A Method to Construct Approximate Fuzzy Voronoi Diagram for Fuzzy Numbers of Dimension Two

In this paper, we propose an approximate "fuzzy Voronoi" diagram(FVD)for fuzzy numbers of dimension two (FNDT) by designing an extension ofcrisp Voronoi diagram for fuzzy numbers. The fuzzy Voronoi sites are defined asfuzzy numbers of dimension two. In this approach, the fuzzy numbers have a convexcontinuous differentiable shape. The proposed algorithm has two stages: in the firststage we use the Fortune’s algorithm in order to construct a "fuzzy Voronoi" diagramfor membership values of FNDTs that are equal to 1. In the second stage, we proposea new algorithm based on the Euclidean distance between two fuzzy numbers in orderto construct the approximate "fuzzy Voronoi" diagram for values of the membershipof FNDTs that are smaller than 1. The experimental results are presented for aparticular shape, the fuzzy ellipse numbers

A potential theory approach to an algorithm of conceptual space partitioning

A potential theory approach to an algorithm of conceptual space partitioningThis paper proposes a new classification algorithm for the partitioning of a conceptual space. All the algorithms which have been used until now have mostly been based on the theory of Voronoi diagrams. This paper proposes an approach based on potential theory, with the criteria for measuring similarities between objects in the conceptual space being based on the Newtonian potential function. The notion of a fuzzy prototype, which generalizes the previous definition of a prototype, is introduced. Furthermore, the necessary conditions that a natural concept must meet are discussed. Instead of convexity, as proposed by Gärdenfors, the notion of geodesically convex sets is used. Thus, if a concept corresponds to a set which is geodesically convex, it is a natural concept. This definition applies, for example, if the conceptual space is an Euclidean space. As a by-product of the construction of the algorithm, an extension of the conceptual space to d-dimensional Riemannian manifolds is obtained. Algorytm podziału przestrzeni konceptualnych przy użyciu teorii potencjałuW niniejszej pracy zaproponowany został nowy algorytm podziału przestrzeni konceptualnej. Dotąd podział taki zazwyczaj wykorzystywał teorię diagramów Voronoi. Nasze podejście do problemu oparte jest na teorii potencjału Miara podobieństwa pomiędzy elementami przestrzeni konceptualnej bazuje na Newtonowskiej funkcji potencjału. Definiujemy pojęcie rozmytego prototypu, który uogólnia dotychczas stosowane definicje prototypu. Ponadto zajmujemy się warunkiem koniecznym, który musi spełniać naturalny koncept. Zamiast wypukłości zaproponowanej przez Gärdenforsa, rozważamy linie geodezyjne w obszarze odpowiadającym danemu konceptowi naturalnemu, otrzymując warunek mówiący, że koncept jest konceptem naturalnym, jeżeli zbiór odpowiadający temu konceptowi jest geodezyjnie wypukły. Ta definicja pokrywa się w przypadku, gdy przestrzenią konceptualną jest przestrzeń euklidesowa. Jako produkt uboczny konstrukcji naszego algorytmu rozważamy dość ogólne przestrzenie konceptualne będące d-wymiarowymi rozmaitościami Reimanna

Doctor of Philosophy

dissertationWith the tremendous growth of data produced in the recent years, it is impossible to identify patterns or test hypotheses without reducing data size. Data mining is an area of science that extracts useful information from the data by discovering patterns and structures present in the data. In this dissertation, we will largely focus on clustering which is often the first step in any exploratory data mining task, where items that are similar to each other are grouped together, making downstream data analysis robust. Different clustering techniques have different strengths, and the resulting groupings provide different perspectives on the data. Due to the unsupervised nature i.e., the lack of domain experts who can label the data, validation of results is very difficult. While there are measures that compute "goodness" scores for clustering solutions as a whole, there are few methods that validate the assignment of individual data items to their clusters. To address these challenges we focus on developing a framework that can generate, compare, combine, and evaluate different solutions to make more robust and significant statements about the data. In the first part of this dissertation, we present fast and efficient techniques to generate and combine different clustering solutions. We build on some recent ideas on efficient representations of clusters of partitions to develop a well founded metric that is spatially aware to compare clusterings. With the ability to compare clusterings, we describe a heuristic to combine different solutions to produce a single high quality clustering. We also introduce a Markov chain Monte Carlo approach to sample different clusterings from the entire landscape to provide the users with a variety of choices. In the second part of this dissertation, we build certificates for individual data items and study their influence on effective data reduction. We present a geometric approach by defining regions of influence for data items and clusters and use this to develop adaptive sampling techniques to speedup machine learning algorithms. This dissertation is therefore a systematic approach to study the landscape of clusterings in an attempt to provide a better understanding of the data