In this paper we present a fuzzy voting scheme for cluster algorithms. This fuzzy voting method allows us to combine several runs of cluster algorithms resulting in a common fuzzy partition. This helps us to overcome instabilities of the cluster algorithms and results in a better clustering. / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

In the paper at hand we apply it to Bayesian statistics to obtain "Fuzzy Bayesian Inference". In the subsequent sections we will discuss a fuzzy valued likelihood function, Bayes' theorem for both fuzzy data and fuzzy priors, a fuzzy Bayes' estimator, fuzzy predictive densities and distributions, and fuzzy H.P.D .-Regions. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

In his paper introducing fuzzy sets, L.A. Zadeh describes the difficulty of assigning some real-world objects to a particular class when the notion of class membership is ambiguous. If exact classification is not obvious, most people approximate using intuition and may reach agreement by placing an object in more than one class. Numbers or ‘degrees of membership’ within these classes are used to provide an approximation that supports this intuitive process. This results in a ‘fuzzy set’. This fuzzy set consists any number of ordered pairs to represent both the class and the class’s degree of membership to provide a formal representation that can be used to model this process. Although the fuzzy approach to reasoning and classification makes sense, it does not comply with two of the basic principles of classical logic. These principles are the laws of contradiction and excluded middle. While they play a significant role in logic, it is the violation of these principles that gives fuzzy logic its useful characteristics. The problem of this representation within a database system, however, is that the class and its degree of membership are represented by two separate, but indivisible attributes. Further, this representation may contain any number of such pairs of attributes. While the data for class and membership are maintained in individual attributes, neither of these attributes may exist without the other without sacrificing meaning. And, to maintain a variable number of such pairs within the representation is problematic. C. J. Date suggested a relation valued attribute (RVA) which can not only encapsulate the attributes associated with the fuzzy set and impose constraints on their use, but also provide a relation which may contain any number of such pairs. The goal of this dissertation is to establish a context in which the relational database model can be extended through the implementation of an RVA to support of fuzzy data on an actual system. This goal represents an opportunity to study through application and observation, the use of fuzzy sets to support imprecise and uncertain data using database queries which appropriately adhere to the relational model. The intent is to create a pathway that may extend the support of database applications that need fuzzy logic and/or fuzzy data.

Data mining is an important process of extracting desirable knowledge from existing databases for specific purposes. Nearly all transactions in real-world databases involve items bought, quantities of the items, and the time periods in which they appear. In the past, temporal quantitative mining was proposed to find temporal quantitative rules from a temporal quantitative database. However, the quantitative values of items are not suitable to human reasoning. To deal with this, the fuzzy set theory was applied to the temporal quantitative mining because of its simplicity and similarity to human reasoning. In this thesis, we thus handle the problem of mining fuzzy temporal association rules from a publication database, and propose three algorithms to achieve it. The three algorithms handle different lifespan definitions, respectively. In the first algorithm, the lifespan of an item is evaluated from the time of the first transaction with the item to the end time of the whole database. In the second algorithm, an additional publication table, which includes the publication date of each item in stores, is given, and thus the lifespan of an item is measured by its entire publication period. Finally in the third algorithm, the lifespan of an item is calculated from the end time of the whole database to its earliest time in the database for the item to be a fuzzy temporal frequent item within the duration. In addition, an effective itemset table structure is designed to store and get information about itemsets and can thus speed up the execution efficiency of the mining process. At last, experimental results on two simulation datasets compare the mined fuzzy temporal quantitative itemsets and rules with and without consideration of lifespans of items under different parameter settings.

With the growing number of large data sets, the necessity of complexity reduction applies today more than ever before. Moreover, some data may also be vague or uncertain. Thus, whenever we have an instrument for data analysis, the questions of how to apply complexity reduction methods and how to treat fuzzy data arise rather naturally. In this thesis, we discuss these issues for the very successful data analysis tool Formal Concept Analysis. In fact, we propose different methods for complexity reduction based on qualitative analyses, and we elaborate on various methods for handling fuzzy data. These two topics split the thesis into two parts. Data reduction is mainly dealt with in the first part of the thesis, whereas we focus on fuzzy data in the second part. Although each chapter may be read almost on its own, each one builds on and uses results from its predecessors. The main crosslink between the chapters is given by the reduction methods and fuzzy data. In particular, we will also discuss complexity reduction methods for fuzzy data, combining the two issues that motivate this thesis. / Komplexitätsreduktion ist eines der wichtigsten Verfahren in der Datenanalyse. Mit ständig wachsenden Datensätzen gilt dies heute mehr denn je. In vielen Gebieten stößt man zudem auf vage und ungewisse Daten. Wann immer man ein Instrument zur Datenanalyse hat, stellen sich daher die folgenden zwei Fragen auf eine natürliche Weise: Wie kann man im Rahmen der Analyse die Variablenanzahl verkleinern, und wie kann man Fuzzy-Daten bearbeiten? In dieser Arbeit versuchen wir die eben genannten Fragen für die Formale Begriffsanalyse zu beantworten. Genauer gesagt, erarbeiten wir verschiedene Methoden zur Komplexitätsreduktion qualitativer Daten und entwickeln diverse Verfahren für die Bearbeitung von Fuzzy-Datensätzen. Basierend auf diesen beiden Themen gliedert sich die Arbeit in zwei Teile. Im ersten Teil liegt der Schwerpunkt auf der Komplexitätsreduktion, während sich der zweite Teil der Verarbeitung von Fuzzy-Daten widmet. Die verschiedenen Kapitel sind dabei durch die beiden Themen verbunden. So werden insbesondere auch Methoden für die Komplexitätsreduktion von Fuzzy-Datensätzen entwickelt.

With the growing number of large data sets, the necessity of complexity reduction applies today more than ever before. Moreover, some data may also be vague or uncertain. Thus, whenever we have an instrument for data analysis, the questions of how to apply complexity reduction methods and how to treat fuzzy data arise rather naturally. In this thesis, we discuss these issues for the very successful data analysis tool Formal Concept Analysis. In fact, we propose different methods for complexity reduction based on qualitative analyses, and we elaborate on various methods for handling fuzzy data. These two topics split the thesis into two parts. Data reduction is mainly dealt with in the first part of the thesis, whereas we focus on fuzzy data in the second part. Although each chapter may be read almost on its own, each one builds on and uses results from its predecessors. The main crosslink between the chapters is given by the reduction methods and fuzzy data. In particular, we will also discuss complexity reduction methods for fuzzy data, combining the two issues that motivate this thesis. / Komplexitätsreduktion ist eines der wichtigsten Verfahren in der Datenanalyse. Mit ständig wachsenden Datensätzen gilt dies heute mehr denn je. In vielen Gebieten stößt man zudem auf vage und ungewisse Daten. Wann immer man ein Instrument zur Datenanalyse hat, stellen sich daher die folgenden zwei Fragen auf eine natürliche Weise: Wie kann man im Rahmen der Analyse die Variablenanzahl verkleinern, und wie kann man Fuzzy-Daten bearbeiten? In dieser Arbeit versuchen wir die eben genannten Fragen für die Formale Begriffsanalyse zu beantworten. Genauer gesagt, erarbeiten wir verschiedene Methoden zur Komplexitätsreduktion qualitativer Daten und entwickeln diverse Verfahren für die Bearbeitung von Fuzzy-Datensätzen. Basierend auf diesen beiden Themen gliedert sich die Arbeit in zwei Teile. Im ersten Teil liegt der Schwerpunkt auf der Komplexitätsreduktion, während sich der zweite Teil der Verarbeitung von Fuzzy-Daten widmet. Die verschiedenen Kapitel sind dabei durch die beiden Themen verbunden. So werden insbesondere auch Methoden für die Komplexitätsreduktion von Fuzzy-Datensätzen entwickelt.

科學研究中，我們常關注變數間是否存在某種相關，及其相關的程度與方向。但傳統的相關分析方法，並不適用於更能表達真實情況的模糊資料。 在統計學中，討論資料之相關性的統計量有許多，本研究旨在針對討論兩變數間之線性關係的皮爾森相關係數 (Pearson Product-Moment Correlation Coefficient)，以模糊統計方法的角度，提出合理的模糊直線相關係數定義，以協助處理區間模糊資料，瞭解模糊資料間的線性關係。 / In the scientific research, we often pay attention to whether there are some relations between two variables, and the strength and direction of a linear relationship. But the traditional statistics method is not suitable for the fuzzy data. There are a lot of statistics of discussing the relevance between two variables. In this study, a modified method, combining Pearson Product-Moment Correlation Coefficient and fuzzy theory, was applied to deal with the fuzzy data, and find the linear relation among them.

本文主要的目的是藉由The Geometer's Sketchpad (GSP)軟體的設計，幫助我們得到一組連續型模糊樣本。另外對於模糊數的無母數檢定我們提供了一個較為一般的方法，可以針對梯型、三角型，區間型的模糊樣本同時進行處理。 藉由利用GSP. 軟體所設計的模糊問卷，可以較清楚地紀錄受訪者的感覺，此外我們所提供之對於模糊數的無母數檢定方法比其他方法較為有效力。 在未來的研究裡，我們仍有一些問題需要解決，呈述如下：當所施測的樣本數很大時，如何有效率的在網路上紀錄受測者所建構的隸屬度函數？ / The purpose of this paper is to develop a methodology for getting a continuous fuzzy data by using the software The Geometer's Sketchpad (GSP). And we propose a general method for nonparametric tests with fuzzy data that can deal with trapezoid, triangular, and interval-valued data simultaneously. Using the fuzzy questionnaire designed by GSP. can help respondents to record their thoughts more precisely. Additionally our method for nonparametric tests with fuzzy data is more powerful than others. Additional research issues for further investigation are expressed by question such as follows: how to record the membership function on line, especially when the sample size is large?

The popularity of spatial databases increases since the amount of the spatial data that need to be handled has increased by the use of digital maps, images from satellites, video cameras, medical equipment, sensor networks, etc. Spatial data are difficult to examine and extract interesting knowledge / hence, applications that assist decision-making about spatial data like weather forecasting, traffic supervision, mobile communication, etc. have been introduced. In this thesis, more natural and precise knowledge from spatial data is generated by construction of fuzzy spatial data cube and extraction of fuzzy association rules from it in order to improve decision-making about spatial data. This involves an extensive research about spatial knowledge discovery and how fuzzy logic can be used to develop it. It is stated that incorporating fuzzy logic to spatial data cube construction necessitates a new method for aggregation of fuzzy spatial data. We illustrate how this method also enhances the meaning of fuzzy spatial generalization rules and fuzzy association rules with a case-study about weather pattern searching. This study contributes to spatial knowledge discovery by generating more understandable and interesting knowledge from spatial data by extending spatial generalization with fuzzy memberships, extending the spatial aggregation in spatial data cube construction by utilizing weighted measures, and generating fuzzy association rules from the constructed fuzzy spatial data cube.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-06T18:25:30Z No. of bitstreams: 1 DissPAL.pdf: 2245333 bytes, checksum: 24abfad37e7d0675d6cef494f4f41d1e (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-12T14:03:53Z (GMT) No. of bitstreams: 1 DissPAL.pdf: 2245333 bytes, checksum: 24abfad37e7d0675d6cef494f4f41d1e (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-12T14:04:01Z (GMT) No. of bitstreams: 1 DissPAL.pdf: 2245333 bytes, checksum: 24abfad37e7d0675d6cef494f4f41d1e (MD5) / Made available in DSpace on 2016-09-12T14:04:09Z (GMT). No. of bitstreams: 1 DissPAL.pdf: 2245333 bytes, checksum: 24abfad37e7d0675d6cef494f4f41d1e (MD5) Previous issue date: 2010-08-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Inductive learning is, traditionally, categorized as supervised and unsupervised. In supervised learning, the learning method is given a labeled data set (classes of data are known). Those data sets are adequate for problems of classification and regression. In unsupervised learning, unlabeled data are analyzed in order to identify structures embedded in data sets. Typically, clustering methods do not make use of previous knowledge, such as classes labels, to execute their job. The characteristics of recently acquired data sets, great volume and mixed attribute structures, contribute to research on better solutions for machine learning jobs. The proposed research fits into this context. It is about semi-supervised fuzzy clustering applied to the generation of sets of fuzzy rules. Semi-supervised clustering does its job by embodying some previous knowledge about the data set. The clustering results are, then, useful for labeling the remaining unlabeled data in the set. Following that, come to action the supervised learning algorithms aimed at generating fuzzy rules. This document contains theoretic concepts, that will help in understanding the research proposal, and a discussion about the context wherein is the proposal. Some experiments were set up to show that this may be an interesting solution for machine learning jobs that have encountered difficulties due to lack of available information about data. / O aprendizado indutivo é, tradicionalmente, dividido em supervisionado e não supervisionado. No aprendizado supervisionado é fornecido ao método de aprendizado um conjunto de dados rotulados (dados que tem a classe conhecida). Estes dados são adequados para problemas de classificação e regressão. No aprendizado não supervisionado são analisados dados não rotulados, com o objetivo de identificar estruturas embutidas no conjunto. Tipicamente, métodos de agrupamento não se utilizam de conhecimento prévio, como rótulos de classes, para desempenhar sua tarefa. A característica de conjuntos de dados atuais, grande volume e estruturas de atributos mistas, contribui para a busca de melhores soluções para tarefas de aprendizado de máquina. É neste contexto em que se encaixa esta proposta de pesquisa. Trata-se da aplicação de métodos de agrupamento fuzzy semi-supervisionados na geração de bases de regras fuzzy. Os métodos de agrupamento semi-supervisionados realizam sua tarefa incorporando algum conhecimento prévio a respeito do conjunto de dados. O resultado do agrupamento é, então, utilizado para rotulação do restante do conjunto. Em seguida, entram em ação algoritmos de aprendizado supervisionado que tem como objetivo gerar regras fuzzy. Este documento contém conceitos teóricos para compreensão da proposta de trabalho e uma discussão a respeito do contexto onde se encaixa a proposta. Alguns experimentos foram realizados a fim de mostrar que esta pode ser uma solução interessante para tarefas de aprendizado de máquina que encontram dificuldades devido à falta de informação disponível sobre dados.