Possibilistic and fuzzy clustering methods for robust analysis of non-precise data

This work focuses on robust clustering of data affected by imprecision. The imprecision is managed in terms of fuzzy sets. The clustering process is based on the fuzzy and possibilistic approaches. In both approaches the observations are assigned to the clusters by means of membership degrees. In fuzzy clustering the membership degrees express the degrees of sharing of the observations to the clusters. In contrast, in possibilistic clustering the membership degrees are degrees of typicality. These two sources of information are complementary because the former helps to discover the best fuzzy partition of the observations while the latter reflects how well the observations are described by the centroids and, therefore, is helpful to identify outliers. First, a fully possibilistic k-means clustering procedure is suggested. Then, in order to exploit the benefits of both the approaches, a joint possibilistic and fuzzy clustering method for fuzzy data is proposed. A selection procedure for choosing the parameters of the new clustering method is introduced. The effectiveness of the proposal is investigated by means of simulated and real-life data

Informational Paradigm, management of uncertainty and theoretical formalisms in the clustering framework: A review

Fifty years have gone by since the publication of the first paper on clustering based on fuzzy sets theory. In 1965, L.A. Zadeh had published “Fuzzy Sets” [335]. After only one year, the first effects of this seminal paper began to emerge, with the pioneering paper on clustering by Bellman, Kalaba, Zadeh [33], in which they proposed a prototypal of clustering algorithm based on the fuzzy sets theory

The legacy of 50 years of fuzzy sets: A discussion

International audienceThis note provides a brief overview of the main ideas and notions underlying fifty years of research in fuzzy set and possibility theory, two important settings introduced by L.A. Zadeh for representing sets with unsharp boundaries and uncertainty induced by granules of information expressed with words. The discussion is organized on the basis of three potential understanding of the grades of membership to a fuzzy set, depending on what the fuzzy set intends to represent: a group of elements with borderline members, a plausibility distribution, or a preference profile. It also questions the motivations for some existing generalized fuzzy sets. This note clearly reflects the shared personal views of its authors

The posterity of Zadeh's 50-year-old paper: A retrospective in 101 Easy Pieces – and a Few More

International audienceThis article was commissioned by the 22nd IEEE International Conference of Fuzzy Systems (FUZZ-IEEE) to celebrate the 50th Anniversary of Lotfi Zadeh's seminal 1965 paper on fuzzy sets. In addition to Lotfi's original paper, this note itemizes 100 citations of books and papers deemed “important (significant, seminal, etc.)” by 20 of the 21 living IEEE CIS Fuzzy Systems pioneers. Each of the 20 contributors supplied 5 citations, and Lotfi's paper makes the overall list a tidy 101, as in “Fuzzy Sets 101”. This note is not a survey in any real sense of the word, but the contributors did offer short remarks to indicate the reason for inclusion (e.g., historical, topical, seminal, etc.) of each citation. Citation statistics are easy to find and notoriously erroneous, so we refrain from reporting them - almost. The exception is that according to Google scholar on April 9, 2015, Lotfi's 1965 paper has been cited 55,479 times

Fuzzy C-ordered medoids clustering of interval-valued data

Fuzzy clustering for interval-valued data helps us to find natural vague boundaries in such data. The Fuzzy c-Medoids Clustering (FcMdC) method is one of the most popular clustering methods based on a partitioning around medoids approach. However, one of the greatest disadvantages of this method is its sensitivity to the presence of outliers in data. This paper introduces a new robust fuzzy clustering method named Fuzzy c-Ordered-Medoids clustering for interval-valued data (FcOMdC-ID). The Huber's M-estimators and the Yager's Ordered Weighted Averaging (OWA) operators are used in the method proposed to make it robust to outliers. The described algorithm is compared with the fuzzy c-medoids method in the experiments performed on synthetic data with different types of outliers. A real application of the FcOMdC-ID is also provided

On the semantics of fuzzy logic

AbstractThis paper presents a formal characterization of the major concepts and constructs of fuzzy logic in terms of notions of distance, closeness, and similarity between pairs of possible worlds. The formalism is a direct extension (by recognition of multiple degrees of accessibility, conceivability, or reachability) of the najor modal logic concepts of possible and necessary truth.Given a function that maps pairs of possible worlds into a number between 0 and 1, generalizing the conventional concept of an equivalence relation, the major constructs of fuzzy logic (conditional and unconditioned possibility distributions) are defined in terms of this similarity relation using familiar concepts from the mathematical theory of metric spaces. This interpretation is different in nature and character from the typical, chance-oriented, meanings associated with probabilistic concepts, which are grounded on the mathematical notion of set measure. The similarity structure defines a topological notion of continuity in the space of possible worlds (and in that of its subsets, i.e., propositions) that allows a form of logical “extrapolation” between possible worlds.This logical extrapolation operation corresponds to the major deductive rule of fuzzy logic — the compositional rule of inference or generalized modus ponens of Zadeh — an inferential operation that generalizes its classical counterpart by virtue of its ability to be utilized when propositions representing available evidence match only approximately the antecedents of conditional propositions. The relations between the similarity-based interpretation of the role of conditional possibility distributions and the approximate inferential procedures of Baldwin are also discussed.A straightforward extension of the theory to the case where the similarity scale is symbolic rather than numeric is described. The problem of generating similarity functions from a given set of possibility distributions, with the latter interpreted as defining a number of (graded) discernibility relations and the former as the result of combining them into a joint measure of distinguishability between possible worlds, is briefly discussed

Outline of a new feature space deformation approach in fuzzy pattern recognition

Sposobnost prepoznavanja oblika je jedno od najznačajnijih svojstava koja karakterišu inteligentno ponašanje bioloških ili veštačkih sistema. Matematičko prepoznavanje oblika predstavlja formalnu osnovu za rešavanje ovog zadatka primenom precizno forumulisanih algoritama, koji su u najvećem delu bazirni na konvencionalnoj matematici. Kod kompleksnih sistema ovakav pristup pokazuje značajne nedostatke, prvenstveno zbog zahteva za obimnim izračunavanjima i nedovoljne robusnosti. Algoritmi koji su bazirani na 'soft computing' metodama predstavljaju dobru alternativu, otvarajući prostor za razvoj efikasnih algoritama za primenu u realnom vremenu, polazeći od činjenice da značenje sadržaja informacija nosi veću vrednost u odnosu na preciznost. U ovom radu izlaže se modifikacija i proširenje 'Subrtactive Clustering' metode, koja se pokazala efikasnom u obradi masivnih skupova oblika u realnom vremenu. Novi pristup koji je baziran prvenstveno na povezivanju parametara algoritma sa informacionim sadržajem prisutnim u skupu oblika koji se obrađuje, daje dodatne stepene slobode i omogućava da proces prepoznavanja bude vođen podacima koji se obrađuju. Predloženi algoritam je verifikovan velikim brojem simulacionih eksperimenata, od kojih su neki navedeni u ovom radu.Pattern recognition ability is one of the most important features that characterize intelligent behavior of either biological or artificial systems. Mathematical pattern recognition is the way to solve this problem using transparent algorithms that are mostly based on conventional mathematics. In complex systems it shows inadequacy, primary due to the needs for extensive computation and insufficient robustness. Algorithms based on soft computing approach offer a good alternative, giving a room to design effective tools for real-time application, having in mind that relevance (significance) prevails precision in complex systems. In this article is modified and extended subtractive clustering method, which is proven to be effective in real-time applications, when massive pattern sets is processed. The new understanding and new relations that connect parameters of the algorithm with the information underlying the pattern set are established, giving on this way the algorithm ability to be data driven to the maximum extent. Proposed algorithm is verified by a number of experiments and few of them are presented in this article

Outline of a new feature space deformation approach in fuzzy pattern recognition

Sposobnost prepoznavanja oblika je jedno od najznačajnijih svojstava koja karakterišu inteligentno ponašanje bioloških ili veštačkih sistema. Matematičko prepoznavanje oblika predstavlja formalnu osnovu za rešavanje ovog zadatka primenom precizno forumulisanih algoritama, koji su u najvećem delu bazirni na konvencionalnoj matematici. Kod kompleksnih sistema ovakav pristup pokazuje značajne nedostatke, prvenstveno zbog zahteva za obimnim izračunavanjima i nedovoljne robusnosti. Algoritmi koji su bazirani na 'soft computing' metodama predstavljaju dobru alternativu, otvarajući prostor za razvoj efikasnih algoritama za primenu u realnom vremenu, polazeći od činjenice da značenje sadržaja informacija nosi veću vrednost u odnosu na preciznost. U ovom radu izlaže se modifikacija i proširenje 'Subrtactive Clustering' metode, koja se pokazala efikasnom u obradi masivnih skupova oblika u realnom vremenu. Novi pristup koji je baziran prvenstveno na povezivanju parametara algoritma sa informacionim sadržajem prisutnim u skupu oblika koji se obrađuje, daje dodatne stepene slobode i omogućava da proces prepoznavanja bude vođen podacima koji se obrađuju. Predloženi algoritam je verifikovan velikim brojem simulacionih eksperimenata, od kojih su neki navedeni u ovom radu.Pattern recognition ability is one of the most important features that characterize intelligent behavior of either biological or artificial systems. Mathematical pattern recognition is the way to solve this problem using transparent algorithms that are mostly based on conventional mathematics. In complex systems it shows inadequacy, primary due to the needs for extensive computation and insufficient robustness. Algorithms based on soft computing approach offer a good alternative, giving a room to design effective tools for real-time application, having in mind that relevance (significance) prevails precision in complex systems. In this article is modified and extended subtractive clustering method, which is proven to be effective in real-time applications, when massive pattern sets is processed. The new understanding and new relations that connect parameters of the algorithm with the information underlying the pattern set are established, giving on this way the algorithm ability to be data driven to the maximum extent. Proposed algorithm is verified by a number of experiments and few of them are presented in this article