A characterization theorem for t-representable n-dimensional triangular norms

n-dimensional fuzzy sets are an extension of fuzzy sets that includes interval-valued fuzzy sets and interval-valued Atanassov intuitionistic fuzzy sets. The membership values of n-dimensional fuzzy sets are n-tuples of real numbers in the unit interval [0,1], called n-dimensional intervals, ordered in increasing order. The main idea in n-dimensional fuzzy sets is to consider several uncertainty levels in the memberships degrees. Triangular norms have played an important role in fuzzy sets theory, in the narrow as in the broad sense. So it is reasonable to extend this fundamental notion for n-dimensional intervals. In interval-valued fuzzy theory, interval-valued t-norms are related with t-norms via the notion of t-representability. A characterization of t-representable interval-valued t-norms is given in term of inclusion monotonicity. In this paper we generalize the notion of t-representability for n-dimensional t-norms and provide a characterization theorem for that class of n-dimensional t-norms. © 2011 Springer-Verlag Berlin Heidelberg.

On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts

Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous extensions of the weighted arithmetic mean and ordered weighted averaging operator also have the absorbent element 〈1,0〉, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the operations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit analogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.

Defining Bonferroni means over lattices

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers' preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.

A penalty-based aggregation operator for non-convex intervals

In the case of real-valued inputs, averaging aggregation functions have been studied extensively with results arising in fields including probability and statistics, fuzzy decision-making, and various sciences. Although much of the behavior of aggregation functions when combining standard fuzzy membership values is well established, extensions to interval-valued fuzzy sets, hesitant fuzzy sets, and other new domains pose a number of difficulties. The aggregation of non-convex or discontinuous intervals is usually approached in line with the extension principle, i.e. by aggregating all real-valued input vectors lying within the interval boundaries and taking the union as the final output. Although this is consistent with the aggregation of convex interval inputs, in the non-convex case such operators are not idempotent and may result in outputs which do not faithfully summarize or represent the set of inputs. After giving an overview of the treatment of non-convex intervals and their associated interpretations, we propose a novel extension of the arithmetic mean based on penalty functions that provides a representative output and satisfies idempotency.

Averaging aggregation functions for preferences expressed as Pythagorean membership grades and fuzzy orthopairs

Rather than denoting fuzzy membership with a single value, orthopairs such as Atanassov's intuitionistic membership and non-membership pairs allow the incorporation of uncertainty, as well as positive and negative aspects when providing evaluations in fuzzy decision making problems. Such representations, along with interval-valued fuzzy values and the recently introduced Pythagorean membership grades, present particular challenges when it comes to defining orders and constructing aggregation functions that behave consistently when summarizing evaluations over multiple criteria or experts. In this paper we consider the aggregation of pairwise preferences denoted by membership and non-membership pairs. We look at how mappings from the space of Atanassov orthopairs to more general classes of fuzzy orthopairs can be used to help define averaging aggregation functions in these new settings. In particular, we focus on how the notion of 'averaging' should be treated in the case of Yager's Pythagorean membership grades and how to ensure that such functions produce outputs consistent with the case of ordinary fuzzy membership degrees.

Incerteza intervalar em otimização e controle

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Inteligencia artificial para la predicción y control del acabado superficial en procesos de fresado a alta velocidad

El objetivo principal de esta tesis doctoral es profundizar en el análisis y diseño de un sistema inteligente para la predicción y control del acabado superficial en un proceso de fresado a alta velocidad, basado fundamentalmente en clasificadores Bayesianos, con el prop´osito de desarrollar una metodolog´ıa que facilite el diseño de este tipo de sistemas. El sistema, cuyo propósito es posibilitar la predicción y control de la rugosidad superficial, se compone de un modelo aprendido a partir de datos experimentales con redes Bayesianas, que ayudar´a a comprender los procesos dinámicos involucrados en el mecanizado y las interacciones entre las variables relevantes. Dado que las redes neuronales artificiales son modelos ampliamente utilizados en procesos de corte de materiales, también se incluye un modelo para fresado usándolas, donde se introdujo la geometría y la dureza del material como variables novedosas hasta ahora no estudiadas en este contexto. Por lo tanto, una importante contribución en esta tesis son estos dos modelos para la predicción de la rugosidad superficial, que se comparan con respecto a diferentes aspectos: la influencia de las nuevas variables, los indicadores de evaluación del desempeño, interpretabilidad. Uno de los principales problemas en la modelización con clasificadores Bayesianos es la comprensión de las enormes tablas de probabilidad a posteriori producidas. Introducimos un m´etodo de explicación que genera un conjunto de reglas obtenidas de árboles de decisión. Estos árboles son inducidos a partir de un conjunto de datos simulados generados de las probabilidades a posteriori de la variable clase, calculadas con la red Bayesiana aprendida a partir de un conjunto de datos de entrenamiento. Por último, contribuimos en el campo multiobjetivo en el caso de que algunos de los objetivos no se puedan cuantificar en números reales, sino como funciones en intervalo de valores. Esto ocurre a menudo en aplicaciones de aprendizaje automático, especialmente las basadas en clasificación supervisada. En concreto, se extienden las ideas de dominancia y frontera de Pareto a esta situación. Su aplicación a los estudios de predicción de la rugosidad superficial en el caso de maximizar al mismo tiempo la sensibilidad y la especificidad del clasificador inducido de la red Bayesiana, y no solo maximizar la tasa de clasificación correcta. Los intervalos de estos dos objetivos provienen de un m´etodo de estimación honesta de ambos objetivos, como e.g. validación cruzada en k rodajas o bootstrap.---ABSTRACT---The main objective of this PhD Thesis is to go more deeply into the analysis and design of an intelligent system for surface roughness prediction and control in the end-milling machining process, based fundamentally on Bayesian network classifiers, with the aim of developing a methodology that makes easier the design of this type of systems. The system, whose purpose is to make possible the surface roughness prediction and control, consists of a model learnt from experimental data with the aid of Bayesian networks, that will help to understand the dynamic processes involved in the machining and the interactions among the relevant variables. Since artificial neural networks are models widely used in material cutting proceses, we include also an end-milling model using them, where the geometry and hardness of the piecework are introduced as novel variables not studied so far within this context. Thus, an important contribution in this thesis is these two models for surface roughness prediction, that are then compared with respecto to different aspects: influence of the new variables, performance evaluation metrics, interpretability. One of the main problems with Bayesian classifier-based modelling is the understanding of the enormous posterior probabilitiy tables produced. We introduce an explanation method that generates a set of rules obtained from decision trees. Such trees are induced from a simulated data set generated from the posterior probabilities of the class variable, calculated with the Bayesian network learned from a training data set. Finally, we contribute in the multi-objective field in the case that some of the objectives cannot be quantified as real numbers but as interval-valued functions. This often occurs in machine learning applications, especially those based on supervised classification. Specifically, the dominance and Pareto front ideas are extended to this setting. Its application to the surface roughness prediction studies the case of maximizing simultaneously the sensitivity and specificity of the induced Bayesian network classifier, rather than only maximizing the correct classification rate. Intervals in these two objectives come from a honest estimation method of both objectives, like e.g. k-fold cross-validation or bootstrap.

Clustering web documents using hierarchical representation with multi-granularity

Web document cluster analysis plays an important role in information retrieval by organizing large amounts of documents into a small number of meaningful clusters. Traditional web document clustering is based on the Vector Space Model (VSM), which takes into account only two-level (document and term) knowledge granularity but ignores the bridging paragraph granularity. However, this two-level granularity may lead to unsatisfactory clustering results with “false correlation”. In order to deal with the problem, a Hierarchical Representation Model with Multi-granularity (HRMM), which consists of five-layer representation of data and a twophase clustering process is proposed based on granular computing and article structure theory. To deal with the zero-valued similarity problemresulted from the sparse term-paragraphmatrix, an ontology based strategy and a tolerance-rough-set based strategy are introduced into HRMM. By using granular computing, structural knowledge hidden in documents can be more efficiently and effectively captured in HRMM and thus web document clusters with higher quality can be generated. Extensive experiments show that HRMM, HRMM with tolerancerough-set strategy, and HRMM with ontology all outperform VSM and a representative non VSM-based algorithm, WFP, significantly in terms of the F-Score.

Algebraic Computations with Hausdorff Continuous Functions

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

On the Approximation of the Generalized Cut Function of Degree p+1 By Smooth Sigmoid Functions

We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p + 1 (GCFP). We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the logistic and the shifted logistic functions. The limiting case of the interval-valued Heaviside step function is also discussed which imposes the use of Hausdorff metric. Numerical examples are presented using CAS MATHEMATICA.

On Musical Self-Similarity : Intersemiosis as Synecdoche and Analogy

Self-similarity, a concept taken from mathematics, is gradually becoming a keyword in musicology. Although a polysemic term, self-similarity often refers to the multi-scalar feature repetition in a set of relationships, and it is commonly valued as an indication for musical coherence and consistency . This investigation provides a theory of musical meaning formation in the context of intersemiosis, that is, the translation of meaning from one cognitive domain to another cognitive domain (e.g. from mathematics to music, or to speech or graphic forms). From this perspective, the degree of coherence of a musical system relies on a synecdochic intersemiosis: a system of related signs within other comparable and correlated systems. This research analyzes the modalities of such correlations, exploring their general and particular traits, and their operational bounds. Looking forward in this direction, the notion of analogy is used as a rich concept through its two definitions quoted by the Classical literature: proportion and paradigm, enormously valuable in establishing measurement, likeness and affinity criteria. Using quantitative qualitative methods, evidence is presented to justify a parallel study of different modalities of musical self-similarity. For this purpose, original arguments by Benoît B. Mandelbrot are revised, alongside a systematic critique of the literature on the subject. Furthermore, connecting Charles S. Peirce s synechism with Mandelbrot s fractality is one of the main developments of the present study. This study provides elements for explaining Bolognesi s (1983) conjecture, that states that the most primitive, intuitive and basic musical device is self-reference, extending its functions and operations to self-similar surfaces. In this sense, this research suggests that, with various modalities of self-similarity, synecdochic intersemiosis acts as system of systems in coordination with greater or lesser development of structural consistency, and with a greater or lesser contextual dependence.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Interval multivariate curve resolution in the dereplication of HPLC-DAD data from Jatropha gossypifolia

Introduction Jatropha gossypifolia has been used quite extensively by traditional medicine for the treatment of several diseases in South America and Africa. This medicinal plant has therapeutic potential as a phytomedicine and therefore the establishment of innovative analytical methods to characterise their active components is crucial to the future development of a quality product. Objective To enhance the chromatographic resolution of HPLC-UV-diode-array detector (DAD) experiments applying chemometric tools. Methods Crude leave extracts from J. gossypifolia were analysed by HPLC-DAD. A chromatographic band deconvolution method was designed and applied using interval multivariate curve resolution by alternating least squares (MCR-ALS). Results The MCR-ALS method allowed the deconvolution from up to 117% more bands, compared with the original HPLC-DAD experiments, even in regions where the UV spectra showed high similarity. The method assisted in the dereplication of three C-glycosylflavones isomers: vitexin/isovitexin, orientin/homorientin and schaftoside/isoschaftoside. Conclusion The MCR-ALS method is shown to be a powerful tool to solve problems of chromatographic band overlapping from complex mixtures such as natural crude samples. Copyright © 2013 John Wiley & Sons, Ltd. Extracts from J. gossypifolia were analyzed by HPLC-DAD and, dereplicated applying MCR-ALS. The method assisted in the detection of three C-glycosylflavones isomers: vitexin/isovitexin, orientin/homorientin and schaftoside/isoschaftoside. The application of MCR-ALS allowed solving problems of chromatographic band overlapping from complex mixtures such as natural crude samples. Copyright © 2013 John Wiley & Sons, Ltd.

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.