Characterizations of bivariate conic, extreme value, and Archimax copulas

Based on a general construction method by means of bivariate ultramodular copulas we construct, for particular settings, special bivariate conic, extreme value, and Archimax copulas. We also show that the sets of copulas obtained in this way are dense in the sets of all conic, extreme value, and Archimax copulas, respectively

Development of Machine Learning Techniques for Diabetic Retinopathy Risk Estimation

La retinopatia diabètica (DR) és una malaltia crònica. És una de les principals complicacions de diabetis i una causa essencial de pèrdua de visió entre les persones que pateixen diabetis. Els pacients diabètics han de ser analitzats periòdicament per tal de detectar signes de desenvolupament de la retinopatia en una fase inicial. El cribratge precoç i freqüent disminueix el risc de pèrdua de visió i minimitza la càrrega als centres assistencials. El nombre dels pacients diabètics està en augment i creixements ràpids, de manera que el fa difícil que consumeix recursos per realitzar un cribatge anual a tots ells. L’objectiu principal d’aquest doctorat. la tesi consisteix en construir un sistema de suport de decisions clíniques (CDSS) basat en dades de registre de salut electrònic (EHR). S'utilitzarà aquest CDSS per estimar el risc de desenvolupar RD. En aquesta tesi doctoral s'estudien mètodes d'aprenentatge automàtic per constuir un CDSS basat en regles lingüístiques difuses. El coneixement expressat en aquest tipus de regles facilita que el metge sàpiga quines combindacions de les condicions són les poden provocar el risc de desenvolupar RD. En aquest treball, proposo un mètode per reduir la incertesa en la classificació dels pacients que utilitzen arbres de decisió difusos (FDT). A continuació es combinen diferents arbres, usant la tècnica de Fuzzy Random Forest per millorar la qualitat de la predicció. A continuació es proposen diverses tècniques d'agregació que millorin la fusió dels resultats que ens dóna cadascun dels arbres FDT. Per millorar la decisió final dels nostres models, proposo tres mesures difuses que s'utilitzen amb integrals de Choquet i Sugeno. La definició d’aquestes mesures difuses es basa en els valors de confiança de les regles. En particular, una d'elles és una mesura difusa que es troba en la qual l'estructura jeràrquica de la FDT és explotada per trobar els valors de la mesura difusa. El resultat final de la recerca feta ha donat lloc a un programari que es pot instal·lar en centres d’assistència primària i hospitals, i pot ser usat pels metges de capçalera per fer l'avaluació preventiva i el cribatge de la Retinopatia Diabètica.La retinopatía diabética (RD) es una enfermedad crónica. Es una de las principales complicaciones de diabetes y una causa esencial de pérdida de visión entre las personas que padecen diabetes. Los pacientes diabéticos deben ser examinados periódicamente para detectar signos de diabetes. desarrollo de retinopatía en una etapa temprana. La detección temprana y frecuente disminuye el riesgo de pérdida de visión y minimiza la carga en los centros de salud. El número de pacientes diabéticos es enorme y está aumentando rápidamente, lo que lo hace difícil y Consume recursos para realizar una evaluación anual para todos ellos. El objetivo principal de esta tesis es construir un sistema de apoyo a la decisión clínica (CDSS) basado en datos de registros de salud electrónicos (EHR). Este CDSS será utilizado para estimar el riesgo de desarrollar RD. En este tesis doctoral se estudian métodos de aprendizaje automático para construir un CDSS basado en reglas lingüísticas difusas. El conocimiento expresado en este tipo de reglas facilita que el médico pueda saber que combinaciones de las condiciones son las que pueden provocar el riesgo de desarrollar RD. En este trabajo propongo un método para reducir la incertidumbre en la clasificación de los pacientes que usan árboles de decisión difusos (FDT). A continuación se combinan diferentes árboles usando la técnica de Fuzzy Random Forest para mejorar la calidad de la predicción. Se proponen también varias políticas para fusionar los resultados de que nos da cada uno de los árboles (FDT). Para mejorar la decisión final propongo tres medidas difusas que se usan con las integrales Choquet y Sugeno. La definición de estas medidas difusas se basa en los valores de confianza de las reglas. En particular, uno de ellos es una medida difusa descomponible en la que se usa la estructura jerárquica del FDT para encontrar los valores de la medida difusa. Como resultado final de la investigación se ha construido un software que puede instalarse en centros de atención médica y hospitales, i que puede ser usado por los médicos de cabecera para hacer la evaluación preventiva y el cribado de la Retinopatía Diabética.Diabetic retinopathy (DR) is a chronic illness. It is one of the main complications of diabetes, and an essential cause of vision loss among people suffering from diabetes. Diabetic patients must be periodically screened in order to detect signs of diabetic retinopathy development in an early stage. Early and frequent screening decreases the risk of vision loss and minimizes the load on the health care centres. The number of the diabetic patients is huge and rapidly increasing so that makes it hard and resource-consuming to perform a yearly screening to all of them. The main goal of this Ph.D. thesis is to build a clinical decision support system (CDSS) based on electronic health record (EHR) data. This CDSS will be utilised to estimate the risk of developing RD. In this Ph.D. thesis, I focus on developing novel interpretable machine learning systems. Fuzzy based systems with linguistic terms are going to be proposed. The output of such systems makes the physician know what combinations of the features that can cause the risk of developing DR. In this work, I propose a method to reduce the uncertainty in classifying diabetic patients using fuzzy decision trees. A Fuzzy Random forest (FRF) approach is proposed as well to estimate the risk for developing DR. Several policies are going to be proposed to merge the classification results achieved by different Fuzzy Decision Trees (FDT) models to improve the quality of the final decision of our models, I propose three fuzzy measures that are used with Choquet and Sugeno integrals. The definition of these fuzzy measures is based on the confidence values of the rules. In particular, one of them is a decomposable fuzzy measure in which the hierarchical structure of the FDT is exploited to find the values of the fuzzy measure. Out of this Ph.D. work, we have built a CDSS software that may be installed in the health care centres and hospitals in order to evaluate and detect Diabetic Retinopathy at early stages

Fuzzy measures and integrals in MCDA

This chapter aims at a unified presentation of various methods of MCDA based onfuzzy measures (capacity) and fuzzy integrals, essentially the Choquet andSugeno integral. A first section sets the position of the problem ofmulticriteria decision making, and describes the various possible scales ofmeasurement (difference, ratio, and ordinal). Then a whole section is devotedto each case in detail: after introducing necessary concepts, the methodologyis described, and the problem of the practical identification of fuzzy measuresis given. The important concept of interaction between criteria, central inthis chapter, is explained in details. It is shown how it leads to k-additivefuzzy measures. The case of bipolar scales leads to thegeneral model based on bi-capacities, encompassing usual models based oncapacities. A general definition of interaction for bipolar scales isintroduced. The case of ordinal scales leads to the use of Sugeno integral, andits symmetrized version when one considers symmetric ordinal scales. Apractical methodology for the identification of fuzzy measures in this contextis given. Lastly, we give a short description of some practical applications.Choquet integral; fuzzy measure; interaction; bi-capacities

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

A Deep Study of Fuzzy Implications

This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents $n$ times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation $I(x,y)=I(x,I(x,y))$, for all $x$, $y\in[0,1]$. In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication $I$ to define a fuzzy $I$-adjunction in $\mathcal{F}(\mathbb{R}^{n})$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction $\mathcal{C}$ on the unit interval and a fuzzy erosion which is defined from a fuzzy implication $I^{'}$ to form a fuzzy $I$-adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction $\mathcal{C}$ on the unit interval and the implication $I$ or the implication $I^{'}$ play important roles in such conditions

MODELFY: A Model-driven Solution for Decision Making based on Fuzzy Information

There exist areas, such as the disease prevention or inclement weather protocols, in which the analysis of the information based on strict protocols require a high level of rigor and security. In this situation, it would be desirable to apply formal methodologies that provide these features. In this scope, recently, it has been proposed a formalism, fuzzy automaton, that captures two relevant aspects for fuzzy information analysis: imprecision and uncertainty. However, the models should be designed by domain experts, who have the required knowledge for the design of the processes, but do not have the necessary technical knowledge. To address this limitation, this paper proposes MODELFY, a novel model-driven solution for designing a decision-making process based on fuzzy automata that allows users to abstract from technical complexities. With this goal in mind, we have developed a framework for fuzzy automaton model design based on a Domain- Specific Modeling Language (DSML) and a graphical editor. To improve the interoperability and functionality of this framework, it also includes a model-to-text transformation that translates the models designed by using the graphical editor into a format that can be used by a tool for data analysis. The practical value of this proposal is also evaluated through a non-trivial medical protocol for detecting potential heart problems. The results confirm that MODELFY is useful for defining such a protocol in a user-friendly and rigorous manner, bringing fuzzy automata closer to domain expert