, j , j 1, k;
0
x
svd T z z , z Z , z 0,1
T
x
, (13)
x x
T0 reshape T , f , f , f , n k k f f f
T
x T x :,:,1 ,...,T x :,:, k (14)
The hidden knowledge that can be "extracted" from the tensor models of FS, includes the following,
33
primarily matrix and tensor invariants. For 2D tensor А main invariants can be de-fined as
I tr A A A A ;
1 11 22 33 1 2 3
1
2 2
2
I tr A tr A 2
1 3 +
; I det A .
1 2 2 3 3 1 2 3
Another possibility for obtaining new knowledge is that FS-1 and 2 types have fundamentally
equivalent 2D tensor models (obtained on the basis of the tensor product of components), which allows
solving fuzzy equations and systems of fuzzy equations, where all variables and coefficients are fuzzy
sets (1 or 2 types) almost one algorithm, the concept of which is given below.
a , a A,
Solution of fuzzy equations axb c , where a , x , b , c - fuzzy variables, a a /
b 0, 1 ; a 0, 1 ; x x / x , x X , x 0, 1 ; b b / b , b B,
c c 0, 1 based on 2 main principles:
c c / , c C ,
conversion of FV into 2D tensor (matrix)
aT
a a :, 1 a :, 2 , x T x x :, 1 x :, 2 , (15)
bT
b b :, 1 b :, 2 , c T c c :, 1 c :, 2 ; (16)
formation of a matrix equation T
a T x T b T c , its solution based on the vectorization
AXB C
procedure (Kronecker algebra) described in [26].
If we limit the case when all FV have the same number of (n) - -levels, then the solution has the
nn
form X ∈ , using the procedure of singular decomposition u s v svd X we can obtain
x , x X , x 0, 1 , which is a concrete solution of the fuzzy equation.
SOP x x /
The following fuzzy equation ax xb c , where a, x, b , c - fuzzy variables is solved similarly.
By converting the FV into a tensor variable, we obtain a matrix equation
T
a T x T x T b T c , (17)
AX XB C
To solve this problem, we apply the operator vec to the left and right sides of the above equation.
Thus, the equation can be written in the form (Im ⊗ A + BT ⊗ In) vec (X) =vec (C) vec (X)= vec
(C) (Im ⊗ A + BT ⊗ In)-1. Next steps: conversion vec (X) into a matrix Х , singular decomposition of
x , x X , x 0, 1 .
the matrix X, and determination of a subset of ordered pair x x /
The computer experiment contained specific tasks (algorithms, programs, interpretation of results)
that must be implemented to achieve the goal: realize mathematical support using MatLab to represent
FS-1 and 2 types of tensor models based on the use of tensor products of these components FS in order
to justify the need and feasibility of the proposed approaches:
In fig. 2 presents a general scheme of a computer experiment: initial FS 2D tensor model 3D
tensor model tensor analysis.
Figure 2: General scheme of computer simulation implementation main tasks.
n2
1. Standard FS with triangular (or Gaussian) MF a , which is presented in matrix
trimf
form, is transformed into a 2D tensor atrimf
a
T trimf
n n
, singular decomposition of
TP
comp . FS
a nev a n2
which svd T trimf allows you to get SOP , whose properties:
trimf
nev a nev
trimf
atrimf , def
F
atrimf def atrimf ;
F
2.
a
Transformation of a 2D tensor model T trimf
nn
standard FS atrimf in the 3D tensor
a
model: T trimf
nn
new T trimf
a p q m
, p q m n n , high-order
reshape
new a b b
singular decomposition HOSVD T trimf
allows you to get SOP b b, ,
1 2
m3 nev a n2
, which in terms of the criteria of claim 1 is equivalent to SOP trimf
.
a /
a , a A, a 0, 1 ; b b
b / , b B ,
3. Given FS a
trimf trapmf
0, 1 ; defined results c a f b c / , c C , 0, 1 , where
b c c
f
,,*, ,/ , calculated tensor models FS a T , b T
a b
accordingly, the calculated
values T
c T a T
b , which by means of singular decomposition are transformed into
f
SOP:
c newc newc / new c , newc C , new c 0, 1 .
T
new c new
Confirm equivalence c , def c def c .
F
F
4. Modeling of 3D data processing using matrix algebra. Given a set of 3D unstructured data, the
procedure reshape () allows you to structure a given set in the form of a 3D tensor - reshape (S, m,
p, n) {A (:,:, 1), A (:,:, 2) ,…, A (:,:, n)}, represented in the form of a set of frontal slices; in the
following figure. this procedure is applied to a separate time series window.
According to Theorem 2.4.1 [27], blkdiag (A) is a block diagonal matrix, which is defined as
follows:
1
A
2
blkdiag A
A (18)
n
A
where A(i) – і-th frontal slice A , i = 1, 2, ..., n3.
Separately, we note that the proposed approach to the analysis of 3D data under uncertainty is
applied to the analysis of 3D fuzzy time series
The next step is a singular decomposition of a block svd blkdiag A diagonal matrix, which
makes it possible to obtain a set of ordered pairs and process the resulting object by TFS methods. Note
that the proposed procedure for converting 3D data to SOP can be used for fuzzy logic systems with
3D data.
a b
Figure 3: a-Example [28-29] representation of a separate window of a 3D time series in the form
of a tensor model (a) - 3-time series, (b) - a tensor model of a window of the TS
5. Modeling the solution of fuzzy equations and systems of fuzzy equations under conditions of
uncertainty, if all parameters of the equation are type-1 FS or type-2 FS.
In [25, 29,32] an example of a real type-2 FS is given, the general form of which is presented in
several formats, some of which are given below (Fig. 4):
Consider this example in order to compare the proximity of defuzzyfied values and F-norms of type-
1 FS and type-2 FS created from the initial FS by the fuzziness of MF. In addition, this example is
important for 2 reasons: 1 - in [29] an example of type-2 FLS is given and it is shown that the yield of
FLS-2 type has a defuzzyfied value def ( y ) for type-1 FS obtained as a reduction of the initial FS type
obtained at the stage of the fuzzification.
According to the notation introduced in [30-33], this FS is also representative; 2 - tensor model type-
2 FS allows you to calculate a subset of ordered pairs or a subset of ordered sequences (analogs), which
allows you to have alternative solutions. For the purpose of transparency of calculations type-2 FS A
is transformed into a set of objects: matrices (secondary FN - ), vectors (primary FN - and US
2 1
x respectively) are shown in Fig. 4 above.
Recall that the type-2 FS A, B are considered as defined on U in the form A x / x,
xX A
B
B
x / x , where x f x u / u, x B g x w / w .
xX A
uJ x wJ x
A B
a b
Figure 4: a - example type-2 FS is presented in the format Moderate [30],
b - 3D form of presentation type-2 FS in Moderate format [30]
In the [29] it is shown that FS B x / x can be represented as a subset of ordered pairs
B
x X
n2 nn
B x x
B
and can in turn be converted to a 2D tensor
B
x x
, singular
decomposition of which allows obtaining a new SOP (or SOS if necessary), endowed with the property
of proximity to the original SOP.
5. Results
Below are FS with a triangular MF, which simulates the statement of approximately 9.5, and the
nearest crisp set, their F-norms and defuzzyfied.
a b c
Figure 5.: a - the initial FS of approximately 9.5 with a triangular MF,
b, c - SOP, formed as a result of singular decomposition of the tensor model FS.
In fig.5. are shown the initial FS of approximately 9.5 with a triangular MF: x x /
x ,
x 0, 1 X=[5:9/8:14]; x y=trimf(x, [5 9 14]); b, c - SOP, formed as a result of sin-
x T x :, 1 x :, 2 99
gular decomposition of the tensor model FS -
FS 9.5 trimf crisp set
5.00 0 5.00 0
6.13 0.28 6.13 0
7.25 0.56 7.25 1
8.38 0.84 8.38 1
9.50 0.90 9.50 1
10.63 0.68 10.63 1
11.75 0.55 11.75 1
12.88 0. 23 12.88 0
14.00 0 14.00 0
F-norm and defuzzyfied value
29.85 9.34 29.85 9.44
Tensor model FS 9.5 trimf
0 1.41 2.81 4.22 4.50 3.38 2.25 1.13 0
0 1.72 3.45 5.17 5.51 4.13 2.76 1.38 0
0 2.04 4.08 6.12 6.53 4.89 3.26 1.63 0
0 2.36 4.71 7.07 7.54 5.65 3.77 1.88 0
0 2.67 5.34 8.02 8.55 6.41 4.28 2.14 0
0 2.99 5.98 8.96 9.56 7.17 4.78 2.39 0
0 3.30 6.61 9.91 10.58 7.93 5.29 2.64 0
0 3.62 7.24 10.86 11.59 8.69 5.79 2.90 0
0 3.94 7.88 11.81 12.60 9.45 6.30 3.15 0
Subset of ordered pairs
porp1 = with sort
porp =
4.50 0 4.50 0
5.51 0.31 5.51 0
6.52 0.63 6.52 0.25
7.54 0.94 7.54 0.31
8.55 1.00 8.55 0.50
9.56 0.75 9.56 0.63
10.58 0.50 10.58 0.75
11.59 0.25 11.59 0.94
12.60 0 12.60 1.00
F-norm and defuzzyfied value
26.88 8.41 26.88 10.43
For comparison, we give similar parameters of the initial FS and SOP: F-norm: 29.85 26.88;
defuzzyfied value: 8.41 (9.34 9.44) 10.43
Norm"s kron prod. of Subset of ordered pairs and Tensor model
x :, 1 x :, 1 porp1 :, 1 porp1 :, 1 porp :, 1 porp :, 1 48.30
F F F
This example confirms the main thesis - a subset of ordered pairs, the result obtained by the singular
decomposition of the tensor model of the initial FS, adequately represents the initial FS.
Type-2 FS. Initial data - an object under uncertainty is represented as a data matrix
disp ('Initial data - object under uncertainty is represented as a data matrix')
% Presented secondary FS – type-2 FS
mu2=
[ 0 0 0 0 1.00 1.00 1.00 0 0 0 0;
0 0 0 0 0.20 0.60 0.80 0 0 0 0;
0 0 0 0.30 0.60 0.20 0.60 0.30 0 0 0;
0 0 0 0.60 0.40 0 0.40 0.60 0 0 0;
0 0 0 0.80 0.20 0 0 1.00 0 0 0;
0 0 0.20 0.80 0 0 0 0.80 0.20 0 0;
0 0 0.40 0.60 0 0 0 0.60 0.30 0 0;
0 .80 0.60 0.30 0 0 0 0.30 0.60 0.80 0;
0 .90 0.80 0 0 0 0 0.80 0.90 0 0;
1.00 1.00 1.00 0 0 0 0 0 1.00 1.00 1.00];
disp ('The specified object is approximated by a standard FS with a triangular MF')
disp ('Comparative evaluation - standard FS with triangular MF')
Universal set
X = [0:10]; Primary MF
mf = [0:0.1:1]; standard triangular MF
y = trimf(X, [0 mean(X) 10]); Standard FS with a triangular MF for comparison is presented
v = [X' y']; as a matrix в 11 2
disp ('NORM and Defuzzyfied value of standard type-1 FS')
[norm(v,'fro') sum(v(:, 1).*v(:, 2))/sum(v(:, 2))]
19.71 5.00
n_kr_v = norm(kron(v(:,1),v(:,2)'),'fro') -norm of the Kron product of the standard FS component
36.18
disp('Implementation of the NORM calculation procedure and defuzzyfied value of type-2 FS')
used calculation formulas given in the work
for i=1:11
[b(:,i)]=mu2(:,i).*mf';
end
b;
bs=sum(b(:,1:11));
s=sum(mu2(:,1:11));
bss=bs./s;
vnew=[X' (1-bss)'];
%************************************************************************
disp('NORM and Defuzzyfied value of FS-2type(type reduction)')
[norm(vnew,'fro')sum(vnew(:,1).*vnew(:,2))/sum(vnew(:,2))]
Implementation of the NORM calculation procedure and defuzzyfied value of type-2 FS
F-NORM and Defuzzyfied value of FS-2 type (type reduction) 19.77 5.02
Universal Membership functions
set Type-2 FS Standard
trimf( ) Type-1 FS
1 2 3
0 0.00 0.00
1.00 0.09 0.20
2.00 0.13 0.40
3.00 0.50 0.60
4.00 0.86 0.80
5.00 0.94 1.00
6.00 0.87 0.80
7.00 0.44 0.60
8.00 0.13 0.40
9.00 0.09 0.20
10.00 0.00 0.00
Note: 1. Universal set common to FS-1 and FS-2 type.
2.Col.2 –type-2 FS with triangular MF, blurring of MF forms type-1 FS.
3.Col.3 – type-1 FS, obtained as a result of the procedure reduced type:
type-2 FS type-1 FS.
Note that the established criteria are -F-norm FS, presented in the form of a matrix with n 2 , and
the defuzzyfied value for both cases of uncertainty representation practically coincide: (19.71, 5.00)
and (19.77, 5.02), although the use of accuracy criteria in modeling uncertainty is a rather contradictory
approach.
The tensor model FS-2 type in MatLab notation has the form:
fs1=[0:0.1:1];
z=kron(X, kron(fs1,mu2(:,1:11))); Tensor (Kroneker-product) model type-2 FS from 121 121
size(z) 11x11x121 – irrational form of representation
z1=reshape(z,121,121); Transformation of the initial KP model into a square matrix
n_kr_Ta 183.79
Formation of a subset of ordered pairs
[u s v]=svd(z1);
disp('1 variant -> Singular decomposition of the type-2 FS tensor model')
disp('Subset of ordered pairs -sort')
Tab_pup_x=sort([abs(u(:,1)*s(1,1))*max(abs(v(:,1))),abs(v(:,1))/max(abs(v(:,1)))]);
disp('F-norm and Defuzzyfied value of SOP')
[norm(Tab_pup_x,'fro') sum(Tab_pup_x(:,1).*Tab_pup_x(:,2))/sum(Tab_pup_x(:,2))]
singular decomposition of the type-2 FS tensor model
The subset of ordered pairs -sort
F-norm and Defuzzyfied value of SOP 47.89 5.75 (*)
NORM of kron product of SOP components
Comparison of norms 183.79 183.79
sparse SOP: NORM and Defazzifited value of sparse SOP 13.60 5.36 (**)
note that the case (*) concerns SOP from 121 2 , the case (**)SOP from 11 2
(sparced set)
z3=reshape(z1,11,11,11,11); Transformation of the initial CD model into a tensor
CP4_ALSLS [32] CANDECOMP/PARAFAC decomposition of a fourth-order tensor(CP4).
[A1,A2,A3,A4]=cp4_alsls(X,R) computes a CANDECOMP/PARAFAC decomposition of a
fourth-order tensor X in Rank-one terms, stored in the factor matrices A1, A2, A3, A4, belonging to
the first, second, third and fourth mode, respectively.
[A1,A2,A3,A4] = cp4_alsls(z3,1)
I I I
СР (CANDECOMP/PARAFAC ) decomposition (factorization) of the tensor Y 1 2 N
J
can be defined as Y a
1 a 2 a
N E .
j j j j
j 1
Recall that the matrix Y is a rank-1 matrix, if and only if Y = uvT, where u and v are nonzero
vectors.
In J
n a n , a n , , a n , n 1, N contain latent components aj as
n
Factor matrices A 1 2 j
columns
The first columns of factor matrices
c=abs([A1 A2 A3 A4])=
0.17 0.66 0 0
0.09 1.44 0.17 0.17
0.31 1.44 0.34 0.34
0.50 0.97 0.51 0.51
0.61 0.22 0.68 0.68
0.73 0.09 0.85 0.85
0.77 0.24 1.02 1.02
0.76 1.36 1.19 1.19
1.38 1.47 1.36 1.36
1.52 1.00 1.53 1.53
2.09 0.66 1.70 1.70
Normalized factor matrices
c1=[c(:,1)*max(c(:,2))*max(c(:,3))*max(c(:,4)) c(:,2)/max(c(:,2)) c(:,3)/max(c(:,3))
c(:,4)/max(c(:,4))]
c1 =
0.72 0.45 0 0
0.38 0.98 0.10 0.10
1.31 0.98 0.20 0.20
2.09 0.66 0.30 0.30
2.59 0.15 0.40 0.40
3.09 0.06 0.50 0.50
3.24 0.16 0.60 0.60
3.20 0.92 0.70 0.70
5.83 1.00 0.80 0.80
6.44 0.68 0.90 0.90
8.84 0.45 1.00 1.00
F-norm of the result norm(c1,'fro') 14.50
Reduced type: matrix reduction с1 114 to SOP с2 11 2 (min(c(1:11,2:4))
c2=[c1(:,1) [ 0 0.1 0.2 0.3 0.15 0.06 0.16 0.7 0.8 0.68 0.45]']
c2 =[0.72 0; 0.38 0.10;1.31 0.20; 2.09 0.30;2.59 0.15; 3.09 0.06;
3.24 0.16; 3.20 0.70; 5.83 0.80; 6.44 0.68; 8.84 0.45]
Calculation of defuzzyfied value and F-norm of SOP
sum(c2(:,1).*c2(:,2))/sum(c2(:,2)) 4.90 14.50
The given object is approximated by standard FS with triangular FN
Comparative evaluation - standard FS with triangular FN
NORM and Defuzzyfied value of standard type-1 FS 19.71 5.00
Implementation of the procedure for calculating the NORM and Defuzzyfied value of type-2 FS
NORM and Defaulted value of type-2 FS (type reduction)
19.77 5.02
Sparsed SOP
NORM and Defuzzyfied value of Sparsed SOP 13.60 5.36
6. Conclusions
1. The theory of FS is now a practically universal apparatus, which is used in almost all cases
where there may be uncertainty. However, the emergence of new problems requires continuous
expansion of the standard theory of fuzzy sets, which is reproduced in the creation of new types of
FS (rough FS, hesitate FS, multiFS, etc.), automation of FS formation processes, including MF,
contradicts the ideology of TFS. However, the main object of TFS is the fuzzy set, which has not
been studied to provide adequate answers to modern requirements, in particular, the urgent need to
process BIG DATA.
2. One of the areas of possible research is tensor modeling of uncertainty, the basis of which is
embedded in the nature of FS - a subset of ordered pairs. Objects that can represent tensors include
vectors and scalars, as well as other tensors. Tensors can take several different forms, such as scalars
and vectors (which are the simplest tensors), double vectors, multiline maps between vector spaces,
and even some operations such as a point product. Tensors are defined independently of any basis,
although their components are often called bases based on a particular coordinate system.
3. The representation of type-1 FS or type-2 FS as a tensor product of components is offered, the
result is a 2D tensor (or 3D tensor in case of large dimension). This approach allows to use of the
possibilities of tensor-matrix analysis to solve problems under uncertainty, along with the TFS
apparatus, realizing the extraction of new knowledge (matrix-tensor invariants, matrix-tensor
decompositions), which significantly expands the range of problems under uncertainty.
4. Based on the concept of extracting hidden knowledge, a method of automatically creating FS
by structuring the initial data set with subsequent tensor decomposition is proposed, the obtained
SOP is endowed with all the properties of MF. If it is impossible to implement the procedure of
structuring IDS, it is proposed on the basis of calculating the values of creating the US vector in
the form and blurring the latter by using special matrices (Toeplitz, Hankel, etc.), matrix (tensor)
decomposition of which will create SOP - analog FS.
5. It is shown that tensor models of standard type-1 FS allow representing this object
simultaneously as a 2D tensor, 3D tensor, type-2 FS with sparse US, and multiFS; in addition, the
standard type-2 FS can be represented as type-1 FS, preserving the properties of the original object
(F-norm, defuzzyfied value). This conclusion allows us to solve fuzzy equations in which the
coefficients and the unknown are fuzzy variables of types 1 and/or 2, at the level of standard matrix
equations, followed by the transformation of the matrix solution into the SOP.
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