A Dynamic Interval-Valued Intuitionistic Fuzzy Sets Applied to Pattern Recognition

We present dynamic interval-valued intuitionistic fuzzy sets (DIVIFS), which can improve the recognition accuracy when they are applied to pattern recognition. By analyzing the degree of hesitancy, we propose some DIVIFS models from intuitionistic fuzzy sets (IFS) and interval-valued IFS (IVIFS). And then we present a novel ranking condition on the distance of IFS and IVIFS and introduce some distance measures of DIVIFS satisfying the ranking condition. Finally, a pattern recognition example applied to medical diagnosis decisionmaking is given to demonstrate the application of DIVIFS and its distances.The simulation results show that the DIVIFS method is more comprehensive and flexible than the IFS method and the IVIFS method.

First, we introduce a series of definitions and construction methods of DIVIFS and propose some DIVIFS models according to IFS and IVIFS.Then, on the basis of the traditional distance measures, DIVIFS is applied to pattern recognition on medical diagnosis.The simulation results show that the method introduced in this paper is more comprehensive and flexible than the IFS method and the IVIFS method.Thus, this paper can provide valuable conclusion for the application research of IFS and IVIFS.The model of DIVIFS is also useful for the generalization from fuzzy reasoning [3] and intuitionistic fuzzy reasoning to interval-valued intuitionistic fuzzy reasoning [29].
Definition 5.  is a universe of discourse.A DIVIFS  in  is an object having the following form: where   (),   (), and   () are the same as Definition 3.
From Definition 3, let all sample data be divided into three parts,  −  () being the firm support party of event , ] −  () representing the firm opposition party of event , and  +  () showing all the absent party.In the absent party,  −  () is the firm absent party, and  +  () −  −  () is the convertible absent party, in which each sample may become either the support party or the opposition party.If there is   () sample supporting event  and   () sample opposing event , we have DIVIFS based on Definition 5.If the proportion of the absent party converted to the support party is  1 () and the proportion of the absent party converted to the opposition party is 1 −  1 (), then the model will become a DIVIFS with single variable, where , and then we will get the other DIVIFS definition as follows.Definition 6.  is a universe of discourse.DIVIFS  in  is an object having the following form where   (),   (), and   () are the same as Definition 5.
For each  0 () ∈ [0, 1] and for each  1 () ∈ [0, 1], we assume that , then we have a special DIVIFS which is derived from IFS. Definition 7. A DIVIFS derived from IFS is given as follows: where   (),   (), and   () are the same as Definition 5 and . This DIVIFS model is also an extension of the IFS model.
According to fuzzy sets [1], the distance measure of FS is composed of membership degree: In [6], Atanassov introduced a distance measure (9) of IFS from membership degree and non-membership degree: Considering the degree range of hesitancy, Szmidt and Kacprzyk presented a novel distance measure (10) according to Definition 1 and Theorem 2 in 2000 [30]:   Mathematical Problems in Engineering In [14], Dengfeng and Chuntian proposed a novel distance model as follows: At the same time, Grzegorzewski [31] and Hung and Yang [32] proposed a Hausdorff distance measure of IFS, respectively: In 2005, Wang and Xin [33] presented a new distance measure which is made up of  At and  P,H : In 2005, Liu [34] introduced some similar distance formulas as mentioned above.

Construction of Distance Measures on DIVIFS
First, we will define a novel essential ranking property of the distance measure on IFS and IVIFS.
In general, a distance measure should conform to the following Theorem 10 according to its properties.If  ⊂  ⊂  ⊂ , then (, ) > (, ).
Obviously, for all the distance formulas above, (P1), (P2), and (P3) can be obtained.However, most of them do not satisfy the ranking property (P4).
From the analysis to the formulas from Xia and Xu above, only (15) basically meets the requirement of (P4); the others only meet the requirement of (P4) in part.Similarly, most of the conventional distance measures of IVIFS do not satisfy property (P4).Therefore, it is necessary to find a distance measure satisfying the necessary ranking property.Considering the component elements of the distance formula of IFS and IVIFS, we can define some appropriate distance measures of DIVIFS (Figure 2).
In Xu's papers [16,17,19,20], many weighted similarity measures are proposed according to standardized distance measures based on the IFS theory and the IVIFS theory.And Xu presented them by defining the appropriate weights of membership function, non-membership function, and hesitancy function.According to Xu, we can also define  the corresponding weighted distance measures on IVIFS as follows: It is easy to prove that ( 11) is equivalent to a special case of (15) when  = 0.5.And ( 15) is also a special case of (34) when  1 =  2 =  3 =  4 =  6 =  8 = 0 and  5 =  7 = 1.To simplifiy the formula, let  5 =  7 and  6 =  8 .Assume that  and  are two DIVIFS, and we have ( Comparing ( 15)-( 21) with ( 34)-( 35), we conclude that Xu's methods are different from the DIVIFS method.Xu defined the measures by constructing the weights of membership function, non-membership function, and hesitancy function, respectively.However, we concentrate on the detachment of hesitancy function and present the DIVIFS method.If we use Xu's methods solely, we can define the weights of certainty and uncertainty, but we cannot analyze the variation of the absent party.Moreover, in Xu's research, and even in all the conventional research, there is no research on the ranking condition (P4).Therefore, the distance measures on DIVIFS above are an important supplement to Xu's methods and the others' .
In order to simplify the calculation, we define the following distance measure: If  = 1, the distance measure is the Hamming distance; if  = 2, it is the Euclidean distance.In this paper, the Euclidean distance on DIVIFS will be used in the simulation of pattern recognition: (, ) (, )

Application to Medical Diagnosis Based on Pattern Recognition
To make a proper diagnosis  for a patient with given values of symptoms , a medical knowledge base is necessary that involves elements described in terms of IFS.We adopt the same data as those in [8,9]: the set of disease diagnoses is  = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}, and the set of symptoms is a universe of discourse  = {temperature, headache, stomach pain, cough, chest pain}.The data are given in Table 1, where each symptom is described as follows: membership function   (), nonmembership function ]  (), and hesitancy function   ().
For (10), the normalized Hamming distance for all symptoms of patient th from diagnosis th is (40), and  Applying the conventional distance measures on IFS and DIVIFS to medical diagnosis, we have the results in Table 6.From Table 6, we conclude that the most accurate results of medical diagnosis are from Xu's equation (15) when  = 0.5, which is also only a special case of (35) when  1 =  2 =  3 =  4 =  6 =  8 = 0 and  5 =  7 = 1.And in 77 individuals, 74% of all the women can be classified correctly.However, using (38), the recognition rate will be 83%.Some results of medical diagnosis have been shown in Table 7.

Conclusion
We propose a method for the evaluation of a degree of agreement in a group of individuals by calculating distances between DIVIFS.By analyzing the variation on the degree of hesitancy, we introduce a novel DIVIFS method derived from IVIFS, which is an effective method for us to construct dynamic IVIFS model from IFS and IVIFS.And then we propose a novel ranking condition for the distance measures on IFS, IVIFS, and DIVIFS.According to the new condition, we show the construction of distance measures on DIVIFS in theory.Finally, we apply some distances on DIVIFS to medical diagnosis based on pattern recognition.The experimental results show that the DIVIFS method and its distance measures are more comprehensive and more effective than the conventional distance measures on IFS and IVIFS.

Figure 1 :
Figure 1: Distance between each patient and each disease on IFS calculated by Xu's methods.

Figure 2 :
Figure 2: Distance between each patient and each disease on DIVIFS.

Table 4 :
Distance between each patient and each disease on IFS calculated by Xu's methods.

Table 5 :
Distance description between each patient and each disease based on DIVIFS.

Table 6 :
The optimal results of medical diagnosis based on the distances of IFS and DIVIFS.

Table 7 :
Comparison between classification of DIVIFS and existing classification calculated by similarity ( = 1 − , where  denotes the standardized distance).Hospital, Southern Medical University.And the set of disease diagnoses is  = {Not successful pregnancy and no symptoms of OHSS, Successful pregnancy and no symptoms of OHSS, Not successful pregnancy and symptoms of OHSS, Successful pregnancy and symptoms of OHSS}, and the set of symptoms is a universe of discourse  = {Basic Hormone FSH, Basic Hormone P, Basic Hormone E2, Basic Hormone T, BMI, Day of HCG E2, Day of HCG endometrial thickness}, where each symptom is described as follows: membership function   (), non-membership function ]  (), and hesitancy function   ().