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.

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.

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.

Fuzzy real option analysis in patent related decision making and patent valuation

The shift towards a knowledge-based economy has inevitably prompted the evolution of patent exploitation. Nowadays, patent is more than just a prevention tool for a company to block its competitors from developing rival technologies, but lies at the very heart of its strategy for value creation and is therefore strategically exploited for economic pro t and competitive advantage. Along with the evolution of patent exploitation, the demand for reliable and systematic patent valuation has also reached an unprecedented level. However, most of the quantitative approaches in use to assess patent could arguably fall into four categories and they are based solely on the conventional discounted cash flow analysis, whose usability and reliability in the context of patent valuation are greatly limited by five practical issues: the market illiquidity, the poor data availability, discriminatory cash-flow estimations, and its incapability to account for changing risk and managerial flexibility. This dissertation attempts to overcome these impeding barriers by rationalizing the use of two techniques, namely fuzzy set theory (aiming at the first three issues) and real option analysis (aiming at the last two). It commences with an investigation into the nature of the uncertainties inherent in patent cash flow estimation and claims that two levels of uncertainties must be properly accounted for. Further investigation reveals that both levels of uncertainties fall under the categorization of subjective uncertainty, which differs from objective uncertainty originating from inherent randomness in that uncertainties labelled as subjective are highly related to the behavioural aspects of decision making and are usually witnessed whenever human judgement, evaluation or reasoning is crucial to the system under consideration and there exists a lack of complete knowledge on its variables. Having clarified their nature, the application of fuzzy set theory in modelling patent-related uncertain quantities is effortlessly justified. The application of real option analysis to patent valuation is prompted by the fact that both patent application process and the subsequent patent exploitation (or commercialization) are subject to a wide range of decisions at multiple successive stages. In other words, both patent applicants and patentees are faced with a large variety of courses of action as to how their patent applications and granted patents can be managed. Since they have the right to run their projects actively, this flexibility has value and thus must be properly accounted for. Accordingly, an explicit identification of the types of managerial flexibility inherent in patent-related decision making problems and in patent valuation, and a discussion on how they could be interpreted in terms of real options are provided in this dissertation. Additionally, the use of the proposed techniques in practical applications is demonstrated by three fuzzy real option analysis based models. In particular, the pay-of method and the extended fuzzy Black-Scholes model are employed to investigate the profitability of a patent application project for a new process for the preparation of a gypsum-fibre composite and to justify the subsequent patent commercialization decision, respectively; a fuzzy binomial model is designed to reveal the economic potential of a patent licensing opportunity.

Linguistic models for decision support

Linguistic modelling is a rather new branch of mathematics that is still undergoing rapid development. It is closely related to fuzzy set theory and fuzzy logic, but knowledge and experience from other fields of mathematics, as well as other fields of science including linguistics and behavioral sciences, is also necessary to build appropriate mathematical models. This topic has received considerable attention as it provides tools for mathematical representation of the most common means of human communication - natural language. Adding a natural language level to mathematical models can provide an interface between the mathematical representation of the modelled system and the user of the model - one that is sufficiently easy to use and understand, but yet conveys all the information necessary to avoid misinterpretations. It is, however, not a trivial task and the link between the linguistic and computational level of such models has to be established and maintained properly during the whole modelling process. In this thesis, we focus on the relationship between the linguistic and the mathematical level of decision support models. We discuss several important issues concerning the mathematical representation of meaning of linguistic expressions, their transformation into the language of mathematics and the retranslation of mathematical outputs back into natural language. In the first part of the thesis, our view of the linguistic modelling for decision support is presented and the main guidelines for building linguistic models for real-life decision support that are the basis of our modeling methodology are outlined. From the theoretical point of view, the issues of representation of meaning of linguistic terms, computations with these representations and the retranslation process back into the linguistic level (linguistic approximation) are studied in this part of the thesis. We focus on the reasonability of operations with the meanings of linguistic terms, the correspondence of the linguistic and mathematical level of the models and on proper presentation of appropriate outputs. We also discuss several issues concerning the ethical aspects of decision support - particularly the loss of meaning due to the transformation of mathematical outputs into natural language and the issue or responsibility for the final decisions. In the second part several case studies of real-life problems are presented. These provide background and necessary context and motivation for the mathematical results and models presented in this part. A linguistic decision support model for disaster management is presented here – formulated as a fuzzy linear programming problem and a heuristic solution to it is proposed. Uncertainty of outputs, expert knowledge concerning disaster response practice and the necessity of obtaining outputs that are easy to interpret (and available in very short time) are reflected in the design of the model. Saaty’s analytic hierarchy process (AHP) is considered in two case studies - first in the context of the evaluation of works of art, where a weak consistency condition is introduced and an adaptation of AHP for large matrices of preference intensities is presented. The second AHP case-study deals with the fuzzified version of AHP and its use for evaluation purposes – particularly the integration of peer-review into the evaluation of R&D outputs is considered. In the context of HR management, we present a fuzzy rule based evaluation model (academic faculty evaluation is considered) constructed to provide outputs that do not require linguistic approximation and are easily transformed into graphical information. This is achieved by designing a specific form of fuzzy inference. Finally the last case study is from the area of humanities - psychological diagnostics is considered and a linguistic fuzzy model for the interpretation of outputs of multidimensional questionnaires is suggested. The issue of the quality of data in mathematical classification models is also studied here. A modification of the receiver operating characteristics (ROC) method is presented to reflect variable quality of data instances in the validation set during classifier performance assessment. Twelve publications on which the author participated are appended as a third part of this thesis. These summarize the mathematical results and provide a closer insight into the issues of the practicalapplications that are considered in the second part of the thesis.

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.

Ilmastonmuutosteknologiat maailman patenttijärjestelmissä: Case autoteollisuus

Tässä työssä tarkastellaan kahden kansainvälisen patenttiluokitusjärjestelmän vihreiden teknologioiden luokitusjärjestelmiä autoteollisuudessa. Työn tarkoitus on tutkia, kuinka paljon vihreän teknologian patenttianalyysin tulokset eroavat toisistaan, jos tutkimuksissa käytetään eri patenttien luokitusjärjestelmiä. Vanhempi järjestelmä, International Patent Classification, on asemansa vakiinnuttanut kansainvälinen patenttienluokitusjärjestelmä. Vasta viime vuosina käyttöön otettu Cooperative Patent Classification on Euroopan ja Yhdysvaltojen patenttijärjestöjen kehittämä patenttien luokitusjärjestelmä. Tutkimusmenetelmissä hyödynnetään patenttianalyysia ja joukko-oppia. Tutkimuksessa vihreiden teknologioiden luokittelumenetelmien vertailukohteille saatiin määrällisesti samankaltaiset tulokset, mutta niiden sisältämät patentit eivät olleet pääsäännöllisesti samoja. Työssä tarkastellaan myös Toyotan, Daimlerin ja Fordin vihreiden autoteknologiapolkujen kehitystä. Varsinkin Toyota ja Daimler tulevat yhä enemmän panostamaan sähkö- ja hybridiautoihin verrattuna polttomoottoriautoihin.

Digital collaborations and entrepreneurship – the role of shareconomy and crowdsourcing in the era of smart city

The thesis begins with the classical cooperation and transfers it to the digital world. This work gives a detailed overview of the young fields of research smart city, shareconomy and crowdsourcing and links these fields with entrepreneurship. The core research aim is the finding of connections between the research fields smart city, shareconomy and crowdsourcing and entrepreneurial activities and the specific fields of application, success factors and conditions for entrepreneurs. The thesis consists of seven peer-reviewed publications. Based on primary and secondary data, the existence of entrepreneurial opportunities in the fields of smart city, shareconomy and crowdsourcing could be confirmed. The first part (publications 1-3) of the thesis are literature reviews to secure the fundamental base for further research. This part consists of newly created definitions and an extreme sharpening of the research fields for the near future. In the second part of the thesis (publications 4-7), empirical field work (in-depth interviews with entrepreneurs) and quantitative analyses (fuzzy set/qualitative comparative analysis and binary logistic regression analysis) contribute to the field of research with additional new insights. Summarizing, the insights are multi-layered: theoretical (e.g. new definitions, sharpening of the research field), methodical (e.g. first time application of the fuzzy set/qualitative comparative analysis in the field of crowdfunding) and qualitative (first time application of in-depth interviews with entrepreneurs in the fields of smart city and shareconomy). The global research question could be answered: the link between entrepreneurship and smart city, shareconomy and crowdfunding could be confirmed, concrete fields of application could be identified and further developments could be touched upon. This work strongly contributes to the young fields of research through much-needed basic work, new qualitative approaches, innovative methods and new insights and offers opportunities for discussion, criticism and support for further research.

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.

Muon detector link system test set-up

CERNin tutkimuskeskuksen rakenteilla olevan hadronikiihdyttimen eräs tarkoitus on todistaa Higgsin bosonin olemassaolo. Higgsin bosonin löytyminen yhtenäistäisi nykyisen hiukkasfysiikan teorian ja antaisi selityksen sille kuinka hiukkaset saavat massansa. Kiihdyttimen CMS koeasema on tarkoitettu erityisesti myonien ilmaisuun. Tämä työ liittyy CMS koeaseman RPC-ilmaisintyypin linkkijärjestelmään, jonka tarkoituksena on käsitellä ilmaisimelta tulevia myonien aiheuttamia signaaleja ja lähettää tiedot tärkeäksi katsotuista törmäystapahtumista tallennettavaksi analysointia varten. Työssä on toteutettu linkkijärjestelmän ohjaus- ja linkkikorteille testiympäristö, jolla voidaan todeta järjestelmän eri osien keskinäinen yhteensopivuus ja toimivuus. Työn alkuosassa esitellään ilmaisimen linkkijärjestelmän eri osat ja niiden merkitykset. Työn loppuosassa käydään läpi eri testimenetelmiä ja analysoidaan niiden antamia tuloksia.

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.

Hybrid LC Filter for Power Electronic Drives: Theory and Implementation

Power electronic converter drives use, for the sake of high efficiency, pulse-width modulation that results in sequences of high-voltage high-frequency steep-edged pulses. Such a signal contains a set of high harmonics not required for control purposes. Harmonics cause reflections in the cable between the motor and the inverter leading to faster winding insulation ageing. Bearing failures and problems with electromagnetic compatibility may also result. Electrical du/dt filters provide an effective solution to problems caused by pulse-width modulation, thereby increasing the performance and service life of the electrical machines. It is shown that RLC filters effectively decrease the reflection phenomena in the cable. Improved (simple, but effective) solutions are found for both differential- and common-mode signals; these solutions use a galvanic connection between the RLC filter star point and the converter DC link. Foil chokes and film capacitors are among the most widely used components in high-power applications. In actual applications they can be placed in different parts of the cabinet. This fact complicates the arrangement of the cabinet and decreases the reliability of the system. In addition, the inductances of connection wires may prevent filtration at high frequencies. This thesis introduces a new hybrid LC filter that uses a natural capacitance between the turns of the foil choke based on integration of an auxiliary layer into it. The main idea of the hybrid LC filter results from the fact that both the foil choke and the film capacitors have the same roll structure. Moreover, the capacitance between the turns (“intra capacitance”) of the foil inductors is the reason for the deterioration of their properties at high frequencies. It is shown that the proposed filter has a natural cancellation of the intra capacitance. A hybrid LC filter may contain two or more foil layers isolated from each other and coiled on a core. The core material can be iron or even air as in the filter considered in this work. One of the foils, called the main foil, can be placed between the inverter and the motor cable. Other ones, called auxiliary foils, may be connected in star to create differential-mode noise paths, and then coupled to the DC link midpoint to guarantee a traveling path, especially for the common-mode currents. This way, there is a remarkable capacitance between the main foil and the auxiliary foil. Investigations showed that such a system can be described by a simple equivalent LC filter in a wide range of frequencies. Because of its simple hybrid construction, the proposed LC filter can be a cost-effective and competitive solution for modern power drives. In the thesis, the application field of the proposed filter is considered and determined. The basics of hybrid LC filter design are developed further. High-frequency behaviour of the proposed filter is analysed by simulations. Finally, the thesis presents experimental data proving that the hybrid LC filter can be used for du/dt of PWM pulses and reduction of common-mode currents.

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