Fuzzy Principal Components Analysis

Due to the large number of characteristics, there is a need to extract the most relevant characteristicsfrom the input data, so that the amount of information lost in this way is minimal, and the classification realized with the projected data set is relevant with respect to the original data. In order to achieve this feature extraction, different statistical techniques, as well as the principal components analysis (PCA) may be used. This thesis describes an extension of principal components analysis (PCA) allowing the extraction ofa finite number of relevant features from high-dimensional fuzzy data and noisy data. PCA finds linear combinations of the original measurement variables that describe the significant variation in the data. The comparisonof the two proposed methods was produced by using postoperative patient data. Experiment results demonstrate the ability of using the proposed two methods in complex data. Fuzzy PCA was used in the classificationproblem. The classification was applied by using the similarity classifier algorithm where total similarity measures weights are optimized with differential evolution algorithm. This thesis presents the comparison of the classification results based on the obtained data from the fuzzy PCA.

Investment Determinants and Future Shape of Russia's Power Generation

The reformation of power sector is still in the process of development. The present day situation in Russian electricity power market ischaracterized as transitional period: competitive electricity market is forming, new companies are being created and the power of government regulation is decreasing. The main aim of the reformation is to attract much-needed private investments to the power sector. The electricity consumption increases very rapidly and power sector has to cope with high demand. The goal of this master's thesis is to analyze the nowadays situation in Russian power sector, such as generation structure, condition of electricity networks, electricity price formation for end-users, shape of fuel sector and investments risks and attraction.The final result of this work is creation of scenario of Russian power sector future shape and analysis of the present day situation.

Java Specification Request development on a mobile device

Nykyaikaiset Java-teknologiaa sisältävät matkapuhelimet kehittyvät vauhdikkaasti prosessoritehon, muistin määrän sekä uusien käyttöjärjestelmäversioiden tarjoamien ominaisuuksien myötä. Laitteiden näyttöjen koko tulee pysymään pienenä,mutta silti moninaista multimediasisältöä äänen, videon ja kuvan osilta voidaanhuomattavasti parantaa JSR 234:n eli kehittyneen multimedialaajennuksen avulla.Erityisesti edistyneet ääniominaisuudet ovat tervetullut lisä, sillä viime aikojen kehitys matkapuhelimissa on saanut aikaan niiden muuntumisen myös kannettavaksi musiikkisoittimiksi. Diplomityössä JSR 234 -spesifikaation tietty osa kehitettiin ympäristössä, joka koostui Series 60 -ohjelmistoalustankolmannesta versiosta sekä Symbian OS v9.1 käyttöjärjestelmästä. Tuloksena syntynyt Java-rajapinta tarjoaa sovelluskehittäjille yksinkertaisemman lähestymistavan Symbianin efektirajapintaan piilottaen samalla alla olevan käyttöjärjestelmänmonimutkaisuuden. Toteutuksen täytyy olla läpikotaisin testattu, jotta voidaan varmentua sen noudattavan tarkkaan JSR 234 -spesifikaatiota. Työssä on esitelty useita eri testausmenetelmiä tarkoituksena saavuttaa projektissa paras mahdollinen laatu.

A Topological Approach to Fuzzy Sets

This thesis presents a topological approach to studying fuzzy setsby means of modifier operators. Modifier operators are mathematical models, e.g., for hedges, and we present briefly different approaches to studying modifier operators. We are interested in compositional modifier operators, modifiers for short, and these modifiers depend on binary relations. We show that if a modifier depends on a reflexive and transitive binary relation on U, then there exists a unique topology on U such that this modifier is the closure operator in that topology. Also, if U is finite then there exists a lattice isomorphism between the class of all reflexive and transitive relations and the class of all topologies on U. We define topological similarity relation "≈" between L-fuzzy sets in an universe U, and show that the class LU/ ≈ is isomorphic with the class of all topologies on U, if U is finite and L is suitable. We consider finite bitopological spaces as approximation spaces, and we show that lower and upper approximations can be computed by means of α-level sets also in the case of equivalence relations. This means that approximations in the sense of Rough Set Theory can be computed by means of α-level sets. Finally, we present and application to data analysis: we study an approach to detecting dependencies of attributes in data base-like systems, called information systems.

A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Fuzzy subgroups, algebraic and topological points of view and complex analysis

Fuzzy subsets and fuzzy subgroups are basic concepts in fuzzy mathematics. We shall concentrate on fuzzy subgroups dealing with some of their algebraic, topological and complex analytical properties. Explorations are theoretical belonging to pure mathematics. One of our ideas is to show how widely fuzzy subgroups can be used in mathematics, which brings out the wealth of this concept. In complex analysis we focus on Möbius transformations, combining them with fuzzy subgroups in the algebraic and topological sense. We also survey MV spaces with or without a link to fuzzy subgroups. Spectral space is known in MV algebra. We are interested in its topological properties in MV-semilinear space. Later on, we shall study MV algebras in connection with Riemann surfaces. In fact, the Riemann surface as a concept belongs to complex analysis. On the other hand, Möbius transformations form a part of the theory of Riemann surfaces. In general, this work gives a good understanding how it is possible to fit together different fields of mathematics.

Mechanical closing of the paperboard package and the effect of the shape of paperboard package to its manufacturability

The main objective of this thesis was to design a hinge and a closing mechanism for the plastic rim of a paperboard package. Of the hinge and closing mechanisms the 3D-models were designed using SolidWorks program and the functionality of the mechanisms was tested with rapid prototype models. When a mechanism that worked was found, the manufacturability of the mechanisms was tested in an injection molding machine with changeable inserts. Another objective of this thesis was to test the effect of the shape of paperboard package to its manufacturability. The effect of the packages shape was tested with plastic tools made for three different shaped trays. Suggestions for further research were made according to the results of the thesis.

Conditions for different autonomy regimes in the world : a fuzzy-set application

Avhandlingen behandlar temat territoriell autonomi ur ett globalt perspektiv. Syftet är dels att kartlägga de territoriella autonomierna i världen och dels att visa hur en ny metod som fuzzy-set kan användas inom ämnesområdet jämförande politik. Forskningsproblemet är att försöka finna de bakgrundsfaktorer som förklarar uppkomsten av territoriell autonomi som sådant. Territoriella autonomier ses som särlösningar inom stater. Dessa regioner har erhållit en specialställning i förhållande till andra regioner inom respektive stat och även i förhållande till centralmakten i övrigt. Regionerna kan därför ses som undantag inom det enhetliga federala, regionala eller decentraliserade systemet inom en viss stat ifråga. En kartläggning visar att det finns 65 specialregioner fördelade på 25 stater i världen. De flesta av dessa utgörs av öar. Resultaten visar att det finns två vägar vilka leder till territoriell autonomi i allmänhet. Den ena vägen är en kombination av etnisk särprägel och liten befolkningsmängd, medan den andra vägen utgörs av kombinationen av historiska orsaker och geografiskt avstånd. Båda vägar är lika giltiga och förutsättningen är en demokratisk miljö.

Solving Leontief Input-Output Model with Trapezoidal Fuzzy Numbers and Gauss-Seidel Algorithm

In this work a fuzzy linear system is used to solve Leontief input-output model with fuzzy entries. For solving this model, we assume that the consumption matrix from di erent sectors of the economy and demand are known. These assumptions heavily depend on the information obtained from the industries. Hence uncertainties are involved in this information. The aim of this work is to model these uncertainties and to address them by fuzzy entries such as fuzzy numbers and LR-type fuzzy numbers (triangular and trapezoidal). Fuzzy linear system has been developed using fuzzy data and it is solved using Gauss-Seidel algorithm. Numerical examples show the e ciency of this algorithm. The famous example from Prof. Leontief, where he solved the production levels for U.S. economy in 1958, is also further analyzed.

A quantitative view on fuzzy numbers

Since its introduction, fuzzy set theory has become a useful tool in the mathematical modelling of problems in Operations Research and many other fields. The number of applications is growing continuously. In this thesis we investigate a special type of fuzzy set, namely fuzzy numbers. Fuzzy numbers (which will be considered in the thesis as possibility distributions) have been widely used in quantitative analysis in recent decades. In this work two measures of interactivity are defined for fuzzy numbers, the possibilistic correlation and correlation ratio. We focus on both the theoretical and practical applications of these new indices. The approach is based on the level-sets of the fuzzy numbers and on the concept of the joint distribution of marginal possibility distributions. The measures possess similar properties to the corresponding probabilistic correlation and correlation ratio. The connections to real life decision making problems are emphasized focusing on the financial applications. We extend the definitions of possibilistic mean value, variance, covariance and correlation to quasi fuzzy numbers and prove necessary and sufficient conditions for the finiteness of possibilistic mean value and variance. The connection between the concepts of probabilistic and possibilistic correlation is investigated using an exponential distribution. The use of fuzzy numbers in practical applications is demonstrated by the Fuzzy Pay-Off method. This model for real option valuation is based on findings from earlier real option valuation models. We illustrate the use of number of different types of fuzzy numbers and mean value concepts with the method and provide a real life application.

Bankruptcy prediction using fuzzy methods

In this thesis, a classi cation problem in predicting credit worthiness of a customer is tackled. This is done by proposing a reliable classi cation procedure on a given data set. The aim of this thesis is to design a model that gives the best classi cation accuracy to e ectively predict bankruptcy. FRPCA techniques proposed by Yang and Wang have been preferred since they are tolerant to certain type of noise in the data. These include FRPCA1, FRPCA2 and FRPCA3 from which the best method is chosen. Two di erent approaches are used at the classi cation stage: Similarity classi er and FKNN classi er. Algorithms are tested with Australian credit card screening data set. Results obtained indicate a mean classi cation accuracy of 83.22% using FRPCA1 with similarity classi- er. The FKNN approach yields a mean classi cation accuracy of 85.93% when used with FRPCA2, making it a better method for the suitable choices of the number of nearest neighbors and fuzziness parameters. Details on the calibration of the fuzziness parameter and other parameters associated with the similarity classi er are discussed.

Feature selection using Fuzzy Entropy measures with Yu's Similarity measure

In this study, feature selection in classification based problems is highlighted. The role of feature selection methods is to select important features by discarding redundant and irrelevant features in the data set, we investigated this case by using fuzzy entropy measures. We developed fuzzy entropy based feature selection method using Yu's similarity and test this using similarity classifier. As the similarity classifier we used Yu's similarity, we tested our similarity on the real world data set which is dermatological data set. By performing feature selection based on fuzzy entropy measures before classification on our data set the empirical results were very promising, the highest classification accuracy of 98.83% was achieved when testing our similarity measure to the data set. The achieved results were then compared with some other results previously obtained using different similarity classifiers, the obtained results show better accuracy than the one achieved before. The used methods helped to reduce the dimensionality of the used data set, to speed up the computation time of a learning algorithm and therefore have simplified the classification task