Fuzzy Measures

People recognize familiar faces in a similar way by using interior facial features (facial regions) such as eyes, nose, mouth, etc. However, the importance of these regions in the realization of face identification and a quantification of the impact of such regions on the recognition process could vary from one region to another. An intuitively appealing observation is that of monotonicity: the more regions are taken into account in the recognition process, the better. From a formal point of view, the relevance of the facial regions and an aggregation of these pieces of experimental evidence can be described in the formal setting of fuzzy measures. Fuzzy measures are of particular interest with this regard given their monotonicity property (which stands in a clear contrast with the more restrictive additivity property inherent to probability-like measures). In this study, we concentrate on the construction of fuzzy measures (more specifically, k-fuzzy measure) and characterize their performance in the problem of face recognition using a collection of experimental data.

Mathematically considered, a Triangular Norm is a kind of binary operation frequently used in the context of Probabilistic Metric Spaces, but also in other very interesting fields, as may be Fuzzy Logic, or in general, in Multi-Valued Logic (MVL). The T-conorm, or S-norm, is a dual concept. Both ideas allow us to generalize the intersection and the union in a Lattice, or disjunction and conjunction in Logic. Also may be very interesting to introduce a special class of real monotone operations. We refer to the so-called Copulas, very useful in many fields. So, we offer now a comprehensive analysis of all these aggregation operators.

Evaluating software quality is an important and essential issue in the development process because it helps to deliver a competitive software product. A decision of selecting the best software based on quality attributes is a type of multi-criteria decision-making (MCDM) processes where interactions among criteria should be considered. This paper presents and develops quantitative evaluations by considering  interactions among criteria in the MCDM problems. The aggregator methods such as Arithmetic Mean
(AM) and Weighted Arithmetic Mean (WAM) are introduced, described and compared to Choquet Integral (CI) approach which is a type of fuzzy measure used as a new method for MCDM. The comparisons are shown by evaluating and ranking software alternatives based on six main quality attributes as identified by the ISO 9126-1 standard. The evaluation experiments depend on real data collected from case studies.

Kernel  methods  for  classification  is  a  well-studied area  in  which  data  are  implicitly  mapped  from  a  lower-
dimensional  space  to  a  higher-dimensional  space  to improve classification  accuracy.  However,  for  most  kernel  methods,  one must  still  choose  a  kernel  to  use  for  the  problem.  Since  there is,  in  general,  no  way  of  knowing  which  kernel  is  the  best, multiple kernel learning
(MKL)  is  a  technique  used  to  learn  the aggregation of a set of valid kernels into a single (ideally) superior
kernel. The aggregation can be done using weighted sums of the pre-computed  kernels,  but  determining  the  summation  weights is  not  a  trivial  task.  A  popular  and  successful  approach  to  this problem  is MKL-group lasso (MKLGL),  where  the  weights  and classification surface are simultaneously solved by iteratively optimizing a min-max optimization until convergence. In this work, we  propose  an
lp-normed genetic algorithm MKL (GAMKLp), which  uses  a  genetic  algorithm  to  learn  the  weights  of  a  set  of pre-computed  kernel  matrices  for  use  with  MKL  classification. We  prove  that  this  approach  is  equivalent  to  a  previously proposed  fuzzy  integral  aggregation  of  multiple  kernels  called fuzzy integral: genetic algorithm (FIGA).  A  second  algorithm, which  we  call
decision-level fuzzy integral MKL (DeFIMKL),
is  also  proposed,  where  a  fuzzy  measure  with  respect  to  the fuzzy  Choquet  integral  is  learned  via  quadratic  programming, and  the  decision  value—viz.,  the  class  label—is  computed  using the  fuzzy Choquet  integral aggregation.  Experiments on  several benchmark  data  sets  show  that  our  proposed  algorithms  can outperform  MKLGL  when  applied  to support vector machine (SVM)-based classification.

In digital image classification the conventional statistical approaches for image classification use only the gray values. Different advanced techniques in image classification like Artificial Neural Networks (ANN), Support Vector Machines (SVM), Fuzzy measures, Genetic Algorithms (GA), Fuzzy support Vector Machines (FSVM) and Genetic Algorithms with Neural Networks are being developed for image classification. Artificial neural networks can handle non-convex decisions. The use of textural features in ANN helps to resolve misclassification. SVM was found competitive with the best available machine learning algorithms in classifying high-dimensional data sets. Fuzzy measures show the detection of textures by analyzing the image by stochastic properties. The fundamental stochastic properties of the image are isolated by different kinds of stochastic methods, by non-linear filtering and by non-parametric methods. Fuzzy support vector machines (FSVM) was proposed to overcome the n-class...

The problem of multiperson and multiobjective decision making is considered. Models based on fuzzy measures are introduced. Moreover the problem of checking coherence of an assessment of scores to pairs (alternative, objective) is faced. The procedures of aggregating scores w.r.t. the objectives and w.r.t. the decision makers are studied. Geometrical interpretations of the concepts introduced and the results obtained are shown, based on hyperstructures and, in particular, join spaces.

MM is one of the central ideas in the nowadays mathematics education. In an earlier paper applying ideas from fuzzy logic we have developed a model formalizing the MM process and we have used the total possibilistic uncertainty as a measure of students' MM capacities. In the present paper we develop two alternative fuzzy measures for MM. The first of them concerns an adaptation for use in a fuzzy environment of the well known Shannon's formula for measuring a system's probabilistic uncertainty. The second one is based on the idea of the center of mass of the represented a fuzzy set figure, that is commonly used in fuzzy logic approach to measure performance. The above (three in total) fuzzy measures for MM are compared to each other and a classroom experiment presented in our earlier paper is reconsidered here illustrating our results in practice.