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.

In this article we exploit the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets (IFSs). We then propose two families of entropy measures for IFSs and also construct the axiom definition and properties. Two definitions of entropy for IFSs proposed by Burillo and Bustince in 1996 and Szmidt and Kacprzyk in 2001 are used. The first one allows us to measure the degree of intuitionism of an IFS, whereas the second one is a nonprobabilistic-type entropy measure with a geometric interpretation of IFSs used in comparison with our proposed entropy of IFSs in the numerical comparisons. The results show that the proposed entropy measures seem to be more reliable for presenting the degree of fuzziness of an IFS. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 443–451, 2006.

This paper presents an alternating optimization clustering procedure called a similarity-based clustering method (SCM). It is an effective and robust approach to clustering on the basis of a total similarity objective function related to the approximate density shape estimation. We show that the data points in SCM can self-organize local optimal cluster number and volumes without using cluster validity functions or a variance-covariance matrix. The proposed clustering method is also robust to noise and outliers based on the influence function and gross error sensitivity analysis. Therefore, SCM exhibits three robust clustering characteristics: 1) robust to the initialization (cluster number and initial guesses), 2) robust to cluster volumes (ability to detect different volumes of clusters), and 3) robust to noise and outliers. Several numerical data sets and actual data are used in the SCM to show these good aspects. The computational complexity of SCM is also analyzed. Some experimental results of comparing the proposed SCM with the existing methods show the superiority of the SCM method.

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.

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.

Multi-attribute decision-making (MADM) is usually used to aggregate fuzzy data successfully. Choosing the best option regarding data is not generally symmetric on the grounds that it does not provide complete information. Since Aczel-Alsina aggregation operators (AOs) have great impact due to their parameter variableness, they have been well applied in MADM under fuzzy construction. Recently, the Aczel-Alsina AOs on intuitionistic fuzzy sets (IFSs), interval-valued IFSs and T-spherical fuzzy sets have been proposed in the literature. In this article, we develop new types of Pythagorean fuzzy AOs by using Aczel-Alsina t-norm and Aczel-Alsina t-conorm. Thus, we give these new operations Aczel-Alsina sum and Aczel-Alsina product on Pythagorean fuzzy sets based on Aczel-Alsina t-norm and Aczel-Alsina t-conorm. We also develop new types of Pythagorean fuzzy AOs including Pythagorean fuzzy Aczel-Alsina weighted averaging and Pythagorean fuzzy Aczel-Alsina weighted geometric operators. We elaborate some characteristics of these proposed Aczel-Alsina AOs on Pythagorean fuzzy sets, such as idempotency, monotonicity, and boundedness. By utilizing the proposed works, we solve an example of MADM in the information of the multinational company under the evaluation of impacts in MADM. We also illustrate the comparisons of the proposed works with previously existing AOs in different fuzzy environments. These comparison results demonstrate the effectiveness of the proposed Aczel-Alsina AOs on Pythagorean fuzzy sets.