scholarly journals Double-Valued Neutrosophic Sets, their Minimum Spanning Trees, and Clustering Algorithm

2018 ◽  
Vol 27 (2) ◽  
pp. 163-182 ◽  
Author(s):  
Ilanthenral Kandasamy
Keyword(s):  
Fuzzy Sets ◽  
Spanning Tree ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Distance Measure ◽  
Clustering Algorithms ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Inconsistent Information ◽  
Intuitionistic Fuzzy

AbstractNeutrosophy (neutrosophic logic) is used to represent uncertain, indeterminate, and inconsistent information available in the real world. This article proposes a method to provide more sensitivity and precision to indeterminacy, by classifying the indeterminate concept/value into two based on membership: one as indeterminacy leaning towards truth membership and the other as indeterminacy leaning towards false membership. This paper introduces a modified form of a neutrosophic set, called Double-Valued Neutrosophic Set (DVNS), which has these two distinct indeterminate values. Its related properties and axioms are defined and illustrated in this paper. An important role is played by clustering in several fields of research in the form of data mining, pattern recognition, and machine learning. DVNS is better equipped at dealing with indeterminate and inconsistent information, with more accuracy, than the Single-Valued Neutrosophic Set, which fuzzy sets and intuitionistic fuzzy sets are incapable of. A generalised distance measure between DVNSs and the related distance matrix is defined, based on which a clustering algorithm is constructed. This article proposes a Double-Valued Neutrosophic Minimum Spanning Tree (DVN-MST) clustering algorithm, to cluster the data represented by double-valued neutrosophic information. Illustrative examples are given to demonstrate the applications and effectiveness of this clustering algorithm. A comparative study of the DVN-MST clustering algorithm with other clustering algorithms like Single-Valued Neutrosophic Minimum Spanning Tree, Intuitionistic Fuzzy Minimum Spanning Tree, and Fuzzy Minimum Spanning Tree is carried out.

scholarly journals Single-Valued Neutrosophic Minimum Spanning Tree and Its Clustering Method

2014 ◽  
Vol 23 (3) ◽  
pp. 311-324 ◽  
Author(s):  
Jun Ye
Keyword(s):  
Machine Learning ◽  
Fuzzy Sets ◽  
Spanning Tree ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Distance Measure ◽  
Intuitionistic Fuzzy Sets ◽  
Clustering Method ◽  
Neutrosophic Sets ◽  
Generalized Distance

AbstractClustering plays an important role in data mining, pattern recognition, and machine learning. Then, single-valued neutrosophic sets (SVNSs) are a useful means to describe and handle indeterminate and inconsistent information, which fuzzy sets and intuitionistic fuzzy sets cannot describe and deal with. To cluster the data represented by single-value neutrosophic information, the article proposes a single-valued neutrosophic minimum spanning tree (SVNMST) clustering algorithm. Firstly, we defined a generalized distance measure between SVNSs. Then, we present an SVNMST clustering algorithm for clustering single-value neutrosophic data based on the generalized distance measure of SVNSs. Finally, two illustrative examples are given to demonstrate the application and effectiveness of the developed approach.

scholarly journals Certain Notions of Neutrosophic Topological K-Algebras

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 234 ◽  
Author(s):  
Muhammad Akram ◽  
Hina Gulzar ◽  
Florentin Smarandache ◽  
Said Broumi
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Point Of View ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Research Article

The concept of neutrosophic set from philosophical point of view was first considered by Smarandache. A single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view and an extension of intuitionistic fuzzy sets. In this research article, we apply the notion of single-valued neutrosophic sets to K-algebras. We introduce the notion of single-valued neutrosophic topological K-algebras and investigate some of their properties. Further, we study certain properties, including C 5 -connected, super connected, compact and Hausdorff, of single-valued neutrosophic topological K-algebras. We also investigate the image and pre-image of single-valued neutrosophic topological K-algebras under homomorphism.

CLUSTERING WITH MINIMUM SPANNING TREES

2011 ◽  
Vol 20 (01) ◽  
pp. 139-177 ◽  
Author(s):  
YAN ZHOU ◽  
OLEKSANDR GRYGORASH ◽  
THOMAS F. HAIN
Keyword(s):  
Spanning Tree ◽  
Clustering Algorithm ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Clustering Algorithms ◽  
Predefined Criterion ◽  
Point Set ◽  
Color Clustering ◽  
Representative Points ◽  
Set Of Points

We propose two Euclidean minimum spanning tree based clustering algorithms — one a k-constrained, and the other an unconstrained algorithm. Our k-constrained clustering algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of a set of representative points and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. Our unconstrained clustering algorithm partitions a point set into a group of clusters by maximally reducing the overall standard deviation of the edges in the Euclidean minimum spanning tree constructed from a given point set, without prescribing the number of clusters. We present our experimental results comparing our proposed algorithms with k-means, X-means, CURE, Chameleon, and the Expectation-Maximization (EM) algorithm on both artificial data and benchmark data from the UCI repository. We also apply our algorithms to image color clustering and compare them with the standard minimum spanning tree clustering algorithm as well as CURE, Chameleon, and X-means.

scholarly journals Intuitionistic Fuzzy Possibilistic C Means Clustering Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
Arindam Chaudhuri
Keyword(s):  
Fuzzy Sets ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Intuitionistic Fuzzy Sets ◽  
Distance Measures ◽  
Mathematical Framework ◽  
Cluster Validity ◽  
Intuitionistic Fuzzy ◽  
Validity Measures ◽  
Interval Valued

Intuitionistic fuzzy sets (IFSs) provide mathematical framework based on fuzzy sets to describe vagueness in data. It finds interesting and promising applications in different domains. Here, we develop an intuitionistic fuzzy possibilistic C means (IFPCM) algorithm to cluster IFSs by hybridizing concepts of FPCM, IFSs, and distance measures. IFPCM resolves inherent problems encountered with information regarding membership values of objects to each cluster by generalizing membership and nonmembership with hesitancy degree. The algorithm is extended for clustering interval valued intuitionistic fuzzy sets (IVIFSs) leading to interval valued intuitionistic fuzzy possibilistic C means (IVIFPCM). The clustering algorithm has membership and nonmembership degrees as intervals. Information regarding membership and typicality degrees of samples to all clusters is given by algorithm. The experiments are performed on both real and simulated datasets. It generates valuable information and produces overlapped clusters with different membership degrees. It takes into account inherent uncertainty in information captured by IFSs. Some advantages of algorithms are simplicity, flexibility, and low computational complexity. The algorithm is evaluated through cluster validity measures. The clustering accuracy of algorithm is investigated by classification datasets with labeled patterns. The algorithm maintains appreciable performance compared to other methods in terms of pureness ratio.

Multicriteria Decision Making Using Double Refined Indeterminacy Neutrosophic Cross Entropy and Indeterminacy Based Cross Entropy

2016 ◽  
Vol 859 ◽  
pp. 129-143 ◽  
Author(s):  
Ilanthenral Kandasamy ◽  
Florentin Smarandache
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Multicriteria Decision Making ◽  
Cross Entropy ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Multicriteria Decision ◽  
Inconsistent Information ◽  
Decision Making Problem ◽  
The Cross

Double Refined Indeterminacy Neutrosophic Set (DRINS) is an inclusive case of the refined neutrosophic set, defined by Smarandache (2013), which provides the additional possibility to represent with sensitivity and accuracy the uncertain, imprecise, incomplete, and inconsistent information which are available in real world. More precision is provided in handling indeterminacy; by classifying indeterminacy (I) into two, based on membership; as indeterminacy leaning towards truth membership (IT) and indeterminacy leaning towards false membership (IF). This kind of classification of indeterminacy is not feasible with the existing Single Valued Neutrosophic Set (SVNS), but it is a particular case of the refined neutrosophic set (where each T, I, F can be refined into T1, T2, ...; I1, I2, ...; F1, F2, ...). DRINS is better equipped at dealing indeterminate and inconsistent information, with more accuracy than SVNS, which fuzzy sets and Intuitionistic Fuzzy Sets (IFS) are incapable of. Based on the cross entropy of neutrosophic sets, the cross entropy of DRINSs, known as Double Refined Indeterminacy neutrosophic cross entropy, is proposed in this paper. This proposed cross entropy is used for a multicriteria decision-making problem, where the criteria values for alternatives are considered under a DRINS environment. Similarly, an indeterminacy based cross entropy using DRINS is also proposed. The double valued neutrosophic weighted cross entropy and indeterminacy based cross entropy between the ideal alternative and an alternative is obtained and utilized to rank the alternatives corresponding to the cross entropy values. The most desirable one(s) in decision making process is selected. An illustrative example is provided to demonstrate the application of the proposed method. A brief comparison of the proposed method with the existing methods is carried out.

Clustering Methods Using Distance-Based Similarity Measures of Single-Valued Neutrosophic Sets

2014 ◽  
Vol 23 (4) ◽  
pp. 379-389 ◽  
Author(s):  
Jun Ye
Keyword(s):  
Machine Learning ◽  
Data Mining ◽  
Fuzzy Sets ◽  
Clustering Algorithm ◽  
Distance Measure ◽  
Similarity Measures ◽  
Intuitionistic Fuzzy Sets ◽  
Clustering Methods ◽  
Neutrosophic Sets ◽  
Generalized Distance

AbstractClustering plays an important role in data mining, pattern recognition, and machine learning. Single-valued neutrosophic sets (SVNSs) are useful means to describe and handle indeterminate and inconsistent information that fuzzy sets and intuitionistic fuzzy sets cannot describe and deal with. To cluster the data represented by single-valued neutrosophic information, this article proposes single-valued neutrosophic clustering methods based on similarity measures between SVNSs. First, we define a generalized distance measure between SVNSs and propose two distance-based similarity measures of SVNSs. Then, we present a clustering algorithm based on the similarity measures of SVNSs to cluster single-valued neutrosophic data. Finally, an illustrative example is given to demonstrate the application and effectiveness of the developed clustering methods.

Neutrosophic Sets and Logic

2017 ◽  
pp. 18-63
Author(s):  
Mumtaz Ali ◽  
Florentin Smarandache ◽  
Luige Vladareanu
Keyword(s):  
Fuzzy Sets ◽  
Approximation Theory ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Hybrid Structures ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Basic Concepts ◽  
Interval Valued

Neutrosophic sets and Logic plays a significant role in approximation theory. It is a generalization of fuzzy sets and intuitionistic fuzzy set. Neutrosophic set is based on the neutrosophic philosophy in which every idea Z, has opposite denoted as anti(Z) and its neutral which is denoted as neut(Z). This is the main feature of neutrosophic sets and logic. This chapter is about the basic concepts of neutrosophic sets as well as some of their hybrid structures. This chapter starts with the introduction of fuzzy sets and intuitionistic fuzzy sets respectively. The notions of neutrosophic set are defined and studied their basic properties in this chapter. Then we studied neutrosophic crisp sets and their associated properties and notions. Moreover, interval valued neutrosophic sets are studied with some of their properties. Finally, we presented some applications of neutrosophic sets in the real world problems.

scholarly journals CciMST: A Clustering Algorithm Based on Minimum Spanning Tree and Cluster Centers

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaobo Lv ◽  
Yan Ma ◽  
Xiaofu He ◽  
Hui Huang ◽  
Jie Yang
Keyword(s):  
Shortest Path ◽  
Spanning Tree ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Geodesic Distance ◽  
Clustering Algorithms ◽  
Cluster Center ◽  
Data Sets ◽  
Intrinsic Structure ◽  
Initialization Algorithm

The minimum spanning tree- (MST-) based clustering method can identify clusters of arbitrary shape by removing inconsistent edges. The definition of the inconsistent edges is a major issue that has to be addressed in all MST-based clustering algorithms. In this paper, we propose a novel MST-based clustering algorithm through the cluster center initialization algorithm, called cciMST. First, in order to capture the intrinsic structure of the data sets, we propose the cluster center initialization algorithm based on geodesic distance and dual densities of the points. Second, we propose and demonstrate that the inconsistent edge is located on the shortest path between the cluster centers, so we can find the inconsistent edge with the length of the edges as well as the densities of their endpoints on the shortest path. Correspondingly, we obtain two groups of clustering results. Third, we propose a novel intercluster separation by computing the distance between the points at the intersection of clusters. Furthermore, we propose a new internal clustering validation measure to select the best clustering result. The experimental results on the synthetic data sets, real data sets, and image data sets demonstrate the good performance of the proposed MST-based method.

Reverse triple I method based on single valued neutrosophic fuzzy inference

2020 ◽  
Vol 39 (5) ◽  
pp. 7071-7083
Author(s):  
Ruirui Zhao ◽  
Minxia Luo ◽  
Shenggang Li
Keyword(s):  
Fuzzy Sets ◽  
Similarity Measure ◽  
Fuzzy Inference ◽  
Intuitionistic Fuzzy Sets ◽  
Neutrosophic Sets ◽  
Inconsistent Information ◽  
Intuitionistic Fuzzy ◽  
Triple I Method

The theory of single valued neutrosophic sets, which is a generalization of intuitionistic fuzzy sets, is more capable of dealing with inconsistent information in practice. In this paper, we propose reverse triple I method under single valued neutrosophic environment. Firstly, we give the definitions of single valued neutrosophic t-representation t-norms and single valued neutrosophic residual implications. Secondly, we develop a formula for calculating single valued neutrosophic residual implications. Then we propose reverse triple I method based on left-continuous single valued neutrosophic t-representation t-norms and its solutions. Lastly, we discuss the robustness of reverse triple I method based on the proposed similarity measure.

Sign in / Sign up

Export Citation Format

Share Document