Fuzzy set theory

A new fuzzy approach for defining multi-purpose criticality of activities in PERT

by Sina Khanmohammadi

In this paper a new fuzzy approach is developed for defining the general criticality of activities where some other... more In this paper a new fuzzy approach is developed for defining the general criticality of activities where some other features such as probability of finishing on time zone, probability of impact, impact threat and ability to retaliate are considered as criticality factors of activities in project management process. In this way the risky situation (vulnerability) of activities are calculated by using fuzzy inference system. Activities are prioritized and classified by means of a fuzzy decision making procedure. The effect of considering such factors on project duration and cost are compared with classic PERT - where only the slack times are considered as criticality factors of activities -by means of Mont Carlo simulation.

Comparative media analysis: Why some fuzzy thinking might help

by James Stanyer

Co-authored with John Downey

Fuzzy Set Theory (or Fuzzy Logic) to Represent the Messy Data of Complex Human (and other) Systems

by J. B. (Jack) Owens

Co-authored with Emery A. Coppola, Jr.

Historians and Human Geographers deal with human systems or subsystems of considerable complexity. This situation... more Historians and Human Geographers deal with human systems or subsystems of considerable complexity. This situation presents a dilemma to those who use computational technologies, which demand a high level of precision to organize, analyze, and visualize information: the more complex the system is, the greater the imprecision of the available data. Historians and geographers often feel that their imprecise, ambiguous, contradictory, messy, largely qualitative information does not “fit” well in the available software categories, and they have trouble discussing the results produced when they work within computational environments because category assignment seems so arbitrary. This dilemma appears dramatically with the use of Geographically-Integrated History (GIH) as a research strategy. In this paper, we introduce fuzzy set theory (or fuzzy logic) as a proven solution for dealing with imprecision in complex systems.

A new fuzzy mathematical model for multi criteria decision making: An application of fuzzy mathematical model in an SWOT analysis

by Dragan Pamucar

Analysis of strengths, weaknesses, opportunities and threats (SWOT) is a method to formulate the strategy. Although... more Analysis of strengths, weaknesses, opportunities and threats (SWOT) is a method to formulate the strategy. Although the SWOT analysis successfully provides the key factors of the problem, it has some drawbacks in selecting appropriate strategy for the evaluation and final decision steps. During recent years, some multiple criteria decision making (MCDM) remove some of these deficiencies, but the nature of these decision usually is very complex and using crisp datais not suitable. In this paper, linguistic variable represented with fuzzy numbers are used to assess the ratings and weights. This paper presents a new fuzzy mathematical model for evaluating the proposed alternatives. Fuzzy linguistic descriptors were used for describing the criteria. In this way, fuzzy logic enables the exploitation of tolerance that exists in imprecision, uncertainty and partial truth of the acquired research results. 
The paper presents a model for designing the organisational structure of transport support authorities in the Serbian Armed Forces. Various organisational structure options are proposed in application of the given model, taking into account the fact that transport authorities should be designed and dimensioned so as to achieve the rudimentary goals and tasks for fulfilment of which they were established. Each task set before the transport authorities requires reliable and top-quality performance in all environmental conditions. Since most of the acquired data is characterized by a high degree of imprecision, subjectivity and uncertainty, fuzzy logic was used for displaying these.

Modeling the skull-face overlay uncertainty using fuzzy sets

by Oscar Ibáñez

O. Ibáñez, O. Cordón, S. Damas, J. Santamaría. Modeling the skull-face overlay uncertainty using fuzzy sets. IEEE Transactions on Fuzzy Systems 19:5 (2011) 946-959 . Impact factor 2010: 2.683. Category: ENGINEERING, ELECTRICAL & ELECTRONIC. Order: 15/247. Q1.

Craniofacial superimposition (CS) is a forensic process where photographs or video shots of a missing person are... more Craniofacial superimposition (CS) is a forensic process where photographs or video shots of a missing person are compared with the skull that is found. By projecting both photographs on top of each other (or, even better, matching a scanned 3-D skull model against the face photo/video shot), the forensic anthropologist can try to establish whether it is the same person. The whole process is influenced by inherent uncertainty, mainly because two objects of different nature (a skull and a face) are involved. In this paper, we extend our previous evolutionary-algorithm-based method to automatically superimpose the 3-D skull model and the 2-D face photo with the aim to overcome the limitations that are associated with the different sources of uncertainty, which are present in the problem. Two different approaches to handle the imprecision will be proposed: weighted and fuzzy-set-theory-based landmarks. The performance of the new proposal is analyzed, considering five skull-face overlay problem instances that correspond to three real-world cases solved by the Physical Anthropology Laboratory, University of Granada, Granada, Spain. The experimental study that is developed shows how the fuzzy-set-based approach clearly outperforms the previous crisp solution. Finally, the proposed method is validated by the comparison of its outcomes with respect to those manually achieved by the forensic experts in nine skull-face overlay problem instances.

Aporetic and Nonaporetic Paradoxes from the Viewpoint of an Axiomatized Contradictorial Fuzzy Set-Theory

by Lorenzo Peña

¿Lógica Combinatoria o Teoría Estándar de Conjuntos?

by Lorenzo Peña

Arbor nº 520 (abril 1989), pp. 33-73. ISSN 0210-1963

KEYWORDS.- set theory, combinatory logic, comprising relation, reflexivity, unlevelling.

RESUMEN.- more KEYWORDS.- set theory, combinatory logic, comprising relation, reflexivity, unlevelling.

RESUMEN.-
Filosóficamente la teoría estándar de conjuntos es insatisfactoria, al no existir ninguna noción presistemática como la que quieren aducir sus adeptos; esa concepción resulta sospechosamente híbrida. La teoría no puede ser verdadera, salvo interpretada de un modo no estándar.
Afortunadamente, hay alternativas preferibles. Entre ellas una teoría combinatoria de conjuntos basada en una lógica no clásica. Esta lógica combinatoria gira en torno a la relación primitiva de abarcar, que puede también guardar reflexivamente un ente consigo mismo. Los conjuntos son, así, abarcamientos que no se sujetan al constreñimiento de la desnivelación.

Fuzzy mereology

by Josh Parsons

draft only (2011)

Some philosophers have attempted to solve metaphysical problems about vagueness by understanding objects with vague... more Some philosophers have attempted to solve metaphysical problems about vagueness by understanding objects with vague boundaries as analogous to fuzzy sets. I formulate such a view and argue that it suffers from a serious lacuna, which I attempt to fill.

LINKING DROUGHTY AND WET WEATHER IN THE JORDAN REGION WITH ATMOSPHERIC CIRCULATION PATTERNS

by Patrick Laux

RESUMEN
El alcance de este estudio es la identificación de patrones de circulación sobre el Mediterráneo... more RESUMEN
El alcance de este estudio es la identificación de patrones de circulación sobre el Mediterráneo oriental los cuales están relacionados de una manera significativa a la precipitación extrema en la cuenca hidrográfica del Jordan. A ese fin, un método multi-objetivo de clasificación a base de lógica fuzzy se pone en practica. El método condiciona los datos de la precipitación y patrones de circulación atmosférica a gran escala diariamente.
Para comenzar, se realiza la clasificación condicional de la precipitación para el período de 1961-1990 usando presión a nivel del mar y potencial geológico en 500hPa, dispuestos por el NCEP/NCAR proyecto de reanálisis. Se comproba la plausibilidad de los patrones de circulación obtenidos para situaciones áridos y mojados. Luego, se realiza un análisis de la frecuencia de su presencia a escala temporal de meses, años y décadas. Por último, se compara la distribución de frecuencia de los patrones de circulación para los años 1961-1990 con la distribución de frecuencia para los años 2011-2040 utilizando ECHAM5 impulsado por A1B. Se presenta y discuta el impacto del cambio climático al cambio de frecuencia en patrones de circulación áridos y mojados.
Palabras clave:
Clasificación multi-objetivo a base de lógica fuzzy, clasificación de patrones de circulación, Mediterráneo oriental, cuenca hidrográfica del Jordan

ABSTRACT
The scope of this study is to identify circulation patterns (CPs) over the Eastern Mediterranean (EM), which are significantly linked to extreme rainfall events in the Jordan region. For this reason, a multi objective fuzzy logic-based classification (MOFRBC) method is applied, which is conditioning rainfall data to large-scale atmospheric circulation patterns on daily time scale.
First, the rainfall conditional classification is performed for the period 1961-1990 using Sea Level Pressure (SLP) and Geopotential Height in 500hPa (GPH500), retrieved from the NCEP/NCAR reanalysis project. The obtained circulation patterns are cross-checked with another objective classification algorithm for this region and a frequency analysis of the occurrence of droughty and wet CPs is performed on interannual and decadal time scale. Second, the CP frequency distribution of the 1961-1990 time slice is then compared to the frequency distribution of the 2011-2040 time slice using the A1B driven ECHAM5. The impact of climate change on the frequency change of droughty and wet circulation patterns is presented and discussed.
Key words:
Multi-objective fuzzy logic-based classification, Eastern Mediterranean, Jordan region

Comparison of different objective atmospheric circulation pattern analyses in the Jordan region

by Patrick Laux

Abstract Different existing circulation pattern (CP) classification algorithms were applied for the Jordan region. The... more Abstract Different existing circulation pattern (CP) classification algorithms were applied for the Jordan region. The obtained daily CP time series were statistically tested for mutual dependency using the χ2-test. The strength of the dependency was assessed by means of the adjusted contingency coefficient and Cramér’s coefficient, separately for the whole-year-round and the seasonal consideration. The persistence of the mutual dependencies was analyzed using a moving-window approach for the period 1961-1990. In order to estimate the possibility of making predictions of a certain CP classification, the Guttman’s λ was calculated. The different methodologies were tested for usability for CP conditional rainfall modelling.
Most of the mutual CP classification approaches are found to be non-independent. The highest correlation occurs between Beck’s and Alpert’s classification approach for the whole-year-round consideration, different results are obtained for the seasonal consideration. The strength of the mutual dependencies between the different classifications is found to depend strongly on the season. The greatest dependencies exist for winter, the lowest for summer. The relationships are found to remain relatively stable over the analyzed period.
The performance of the different classification schemes for rainfall modelling within the Jordan region was analyzed. Except for two observation stations in the southern part of the research area, all the CP conditional approaches were superior to the unconditional rainfall modelling. The semi-objective Alpert CP classification was found to perform slightly better than the fully objective methodologies.
Keywords circulation pattern analysis; Jordan region; χ2-test; Cramer’s V; adjusted contingency coefficient; Guttman’s λ

Towards a complete fuzzification of supervised classification for remote sensing digital image processing

by Minhe Ji

Doctoral dissertation

In characterizing biophysical phenomena through remotely sensed image data, a fuzzy partition matrix has proven more... more In characterizing biophysical phenomena through remotely sensed image data, a fuzzy partition matrix has proven more representative of the real situation than the conventional thematic model. The geographical concepts of concentration and diffusion, important to many areas of geographical research and applications, can be readily supported with such a data model. In a statistical supervised classification, however, a fuzzy partition matrix based on spectral signatures derived through a conventional two-value logic is a faulty representation of the real situation. In order to improve the accuracy of a fuzzy partition matrix as well as a conventional thematic map, an adequate fuzzy approach must be incorporated into supervised training, which is the determinant of the final accuracy level for a classification.

This dissertation was attempted to achieve a complete fuzzy supervised classification by fuzzifying both training stage and error analysis stage of the conventional maximum likelihood procedure. It is reasonable to assume that a certain degree of uncertainty exists in both human knowledge about the training site and spectral coherence of the training pixels delineated by the analyst with respect to the class in definition. Fuzzy training copes with the data uncertainty in the early stage of image classification in order to derive statistical parameters that are closer to real ones. It is also a design to ensure a proper transmission of uncertainty information from the raw image data to the final product. A self-organizing iterative fuzzy parameter estimator (ISOFPE) was developed by modifying Bezdek's Fuzzy c-Means engine. Fuzzy error analysis was performed through the construction of a fuzzy concentration table and the computation of a non-fuzziness index for each class. Both synthetic data and Landsat Thematic Mapper imagery were used to test the fuzzy procedure. In this study, classification results based on the fuzzified spectral signatures were favorable over those with the conventional training method. Major findings include a statistically significant improvement in the overall classification accuracy and the accuracy of the classes with a high degree of confusion, and a fuzzy membership distribution that is more realistic and consistent to human knowledge. Future directions include class-based fuzzification, fuzzy Kappa coefficients, and the incorporation of contextual information into fuzzy image segmentation.

Using fuzzy sets to improve cluster labelling in unsupervised classification

by Minhe Ji

This paper presents a fuzzy set approach to improving the cluster labelling process for unsupervised classification.... more This paper presents a fuzzy set approach to improving the cluster labelling process for unsupervised classification. Assuming normality for spectral data distribution, this approach represents both spectral clusters and the spectral signatures of information classes as fuzzy sets, with their fuzzy membership functions being defined using spectral means and variances. A fuzzy similarity index, defined through a fuzzy binary relation between each cluster fuzzy set and its corresponding class fuzzy set, is derived by applying a max-min fuzzy composition to the paired fuzzy sets. A second max-min fuzzy composition was performed to determine the final labelling of individual clusters based on these indices. Using a Thematic Mapper (TM) image of Imperial Valley, California as the test dataset, the approach was compared with both a human interpretation and a supervised classification resulting from the maximum-likelihood classifier. Both the process and the results indicated that the fuzzy approach surpassed the human interpretation in speed and equated the supervised classification in accuracy. In comparison to the Bayesian maximum-likelihood classifier, this approach supports a quick identification of mixed clusters and missing classes and incomplete classification. In addition, the fuzzy labelling approach provides a great potential for rapid a posteriori cluster validation.

Phenomenological Sociography and Time Travel

by Peter Theodoropoulos

Related to questions of Time, structure and aspectual dimensions of relational time. Related theories that I am working on to discuss this are syntactogenerics and syntactogenesis, Syntacto generics and Semanto Generics, Syntactogenesis and Semantogenesis relations, and the nature of time as emminative, immanent and transcendent meaning both virtual and ontological. Please offer your input if you so wish.

What is time? How is time non linear? What are the relational or assemblages of subjective structure, how can we... more What is time? How is time non linear? What are the relational or assemblages of subjective structure, how can we create a taxonomy of types of relational time? By what features does time seem linear, and what is the nature of conditioning, or ritual space that instantiates regularities IN the mind? How is this relation both physical and immaterial? IS there such thing as the immaterial? Can time be said to be Immaterial and materializable? Is time instantiative or phasic, normative and bound to rules or flexible? How do we understand distinctions in subjective and objective time, how can we quantify and compare and contrast variables of each? Is there such thing as a phase shift in which ontological bias may be met with nomological or noumenal objectivity, if so what are the expectations, means of discernment and systems available to explore critique, modify and experiment with time relations? What, finally, are some means to codify and structure simulations of these relations utilizing electromagnetic fields that may interface with the Neural correlates of Consciousness study to explore results? What should our ethical considerations with regards to this sort of study be? How was your day?

C1 fuzzy manifolds

by David Foster

This paper introduces the notion of a C^1 fuzzy manifold as a natural development of the notions of a fuzzy... more This paper introduces the notion of a C^1 fuzzy manifold as a natural development of the notions of a fuzzy topological vector space and of a fuzzy derivative of a fuzzy continuous mapping between fuzzy topological vector spaces. First, a fuzzy atlas of class C^1 on a set is constructed and shown to yield a fuzzy topology that is compatible with the fuzzy atlas. The structure of a C^1 fuzzy manifold on the set then follows. Next, it is shown that the product of two fuzzy manifolds is a fuzzy manifold, and that the composition of two fuzzy differentiable mappings between fuzzy manifolds is fuzzy differentiable. Finally, the notions of a tangent vector and of a tangent space at a point in a fuzzy manifold are formulated, and the tangent space is shown to be a vector space.

Differentiation of fuzzy continuous mappings on fuzzy topological vector spaces

by David Foster

In a classic paper [1] , Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Chang [2], Wong [3],... more In a classic paper [1] , Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Chang [2], Wong [3], Lowen [4], and others developed a theory of fuzzy topological spaces and Rosenfeld [5] initiated a theory of fuzzy groups. These were brought together by Foster [6] to form the elements of a theory of fuzzy topological groups. Starting with a vector space E, a structure for fuzzy vector spaces and fuzzy topological vector spaces was proposed by Katsaras and Liu [7]. In this paper, we develop the theory of fuzzy topological vector spaces further and introduce the notion of the differentiability of fuzzy continuous mappings defined on fuzzy topological vector spaces. The properties of derivatives and formal rules of derivation are also briefly discussed. We point out that our approach does not depend upon the imposition of a norm on the space E. In particular, the derivative defined here should be distinguished from the differential of a "fuzzy function" described by Puri and Ralescu [8l which relates to mappings from an open subset of a normed space into a subset of fuzzy sets defined on a reflexive Banach space.

Fuzzy topological groups

by David Foster

In his classic paper [1] of 1965, Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Subsequently,... more In his classic paper [1] of 1965, Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Subsequently, Chang [2[, Wong [3], Lowen [4] and others applied some basic concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. In an analogous application with groups, Rosenfeld [5] formulated the elements of a theory of fuzzy groups. In the present paper, we bring together the structure of a fuzzy topological space and that of a fuzzy group to form a combined structure, that of a fuzzy topological group. Homomorphic images and inverse images, quotients and products of fuzzy topological groups are also briefly examined. Notation for fuzzy sets follows that of Zadeh [l].

Classical and fuzzy differential methods in shape analysis

by David Foster

This study considers four means of defining differential operators for extracting local aspects of shape in... more This study considers four means of defining differential operators for extracting local aspects of shape in ill-specified environments: fuzzy differentiation as kernel smoothing; differentiation in the sense of weak or generalized derivatives; differentiation for fuzzy functions between normed spaces; and fuzzy differentiation for mappings between fuzzy manifolds. More consideration is given to the last, norm-free approach, which involves the notions of an abstract fuzzy topological vector space, fuzzy differentiation between fuzzy topological vector spaces, fuzzy atlases, and tangent vectors of fuzzy manifolds.

Elements of a fuzzy geometry for visual space

by David Foster

This study introduces the notions of fuzzy location and fuzzy proximity to capture the imprecision associated with... more This study introduces the notions of fuzzy location and fuzzy proximity to capture the imprecision associated with judgements of absolute and relative visual position. These notions are used to establish the elements of a fuzzy geometry for visual space, including the fuzzy betweenness of points, the fuzzy orientation of a pair of points, and the fuzzy collinearity of three or more points. Fuzzy orientation and fuzzy collinearity are, in turn, used to define the fuzzy straightness of a curve and the fuzzy tangency of two curves.

Comparative analysis of fuzzy approaches to remote sensing image classification

by Minhe Ji

Co-authored with Wen Chen

This paper compares four commonly used fuzzy analytical methods for remote sensing digital image classification, i.e.... more This paper compares four commonly used fuzzy analytical methods for remote sensing digital image classification, i.e. fuzzy c-means, semi-supervised fuzzy cluster labeling, fuzzy nearest neighbor, and object-oriented fuzzy classifiers. Merits and weak points of each method were examined through a case study with a multispectral high-resolution airborne digital image of urban settings. Results showed that the fuzzy labeling approach produced the highest quality, which was followed by the object-oriented fuzzy classifier. As the former combines merits of supervised and unsupervised classifications, the latter takes the full account of contextual and spatial features.

GIS and fuzzy sets for the land suitability analysis

by Beniamino Murgante

Murgante B., Las Casas G. (2004) “G.I.S. and Fuzzy Sets for the Land Suitability Analysis”, Lecture Notes in Computer Science LNCS vol. 3044, pp. 1036-1045 ISSN: 0302-9743. Springer-Verlag, Berlin. Impact Factor: 0.513. doi:10.1007/978-3-540-24709-8_109

This paper reports about uncertainty in defining boundaries, which assume an institutional significance when... more This paper reports about uncertainty in defining boundaries, which assume an institutional significance when transposed in planning prescription. Every discipline involved in environmental planning uses different approaches to represent its own vision of reality. Geological sciences or hydraulics evaluate risks by consistent mathematical models which are relevantly different to non linear models emploied in the field of ecology, and at the same time information about significance and value of cultural heritage in a given environment does not easily correspond to a value attribution. These questions represent an interesting field of research, related with the different character of information deriving from different disciplinary approaches, and with the more appropriate way of combining the same information. Different ways of managing values correspond to different ways of giving information. The result is a set of discrete representations of the physical space which correspond to a set of different values referring to areas which are considered homogeneous according to each disciplinary point of view, but very difficult to combine to create landscape units according to the whole of disciplines. The present paper illustrates a reflection on a G.I.S. application in a land suitability study on a sub-regional area of Southern Italy. Emerging questions are related to the need to combine contributions of all environmental information which are represented at different scales, with different interpretative models, with different precision of identification of landscape unit, etc.