L-Fuzzy Relations in Coq

Heyting categories, a variant of Dedekind categories, and Arrow categories provide a convenient framework for expressing and reasoning about fuzzy relations and programs based on those methods. In this thesis we present an implementation of Heyting and arrow categories suitable for reasoning and program execution using Coq, an interactive theorem prover based on Higher-Order Logic (HOL) with dependent types. This implementation can be used to specify and develop correct software based on L-fuzzy relations such as fuzzy controllers. We give an overview of lattices, L-fuzzy relations, category theory and dependent type theory before describing our implementation. In addition, we provide examples of program executions based on our framework.

An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases

Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation.

L-Fuzzy Structured Query Language

Lattice valued fuzziness is more general than crispness or fuzziness based on the unit interval. In this work, we present a query language for a lattice based fuzzy database. We define a Lattice Fuzzy Structured Query Language (LFSQL) taking its membership values from an arbitrary lattice L. LFSQL can handle, manage and represent crisp values, linear ordered membership degrees and also allows membership degrees from lattices with non-comparable values. This gives richer membership degrees, and hence makes LFSQL more flexible than FSQL or SQL. In order to handle vagueness or imprecise information, every entry into an L-fuzzy database is an L-fuzzy set instead of crisp values. All of this makes LFSQL an ideal query language to handle imprecise data where some factors are non-comparable. After defining the syntax of the language formally, we provide its semantics using L-fuzzy sets and relations. The semantics can be used in future work to investigate concepts such as functional dependencies. Last but not least, we present a parser for LFSQL implemented in Haskell.

Análise dos indicadores biométricos e nutricionais da cultura da alface (Lactuca sativa L.) irrigada com água tratada magneticamente utilizando modelagem fuzzy

Pós-graduação em Agronomia (Irrigação e Drenagem) - FCA

Sviluppo di un approccio congiunto fuzzy-Bayesiano per l'analisi e la modellizzazione degli ecosistemi: applicazione ad ecosistemi marini costieri.

Nell’attuale contesto di aumento degli impatti antropici e di “Global Climate Change” emerge la necessità di comprenderne i possibili effetti di questi sugli ecosistemi inquadrati come fruitori di servizi e funzioni imprescindibili sui quali si basano intere tessiture economiche e sociali. Lo studio previsionale degli ecosistemi si scontra con l’elevata complessità di questi ultimi in luogo di una altrettanto elevata scarsità di osservazioni integrate. L’approccio modellistico appare il più adatto all’analisi delle dinamiche complesse degli ecosistemi ed alla contestualizzazione complessa di risultati sperimentali ed osservazioni empiriche. L’approccio riduzionista-deterministico solitamente utilizzato nell’implementazione di modelli non si è però sin qui dimostrato in grado di raggiungere i livelli di complessità più elevati all’interno della struttura eco sistemica. La componente che meglio descrive la complessità ecosistemica è quella biotica in virtù dell’elevata dipendenza dalle altre componenti e dalle loro interazioni. In questo lavoro di tesi viene proposto un approccio modellistico stocastico basato sull’utilizzo di un compilatore naive Bayes operante in ambiente fuzzy. L’utilizzo congiunto di logica fuzzy e approccio naive Bayes è utile al processa mento del livello di complessità e conseguentemente incertezza insito negli ecosistemi. I modelli generativi ottenuti, chiamati Fuzzy Bayesian Ecological Model(FBEM) appaiono in grado di modellizare gli stati eco sistemici in funzione dell’ elevato numero di interazioni che entrano in gioco nella determinazione degli stati degli ecosistemi. Modelli FBEM sono stati utilizzati per comprendere il rischio ambientale per habitat intertidale di spiagge sabbiose in caso di eventi di flooding costiero previsti nell’arco di tempo 2010-2100. L’applicazione è stata effettuata all’interno del progetto EU “Theseus” per il quale i modelli FBEM sono stati utilizzati anche per una simulazione a lungo termine e per il calcolo dei tipping point specifici dell’habitat secondo eventi di flooding di diversa intensità.

A classification methodology for the risk of weed infestation using fuzzy logic

Despite modern weed control practices, weeds continue to be a threat to agricultural production. Considering the variability of weeds, a classification methodology for the risk of infestation in agricultural zones using fuzzy logic is proposed. The inputs for the classification are attributes extracted from estimated maps for weed seed production and weed coverage using kriging and map analysis and from the percentage of surface infested by grass weeds, in order to account for the presence of weed species with a high rate of development and proliferation. The output for the classification predicts the risk of infestation of regions of the field for the next crop. The risk classification methodology described in this paper integrates analysis techniques which may help to reduce costs and improve weed control practices. Results for the risk classification of the infestation in a maize crop field are presented. To illustrate the effectiveness of the proposed system, the risk of infestation over the entire field is checked against the yield loss map estimated by kriging and also with the average yield loss estimated from a hyperbolic model.

An integrated recycling approach for GFRP pultrusion wastes: recycling and reuse assessment into new composite materials using Fuzzy Boolean Nets

In this study, efforts were made in order to put forward an integrated recycling approach for the thermoset based glass fibre reinforced polymer (GPRP) rejects derived from the pultrusion manufacturing industry. Both the recycling process and the development of a new cost-effective end-use application for the recyclates were considered. For this purpose, i) among the several available recycling techniques for thermoset based composite materials, the most suitable one for the envisaged application was selected (mechanical recycling); and ii) an experimental work was carried out in order to assess the added-value of the obtained recyclates as aggregates and reinforcement replacements into concrete-polymer composite materials. Potential recycling solution was assessed by mechanical behaviour of resultant GFRP waste modified concrete-polymer composites with regard to unmodified materials. In the mix design process of the new GFRP waste based composite material, the recyclate content and size grade, and the effect of the incorporation of an adhesion promoter were considered as material factors and systematically tested between reasonable ranges. The optimization process of the modified formulations was supported by the Fuzzy Boolean Nets methodology, which allowed finding the best balance between material parameters that maximizes both flexural and compressive strengths of final composite. Comparing to related end-use applications of GFRP wastes in cementitious based concrete materials, the proposed solution overcome some of the problems found, namely the possible incompatibilities arisen from alkalis-silica reaction and the decrease in the mechanical properties due to high water-cement ratio required to achieve the desirable workability. Obtained results were very promising towards a global cost-effective waste management solution for GFRP industrial wastes and end-of-life products that will lead to a more sustainable composite materials industry.

Regulación del proceso de secado de embutidos crudos-curados mediante la aplicación de lógica borrosa (fuzzy logic)

El presente proyecto tenía como objetivo final el desarrollo de un sistema de control basado en Lógica Fuzzy que permita que el proceso de secado tenga una regulación continua y con una menor dependencia de la experiencia del personal experto, evitando además la formación de encostrado. Asimismo, se plantearon una serie de objetivos parciales, cuya consecución permitiría, además de alcanzar el objetivo final descrito, obtener un conocimiento científico adicional. Por ello, a continuación se resumen los resultados en relación con los objetivos parciales propuestos. Como paso previo, antes de abordar los objetivos planteados se diseñó y construyó un equipo experimental de secado, donde se controló de forma precisa la temperatura, la humedad relativa y la velocidad del aire.

Fuzzy set Theory and Quantum Chemistry

Es discuteixen breument algunes consideracions sobre l'aplicació de la Teoria delsConjunts difusos a la Química quàntica. Es demostra aqui que molts conceptes químics associats a la teoria són adequats per ésser connectats amb l'estructura dels Conjunts difusos. També s'explica com algunes descripcions teoriques dels observables quàntics espotencien tractant-les amb les eines associades als esmentats Conjunts difusos. La funciódensitat es pren com a exemple de l'ús de distribucions de possibilitat al mateix temps queles distribucions de probabilitat quàntiques

Measures of fit in multiple correspondence analysis of crisp and fuzzy coded data

When continuous data are coded to categorical variables, two types of coding are possible: crisp coding in the form of indicator, or dummy, variables with values either 0 or 1; or fuzzy coding where each observation is transformed to a set of "degrees of membership" between 0 and 1, using co-called membership functions. It is well known that the correspondence analysis of crisp coded data, namely multiple correspondence analysis, yields principal inertias (eigenvalues) that considerably underestimate the quality of the solution in a low-dimensional space. Since the crisp data only code the categories to which each individual case belongs, an alternative measure of fit is simply to count how well these categories are predicted by the solution. Another approach is to consider multiple correspondence analysis equivalently as the analysis of the Burt matrix (i.e., the matrix of all two-way cross-tabulations of the categorical variables), and then perform a joint correspondence analysis to fit just the off-diagonal tables of the Burt matrix - the measure of fit is then computed as the quality of explaining these tables only. The correspondence analysis of fuzzy coded data, called "fuzzy multiple correspondence analysis", suffers from the same problem, albeit attenuated. Again, one can count how many correct predictions are made of the categories which have highest degree of membership. But here one can also defuzzify the results of the analysis to obtain estimated values of the original data, and then calculate a measure of fit in the familiar percentage form, thanks to the resultant orthogonal decomposition of variance. Furthermore, if one thinks of fuzzy multiple correspondence analysis as explaining the two-way associations between variables, a fuzzy Burt matrix can be computed and the same strategy as in the crisp case can be applied to analyse the off-diagonal part of this matrix. In this paper these alternative measures of fit are defined and applied to a data set of continuous meteorological variables, which are coded crisply and fuzzily into three categories. Measuring the fit is further discussed when the data set consists of a mixture of discrete and continuous variables.

Fuzzy coding in constrained ordinations

Canonical correspondence analysis and redundancy analysis are two methods of constrained ordination regularly used in the analysis of ecological data when several response variables (for example, species abundances) are related linearly to several explanatory variables (for example, environmental variables, spatial positions of samples). In this report I demonstrate the advantages of the fuzzy coding of explanatory variables: first, nonlinear relationships can be diagnosed; second, more variance in the responses can be explained; and third, in the presence of categorical explanatory variables (for example, years, regions) the interpretation of the resulting triplot ordination is unified because all explanatory variables are measured at a categorical level.

Biplots of fuzzy coded data

A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well-known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure of fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data.

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.

Change points detection in crime-related time series: an on-line fuzzy approach based on a shape space representation

The extension of traditional data mining methods to time series has been effectively applied to a wide range of domains such as finance, econometrics, biology, security, and medicine. Many existing mining methods deal with the task of change points detection, but very few provide a flexible approach. Querying specific change points with linguistic variables is particularly useful in crime analysis, where intuitive, understandable, and appropriate detection of changes can significantly improve the allocation of resources for timely and concise operations. In this paper, we propose an on-line method for detecting and querying change points in crime-related time series with the use of a meaningful representation and a fuzzy inference system. Change points detection is based on a shape space representation, and linguistic terms describing geometric properties of the change points are used to express queries, offering the advantage of intuitiveness and flexibility. An empirical evaluation is first conducted on a crime data set to confirm the validity of the proposed method and then on a financial data set to test its general applicability. A comparison to a similar change-point detection algorithm and a sensitivity analysis are also conducted. Results show that the method is able to accurately detect change points at very low computational costs. More broadly, the detection of specific change points within time series of virtually any domain is made more intuitive and more understandable, even for experts not related to data mining.

Estimativa das condições de conforto térmico para avicultura de postura usando a teoria dos conjuntos Fuzzy

Neste trabalho, foi utilizada uma ferramenta matemática promissora na análise de sistemas e/ou processos, particularmente na área de produção animal. Essa ferramenta é a desenvolvida segundo a abordagem da teoria dos Conjuntos Fuzzy e, neste caso específico, permitiu a análise da composição das variáveis climáticas independentes, como temperatura de bulbo seco e umidade relativa do ar, que influenciam na variável dependente denominada conforto térmico das aves. Foi realizada a construção de regras baseadas na intuição humana, segundo o conhecimento de especialistas da área, a partir do que é possível simular cenários distintos para o suporte à decisão de construção de galpões para abrigo a animais. Neste trabalho, foi estimado o conforto térmico para alojamento de aves poedeiras em produção. Os resultados foram analisados, usando-se o ambiente de computação científica MATLAB 6.5, o que pode ser realizado iterativamente a cada cenário gerado. Com base nos resultados obtidos, pode-se analisar as condições de conforto para distintas composições das variáveis de entrada.