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Journal ArticleDOI

Validity index for crisp and fuzzy clusters

TL;DR: A cluster validity index and its fuzzification is described, which can provide a measure of goodness of clustering on different partitions of a data set, and results demonstrating the superiority of the PBM-index in appropriately determining the number of clusters are provided.
About: This article is published in Pattern Recognition.The article was published on 2004-03-01. It has received 710 citations till now. The article focuses on the topics: Fuzzy clustering & Correlation clustering.
Citations
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Journal ArticleDOI
01 Jan 2008
TL;DR: Differential evolution has emerged as one of the fast, robust, and efficient global search heuristics of current interest as mentioned in this paper, which has been applied to the automatic clustering of large unlabeled data sets.
Abstract: Differential evolution (DE) has emerged as one of the fast, robust, and efficient global search heuristics of current interest. This paper describes an application of DE to the automatic clustering of large unlabeled data sets. In contrast to most of the existing clustering techniques, the proposed algorithm requires no prior knowledge of the data to be classified. Rather, it determines the optimal number of partitions of the data "on the run." Superiority of the new method is demonstrated by comparing it with two recently developed partitional clustering techniques and one popular hierarchical clustering algorithm. The partitional clustering algorithms are based on two powerful well-known optimization algorithms, namely the genetic algorithm and the particle swarm optimization. An interesting real-world application of the proposed method to automatic segmentation of images is also reported.

700 citations


Cites background from "Validity index for crisp and fuzzy ..."

  • ...Since the given data set can be partitioned in a number of ways, maintaining all of the aforementioned properties, a fitness function (some measure of the adequacy of the partitioning) must be defined....

    [...]

Journal ArticleDOI
TL;DR: The fundamental concepts of cluster validity are introduced, and a review of fuzzy cluster validity indices available in the literature are presented, and extensive comparisons of the mentioned indices are conducted in conjunction with the Fuzzy C-Means clustering algorithm.

489 citations


Cites background or result from "Validity index for crisp and fuzzy ..."

  • ...proposed a validity index called the PBM index [37]....

    [...]

  • ...And we must state that the Dataset_4_3, Dataset_5_2, Dataset_6_2 and Dataset_10_2 data sets are similar to data sets used in [3,37,38]....

    [...]

Journal ArticleDOI
TL;DR: An up-to-date review of all major nature inspired metaheuristic algorithms employed till date for partitional clustering and key issues involved during formulation of various metaheuristics as a clustering problem and major application areas are discussed.
Abstract: The partitional clustering concept started with K-means algorithm which was published in 1957. Since then many classical partitional clustering algorithms have been reported based on gradient descent approach. The 1990 kick started a new era in cluster analysis with the application of nature inspired metaheuristics. After initial formulation nearly two decades have passed and researchers have developed numerous new algorithms in this field. This paper embodies an up-to-date review of all major nature inspired metaheuristic algorithms employed till date for partitional clustering. Further, key issues involved during formulation of various metaheuristics as a clustering problem and major application areas are discussed.

457 citations


Additional excerpts

  • ...index [295,296], root-mean-square standard deviation index [295,296], RS index [297], PBM index [300] and SV index [301]....

    [...]

Journal ArticleDOI
TL;DR: A model-based method for predicting RUL of machinery is proposed and the effectiveness of the proposed method is identified, using vibration signals from accelerated degradation tests of rolling element bearings.
Abstract: Remaining useful life (RUL) prediction allows for predictive maintenance of machinery, thus reducing costly unscheduled maintenance. Therefore, RUL prediction of machinery appears to be a hot issue attracting more and more attention as well as being of great challenge. This paper proposes a model-based method for predicting RUL of machinery. The method includes two modules, i.e., indicator construction and RUL prediction. In the first module, a new health indicator named weighted minimum quantization error is constructed, which fuses mutual information from multiple features and properly correlates to the degradation processes of machinery. In the second module, model parameters are initialized using the maximum-likelihood estimation algorithm and RUL is predicted using a particle filtering-based algorithm. The proposed method is demonstrated using vibration signals from accelerated degradation tests of rolling element bearings. The prediction result identifies the effectiveness of the proposed method in predicting RUL of machinery.

358 citations


Cites methods from "Validity index for crisp and fuzzy ..."

  • ...In this algorithm, a PBM index, which is commonly used for verifying the cluster validity [35], is employed to determine an appropriate number of clusters....

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Journal IssueDOI
TL;DR: An alternative, possibly complementary methodology for comparing clustering validity criteria is described and an extensive comparison of the performances of 40 criteria over a collection of 962,928 partitions derived from five well-known clustering algorithms and 1080 different data sets of a given class of interest is made.
Abstract: Many different relative clustering validity criteria exist that are very useful in practice as quantitative measures for evaluating the quality of data partitions, and new criteria have still been proposed from time to time. These criteria are endowed with particular features that may make each of them able to outperform others in specific classes of problems. In addition, they may have completely different computational requirements. Then, it is a hard task for the user to choose a specific criterion when he or she faces such a variety of possibilities. For this reason, a relevant issue within the field of clustering analysis consists of comparing the performances of existing validity criteria and, eventually, that of a new criterion to be proposed. In spite of this, the comparison paradigm traditionally adopted in the literature is subject to some conceptual limitations. The present paper describes an alternative, possibly complementary methodology for comparing clustering validity criteria and uses it to make an extensive comparison of the performances of 40 criteria over a collection of 962,928 partitions derived from five well-known clustering algorithms and 1080 different data sets of a given class of interest. A detailed review of the relative criteria under investigation is also provided that includes an original comparative asymptotic analysis of their computational complexities. This work is intended to be a complement of the classic study reported in 1985 by Milligan and Cooper as well as a thorough extension of a preliminary paper by the authors themselves. Copyright © 2010 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 3: 209-235, 2010

243 citations


Cites background from "Validity index for crisp and fuzzy ..."

  • ...The criterion known as PBM [30] is also based on the within-group and between-group distances:...

    [...]

References
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Book
01 Sep 1988
TL;DR: In this article, the authors present the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields, including computer programming and mathematics.
Abstract: From the Publisher: This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required

52,797 citations

01 Jan 1989
TL;DR: This book brings together the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields.
Abstract: From the Publisher: This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields. Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs. No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required.

33,034 citations

Book
01 Jan 1982
TL;DR: In this article, the authors present an overview of the basic concepts of multivariate analysis, including matrix algebra and random vectors, as well as a strategy for analyzing multivariate models.
Abstract: (NOTE: Each chapter begins with an Introduction, and concludes with Exercises and References.) I. GETTING STARTED. 1. Aspects of Multivariate Analysis. Applications of Multivariate Techniques. The Organization of Data. Data Displays and Pictorial Representations. Distance. Final Comments. 2. Matrix Algebra and Random Vectors. Some Basics of Matrix and Vector Algebra. Positive Definite Matrices. A Square-Root Matrix. Random Vectors and Matrices. Mean Vectors and Covariance Matrices. Matrix Inequalities and Maximization. Supplement 2A Vectors and Matrices: Basic Concepts. 3. Sample Geometry and Random Sampling. The Geometry of the Sample. Random Samples and the Expected Values of the Sample Mean and Covariance Matrix. Generalized Variance. Sample Mean, Covariance, and Correlation as Matrix Operations. Sample Values of Linear Combinations of Variables. 4. The Multivariate Normal Distribution. The Multivariate Normal Density and Its Properties. Sampling from a Multivariate Normal Distribution and Maximum Likelihood Estimation. The Sampling Distribution of 'X and S. Large-Sample Behavior of 'X and S. Assessing the Assumption of Normality. Detecting Outliners and Data Cleaning. Transformations to Near Normality. II. INFERENCES ABOUT MULTIVARIATE MEANS AND LINEAR MODELS. 5. Inferences About a Mean Vector. The Plausibility of ...m0 as a Value for a Normal Population Mean. Hotelling's T 2 and Likelihood Ratio Tests. Confidence Regions and Simultaneous Comparisons of Component Means. Large Sample Inferences about a Population Mean Vector. Multivariate Quality Control Charts. Inferences about Mean Vectors When Some Observations Are Missing. Difficulties Due To Time Dependence in Multivariate Observations. Supplement 5A Simultaneous Confidence Intervals and Ellipses as Shadows of the p-Dimensional Ellipsoids. 6. Comparisons of Several Multivariate Means. Paired Comparisons and a Repeated Measures Design. Comparing Mean Vectors from Two Populations. Comparison of Several Multivariate Population Means (One-Way MANOVA). Simultaneous Confidence Intervals for Treatment Effects. Two-Way Multivariate Analysis of Variance. Profile Analysis. Repealed Measures, Designs, and Growth Curves. Perspectives and a Strategy for Analyzing Multivariate Models. 7. Multivariate Linear Regression Models. The Classical Linear Regression Model. Least Squares Estimation. Inferences About the Regression Model. Inferences from the Estimated Regression Function. Model Checking and Other Aspects of Regression. Multivariate Multiple Regression. The Concept of Linear Regression. Comparing the Two Formulations of the Regression Model. Multiple Regression Models with Time Dependant Errors. Supplement 7A The Distribution of the Likelihood Ratio for the Multivariate Regression Model. III. ANALYSIS OF A COVARIANCE STRUCTURE. 8. Principal Components. Population Principal Components. Summarizing Sample Variation by Principal Components. Graphing the Principal Components. Large-Sample Inferences. Monitoring Quality with Principal Components. Supplement 8A The Geometry of the Sample Principal Component Approximation. 9. Factor Analysis and Inference for Structured Covariance Matrices. The Orthogonal Factor Model. Methods of Estimation. Factor Rotation. Factor Scores. Perspectives and a Strategy for Factor Analysis. Structural Equation Models. Supplement 9A Some Computational Details for Maximum Likelihood Estimation. 10. Canonical Correlation Analysis Canonical Variates and Canonical Correlations. Interpreting the Population Canonical Variables. The Sample Canonical Variates and Sample Canonical Correlations. Additional Sample Descriptive Measures. Large Sample Inferences. IV. CLASSIFICATION AND GROUPING TECHNIQUES. 11. Discrimination and Classification. Separation and Classification for Two Populations. Classifications with Two Multivariate Normal Populations. Evaluating Classification Functions. Fisher's Discriminant Function...nSeparation of Populations. Classification with Several Populations. Fisher's Method for Discriminating among Several Populations. Final Comments. 12. Clustering, Distance Methods and Ordination. Similarity Measures. Hierarchical Clustering Methods. Nonhierarchical Clustering Methods. Multidimensional Scaling. Correspondence Analysis. Biplots for Viewing Sample Units and Variables. Procustes Analysis: A Method for Comparing Configurations. Appendix. Standard Normal Probabilities. Student's t-Distribution Percentage Points. ...c2 Distribution Percentage Points. F-Distribution Percentage Points. F-Distribution Percentage Points (...a = .10). F-Distribution Percentage Points (...a = .05). F-Distribution Percentage Points (...a = .01). Data Index. Subject Index.

11,697 citations

Journal ArticleDOI
TL;DR: In this article, the authors present an overview of the basic concepts of multivariate analysis, including matrix algebra and random vectors, as well as a strategy for analyzing multivariate models.
Abstract: (NOTE: Each chapter begins with an Introduction, and concludes with Exercises and References.) I. GETTING STARTED. 1. Aspects of Multivariate Analysis. Applications of Multivariate Techniques. The Organization of Data. Data Displays and Pictorial Representations. Distance. Final Comments. 2. Matrix Algebra and Random Vectors. Some Basics of Matrix and Vector Algebra. Positive Definite Matrices. A Square-Root Matrix. Random Vectors and Matrices. Mean Vectors and Covariance Matrices. Matrix Inequalities and Maximization. Supplement 2A Vectors and Matrices: Basic Concepts. 3. Sample Geometry and Random Sampling. The Geometry of the Sample. Random Samples and the Expected Values of the Sample Mean and Covariance Matrix. Generalized Variance. Sample Mean, Covariance, and Correlation as Matrix Operations. Sample Values of Linear Combinations of Variables. 4. The Multivariate Normal Distribution. The Multivariate Normal Density and Its Properties. Sampling from a Multivariate Normal Distribution and Maximum Likelihood Estimation. The Sampling Distribution of 'X and S. Large-Sample Behavior of 'X and S. Assessing the Assumption of Normality. Detecting Outliners and Data Cleaning. Transformations to Near Normality. II. INFERENCES ABOUT MULTIVARIATE MEANS AND LINEAR MODELS. 5. Inferences About a Mean Vector. The Plausibility of ...m0 as a Value for a Normal Population Mean. Hotelling's T 2 and Likelihood Ratio Tests. Confidence Regions and Simultaneous Comparisons of Component Means. Large Sample Inferences about a Population Mean Vector. Multivariate Quality Control Charts. Inferences about Mean Vectors When Some Observations Are Missing. Difficulties Due To Time Dependence in Multivariate Observations. Supplement 5A Simultaneous Confidence Intervals and Ellipses as Shadows of the p-Dimensional Ellipsoids. 6. Comparisons of Several Multivariate Means. Paired Comparisons and a Repeated Measures Design. Comparing Mean Vectors from Two Populations. Comparison of Several Multivariate Population Means (One-Way MANOVA). Simultaneous Confidence Intervals for Treatment Effects. Two-Way Multivariate Analysis of Variance. Profile Analysis. Repealed Measures, Designs, and Growth Curves. Perspectives and a Strategy for Analyzing Multivariate Models. 7. Multivariate Linear Regression Models. The Classical Linear Regression Model. Least Squares Estimation. Inferences About the Regression Model. Inferences from the Estimated Regression Function. Model Checking and Other Aspects of Regression. Multivariate Multiple Regression. The Concept of Linear Regression. Comparing the Two Formulations of the Regression Model. Multiple Regression Models with Time Dependant Errors. Supplement 7A The Distribution of the Likelihood Ratio for the Multivariate Regression Model. III. ANALYSIS OF A COVARIANCE STRUCTURE. 8. Principal Components. Population Principal Components. Summarizing Sample Variation by Principal Components. Graphing the Principal Components. Large-Sample Inferences. Monitoring Quality with Principal Components. Supplement 8A The Geometry of the Sample Principal Component Approximation. 9. Factor Analysis and Inference for Structured Covariance Matrices. The Orthogonal Factor Model. Methods of Estimation. Factor Rotation. Factor Scores. Perspectives and a Strategy for Factor Analysis. Structural Equation Models. Supplement 9A Some Computational Details for Maximum Likelihood Estimation. 10. Canonical Correlation Analysis Canonical Variates and Canonical Correlations. Interpreting the Population Canonical Variables. The Sample Canonical Variates and Sample Canonical Correlations. Additional Sample Descriptive Measures. Large Sample Inferences. IV. CLASSIFICATION AND GROUPING TECHNIQUES. 11. Discrimination and Classification. Separation and Classification for Two Populations. Classifications with Two Multivariate Normal Populations. Evaluating Classification Functions. Fisher's Discriminant Function...nSeparation of Populations. Classification with Several Populations. Fisher's Method for Discriminating among Several Populations. Final Comments. 12. Clustering, Distance Methods and Ordination. Similarity Measures. Hierarchical Clustering Methods. Nonhierarchical Clustering Methods. Multidimensional Scaling. Correspondence Analysis. Biplots for Viewing Sample Units and Variables. Procustes Analysis: A Method for Comparing Configurations. Appendix. Standard Normal Probabilities. Student's t-Distribution Percentage Points. ...c2 Distribution Percentage Points. F-Distribution Percentage Points. F-Distribution Percentage Points (...a = .10). F-Distribution Percentage Points (...a = .05). F-Distribution Percentage Points (...a = .01). Data Index. Subject Index.

10,148 citations