Comparación entre el Índice de Yager y el Centroide para Reducción de tipo de un Número Difuso Tipo-2 de Intervalo

Context: There is a need for ranking and defuzzification of Interval Type-2 fuzzy sets (IT2FS), in particular Interval Type-2 fuzzy numbers (IT2FN). To do so, we use the classical Yager Index Rank (YIR) for fuzzy sets to IT2FNs in order to find an alternative to the centroid of an IT2FN.Method: We use a simulation strategy to compare the results of the centroid and the YIR of an IT2FN. This way, we simulate 1000 IT2FNs of the following three kinds: gaussian, triangular, and non symmetrical in order to compare their centroids and YIRs.Results: After performing the simulations, we compute some statistics about its behavior such as the degree of subsethood, equality and the size of the Footprint of Uncertainty (FOU) of an IT2FN. A description of the obtained results shows that the YIR is less wide than centroid of an IT2FN.Conclusions: In general, YIR is less complex to obtain than the centroid of an IT2FN, which is highly desirable in practical applications such as fuzzy decision making and control. Some other properties regarding its size and location are also discussed.Contexto: Hay una necesidad por defuzzificar y rankear Conjuntos Difusos Tipo-2 de Intervalo (IT2FS), en particular Números Difusos Tipo-2 de Intervalo (IT2FN). Para ello, usamos el Índice de Yager (YIR) para conjuntos difusos aplicado a IT2FNs con el fin de encontrar una alternativa al centroide de un IT2FN.Método: Usamos una estrategia de simulación para comparar los resultados del centroide y del YIR de un IT2FN. Así pues, simulamos 1000 IT2FNs de cada uno de los siguientes tres tipos: gausianos, triangulares y asimétricos para comparar sus centroides y YIRs.Resultados: Después de realizar las simulaciones, se calculan algunas estadísticas de su comportamiento como el grado de cobertura y de igualdad relativas del YIR respecto al centroide así como el tamaño de la Huella de Incertidumbre (FOU) de un IT2FN. La descripción de los resultados obtenidos muestra que el YIR es menos amplio que el centroide.Conclusiones: En general, el YIR es menos complejo de obtener que el centroide de un IT2FN, lo cual es altamente deseable en aplicaciones prácticas como toma de decisiones y control. Otras propiedades relacionadas con su tamaño y ubicación también son discutidas

Equilibrium linguistic computation method for fuzzy group decision making

This paper proposes an equilibrium computation method using linguistic variables based on the conflicting bifuzzy sets. The linguistic terms were defined and associated with the triangular fuzzy number as well as the labeling system in the early stages. Then, the negation operator was introduced and the bifuzzy approaches were employed to derive the aggregation equilibrium linguistic judgement for evaluation process

The Classroom as Community: Ideas From an Early Childhood Teacher

I have based my views on my l953-97 experiences as a preschool teacher, administrator, and consultant—in cooperative preschools, Head Start programs, a college lab school, and child day care centers, including special education classrooms. Children in these settings have come from diverse economic and ethnic homes and neighborhoods. Staff members in centers for severely disabled children, as well as those with extremely limited budgets, may feel that particular realities prevent their adoption of some practices described here—such as class trips or purchase of quality materials, which can be expensive. I know how some teachers have to modify their programs for practical reasons and yet how ingenious they are in upholding high standards. I really believe that the basic philosophy in this paper applies to all facilities for children. All children attending childcare programs benefit from respectful teaching and they all belong to classroom communities, whether they are in family day care or in large inclusive urban centers. My hope is that they will enjoy learning to be together, in whatever setting they find themselves; that they will thrive as individuals; and that they will take good care of each other

The Decision Tree Aided Neuro-Fuzzy Inference Characterization of the Stochastic Hydrology of the Tana Alluvial Aquifer

The Tana Alluvial Aquifer is the name given to the little-understood aquifer which is active in the areas bordering the River Tana Flow course as the river weaves its way through the sedimentary plains of Balambala, Garissa, Fafi and Ijara and, finally, into the Tana Delta areas, with the common denominator being the proximity to the Lower Tana catchment, especially the riparian corridor of the River itself, and beyond. The aquifer may extend to between five to fifteen kilometers away from the river channels course way, and at times, it may be felt even 20 kilometers away. The geology of the locality is heterogeneous and comprise sediments whose soil mechanics may not be easily deciphered, since some areas close to the river have very fresh water while others are saline (Bura East in Fafi Sub County easily comes to mind here).  There are areas far from the river but bearing fresh water (Mulanjo comes to mind). In some areas, sites close to the river discharge low yield figures, whereas those located farther afield discharge favorably. The water quality and discharge are therefore stochastic variables, subject to chance occurrence. In view of this inconsistency, and on the account of data scarcity, the neuro-fuzzy inference algorithm was developed to map the Universe of Discourse of the Tana Alluvial Aquifer, aka the T.A.A., as it relates to the longitudes, latitudes, depths, and discharges of the aquifers in the study area. The mapping was with respect to aquifer discharge, the variable used to characterize an aquifer, in terms of Transmissivity and Hydraulic Conductivity, thereby defining aquifer recharge propensity. Membership functions were developed using the trapezoidal membership family, and fuzzy rules were appropriately evolved from the fuzzified aquifer data, before finally employing the Sugeno inference engines (in Python) to make predictions of discharge, at each of the T.A.A. aquifer subsets mapped for fresh, saline, hard and brackish water species. The accuracy in the outputs achieved in the areas mapped vindicated the power of the neuro-fuzzy inference systems, as the accuracy oscillated between 92 and 99 percent, when the discharge values predicted were compared with the actual known discharge values of the wells mapped. The water quality class characterization was then undertaken using the decision tree (DT) algorithm in python which gave rise to a 100 percent prediction accuracy. The same DT algorithm could not successfully predict the discrete values of aquifer discharge or EC values, with as much accuracy (but performed excellently with salinity class data), and that was why fuzzy logic was employed. The study vindicated the use of the DT and Fuzzy Logic Algorithms as simple, yet powerful analytical tools, in characterizing the Stochastic Hydrology of the Tana Alluvial Aquifer.

Trial by Traditional Probability, Relative Plausibility, or Belief Function?

Almost incredible is that no one has ever formulated an adequate model for applying the standard of proof. What does the law call for? The usual formulation is that the factfinder must roughly test the finding on a scale of likelihood. So, the finding in a civil case must at least be more likely than not or, for the theoretically adventuresome, more than fifty percent probable. Yet everyone concedes that this formulation captures neither how human factfinders actually work nor, more surprisingly, how theory tells us that factfinders should work. An emerging notion that the factfinder should compare the plaintiff’s story to the defendant’s story might be a step forward, but this relative plausibility conjecture has its problems. I contend instead that the mathematical theory of belief functions provides an alternative without those problems, and that the law in fact conforms to this theory. Under it, the standards of proof reveal themselves as instructions for the factfinder to compare the affirmative belief in the finding to any belief in its contradiction, but only after setting aside the range of belief that imperfect evidence leaves uncommitted. Accordingly, rather than requiring a civil case’s elements to exceed fifty percent or comparing best stories, belief functions focus on whether the perhaps smallish imprecise belief exceeds its smallish imprecise contradiction. Belief functions extend easily to the other standards of proof. Moreover, belief functions nicely clarify the workings of burdens of persuasion and production

A review of applications of fuzzy sets to safety and reliability engineering

Safety and reliability are rigorously assessed during the design of dependable systems. Probabilistic risk assessment (PRA) processes are comprehensive, structured and logical methods widely used for this purpose. PRA approaches include, but not limited to Fault Tree Analysis (FTA), Failure Mode and Effects Analysis (FMEA), and Event Tree Analysis (ETA). In conventional PRA, failure data about components is required for the purposes of quantitative analysis. In practice, it is not always possible to fully obtain this data due to unavailability of primary observations and consequent scarcity of statistical data about the failure of components. To handle such situations, fuzzy set theory has been successfully used in novel PRA approaches for safety and reliability evaluation under conditions of uncertainty. This paper presents a review of fuzzy set theory based methodologies applied to safety and reliability engineering, which include fuzzy FTA, fuzzy FMEA, fuzzy ETA, fuzzy Bayesian networks, fuzzy Markov chains, and fuzzy Petri nets. Firstly, we describe relevant fundamentals of fuzzy set theory and then we review applications of fuzzy set theory to system safety and reliability analysis. The review shows the context in which each technique may be more appropriate and highlights the overall potential usefulness of fuzzy set theory in addressing uncertainty in safety and reliability engineering

A Theory of Factfinding: The Logic for Processing Evidence

Academics have never agreed on a theory of proof. The darkest corner of anyone’s theory has concerned how legal decisionmakers logically should find facts. This Article pries open that cognitive black box. It does so by employing multivalent logic, which enables it to overcome the traditional probability problems that impeded all prior attempts. The result is the first-ever exposure of the proper logic for finding a fact or a case’s facts. The focus will be on the evidential processing phase, rather than the application of the standard of proof as tracked in my prior work. Processing evidence involves (1) reasoning inferentially from a piece of evidence to a degree of belief and of disbelief in the element to be proved, (2) aggregating pieces of evidence that all bear to some degree on one element in order to form a composite degree of belief and of disbelief in the element, and (3) considering the series of elemental beliefs and disbeliefs to reach a decision. Zeroing in, the factfinder in step #1 should connect each item of evidence to an element to be proved by constructing a chain of inferences, employing multivalent logic’s usual rules for conjunction and disjunction to form a belief function that reflects the belief and the disbelief in the element and also the uncommitted belief reflecting uncertainty. The factfinder in step #2 should aggregate, by weighted arithmetic averaging, the belief functions resulting from all the items of evidence that bear on any one element, creating a composite belief function for the element. The factfinder in step #3 does not need to combine elements, but instead should directly move to testing whether the degree of belief from each element’s composite belief function sufficiently exceeds the corresponding degree of disbelief. In sum, the factfinder should construct a chain of inferences to produce a belief function for each item of evidence bearing on an element, and then average them to produce for each element a composite belief function ready for the element-by-element standard of proof. This Article performs the task of mapping normatively how to reason from legal evidence to a decision on facts. More significantly, it constitutes a further demonstration of how embedded the multivalent-belief model is in our law