Pythagorean fuzzy sets (PFSs) were proposed by Yager in 2013 to treat imprecise and vague information in daily life more rigorously and efficiently with higher precision than intuitionistic fuzzy ...sets. In this paper, we construct new distance and similarity measures of PFSs based on the Hausdorff metric. We first develop a method to calculate a distance between PFSs based on the Hasudorff metric, along with proving several properties and theorems. We then consider a generalization of other distance measures, such as the Hamming distance, the Euclidean distance, and their normalized versions. On the basis of the proposed distances for PFSs, we give new similarity measures to compute the similarity degree of PFSs. Some examples related to pattern recognition and linguistic variables are used to validate the proposed distance and similarity measures. Finally, we apply the proposed methods to multicriteria decision‐making by constructing a Pythagorean fuzzy Technique for Order Preference by Similarity to an Ideal Solution and then present a practical example to address an important issue related to social sector. Numerical results indicate that the proposed methods are reasonable and applicable and also that they are well suited in pattern recognition, linguistic variables, and multicriteria decision‐making with PFSs.
The k-means algorithm is generally the most known and used clustering method. There are various extensions of k-means to be proposed in the literature. Although it is an unsupervised learning to ...clustering in pattern recognition and machine learning, the k-means algorithm and its extensions are always influenced by initializations with a necessary number of clusters a priori. That is, the k-means algorithm is not exactly an unsupervised clustering method. In this paper, we construct an unsupervised learning schema for the k-means algorithm so that it is free of initializations without parameter selection and can also simultaneously find an optimal number of clusters. That is, we propose a novel unsupervised k-means (U-k-means) clustering algorithm with automatically finding an optimal number of clusters without giving any initialization and parameter selection. The computational complexity of the proposed U-k-means clustering algorithm is also analyzed. Comparisons between the proposed U-k-means and other existing methods are made. Experimental results and comparisons actually demonstrate these good aspects of the proposed U-k-means clustering algorithm.
Fuzzy clustering algorithms generally treat data points with feature components under equal importance. However, there are various datasets with irrelevant features involved in clustering process ...that may cause bad performance for fuzzy clustering algorithms. That is, different feature components should take different importance. In this paper, we present a novel method for improving fuzzy clustering algorithms that can automatically compute individual feature weight, and simultaneously reduce these irrelevant feature components. In fuzzy clustering, the fuzzy c-means (FCM) algorithm is the best known. We first consider the FCM objective function with feature-weighted entropy, and construct a learning schema for parameters, and then reduce these irrelevant feature components. We call it a feature-reduction FCM (FRFCM). During FRFCM processes, a new procedure for eliminating irrelevant feature(s) with small weight(s) is created for feature reduction. The computational complexity of FRFCM is also analyzed. Some numerical and real datasets are used to compare FRFCM with various feature-weighted FCM methods in the literature. Experimental results and comparisons actually demonstrate these good aspects of FRFCM with its effectiveness and usefulness in practice.
•We construct a robust learning-based fuzzy c-means (FCM) framework, called the robust-learning FCM (RL-FCM) algorithm.•The proposed RL-FCM can automatically find the best number of clusters, without ...any initialization and parameter selection with free of the fuzziness index m.•The computational complexity of the proposed RL-FCM algorithm is analyzed.•The experimental results and comparisons actually demonstrate these good aspects of RL-FCM where it exhibits three robust characteristics.
In fuzzy clustering, the fuzzy c-means (FCM) algorithm is the most commonly used clustering method. Various extensions of FCM had been proposed in the literature. However, the FCM algorithm and its extensions are usually affected by initializations and parameter selection with a number of clusters to be given a priori. Although there were some works to solve these problems in FCM, there is no work for FCM to be simultaneously robust to initializations and parameter selection under free of the fuzziness index without a given number of clusters. In this paper, we construct a robust learning-based FCM framework, called a robust-learning FCM (RL-FCM) algorithm, so that it becomes free of the fuzziness index m and initializations without parameter selection, and can also automatically find the best number of clusters. We first use entropy-type penalty terms for adjusting bias with free of the fuzziness index, and then create a robust learning-based schema for finding the best number of clusters. The computational complexity of the proposed RL-FCM algorithm is also analyzed. Comparisons between RL-FCM and other existing methods are made. Experimental results and comparisons actually demonstrate these good aspects of the proposed RL-FCM where it exhibits three robust characteristics: 1) robust to initializations with free of the fuzziness index, 2) robust to (without) parameter selection, and 3) robust to number of clusters (with unknown number of clusters).
•Fuzzy c-means (FCM) clustering had been extended for handling multi-view data.•We propose a novel multi-view FCM (MVFCM) clustering algorithm with view and feature weights based on collaborative ...learning, called Co-FW-MVFCM.•The proposed Co-FW-MVFCM contains a two-step schema that includes a local step and a collaborative step.•The Co-FW-MVFCM can give feature reduction to exclude redundant feature components during clustering processes.•Comparisons among Co-FW-MVFCM and existing MVFCM algorithms actually demonstrate the effectiveness and usefulness of Co-FW-MVFCM.
Fuzzy c-means (FCM) clustering had been extended for handling multi-view data with collaborative idea. However, these collaborative multi-view FCM treats multi-view data under equal importance of feature components. In general, different features should take different weights for clustering real multi-view data. In this paper, we propose a novel multi-view FCM (MVFCM) clustering algorithm with view and feature weights based on collaborative learning, called collaborative feature-weighted MVFCM (Co-FW-MVFCM). The Co-FW-MVFCM contains a two-step schema that includes a local step and a collaborative step. The local step is a single-view partition process to produce local partition clustering in each view, and the collaborative step is sharing information of their memberships between different views. These two steps are then continuing by an aggregation way to get a global result after collaboration. Furthermore, the embedded feature-weighted procedure in Co-FW-MVFCM can give feature reduction to exclude redundant/irrelevant feature components during clustering processes. Experiments with several data sets demonstrate that the proposed Co-FW-MVFCM algorithm can completely identify irrelevant feature components in each view and that, additionally, it can improve the performance of the algorithm. Comparisons of Co-FW-MVFCM with some existing MVFCM algorithms are made and also demonstrated the effectiveness and usefulness of the proposed Co-FW-MVFCM clustering algorithm.
The theory of complex spherical fuzzy sets (CSFSs) is a mixture of two theories, i.e., complex fuzzy sets (CFSs) and spherical fuzzy sets (SFSs), to cope with uncertain and unreliable information in ...realistic decision-making situations. CSFSs contain three grades in the form of polar coordinates, e.g., truth, abstinence, and falsity, belonging to a unit disc in a complex plane, with a condition that the sum of squares of the real part of the truth, abstinence, and falsity grades is not exceeded by a unit interval. In this paper, we first consider some properties and their operational laws of CSFSs. Additionally, based on CSFSs, the complex spherical fuzzy Bonferroni mean (CSFBM) and complex spherical fuzzy weighted Bonferroni mean (CSFWBM) operators are proposed. The special cases of the proposed operators are also discussed. A multi-attribute decision making (MADM) problem was chosen to be resolved based on the proposed CSFBM and CSFWBM operators. We then propose the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method based on CSFSs (CSFS-TOPSIS). An application example is given to delineate the proposed methods and a close examination is undertaken. The advantages and comparative analysis of the proposed approaches are also presented.
A similarity measure is a useful tool for determining the similarity between two objects. Although there are many different similarity measures among the intuitionistic fuzzy sets (IFSs) proposed in ...the literature, the Jaccard index has yet to be considered as way to define them. The Jaccard index is a statistic used for comparing the similarity and diversity of sample sets. In this study, we propose a new similarity measure for IFSs induced by the Jaccard index. According to our results, proposed similarity measures between IFSs based on the Jaccard index present better properties. Several examples are used to compare the proposed approach with several existing methods. Numerical results show that the proposed measures are more reasonable than these existing measures. On the other hand, measuring the similarity between IFSs is also important in clustering. Thus, we also propose a clustering procedure by combining the proposed similarity measure with a robust clustering method for analyzing IFS data sets. We also compare the proposed clustering procedure with two clustering methods for IFS data sets.
In this paper, the novel approach of complex T-spherical fuzzy sets (CTSFSs) and their operational laws are explored and also verified with the help of examples. CTSFS composes the grade of truth, ...abstinence, and falsity with a condition that the sum of q-power of the real part (also for imaginary part) of the truth, abstinence, and falsity grades cannot be exceeded from a unit interval. Additionally, to examine the interrelationships among the complex T-spherical fuzzy numbers (CTSFNs), we propose two aggregation operators, called complex T-spherical fuzzy weighted averaging (CTSFWA) and complex T-spherical fuzzy weighted geometric (CTSFWG) operators. A multi-attribute decision making (MADM) problem is resolved based on CTSFNs by using the proposed CTSFWA and CTSFWG operators. To examine the proficiency and reliability of the explored works, we use an example to make comparisons between the proposed operators and some existing operators. Based on the comparison results, the proposed CTSFWA and CTSFWG operators are well suited in the fuzzy environment with legitimacy and prevalence by contrasting other existing operators.
Since social media, virtual communities and networks rapidly grow, multiview data become more popular. In general, multiview data always contain different feature components in different views. ...Although these data are extracted in different ways (views) from diverse settings and domains, they are used to describe the same samples, which make them highly related. Hence, applying (single-view) clustering methods for multiview data poses difficulty in achieving desirable clustering results. Thus, multiview clustering methods should be developed that will utilize available multiview information. Most of multiview clustering techniques currently use k-means due to its conceptual simplicity, and use fuzzy c-means (FCM) that the datapoints can belong to more than one cluster based on their membership degrees from 0 to 1. However, the use of k-means or FCM may degrade its performance due to the presence of noise and outliers, especially on large or high-dimensional datasets. The constraint imposed on the membership degrees of k-means and FCM tends to assign a corresponding high membership value to an outlier or a noisy data point. To address these drawbacks, possibilistic c-means (PCM) relaxes the membership constraint of k-means and FCM so that outliers and noisy datapoints can be properly identified. On the other hand, there are various extensions of k-means and FCM for multiview data, but no extension of PCM for multiview data was made in the literature. Thus, we use PCM in our proposed multiview clustering model. In this article, we propose novel weighted multiview PCM algorithms designed for clustering multiview data as well as view and feature weights on PCM approaches, called W-MV-PCM and W-MV-PCM with L2 regularization (W-MV-PCM-L2). In multiview clustering, different views may vary with respect to its importance and each view may contain some irrelevant features. In the proposed algorithms, a learning scheme is constructed to compute for the view weights, and feature weights within each view. This scheme will be able to identify the importance of each view and, at the same time, it will also identify and select relevant features in each view. Comparisons of W-MV-PCM-L2 with existing multiview clustering algorithms are made on both synthetic and real datasets. The experimental results are evaluated using accuracy rate (AR) and external validity indexes, such as Rand index (RI) and normalized mutual information (NMI). The proposed W-MV-PCM-L2 algorithm with comparisons of existing algorithms under criteria of AR, RI, and NMI shows that it is a feasible and effective multiview clustering algorithm.
Aczel-Alsina t-norm (TN) and t-conorm (TCN) were proposed by Aczel and Alsina in 1982 are more flexible than the other TN and TCN. Since Aczel-Alsina TN and TCN have a great impact due to the ...variableness of involved parameters, they have good applications in multi-attribute decision making (MADM) under fuzzy sets (FSs) construction. Recently, Senapati et al. (2022) developed Aczel-Alsina aggregation operators (AOs) under intuitionistic FSs (IFSs) and interval-valued IFSs (IVIFSs) with their applications in solving IFS and IVIFS MADM problems. We know that T-spherical FSs (TSFSs) are a recently developed approach to uncertain information with less information loss and more reliability than IFSs and IVIFSs. In this paper, we develop these AOs on TSFSs as a new approach to solve MADM problems by using Aczel-Alsina TN and Aczel-Alsina TCN under T-spherical fuzzy (TSF) information. Furthermore, the basic operations of TSF numbers (TSFNs) are developed and exemplified. Based on these operations, two types of AOs, i.e., TSF Aczel-Alsina weighted average (TSFAAWA), and TSF Aczel-Alsina weighted geometric (TSFAAWG) operators, are introduced and investigated. The reliability and accuracy of the newly developed AOs are tested numerically and theoretically by the induction methods. To further give applications and also study the sensitivity of these TSF Aczel-Alsina operators, the problem of project evaluation using these proposed operators is comprehensively observed. The results obtained by using these TSF Aczel-Alsina operators are compared with some previously existing AOs of TSFSs. According to comparison results, we observe the reliability and efficiency of the proposed methods.
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