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This course - the second in a series of three - discusses several approaches to the first and third problems of clustering identified in module I - viz., pre-clustering tendency assessment and post-clustering cluster validation. The target audience comprises advanced undergraduate and graduate students majoring in engineering and science, and practicing engineers and scientists interested in either research about or applications of clustering to real world problems such as data mining, image analysis and bioinformatics. Some of subject matter in this course is available in textbooks (most notably some of the material about cluster validity functionals), and some of the subject matter is the object of (my) current research. The references contain pointers to some excellent papers on these topics, and on a number of related or competitive methods that have been proposed and studied by others. I begin with a simple numerical example that establishes the necessity for both assessment and validity. Then, I discuss the visual assessment of tendency family of algorithms (VAT, sVAT and coVAT). These algorithms produce images that enable a user to make useful guesses about the number of clusters to seek in relational data before proceeding with a partitioning method for finding the clusters. Since object data can always be converted to relational form by computing pair wise distances, these methods are well defined for all types of unlabeled numerical data. The coVAT algorithm provides a means for estimating the number of clusters in each of the four problems associated with rectangular relational data: row clusters, column clusters, joint (pure) clusters, and mixed co-clusters. The second half of this course presents some examples of cluster validation using scalar measures or indices of cluster validity. Several examples from each of the three major categories (crisp, fuzzy and probabilistic) of indices are presented. This course concludes with a numerical example that c mpares 23 indices of all three types on clusters in 12 sets of data drawn from mixtures of Gaussian distributions having either 3 or 6 components. (SOME) indices of all three types do pretty well in this example, while others do very badly. I don't think this problem has a general "solution", but since we use clustering in many, many applications, we keep trying to find good indices to validate algorithmic outputs.

Fuzzy logic.

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Fuzzy logic

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The present book contains 14 papers accepted and published in the Special Issue of the MDPI Mathematics journal, entitled "Fuzzy Logic and Soft Computing - Dedicated to the Centenary of the Birth of Lotfi A. Zadeh (1921-2017)", which covers a wide range of topics connected to the theory and applications of fuzzy logic and soft computing. More precisely, these topics include: mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers; soft computing methods in a fuzzy environment; fuzzy inner product spaces; M-hazy vector spaces; interval ranges of fuzzy sets; arithmetic operations of fuzzy sets; gradual numbers; Interval-valued fuzzy soft sets; interval-valued fuzzy soft topology; preference relationships; L-fuzzy sub-effect algebras; Fuzzy differential subordinations; m-polar fuzzy set in semigroups; practical use of the marks; the fuzzy logic approach for decision-making systems in management control; multigranulation rough sets; the multigranulation roughness of an intuitionistic fuzzy set; decision-making algorithms, etc. We hope that this book will be useful for those who work in the domains of fuzzy logic and soft computing or for those who want to familiarize themselves with the most advanced knowledge in the field of fuzzy mathematics.

Fuzzy logic.

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This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. It aims to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named ‘fuzzy inference’ can be naturally understood as logical deduction. It is for mathematicians, logicians, computer scientists, specialists in artificial intelligence and knowledge engineering, and developers of fuzzy logic.

Fuzzy logic

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Fuzzy logic.