Models of Relative Nearness Incidence Based on Standard Distance Entropy

The model of grey nearness incidence cannot reflect the nearness degree of sequences correctly. Therefore, the model of relative nearness incidence of sequences, curves, and surfaces is suggested based on standard distance entropy to remove the current bottlenecks and its properties are studied. At last, three cases are exemplified to demonstrate the validity and practicability of relative nearness incidence. The proposed models have enriched the theory of grey nearness incidence, filled the defects of grey nearness incidence.


Introduction
In 1982, Deng created grey systems theory [1][2][3], and it has been a far-reaching development in the past 30 years.The subjects of grey systems theory is with part of the information known and part of the information unknown.Therefore, grey systems theory has a very broad prospect of development.In recent years, grey system theory has had successful application in many scientific fields, and it won wide recognition and attention.
Previous studies of the grey incidence model are mainly focused on time sequences.However, there is very few research on the grey incidence model of curves or surfaces.
In addition, many grey incidence models do not satisfy the symmetry.Finally, the grey nearness incidence may be 1 for different sequences; this is obviously unreasonable [12].Therefore, this paper will construct the grey incidence model of sequences, curves, and surfaces based on the abovementioned defects.

The Model of Grey Nearness Incidence
With the development of grey incidence theory, Liu et al. proposed the model of grey nearness incidence in 2010; the basic idea is as follows [12].
The system behavior sequence   = (  (1),   (2), . . .,   ()), where  = 1, 2, . . ., .   and   denote the crease of   and   else.Suppose Definition 1. Suppose the length of   and   is equal,   −   as shown above; then the grey nearness incidence of   and   is defined as Theorem 2. Suppose the length of   and   is equal; then (notes: part of [12] seems to be missing, now we will correct it) However, there are many defects.The grey nearness incidence may be 1 for different sequences.We will use Example 3 to illustrate it.
Example 3. The sequences are as follows: The corresponding crease of each sequence is shown in Figure 1.We know the grey nearness incidence is  12 = 1 based on the above-mentioned model.
This result is clearly not consistent with the fact.So the model of grey nearness incidence cannot reflect the nearness degree of sequences correctly.Therefore, the model of relative nearness incidence of sequences, curves, and surfaces is suggested based on standard distance entropy, and they can solve the above-mentioned problem.
During the proof process of Theorem 5, we know the closer the value of  and (1/2), the larger the value of ().Therefore, when  =  > 0,   achieves the maximum value, and the maximum value is ln 2. So we obtain the property (4).Definition 6. Suppose  and  are positive numbers; then the standard distance entropy of  and  is defined as From the definition of standard distance entropy, we know 0 <   ≤ 1, and the closer the value of  and , the larger the value of   .So we can use   to express the relative nearness degree of  and .
The properties of standard distance entropy can obtained from the properties of distance entropy, so we will not repeat it.

The Models of Relative Nearness Incidence
Based on Standard Distance Entropy where   () means the standard distance entropy of   () and   ().

Examples
In order to illustrate the validity and practicability of the relative nearness incidence, we have the following 3 examples.

(17)
As seen in the above, we know Then So the relative nearness incidence of  1 and  2 is weaker than the relative nearness incidence of  1 and  3 .
In order to compare the relative nearness incidence of sequences with the relative degree of grey incidence of sequences, we put the data of Example 2 into the formula of relative degree of grey incidences.It is easy to know that  12 = 0.8425 and  13 = 0.9844.So Then the relative degree of grey incidences of  1 and  2 is weaker than the relative degree of grey incidences of  1 and  3 .Therefore, the relative nearness incidence of sequences and the relative degree of grey incidences of sequences are harmonious.

Conclusion
The model of grey nearness incidence proposed by Liu et al. cannot reflect the nearness degree of sequences correctly.Therefore, the model of relative nearness incidence of sequences, curves, and surfaces is suggested based on the standard distance entropy to remove the current bottlenecks.The proposed models have enriched the theory of grey nearness incidence, filled the defects of grey nearness incidence.However, only two positive numbers can calculate its standard distance entropy.Therefore, there are many defects of relative nearness incidence based on standard distance entropy; our future research will focus on it.

Figure 1 :
Figure 1: The corresponding crease of each sequence.