A novel approach for fuzzy clustering based on neutrosophic association matrix

https://doi.org/10.1016/j.cie.2018.11.007Get rights and content

Highlights

  • We proposed a new fuzzy clustering algorithm based on the neutrosophic set.

  • Data are fuzzified to create neutrosophic association and equivalence matrix.

  • Lambda-cutting matrix is used to determine the clusters.

  • It was experimentally validated on benchmark datasets of UCI Machine Learning.

  • It has better clustering quality than other relevant algorithms.

Abstract

This paper proposes a fuzzy clustering algorithm through neutrosophic association matrix. In the first step, data are fuzzified into neutrosophic sets to create neutrosophic association matrix. By deriving a finite sequence of neutrosophic association matrices, the neutrosophic equivalence matrix is generated. Finally, the lambda-cutting is performed over the neutrosophic equivalence matrix to derive the final lambda-cutting matrix which is used to determine the clusters. Experimental results on several benchmark datasets using different clustering criteria show the advantage of the proposed clustering over the existing algorithms.

Introduction

In practice, data are often uncertain, inconsistency and uncompleted. To handle this problem, fuzzy set was proposed by Zadeh (1965) in which uncertainty is modeled as an elemental dependence of a set. Fuzzy sets have showed meaningful applications in many fields of study (Nguyen et al., 2018, Ye and Du, 2017). One of the most essential utilization regarding the fuzzy set is the representation of information such as “non-membership” and “hesitancy”. For example, when diagnosing a patient, the doctor often concludes the patient's illness rate corresponds to the disease rather than indicating a complete or unspecified illness. There are several extensions of traditional fuzzy set have been proposed such as intuitionistic fuzzy sets (Atanassov, 1986) and neutrosophic fuzzy set (Smarandache, 1998). Neutrosophic set is the generalization of fuzzy set, intuitionistic fuzzy set and others. Neutrosophic set has been studied and applied in various fields such as the medical diagnosis (Mondaland and Pramanik, 2015), decision support systems (Pramanik and Chackrabarti, 2013), robots (Smarandache and Vladareanu, 2014), social and educational information analyzes, etc.

Clustering is an important concept along with fuzzy set theory. Several clustering algorithms based on fuzzy set have been proposed such as: Fuzzy C-Means (FCM) (Bezdek, Ehrlich, & Full, 1984), the methods proposed by Ye and Fu, 2016, Ye and Fu, 2016, Ye and Smarandache, 2016, Ye and Zhang, 2014, Ye, 2014, Ye, 2016, Ye, 2017, Ye, 2018. Recently, neutrosophic association matrix usually is utilized as a tool in many fuzzy clustering algorithms. For the fuzzy clustering algorithm based on neutrosophic association matrix, the most important step is to evaluating the similarities in order to divide the elements into clusters. Ye and Smarandache (2016) proposed three types of measures including Jaccard, Dice and Cosine which then be used in multi-criteria decision making with simple neutrosophic dataset. In Ye, 2014, Ye and Zhang, 2014; Ye continued to propose new neutrosophic fuzzy modification methods for decision-makers by combining above similar measures. On the other hand, Ma, Wang, Wang, and Wu (2015) investigate the similar measures of tangential function for medical applications. Other studies on neutrosophicfuzzy clustering algorithms can be found in Kuo et al., 2018, Wu et al., 2017, Ye and Fu, 2016, Ye and Zhang, 2014, Ye, 2016.

This article proposes a new fuzzy clustering using neutrosophic association matrix. The first step of the algorithm is to construct a neutrosophic association matrix from the data in the dataset. After that, a neutrosophic equivalent matrix is constructed from neutrosophic association matrix. Finally, the lambda-cutting matrix is built based on neutrosophic equivalent matrix by the lambda-cutting step. The result clusters are defined based on the lambda-cutting matrix.

Section 2 presents some background information and proposes a new neutrosophic clustering method though detailed analysis. Section 3 shows the experimental result of proposed algorithm in comparison with other relevant methods on real data sets. Conclusions are in the Section 4.

Section snippets

Background of neutrosophic set

Let ε0 be a infinitesimal number (Smarandache, 1998), i.e., for all positive integers n one has ε<1n. Let 1+=1+ε, where “1” and “ε” are its standard and non-standard parts respectively. Similarly, 0-=0-ε, and ]0-,1+[ is a non-standard unit interval.

A neutrosophic set Ain the universe Xis characterized by a truth, indeterminacy, and falsehood membership functions <TA(x), IA(x), FA(x)> such that TAx,IAx,FAx:X]0-,1+[ and 0TAx+IAx+FAx3 (Smarandache, 1998).

Suppose that A1=x;T1x;I1x;F1x|xX and

Experimental environments

The proposed algorithm has been implemented in addition to the methods of Ye, 2014, Ye, 2016, Huang, 2016 in Matlab 2015a programming language with a PC with CPU Intel(R) Core (TM) i5-2520 [email protected] GHz, 4096 MB RAM, windows 7 Professional 64 bits.

In order to perform the evaluation, two kinds of datasets have been used. The first dataset is the set of EPPO standard dataset which is taken from EPPO Global Database. It provides a large dataset for variety types as agriculture, forestry and plan

Conclusions

This paper proposed a new fuzzy clustering algorithm based on association matrix using the neutrosophic set. After constructing a neutrosophic association matrix from the data, a neutrosophic equivalent matrix is designed based on association matrix. The next step is to construct the lambda-cutting matrix based on neutrosophic equivalent matrix by the lambda-cutting step. Finally, the clusters are defined on the basis of lambda-cutting matrix. To assess the quality of clusters, different

Acknowledgement

The authors would like to thank Editor-in-chiefs, Associate editor and the anonymous referees for their helpful comments and valuable suggestions, which have greatly improved the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.05-2018.02.

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