The Intuitionistic Fuzzy Linguistic Cosine Similarity Measure and Its Application in Pattern Recognition

. We propose the cosine similarity measures for intuitionistic fuzzy linguistic sets (IFLSs) and interval-valued intuitionistic fuzzy linguistic sets (IVIFLSs), which are expressed by the linguistic scale function based on the cosine function. Then, the weighted cosine similarity measure and the ordered weighted cosine similarity measure for IFLSs and IVIFLSs are introduced by taking into account the importance of each element, and the properties of the cosine similarity measures are also given. The main advantage of the proposed cosine similarity measures is that the decision-makers can flexibly select the linguistic scale function depending on the actual semantic situation. Finally, we present the application of the cosine similarity measures for intuitionistic fuzzy linguistic term sets and interval-valued intuitionistic fuzzy linguistic term sets to pattern recognition and medical diagnosis, and the existing cosine similarity measures are compared with the proposed cosine similarity measures by the illustrative example.


Introduction
The fuzzy set was proposed by Zadeh [1] and has achieved a great success in various fields, which is considered to be an effective tool to solve the decision-making problems, pattern recognition, and fuzzy inference [2][3][4].Since the fuzzy set was put forward, it was extended in different aspects.One of the generalizations of fuzzy set is intuitionistic fuzzy set (IFS), which was introduced by Atanassov [5].A typical feature of IFS is that the membership relations are represented by the membership degree and nonmembership degree, respectively.However, due to the fuzziness and uncertainty in the multiple criteria decision-making problems, it is difficult to use the exact values to present qualitative evaluation.At this time, people often provide their opinions in linguistic term sets.In some practical decision-making problems, the decision-maker regards the linguistic information as the values of linguistic variables; that is to say, the values of the variables are not represented by numerical values but are represented by linguistic values, such as "good," "better," "fair," "slightly worse," and "poor."Up to now, many people have studied the problem of linguistic multiple criteria decisionmaking, Herrera and Verdegay [6] proposed the linguistic assessments in group decision-making (GDM) problem in 1993, then Herrera et al. [7] proposed a consensus model for group decision-making based on linguistic evaluations information, and Herrera et al. [8] considered several group decision-making processes using linguistic ordered weighted averaging (LOWA) operator.Later, Xu [9] proposed a group decision-making method based on the uncertain linguistic ordered weighted geometric (LOWG) operators and the induced uncertain LOWG operators.Furthermore, Xu [10] presented the linguistic hybrid aggregation (LHA) operator and applied it to group decision-making.
However, in some practical decision-making problems, the decision-makers may have some indeterminacy in their linguistic evaluation; they cannot express their preferences by using only membership degree of a linguistic term.Then Wang et al. [11] proposed the intuitionistic fuzzy linguistic aggregation operators and applied them to multicriteria group decision-making problems.For example, ⟨ 3 , (0.2, 0.6)⟩ is an intuitionistic fuzzy linguistic number (IFLN), 0.2 is the membership degree of the linguistic term  3 , and 0.6 is the nonmembership degree of the linguistic term  3 .The intuitionistic fuzzy linguistic sets (IFLSs) have made great progress in describing linguistic information and to some extent it can be regarded as an innovative construct.The research on this field has been growing rapidly [12][13][14][15].
On the other hand, similarity measure is an important topic in the fuzzy set theory, and it is widely used in some fields [16][17][18][19], such as pattern recognition, medical diagnosis, and citation analysis.One of the important similarity measures is the cosine similarity measure, which is defined in vector space.Ye [20] proposed the cosine similarity measure and the weighted cosine similarity measure between IFSs.Zhou et al. [21] presented the intuitionistic fuzzy ordered weighted cosine similarity measure and applied it to the group decision-making problem about the choice of investment plan.Liu et al. [22] presented the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVI-FOWCS) measure and applied it to the investment decisionmaking.As far as we know, the study of cosine similarity measures of intuitionistic fuzzy set has not been discussed.In the following, we will propose the cosine similarity measures of IFLSs and IVIFLSs.The main characteristics of the cosine similarity measures that we can calculate are based on linguistic term set by the linguistic scale function.Linguistic scale function between IFLSs and IVIFLSs was introduced by Wang et al. [23], which was used to calculate the Hausdorff distance between hesitant fuzzy linguistic numbers (HFLNs); it can assign different semantic values to the linguistic terms under different circumstances and improve the flexibility of the proposed cosine similarity measures.
The rest of the paper is organized as follows.In Section 2, some basic concepts of LTSs, IFSs, IFLSs, and linguistic scale functions are briefly reviewed.In Section 3, we first introduce the cosine similarity measures between IFLSs and then discussed some related properties.Furthermore, the weighted cosine similarity measure between IFLSs, the ordered weighted cosine similarity measure between IFLSs, and the ordered weighted cosine similarity measure between IVIFLSs are analyzed.In Section 4, we give the application of the proposed cosine similarity measures between IFLSs and IVIFLSs on pattern recognition and medical diagnosis and then make comparison analysis with the existing cosine similarity measures.The conclusions are given in Section 5.

Preliminaries
In this section, we will review and discuss some related basic concepts, including linguistic term sets (LTSs), intuitionistic fuzzy sets (IFSs), intuitionistic fuzzy linguistic sets (IFLSs), linguistic scale functions, the ordered weighted averaging (OWA) operator, and the cosine similarity measure between fuzzy sets.

Linguistic Term Set.
In some practical problems, the information expressed by the numerical values may bring inconvenience.At this time it is suitable to use linguistic term set to express information.
Definition 1 (Herrera and Verdegay [6]).Suppose that  = {  |  = 0, 1, . . ., } is a finite and totally ordered discrete term set, where   represents a possible value for a linguistic variable; it satisfies the following characteristics: For example, a set of seven terms  could be given as follows: The discrete linguistic term  cannot usually adapt to the aggregated results.In order to represent these results accurately, Xu [10] extended the discrete term set  to the continuous term set  = {  |  ∈ [0, ]}, where  ( > ) is a sufficiently large positive integer.
On the basis of intuitionistic fuzzy set and linguistic term set, Wang et al. [11] presented the following intuitionistic fuzzy linguistic term set.
2.4.Linguistic Scale Functions.One of the advantages of linguistic term set is that it can express uncertain information flexibly in practical problems, but if we use the subscript of linguistic terms directly in the process of operations, it may lose this advantage.The most important thing is to find effective tools to transform linguistic terms to numerical values.As we all know, linguistic scale function (Wang et al. [23]) is a mapping from linguistic term set   to the real value   .The linguistic scale function can assign different semantic values to the linguistic terms under different circumstances.In practice, the linguistic scale functions are very popular because they are very flexible and they can give more deterministic results based on different semantics.Definition 4 (Wang et al. [23]).Let  = {  |  = 0, 1, . . ., 2} be a linguistic term.If   is a numeric value between 0 and 1, then the linguistic scale function  can be defined as follows: where The linguistic scale function is strictly monotonously increasing function with respect to the subscript of   ; in fact, the function value   represents the semantics of the linguistic terms.Now we introduce three kinds of linguistic scale functions as follows: The evaluation scale of the linguistic information expressed by  1 (  ) is divided on average.
For linguistic scale function  2 (  ), the absolute deviation between adjacent language sets will increase when we extend it from the middle of the given set of language to both ends. ( For linguistic scale function  3 (  ), the absolute deviation between adjacent language sets will decrease when we extend it from the middle of the given linguistic term set to both ends.
The above linguistic scale functions can be developed to  * :  →  + (where  + is a nonnegative real number), which is a continuous and strictly monotonically increasing function.
2.5.The OWA Operator.The ordered weighted averaging (OWA) operator (Yager [24]) is an aggregation operator that includes the minimum, the average, and the maximum as special cases, which is defined as follows.

Cosine Similarity Measure for Intuitionistic Fuzzy Linguistic Term Sets.
At first, we will present cosine similarity measure for intuitionistic fuzzy linguistic term sets, which includes not only the membership degree and nonmembership degree of the IFLSs but also the linguistic scale function  * .
If we consider the weight of different element   ∈ , now we introduce the intuitionistic fuzzy linguistic weighted cosine similarity measure  IFLWS , which can be defined as follows.
. .,   }, and let  * be a linguistic scale function.Then the weighted cosine similarity measure for intuitionistic fuzzy linguistic term sets between  and  can be defined as follows: where is the weight of   ∈ , and ∑  =1   = 1 (0 ≤   ≤ 1).
Based on the idea of the OWA operator, we present the intuitionistic fuzzy ordered weighted cosine similarity measure  IFLOWS (, ) between intuitionistic fuzzy linguistic term sets  and  as follows.

Cosine Similarity Measure for Interval-Valued Intuitionistic Fuzzy Linguistic Term Sets.
In the intuitionistic fuzzy linguistic set  = {⟨(, () ),   (), ]  ()⟩ |  ∈ },   () represents the membership degree of the element  to  () , and ]  () represents the nonmembership degree of the element  to  () ; they are all precise values in [0, 1].But in some circumstances, it is difficult to provide the precise membership degree and nonmembership degree of the element  to  () .Atanassov and Gargov [26,27] proposed the interval-valued intuitionistic fuzzy linguistic term set (IVIFLS), and the definition of interval-valued intuitionistic fuzzy linguistic term set is given as follows.
Next we go on studying the weighted cosine similarity measure between IVIFLSs; it can be defined as follows.

Applications of the Cosine Similarity Measure
In this section, we will apply the cosine similarity measures of IFLSs and IVIFLSs to pattern recognition and medical diagnosis.
From the result shown in Table 1, we can conclude that the pattern  belongs to the pattern  3 .
To illustrate the influence of the linguistic scale function  * on decision-making, we utilize the different linguistic scaling functions in the proposed cosine similarity measures. Let and  = 1.4, = 3, then the results are listed in Table 2.If and  = 3,   =   = 0.8, the results of the cosine similarity measure are shown in Table 3.
As we can see from Tables 2 and 3, we know that the pattern  should be classified in  3 in most cases.It is a little different when the linguistic scale function  * =  2 (  ) (the linguistic scale function which can be considered as the actual semantic situation), then the decision-makers can select the appropriate linguistic scale function  * according to their interests.

Interval-Valued Intuitionistic Fuzzy Cosine Similarity
Measure for Medical Diagnosis.In this subsection, we will utilize the interval-valued intuitionistic fuzzy cosine similarity measure to discuss the medical diagnosis.In fact, it is also a pattern recognition problem.
From the results of Table 4, we can see that the diagnosis of the patient  is  2 (typhoid).
To illustrate the influence of the linguistic scale function  * on decision-making, we utilize the different linguistic scaling functions  2 (  ) and  3 (  ) (the parameters are the same as Example 2) in the cosine similarity measures between IVIFLSs, and the results are shown in Tables 5 and 6.In these cases, we still find that the diagnosis of the patient  should be classified in  2 (typhoid).

Comparison Analysis with Existing Cosine Similarity
Measure.To illustrate the validity and advantage of the proposed cosine similarity measure for pattern recognition and medical diagnosis, we now use the existing cosine similarity measure between intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets (Ye [20]) for comparison analysis.
The results of Table 7 show that the pattern   can be classified in    3 ; this shows the effectiveness of the proposed cosine similarity measures in this paper.(33) The weight vector of ( 1 ,  2 ,  3 ,  4 ) is (0.25, 0.4, 0.15, 0.2).Using the cosine similarity measure between IVIFSs in [22], we can calculate the interval-valued intuitionistic fuzzy cosine similarity measure  IVIFS , the interval-valued intuitionistic fuzzy weighted cosine similarity measure  IVIFWS , and the interval-valued intuitionistic fuzzy ordered weighted cosine similarity measure  IVIFOWS between   and    ( = 1, 2, 3, 4), respectively, and the results are shown in Table 8.
As we can see from Table 8, the patient   is assigned to the diagnosis   2 (typhoid), and the result is the same as the method that we presented in this paper.According to the comparative analysis of Sections 4.1 and 4.2, the cosine similarity measures in this paper have the following advantage.It is reasonable that the proposed cosine similarity measures between IFLSs and IVIFLSs are defined based on linguistic scale function  * , the decision-makers can flexibly select the linguistic scale function  * depending on Complexity their preferences and the actual semantic situations, and the decision-maker can employ it to address practical decisionmaking problems with precision.

Conclusions
In this paper, we propose the cosine similarity measures for IFLSs and IVIFLSs, which are expressed by the cosine function based on linguistic scale function.Then, the weighted cosine similarity measure and the ordered weighted cosine similarity measure for intuitionistic fuzzy linguistic term sets are introduced.The main advantage of the proposed cosine similarity measures is that the decision-makers can flexibly select the linguistic scale function depending on the actual semantic situation.Furthermore, we present the application of the cosine similarity measures for intuitionistic fuzzy linguistic term sets and interval-valued intuitionistic fuzzy linguistic term sets to pattern recognition and medical diagnosis, and the existing cosine similarity measures are compared with the proposed cosine similarity measures by the illustrative example.In future research, the developed cosine similarity measures will be extended to the intuitionistic fuzzy uncertain linguistic sets and it can be applied in other related decision-making.

Table 7 :
Cosine similarity measures for intuitionistic fuzzy sets.