Rami Al-Hmouz

Fuzzy decision making and consensus: challenges

Group decision making is part of every organizational life. It is a type of participatory process in which multiple decision makers acting collectively, analyze problems, consider and evaluate several alternatives, and select from among the alternatives a solution. In such a situation, an important issue is the level of agreement or consensus achieved among the group of decision makers before obtaining the solution. In the beginning, consensus was meant as a full and unanimous agreement. Regrettably, this stringent concept of consensus in many cases is a utopia. As a result, and from a pragmatic point of view, it makes more sense to speak about a degree of consensus and, here, the theory of fuzzy sets has delivered new tools for the analysis of such imprecise phenomena like consensus. Given the significance of reaching an accepted solution by all the decision makers, consensus is a major aim of group decision making problems and, in such a way, it has obtained a great attention in the literature. However, there still exist several dares which have to be tackled by the researchers. The purpose of this paper is to bring out several issues that represent challenges that have to be faced.

Soft consensus measures in group decision making using unbalanced fuzzy linguistic information

An important question in group decision-making situations is how to estimate the consensus achieved within the group of decision makers. Dictionary meaning of consensus is a general and unanimous agreement among a group of individuals. However, most of the approaches deal with a more realistic situation of partial agreement. Defining a partial agreement of decision makers as a consensus up to some degree, the following question is how to obtain that soft degree of consensus. To do so, different approaches, in which the decision makers express their opinions by using symmetrical and uniformly distributed linguistic term sets, have been proposed. However, there exist situations in which the opinions are represented using unbalanced fuzzy linguistic term sets, in which the linguistic terms are not uniform and symmetrically distributed around the midterm. The aim of this paper was to study how to adapt the existing approaches obtaining soft consensus measures to handle group decision-making situations in which unbalanced fuzzy linguistic information is used. In addition, the advantages and drawbacks of these approaches are analyzed.

Description and prediction of time series

Description and prediction of time series.Information granules based on time windows, amplitude and change of amplitude.Fuzzy relations to predict amplitude and change of amplitude.Several models of granular time series.Experiment results for public data. In this paper, we address problems of description and prediction of time series by developing architectures of granular time series. Granular time series are models of time series formed at the level of information granules expressed in the representation space and time. With regard to temporal granularity, time series is split into temporal windows leading in this way to the formation of temporal information granules. Information granules are also quantified and constructed over the space of amplitude and change of amplitude of the series collected over time windows. In the description of time series we involve clustering techniques and build information granules in the representation space (viz. the space of amplitude and change of amplitude) of the temporal data. Fuzzy relations forming the essence of the prediction model are optimized using particle swarm optimization. Experimental results are reported for a number of publicly available time series.

The Design of Free Structure Granular Mappings: The Use of the Principle of Justifiable Granularity

The study introduces a concept of mappings realized in presence of information granules and offers a design framework supporting the formation of such mappings. Information granules are conceptually meaningful entities formed on a basis of a large number of experimental input-output numeric data available for the construction of the model. We develop a conceptually and algorithmically sound way of forming information granules. Considering the directional nature of the mapping to be formed, this directionality aspect needs to be taken into account when developing information granules. The property of directionality implies that while the information granules in the input space could be constructed with a great deal of flexibility, the information granules formed in the output space have to inherently relate to those built in the input space. The input space is granulated by running a clustering algorithm; for illustrative purposes, the focus here is on fuzzy clustering realized with the aid of the fuzzy C-means algorithm. The information granules in the output space are constructed with the aid of the principle of justifiable granularity (being one of the underlying fundamental conceptual pursuits of Granular Computing). The construct exhibits two important features. First, the constructed information granules are formed in the presence of information granules already constructed in the input space (and this realization is reflective of the direction of the mapping from the input to the output space). Second, the principle of justifiable granularity does not confine the realization of information granules to a single formalism such as fuzzy sets but helps form the granules expressed any required formalism of information granulation. The quality of the granular mapping (viz. the mapping realized for the information granules formed in the input and output spaces) is expressed in terms of the coverage criterion (articulating how well the experimental data are “covered” by information granules produced by the granular mapping for any input experimental data). Some parametric studies are reported by quantifying the performance of the granular mapping (expressed in terms of the coverage and specificity criteria) versus the values of a certain parameters utilized in the construction of output information granules through the principle of justifiable granularity. The plots of coverage-specificity dependency help determine a knee point and reach a sound compromise between these two conflicting requirements imposed on the quality of the granular mapping. Furthermore, quantified is the quality of the mapping with regard to the number of information granules (implying a certain granularity of the mapping). A series of experiments is reported as well.

Hierarchical Granular Clustering: An Emergence of Information Granules of Higher Type and Higher Order

In this study, we introduce a concept of hierarchical granular clustering and establish its algorithmic framework. We show that the proposed model naturally gives rise to information granules that are both of higher order and higher type, offering a compelling justification behind their emergence. In a concise way, we can capture the overall architecture of information granules as a hierarchy exhibiting conceptual layers of increasing abstraction: numeric data → information granules → information granules of type-2, order-2 → ... information granules of higher type/order. The elevated type of information granules is reflective of the visible hierarchical facet of processing and the inherent diversity of the individual locally revealed structures in data. While the concept and the methodology deliver some general settings, the detailed algorithmic aspects are discussed in detail when using fuzzy clustering realized by means of fuzzy c-means. Furthermore, for illustrative purposes, we mainly focus on interval-valued fuzzy sets and granular interval fuzzy sets arising at the higher level of the hierarchy. Higher type fuzzy sets are formed with the help of the principle of justifiable granularity. The conceptually sound hierarchy is established in a general way, which makes it equally applicable to various formalisms of representation of information granules. Experiments are reported for synthetic and publicly available datasets.

Designing granular fuzzy models: a hierarchical approach to fuzzy modeling

In this study, we elaborate on a distributed fuzzy modeling and ensuing granular fuzzy modeling. Such modeling is realized in the presence of separate and locally available data while the ensuing fuzzy rule-based models constructed on their basis are regarded as individual sources of knowledge. In virtue of an inherent diversity of these sources (models) and in an attempt to quantify it, a global model being formed at the higher level of hierarchy is becoming more abstract than those at the lower level is referred to as a granular fuzzy model. An essential concept of this class of models is introduced and their enhanced functionality is studied. Furthermore, we show interesting linkages of these models with type-2 fuzzy models studied in the literature. We highlight a number of arguments motivating a need and justifiable relevance of higher type of information granules. A detailed discussion on fuzzy rule-based models exhibiting an interesting aspect of an incremental format of the rules (whose rules capture an incremental description of input–output relationships formed with respect to some simple reference model (say, constant or linear) is presented. The design practice of the models is elaborated on by highlighting in this context the use of augmented fuzzy clustering. The construction of a granular fuzzy model is guided by the principle of justifiable granularity using which we show how granular parameters of the models are formed. The performance of the model is quantified with respect to the two criteria, namely coverage of experimental data and specificity of granular results. Experimental studies are reported for both synthetic and publicly available data sets.

Description and classification of granular time series

The study is concerned with a concept and a design of granular time series and granular classifiers. In contrast to the plethora of various models of time series, which are predominantly numeric, we propose to effectively exploit the idea of information granules in the description and classification of time series. The numeric (optimization-oriented) and interpretation abilities of granular time series and their classifiers are highlighted and quantified. A general topology of the granular classifier involving a formation of a granular feature space and the usage of the framework of relational structures (relational equations) in the realization of the classifiers is presented. A detailed design process is elaborated on along with a discussion of the pertinent optimization mechanisms. A series of experiments is covered leading to a quantitative assessment of the granular classifiers and their parametric analysis.

An expansion of fuzzy information granules through successive refinements of their information content and their use to system modeling

We discuss a way of building, refining, and generalizing information granules.Refinement of information granules is realized by assessing their information content.General optimization procedure of successive refinement of information granules is provided.Conditional Fuzzy C-Means is discussed as an efficient vehicle to realize information granulation.Discussed are granular decision trees as generic models with information granules. This study is concerned with a fundamental problem of expanding (refining) information granules being treated as functional entities playing a pivotal role in Granular Computing and ensuing constructs such as granular models, granular classifiers, and granular predictors. We formulate a problem of refinement of information granules as a certain optimization task in which a selected information granule is refined into a family of more detailed (precise, viz. more specific) information granules so that a general partition requirement becomes satisfied. As the ensuing information granules are directly linked with the more general information granule positioned at the higher level of hierarchy, the partition criterion is conditional by being implied (conditioned) by the description of the granule positioned one level up in the hierarchy. A criterion guiding a refinement of information granules is formulated and made fully reflective of the nature of the problem (being of regression-like or of classification character), which leads to a distinct way in which the diversity of information granules is articulated and quantified. With regard to the detailed algorithmic setting, we discuss the use of a so-called conditional Fuzzy C-Means and show how information granules (fuzzy sets) are formed in a successive manner. The method helps highlight the ensuing calculations of the resulting membership functions and reveal how the detailed structure of the data is captured. A number of numeric studies in the realm of system modeling are provided to demonstrate the performance of the approach and highlight the nature of the resulting information granules along with the performance of the fuzzy models in which these information granules are used.

Distributed proximity-based granular clustering: towards a development of global structural relationships in data

The study is focused on a development of a global structure in a family of distributed data realized on a basis of locally discovered structures. The local structures are revealed by running fuzzy clustering (Fuzzy C-Means), whereas building a global view is realized by forming global proximity matrices on a basis of the local proximity matrices implied by the partition matrices formed for the individual data sets. To capture the diversity of local structures, a global perspective at the structure of the data is captured in terms of a granular proximity matrix, which is built by invoking a principle of justifiable granularity with regard to the aggregation of individual proximity matrices. The three main scenarios are investigated: (a) designing a global structure among the data through building a granular proximity matrix, (b) refining a local structure (expressed in the form of a partition matrix) by engaging structural knowledge conveyed at the higher level of the hierarchy and provided in the form of the granular proximity matrix, (c) forming a consensus-building scheme and updating all local structures with the aid of the proximity dependences available at the upper layer of the hierarchy. While the first scenario delivers a passive approach to the development of the global structure, the two others are of an active nature by facilitating a structural feedback between the local and global level of the hierarchy of the developed structures. The study is illustrated through a series of experiments carried out for synthetic and publicly available data sets.

License plate localization using a statistical analysis of Discrete Fourier Transform signal

Abstract License Plate Recognition (LPR) is a well-known problem and it has developed as a coherent framework. Research continues on the topic due to the diversity of license plates and outdoor illumination conditions which require attention. One of the most important steps in LPR is the localization part where license plates are extracted from video captured images. In this article we introduce a new approach of plate localization using a statistical analysis of Discrete Fourier Transform of the plate signal. The plate signal is represented by five statistics: strength of the signal, normalized maximum amplitude, frequency of maximum amplitude, frequency center and frequency spread. Combining with the color-based histogram thresholding, the method achieves 97.27% accuracy using plate signals from binary images. Comparative analysis is also reported.