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

DEA-Based Incentive Regimes in Health-Care Provision

A major challenge to legislators, insurance providers and municipalities will be how to manage the reimbursement of health-care on partially open markets under increasing fiscal pressure and an aging population. Although efficiency theoretically can be obtained by private solutions using fixed-payment schemes, the informational rents and production distortions may limit their implementation. The healthcare agency problem is characterized by (i) a complex multi-input multi-output technology, (ii) information uncertainty and asymmetry, and (iii) fuzzy social preferences. First, the technology, inherently nonlinear and with externalities between factors, yield parametric estimation difficult. However, the flexible production structure in Data Envelopment Analysis (DEA) offers a solution that allows for the gradual and successive refinement of potentially nonconvex technologies. Second, the information structure of healthcare suggests a context of considerable asymmetric information and considerable uncertainty about the underlying technology, but limited uncertainty or noise in the registration of the outcome. Again, we shall argue that the DEA dynamic yardsticks (Bogetoft, 1994, 1997, Agrell and Bogetoft, 2001) are suitable for such contexts. A third important characteristic of the health sector is the somewhat fuzzy social priorities and the numerous potential conflicts between the stakeholders in the health system. Social preferences are likely dynamic and contingent on the disclosed information. Similarly, there are several potential hidden action (moral hazard) and hidden information (adverse selection) conflicts between the different agents in the health system. The flexible and transparent response to preferential ambiguity is one of the strongest justifications for a DEA-approach. DEA yardstick regimes have been successfully implemented in other sectors (electricity distribution) and we present an operalization of the power-parameter p in an pseudo-competitive setting that both limits the informational rents and incites the truthful revelation of information. Recent work (Agrell and Bogetoft, 2002) on strategic implementation of DEA yardsticks is commented in the healthcare context, where social priorities change the tradeoff between the motivation and coordination functions of the yardstick. The paper is closed with policy recommendations and some areas of further work.Data Envelopment Analysis, regulation, health care systems, efficiency, Health Economics and Policy,

A General Framework for Multivariate Analysis with Optimal Scaling: The R Package aspect

In a series of papers De Leeuw developed a general framework for multivariate analysis with optimal scaling. The basic idea of optimal scaling is to transform the observed variables (categories) in terms of quantifications. In the approach presented here the multivariate data are collected into a multivariable. An aspect of a multivariable is a function that is used to measure how well the multivariable satisfies some criterion. Basically we can think of two different families of aspects which unify many well-known multivariate methods: Correlational aspects based on sums of correlations, eigenvalues and determinants which unify multiple regression, path analysis, correspondence analysis, nonlinear PCA, etc. Non-correlational aspects which linearize bivariate regressions and can be used for SEM preprocessing with categorical data. Additionally, other aspects can be established that do not correspond to classical techniques at all. By means of the R package aspect we provide a unified majorization-based implementation of this methodology. Using various data examples we will show the flexibility of this approach and how the optimally scaled results can be represented using graphical tools provided by the package.