The present paper is devoted to the linear regression for a fuzzy set of fuzzy data samples. This model allows one to take into account the data of different quality. It is shown that regression parameters are type-2 fuzzy sets. Furthermore, the corresponding type-2 membership functions are given. The decomposition approach is used to investigate the T2FSs of linear regression parameters. It is shown that each T2FS of regression parameter can be decomposed according to secondary membership grades into a finite collection of fuzzy numbers. Each of these fuzzy number is the corresponding fuzzy regression parameter for a set of data numbers. This set is the corresponding α-cut of the original fuzzy set of fuzzy data samples. The illustrative example is given.
The classical regression analysis is based on crisp data and a crisp relationship between the dependent variable and the independent variables. In practice, there are many situations in which observations cannot be measured as crisp quantities, because the information is often fuzzy, incomplete, linguistic or noisy. Fuzzy regression analysis is a non-statistical method based on a fuzzy set theory rather than probability theory (see [1]). In a general model of a fuzzy regression both input and output are fuzzy. In this regard, the fuzzy regression model contains fuzzy parameters instead of error terms.
The three main fields can be distinguished in fuzzy regression analysis. These are possibilistic regression analysis, fuzzy least squares methods and machine learning techniques. The probabilistic approach in fuzzy regression analysis was first proposed by Tanaka et al. [2]. Unlike conventional regression analysis, where deviations between observed and predicted values reflect a measurement error, deviations in fuzzy regression reflect the uncertainty of the system structure expressed by fuzzy parameters of the regression model. Fuzzy parameters of the model are considered to be distributions of possibilities and determined by solving a linear programming problem that allows one to minimize fuzzy deviations subject to membership degrees constraints. Since the membership functions (MFs) of fuzzy sets (FSs) can be viewed as probability distributions, this approach was called ‘possibilistic regression analysis.’ The possibilistic approach was explored and improved by many authors. Reviews of possibilistic regression analysis can be found in D’Urso [3].
Fuzzy regression analysis was also considered from the viewpoint of generalizing the classical least squares method to the case of fuzzy data. The idea of this approach is to minimize in some sense the different distance measures between the predicted fuzzy values and the given fuzzy data. Fuzzy least squares methods were first proposed by Celmins [4]. Later, this approach was significantly developed by many researchers. In [1] one can find qualitative reviews on the fuzzy least squares and fuzzy least absolute methods.
Machine learning techniques made it possible to generalize fuzzy regression analysis through the use of genetic algorithms, neural networks, and support vector machines. Relevant references can be found in Chukhrova and Johannssen [1] and Hastie et al. [5].
Often, when solving applied problems, data of different quality can be used [6]. For example, according to [7], wind tunnel experiments provide high simulation accuracy (a source of high-fidelity data). Also, experiments based on computational physical models have a higher error (source of low fidelity data). In some applications, a significantly more accurate regression model can be built if lowprecision data are also used. In this case, the problem arises of constructing a regression based on data of different quality.
Using data of different quality with the aim of improving model accuracy is not a new concept. For instance, Hevesi et al. [8] predict average annual precipitation values near a potential nuclear waste disposal site using a set of precipitation measurements from the region along with more easily obtainable elevation map of the area. Kennedy and O’Hagan [9] approach the subject from the perspective of model construction using data resulting from computational simulations of varying fidelities and costs. In [7], to construct a Gaussian regression model, the problem of planning an experiment is solved with the choice of the ratio between the sizes of samples of low-precision and high-precision data. Also, to process data with a variable degree of certainty the methods of transfer learning [10], space mapping [11], and others [12] are used.
Data quality (degree of usability of data) is a complex concept. It is characterized by objectivity, integrity, relevance, measurability, controllability, etc. In some cases, the quality of data may not be crisp defined [13]. According to [14], where using data without expert knowledge, the choice of a representative sample becomes an NP-complete problem. Therefore, samples have to be found within a reasonable time, and this justifies the use of fuzzy methods that formalize expert knowledge expressed in natural language words. For instance, in the framework of quality control fuzzy expert assessments are used in [15] to construct acceptance sampling plans (how many units can be selected from a consignment and how many defective units are allowed in this sample).
In this article, we intend to investigate the method of constructing a fuzzy linear regression in the case when fuzzy data samples are of different qualities. Furthermore, the degrees of membership to a FS are known for these samples. This leads to a possibility that the regression takes into account fuzziness of the quality assessments of different samples, rather than just uncertainty of data. Examples of such FSs data samples could be: ‘High quality data samples’, ‘Questionable data samples’, ‘Actual data samples’, etc.
The main result of the article justifies the fact that a FS of different quality samples of type-1 fuzzy data generates a type-2 fuzzy regression model. In this model, the regression parameters are T2FSs on the real line with constant secondary grades. Although, in general, a T2FS is a rather complicated mathematical object, T2FSs with constant secondary grades are simple enough for practical use. This feature allows us to decompose this set by secondary grades into a collection of corresponding fuzzy numbers. Each of them represents the corresponding fuzzy regression parameter for a crisp set of data samples. The set in the focus is the α-cut of the original fuzzy set of data samples. We note that the well-known type 2 fuzzy regression models use crisp collections of type 2 data sets, while the model proposed in the article uses a fuzzy collection of type 1 fuzzy data samples. This is a principal difference between them. It should also be added that this article continues the line of research in the field of mathematical operations with a fuzzy set of operands, first introduced in the context of intersections and unions of fuzzy sets [16, 17].
In this section, we briefly review some existing theories and definitions.
The article focuses on a linear regression in the case when data samples form a FS. We stress that one could have exploited different known methods of constructing a linear regression for fuzzy data. As an alternative, we intend to modify the method [18] based on the extension principle.
Let K {1,..., K } be the set of indices of data samples {yi , xi1,..., xip}, i K , where K is the cardinality of K . A crisp statistical linear regression has the form
yi xi ( (K ))T i , i K , (
transpose, variables (factors, regressors) xil , l 1,..., p ; i is the independent normal random variable. The T p is is the the number vector of
of independent variables,
regression parameters. Let
y(K ) X (K )( (K ))T (K ) be the matrix notation of equations (
ˆ(K ) [ X T (K ) X (K )]1 X T (K ) y(K ) .
Assume that the data is fuzzy. To generalize formula (
fl ( X (K ), y(K )) ˆl (K ) the l -th element, l 0,..., p of the estimate ˆ(K ) . We also denote by
X (K ) {xis}iK,s0,...p , y(K ) ( yi )TiK
1, r [l (K )] ;
1[l (K )]u (r) 0, r [l (K )]uu ;
r
, u [0,1] . According to [18] and Remark 1, formula (
{ xis (xis ), yi ( yi )}: r fl ( X (K ), y(K )), X (K ) {xis}iK;s0,..., p p K , y(K ) {yi}iK
K }
by Zadeh’s extension principle [20]. Here, fl ( X (K ), y(K )) is the l-th element of the vector
ˆ(K ) [ X T (K ) X (K )]1 X T (K ) y(K ) by (
u[0,1] where [l (K )]u is the u -cut of the FN l (K ) . This u -cut is the set [l (K )]u {r : l (K ) (r) u} (K ) [ X T (K ) X (K )]1 X T (K ) y (K ) according to the least squares method [18]. For each l 0,..., p , l (K ) fl ( X (K ), y(K )) {(r,l (K ) (r)) : r } is the FN with the MF with the MF l (K ) has the form the matrix of independent variables and the vector of dependent variables, respectively, where for each i K , xi0 1 is the crisp number which is equal to 1; xis , s 1,..., p and yi are fuzzy numbers (FNs) with the MFs xis (xis ) , xis , s 1,..., p and yi ( yi ) , yi , i K , respectively. Here, is the real line.
Remark 1. Recall that a FN is a normal FS on concave MF (for example, see [19]).
The vector (K ) (l (K ))l0,..., p
with the upper semicontinuous and
quasiof fuzzy parameters of the regression has the form
(
too. Equality (8) entails [l (K )]u { fl ( X (K ), y(K )) : xis [xis ]u , s 0,..., p; yi [ yi ]u , i K} , where for each i K , the u -cuts of the FNs xis , s 0,... p and yi are closed intervals ]D ,[xis ]uH ], s 0,..., p and [ yi ]u [[ yi ]uD ,[ yi ]uH ] , [xil ]u [[xis u
. Since l (K ) is a FN, then its u -cut [l (K )]u is the interval
[l (K )]uD min{ fl ( X (K ), y(K )) : xis [xis ]u , s 0,..., p; yi [ yi ]u , i K} ,
[l (K )]uH max{ fl ( X (K ), y(K )) : xis [xis ]u , , s 0,..., p; yi [ yi ]u , i K} .
Thus, (
l (K ) (r) max{u [0,1]: [l (K )]uD r [l (K )]uH } , r l (K ) {(r,u) : r [[l (K )]uL ,[l (K )]uH ],u [0,1]} {([[l (K )]uL ,[l (K )]uH ],u) : u [0,1]} , respectively. Problems (10) and (11) are rather complicated. In view of this, it is suggested in [18] to use, for l 0,..., p , the approximate value of the l -th fuzzy parameter
l (K ) {([[l (K )]uL ,[l (K )]uH ],u) : u [0,1]} , l (K ) (r) max{u [0,1] : [l (K )]uD r [l (K )]uH } , r fl ([ X (K )]uH ,[ y(K )]uD ), fl ([ X (K )]uH ,[ y(K )]uH )}, [l (K )]uD min{ fl ([ X (K )]uD ,[ y(K )]uD ), fl ([ X (K )]uD ,[ y(K )]H , u fl ([ X (K )]uH ,[ y(K )]uD ), fl ([ X (K )]uH ,[ y(K )]uH )}. [l (K )]uH max{ fl ([ X (K )]uD ,[ y(K )]uD ), fl ([ X (K )]uD ,[ y(K )]H , u where Here, the matrices [ X (K )]uD , [ X (K )]uH and the vectors [ y(K )]uD , [ y (K )]uH are comprised in of the elements [xis ]uD ,[xis ]uH , s 0,..., p; [ yi ]uD ,[ yi ]uH , i K , respectively (see (9)). It is clear that [l (K )]uD [l (K )]uD , [l (K )]uH [l (K )]uH . Therefore, the inclusion [l (K )]u [l (K )]u is hold for the closed interval [l (K )]u [[l (K )]uD ,[l (K )]uH ] . This entails l (K ) (r) l (K ) (r) and thereupon the value l (K ) (r) is a lower bound of l (K ) (r) for r . 2.2.
Zadeh [22] introduced the notion of type-2 fuzzy set (T2FS) as a generalization of type-1 fuzzy set (T1FS) (that is, an ordinary FS). Unlike a T1FS, the membership degree of elements in a T2FS is a FS on closed interval [0, 1]. Based on the ideas of Karnik and Mendel [23], Mendel and John [24] gave a different definition. The T2FS C on a set X is the collection
C {((x,u),C (x,u)) : x X , u J x [0,1]} , where C (x,u) is a type-2 membership function (T2MF), J x is the set of primary membership degrees u of x X to C . The value C (x,u) is a crisp number from the closed interval [0, 1] which is called the secondary grade of the pair (x,u) .
Remark 2. According to the comments of Harding et al. [25] and Aisbett et al. [26], one has to define the T2MF C (x,u) for all x X and u [0,1] . To this end, one should put C (x,u) 0 for all u J x , x X .
Remark 3. The primary membership degree u is deemed as the degree of manifestation of some property (which determines the given FS) of x X . According to [27], we interpret the secondary grade C (x,u) as the degree of truth of the corresponding primary degree u of this property for x.
Following [24], we define embedded , , (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) C {((x,u),C (x,u)) : x X ,u [0,1]} . Assume that ux Ce1 (x) [0,1] is a unique primary degree of membership, for each x X , where Ce1 (x) , x X is the
MF of the
T1FS Ce1 {(x,Ce1 (x)) : x X }. This T1FS is called embedded in the T2FS C . We define the embedded T2FS C e2 in C in the form C e2 {((x,ux ),Ce2 (x,ux )) : x X } with Ce2 (x,ux )) C (x,Ce1 (x))) for all x X .
Remark 4. According to [24], each element of the type-2 fuzzy collection C {((x,u),C (x,u)) : x X ,u [0,1]} is interpreted as a subset. Thus, the collection is represented as the classical union of its elements in the sense of T1FSs. In [24], Mendel and John stated that each T2FS can be represented as a collection of all possible embedded T2FSs. 2.3.
We shall need one special case of a T2FS to be defined according to [28, 29, 30]. Let A { C (x,u) : C (x,u) 0, x X ,u [0,1]} be the set of all possible positive values of secondary grades for the T2FS C {((x,u),C (x,u)) : x X , u [0,1]} . Assume that the set A is finite.
Definition 1. We say that an embedded T2FS C e2 ( ) {((x,ux ),Ce2 ( ) (x,ux )) : x X } in the T2FS has a constant secondary grade A if, for each x X , the unique primary degree C ux Ce1( ) (x) [0,1] exists for which Ce2 ( ) (x,ux ) , i.e. C e2 ( ) {((x,Ce1( ) (x)), ) : x X } .
Here Ce1( ) (x) , x X is the MF of the embedded T1FS C e1( ) {(x,Ce1( ) (x)) : x X } in the T2FS C .
Remark 5. Obviously, for the T2FS C and each A , there is the unique embedded T1FS C e1( ) {(x,Ce1( ) (x)) : x X } which is corresponding to the embedded T2FS C e2 ( ) with constant secondary grade . Hence, {((x,Ce1( ) (x)), ) : x X } .
C e2 ( ) {(Ce1( ), )} {({(x,Ce1( ) (x)) : x X }, )}
Let N {1,..., n} be the set of indices of fuzzy data samples {yi , xi1,..., xip}, i N in the form of
FNs with the MFs xis (xis ) , xis , s 1,..., p , yi ( yi ) , yi , i N , respectively. The matrix
X (N ) of independent variables and the vector y(N ) of dependent variables are given by formula
(
l (K ) {(r,l (K ) (r)) : r }, l (K ) (r) Ulr (K ) , r supp(l (K )) {r :Ulr (K ) 0} , where supp(l (K )) is the support of the FN l (K ) . Next, we generalize formulae (19) and (20) to the case of the FS I of sample indices. We denote by Bl the l -th regression parameter, and by MBl (r) , r its MF for the FS I of data samples indices. In this case, the value of the MF MBl (r) for each fixed r r * coincides with the image U lr* (I ) of the FS I under Ulr* , i.e. M Bl (r*) Ulr* (I ) . According to Zadeh’s extension principle [20], the image of the FS I under the mapping Ulr* : 2N [0,1] (see (18)) is the FS Ulr*(I ) {(u,Ulr*(I ) (u)) : u [0,1]} with the MF Ulr*(I ) (u) max{ [0,1] : u Ulr* (I ( ))} , u [0,1] . Here, I ( ) { j N : I ( j) } is the -cut, [0,1] of the FS I {( j, I ( j)) : j N} of sample indices;
Ulr* (I ( )) l (I ( )) (r*) , is the image of the -cut I ( ) , [0,1] of the FS I of the samples indices in the mapping Ulr* (see (18)). The value Ulr* (I ( )) is equal to the MF value l (I ( )) (r*) of the l -th fuzzy parameter l (I ( )) for the set I ( ) of sample indices.
Remark 6. Let A { I ( j) : j N} be the set of membership degrees values of the fuzzy set I {( j, I ( j)) : j N} of sample indices. Note that the cardinality of the set A is |A| ≤ n. The situation |A| < n may occur if the degrees of membership I ( j) coincide for different indices j N of samples. It is clear that while obtaining -cuts I ( ) { j N : I ( j) } of the fuzzy set I we can assume that A rather than [0,1] .
Thus, according to (20) and (21), for fixed r r * , values of M Bl (r*) form the T1FS {(u, MBl (r*) (u)) : u [0,1]} on [0, 1] with the MF MBl (r*) (u) Ulr*(I ) (u) max{ A : u Ulr* (I ( ))}, u supp(M Bl (r*)) {u [0,1] : u Ulr* (I ( )), A} . Then (22) entails MBl (r*) (u) max{ A : u l (I ( )) (r*)}, (23) where u supp(M Bl (r*)) {u [0,1]: u l (I ( )) (r*), A}. Therefore, we conclude that Bl is a FS on with the MF whose values form T1FS on [0,1] . Then, according to [22], Bl is the T2FS on . In the manner of vertical slices [27] the T2FS Bl on has the form:
Bl {(r, M Bl (r)) : r R} {(r,{(u, MBl (r) (u)) : u Jr }) : r } , (24) where MBl (r) (u) , u [0,1] is the MF of the T1FS M Bl (r) {{(u, MBl (r) (u)) : u [0,1]} of values of fuzzy degree of membership of the element r to the T2FS Bl and Jr supp(M Bl (r)) is the set of primary membership degrees, where supp(M Bl (r)) is the support of the T1FS MBl (r) for r . According to Section 2.2, we can also characterize the T2FS Bl of the l -th regression parameter by means of the T2MF Bl (r,u) MBl (r) (u), u Jr and Bl (r,u) 0, u Jr . This conclusion allows us to introduce the following notion.
Definition 2. By the regression parameter with index l 0,..., p for the FS I of sample indices is meant the T2FS
Bl {((r,u), Bl (r,u)) : u [0,1], r } , (25) with the T2MF
Here,
max{ A : u l (I ( )) (r)}, u Jr ; Bl (r,u) 0, u Jr .
Jr {u [0,1] : u l (I ( )) (r), A}, 4. Regression for a fuzzy set of sample indices 4.1. Decomposition of T2FSs of regression parameters is the set of primary membership degrees u [0,1] with strictly positive secondary grades Bl (r,u) which coincides with the support supp(M Bl (r)) (see (23)) of the T1FS MBl (r) of fuzzy membership degrees of the element r
; is the MF of the l -th fuzzy parameter l (I ( )) (r) max{u [0,1] : [l (I ( ))]uD r [l (I ( ))]uH } ,
l (I ( )) {(r,l (I ( )) (r)) : r } , for the set I ( ) of sample indices (see (19)-(20) for K I ( ) ); fl ([ X (I ( ))]uH ,[ y(I ( ))]uD ), fl ([ X (I ( ))]uH ,[ y(I ( ))]uH )} [l (I ( ))]uD min{ fl ([ X (I ( ))]uD ,[ y(I ( ))]uD ), fl ([ X (I ( ))]uD ,[ y(I ( ))]uH , and [l (I ( ))]uH max{ fl ([ X (I ( ))]uD ,[ y(I ( ))]uD ), fl ([ X (I ( ))]uD ,[ y(I ( ))]uH , fl ([ X (I ( ))]uH ,[ y(I ( ))]uD ), fl ([ X (I ( ))]uH ,[ y(I ( ))]uH )} are the estimates of the lower and upper bounds (see (15)-(16) for K I ( ) ) of the u -cut [l (I ( ))]u of the FN l (I ( )) ;
I ( ) { j N : I ( j) } (32) is the -cut of the FS I of sample indices;
A is the set of the membership degrees values I ( j), j N of the FS I {( j, I ( j)) : j N} of sample indices (see Remark 6). According to (26), the set A includes all possible positive values of secondary grades for the T2FS Bl of the l -th regression parameter. (26) (27) (28) (29) (30) (31) (33)
For each l -th regression parameter, l 0,..., p , we apply a decomposition approach to represent the T2FS Bl in a more convenient form. Theorem 1 justifies the representation of the T2FS Bl in the form of a collection of the embedded T2FSs with constant secondary grades.
Theorem 1. The T2FS Bl of the l -th regression parameter is represented in the form of the collection Bl {Ble2 (I ( )) : A} of embedded T2FSs
Ble2 (I ( )) {(l (I ( )), ))} , with the constant secondary grades A . For each A , the embedded
T1FS l (I ( )) {(r,l (I ( )) (r)) : r } is the FN which is the l -th fuzzy parameter for the set I ( ) { j N : I ( j) } of sample indices, with the MF l (I ( )) (r) in form (28).
Proof. According to (25), the T2FS of the regression parameter with the index l 0,..., p has the form Bl {((r,u), Bl (r,u)) : u [0,1], r } . Hence,
Bl {{((r,u), max{ A : u l (I ( )) (r)}) : u Jr } {((r,u), 0) : u Jr }: r } , by (26). Remark 4 allows us to ignore the pairs (r,u) that have secondary grades equal to 0. Thus, (34)
Bl {((r,u), max{ A : u l (I ( )) (r)}) : u J x , r } , and thereupon
Bl {((r,l (I ( )) (r)) : A) : r } by (27). Note that the collection {(l (I ( )) (r), ) : A} is the T1FS which is formed by the unique value u l (I ( )) (r) of the fuzzy degree of membership of r . The different values A may correspond to u l (I ( )) (r) . Therefore, the equality {(l (I ( )) (r), ) : A} {(u, max{ : u l (I ( )) (r)})} holds true. Further, A regrouping elements yields Bl {(r, (l (I ( )) (r), )) : A, r } {{((r,l (I ( )) (r)), ) : r }: A} . (35) (36) Finally, by virtue of formula (29) we conclude that Bl {(l (I ( )), ) : A} .
The proof of Theorem 1 is complete. 4.2.
First, we construct the set A { I ( j) : j N} of membership degrees values of the FS I {( j, I ( j)) : j N} of sample indices. For each A , according to (32), we construct the -cut I ( ) { j N : I ( j) } of the FS I . Further, for each l 0,..., p , we use the representation of the T2FS Bl in the form of a collection of embedded T2FSs with constant secondary grades (see Theorem 1). This leads to the following sequence of calculations for each A .
We construct the embedded T1FS l (I ( )) {(r,l (I ( )) (r)) : r } . This is the FN, which is the l -th fuzzy parameter of the regression for the set I ( ) of sample indices. To construct the FN l (I ( )) one can use any known methods. An application of the method worked out in Section 2.1 of [18] yields l (I ( )) {([[l (I ( ))]uD ,[l (I ( ))]uH ],u) : u [0,1]} (see (13), (30), (31) for K I ( ) ) with the MF l (I ( )) (rl ) max{u [0,1] : [l (I ( ))]uD rl [l (I ( ))]uH } , rl . (37) Formula (37) is justified by (14) with K I ( ) . Then, the corresponding embedded T2FS with the constant secondary grade has the form Ble2 (I ( )) {(l (I ( )), ))} according to (33).
Once all embedded T2FSs Ble2 (I ( )) with constant secondary grades A have been obtained, the resulting
T2FS of the l -th regression parameter has the form Bl {Ble2 (I ( )) : A} {(l (I ( )), )) : A} by Theorem 1. According to Remark 3, the T2FSs Bl can be interpreted as follows. For fuzzy data of different quality, the l -th regression parameter Bl is equal to the l -th parameter l (I ( )) of the regression for the corresponding crisp set I ( ) of data samples with the degree of truth being equal to , A .
This example is devised to illustrate our approach. We stress that this example does not use real data. All data are given to the second decimal place.
To conduct the historical research, we need to find out the dependence of weight of a middle-aged person on he or her tall in the 10th century. The data borrowed from four authentic historical documents are located in lines 1-4 of Table 1. There is a reason to believe that the values in line 3 are slightly less reliable than the rest. We assume that in those days the height and the weight were measured with an accuracy to ±1Kg and ±5%, respectively. We denote by a (aL , aC , aU ) a ‘triangular’ FN with the MF (r a L ) / (aC a L ), r [a L aC ]; a (r) (a R r) / (a R aC ), r [aC a R ]; 0, otherwise. (38) yi1 ( yiL , yiC , yiU ) , i {1,..., 4} . The fuzzy data xi1 and yi be denoted as xi1 appr( xiC1 ) and yi appr( yiC ) , and can be interpreted as ‘approximately xiC1 and yiC ’, respectively. The dependence of the weight on the height for majority of data in rows 1-4 of Table 1 may seem strange, since weight gain decreases as the height increases. We assume that this can be explained by the small sample size and its non-representativeness.
Table 1 Weight depending on height i 1 2 3 4 5 6 7 8
Height xi1
To solve this problem, we supplement the data sample. We know from the medical sources [31] that
the Broca’s formula for determining the height and the optimal weight has the form W = H − 100,
where H is the height in cm and W is the optimal weight in Kg. According to this formula, we
calculate the optimal weights for four different heights and place these data in rows 5-8 of Table 1.
Thus, we have three sources of data of different quality. The high-quality source is historical, which
we fully trust. We evaluate the degree of reliability of data from this source. The results which are
located in rows 1,2,4 of Table 1 have the values of the degree of reliability are equal to 1. The degree
of less reliable data from the second source of historical data which are located in line 3 of Table 1 is
estimated at 0,9. The third source of data is medical. It is known that modern persons are taller than
persons living in the 10th century and having the same weight. As a consequence, the degree of
reliability of data from a medical source is not very high (for example, we estimate it as 0,7), since
this source only provides information on the ratio of the weight and the height for modern people.
Thus, we construct the FS ‘Reliable data samples’
I {(1;1), (2;1), (3;0,9), (4;1), (5;0, 7), (6;0, 7), (7;0, 7), (8;0, 7)} on the set N {1,...,8} of data sample
indexes with the MF values I (
In view of Remark 6, the set of membership degrees values of the FS I takes the form
A {0, 7;0,9;1}. For 1, 0 ; 0, 9 and 0, 7 , according to (32), we construct the
corresponding α-cuts I (
For 1, we get embedded T1FSs 0 (I (
T2FSs
B0e2 (I (
For 0, 9 , we get embedded T1FSs
0 (I (0, 9)) (68, 47; 64, 4; 60, 41) appr(64, 4) and 1(I (0, 9)) (0, 77; 0,81; 0,85) appr(0,81) in the T2FSs B0e2 (I (0,9)) {(0 (I (0, 9)); 0, 9))} and B1e2 (I (0,9)) {(1(I (0,9));0,9))} , respectively, with the degree of truth being equal to 0,9.
For 0, 7 , we get embedded T1FSs 0 (I (
According to Theorem 1, the resulting T2FSs of regression parameters have the forms B0 {(appr(82, 2);1), (appr(64, 4); 0,9), (appr(77);0, 7)} ,
B1 {(appr(0,91);1), (appr(0,81); 0,9), (appr(0,9); 0, 7)}. (39) (40)
Let us consider a graphical interpretation obtained regression with T2FSs parameters. On Figure 3
thin solid lines demonstrate the graphs of the regression functions
y (1(I (
y (1(I (
Figure 3 allows us to draw the following conclusions. Regressions corresponding to data of different quality may differ from each other and express a different relationship between the independent variables and predicted output. Therefore, the resulting regression with T2FSs of parameters should be taken as a whole as a collection of regressions corresponding to -cuts I ( ), [0,1] of the FS I of data sample indices. Only in this case we get an idea about dependence between the regression and the quality of the data used. Sometimes understanding this is important.
If we need some ‘maximizing’ regression, which has parameters values with the primary membership degrees equal to 1, then it should be considered as a collection of regressions corresponding to -cuts I ( ), [0,1] of the FS I of data sample indices. For these regressions, we take the parameters values with the membership degrees equal to 1. In Example 1, these are regressions with graphs depicted by thick lines in Figure 3.
The regression for fuzzy data of different quality can be represented as a set of fuzzy regressions with the degree of truth A . These fuzzy regressions correspond to data samples of different quality with indices belonging to -cuts I ( ), A of the FS I . Thus, the degree of truth of the regression is determined by the quality of the data for which it is constructed. The higher the quality of data samples, the greater the degree of membership of their indices to the FS I and the higher the degree of truth of the corresponding fuzzy regression in the collection that forms the regression for fuzzy data of different quality. On the other hand, increasing the amount of low-quality data reduces the degree of truth of the corresponding fuzzy regression in the collection.
According to the proposed approach, the linear regression parameters for fuzzy data of different quality are T2FSs with constant secondary grades. Although, in general, a T2FS is a rather complicated mathematical object, T2FSs with constant secondary grades are simple enough for practical use. Therefore, to represent these sets in a form which is convenient for understanding and applications, we have used a decomposition method. The results obtained have allowed us to decompose the T2FSs of linear regression parameters according to secondary grades into finite sets of FNs. These are fuzzy parameters of the regressions which correspond to different -cuts of FSs of data sample indices. One direction of future investigation of regressions with a fuzzy set of data sample indices may be related to the development of a similar approach for possibilistic regression analysis.
This work has been supported by Ministry of Education and Science of Ukraine: R&D Project 0122U001844 for the period of 2022-2024 at Taras Shevchenko National University of Kyiv.
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