Fuzzy Integral

In actual environmental investment for industry, the stakeholders are often required to evaluate the investment strategies according to their own subjective preferences in terms of numerical values from various criteria, such as economic e ectiveness, technique feasibility and environmental regulation. Thus, this situation can be regarded as a fuzzy multiple criteria decision-making (MCDM) problem, so the fuzziness and uncertainty of subjective perception should be considered. This paper proposes an alternative approach, the non-additive fuzzy integral, to cope with evaluation of fuzzy MCDM problems particularly while there is dependence among considered criteria. To illustrate the proposed procedure, the sustainable development strategy for aquatic product processors in Taiwan is investigated. In this paper we employ triangular fuzzy numbers to represent the decision makers' subjective preferences on the considered criteria, as well as for the criteria measurements to evaluate a sustainable development planning case for industry. Firstly, in this study we employ factor analysis to extract four independent common factors from those criteria. Secondly, we construct the evaluation frame using AHP composed of the above four common factors with twelve evaluated criteria, and then derive the relative weights with respect to considered criteria. Thirdly, the synthetic utility value corresponding to each sustainable development strategy is aggregated by the fuzzy weights with fuzzy performance values, and the best investment strategies can then be decided. Through this study, we successfully demonstrate that the non-additive fuzzy integral is an e ective evaluation and appears to be more appropriate than the traditional simple additive weighted method, especially when the criteria are dependent. ?

Internet evolution has affected all industrial and commercial activity and accelerated e-learning growth. Due to cost, time, or flexibility for designer courses and learners, e-learning has been adopted by corporations as an alternative training method. E-learning effectiveness evaluation is vital, and evaluation criteria are diverse. A large effort has been made regarding e-learning effectiveness evaluation; however, a generalized quantitative evaluation model, which considers both the interaffected relation between criteria and the fuzziness of subjective perception concurrently, is lacking. In this paper, the proposed new novel hybrid MCDM model addresses the independent relations of evaluation criteria with the aid of factor analysis and the dependent relations of evaluation criteria with the aid of DEM-ATEL. The AHP and the fuzzy integral methods are used for synthetic utility in accordance with subjective perception environment. Empirical experimental results show the proposed model is capable of producing effective evaluation of e-learning programs with adequate criteria that fit with respondent's perception patterns, especially when the evaluation criteria are numerous and intertwined.

A mechatronic system needs an integrated, concurrent, and system-based design approach due to the existence of interactions among its subsystems, and also the existence of interactions between the criteria involved in a realistic evaluation of a mechatronic product. This paper presents a systematic methodology for a detailed mechatronic design based on a mechatronic design quotient (MDQ). MDQ is a multicriteria index, reflecting a system-based evaluation of a mechatronic design, which is calculated using soft computing techniques, thereby accommodating interactions between criteria and human experience. A niching genetic algorithm is utilized to explore the huge search space raised due to concurrent and integrated design approach, with the aim to find the elite representatives of different possible configurations. To demonstrate the method, it is applied to an industrial fish cutting machine called the Iron Butcher-an electromechanical system that falls into the class of mixed or multidomain systems.

Vendor selection is an evaluation process that is based on many criteria that uses inaccurate or uncertain data. But while the criteria are often numerous and the relationships between higher-level criteria and lower-level sub-criteria are complex, most conventional decision models cannot help us clarify the interrelationships among the sub-criteria. Our proposed integrated fuzzy multiple criteria decision making (MCDM) method addresses this issue within the context of the vendor selection problem. First, we use triangular fuzzy numbers to express the subjective preferences of evaluators. Second, we use interpretive structural modeling (ISM) to map out the relationships among the sub-criteria. Third, we use the fuzzy analytical hierarchy process (AHP) method to compute the relative weights for each criterion, and we use non-additive fuzzy integral to obtain the fuzzy synthetic performance of each common criterion. Fourth, the best vendor is determined according to the overall aggregating score of each vendor using the fuzzy weights with fuzzy synthetic utilities. Fifth, we use an empirical example to show that our proposed method is preferred to the traditional method, especially when the sub-criteria are interdependent. Finally, our results provide valuable suggestions to vendors on how to improve each sub-criterion so that they can bridge the gap between actual and aspired performance values in the future.

This paper describes a fuzzy hierarchical analytic approach to determine the weighting of subjective judgments. In addition, it presents a nonadditive fuzzy integral technique to evaluate a green engineering industry case as a fuzzy multicriteria decision-making (FMCDM) problem. When the invest-ment strategies are evaluated from various aspects, such as economic effectiveness, technical feasibility, and environmental regulation, it can be regarded as an FMCDM problem. Since stakeholders cannot clearly estimate each considered criterion in terms of numerical values for the anticipated alternatives/strategies, fuzziness is considered to be applicable. Consequently, this paper uses triangular fuzzy numbers to establish weights and anticipated achievement values. By ranking fuzzy weights and fuzzy synthetic utility values, we can determine the relative importance of criteria and decide the best strategies. This paper applies what is called a fuzzy measure and nonadditive fuzzy integral technique to evaluate the synthetic performance of green engineering strategies for aquatic products processors in Taiwan. In addition, we demonstrate that the nonadditive fuzzy integral is an effective evaluation and appears to be appropriate, especially when the criteria are not independent.

In this paper, a novel direct torque algorithm for IPMSM was analysed and simulated. The algorithm uses the output of two hysteresis controllers to determine two adjacent active vectors. It also uses the magnitude of the torque error to approximate the required average voltage level and then, together with the reflected stator flux position to select the switching time required for the selected vectors. The selection of the switching time utilizes the new table structure, which simplifies the calculation. The simulation of the algorithm shows adequate dynamic torque and considerable torque ripples reduction as well as lower harmonic current. APPENDIX See .

This study aims to determine the services provided by TSPs for motivating the consumers to purchase new e-era mobile phones and find their corresponding functions of products, and to plan new e-era mobile phones based on customers' attributes for satisfying the customer needs. This study uses four aspects (image of product, price of product, function of product and derived function of product) and 14 criteria to determine customer's motivations of purchase and needs on new e-era mobile phones. We also combined AHP (Analytic hierarchy process) and FIM (Fuzzy integral method) with MCDM techniques to construct the value-created evaluation model for planning the new e-era mobile phones. This study selects 27 styles of 3G mobile phones already sold on the market, and uses the different customer's attributes data to evaluate price's satisfaction (product's price) and value's satisfaction (product's image, product's function and product's derived function) for performance of these 3G mobile phones in a 2-axle evaluation map based on mass customization. This study shows three different customer's attributes (male, female, both) to measure the satisfaction performance of 3G mobile phones in a 2-axle development strategy map using data from people that are surveyed randomly by 21-30 years old. The development strategy map will provide some new strategy view for developing new e-era mobile phone market based on value and price-satisfied portfolios.

In [Z. Ding, H. Bunke, M. Schneider, A. Kandel, Fuzzy timed Petri net -definitions, properties and applications, Math. Comput. Modelling (2005) (in press)], we posed two fuzzy timed Petri Net models. Based on the mark changing rate, they can be classified either as a discrete Fuzzy Timed Petri Net model (discrete-FTPN), or as a continuous Fuzzy Timed Petri Net model (continuous-FTPN). In this paper, we present an algorithm developed to compute reachable states for discrete-FTPN models. We also present properties of the continuous-FTPN model, which are used to describe the system's behavior. From the investigation presented in this paper, we conclude that it is easier to implement a discrete-FTPN model, but for a theoretical study the continuous-FTPN model is better.

Computational intelligence techniques are very useful tools for solving problems that involve understanding, modeling, and analysis of large data sets. One of the numerous fields where computational intelligence has found an extremely important role is finance. More precisely, optimization issues of one's financial investments, to guarantee a given return, at a minimal risk, have been solved using intelligent techniques such as genetic algorithm, rule-based expert system, neural network, and support-vector machine. Even though these methods provide good and usually fast approximation of the best investment strategy, they suffer some common drawbacks including the neglect of the dependence among among criteria characterizing investment assets (i.e. return, risk, etc.), and the assumption that all available data are precise and certain. To face these weaknesses, we propose a novel approach involving utility-based multi-criteria decision making setting and fuzzy integration over intervals.

Appropriate risk allocation and sharing are significant critical success factors for public-private partnership projects, but evidence suggests that poor risk allocation practices prevail. This signifies the need to develop a robust model for assisting stakeholders in risk allocation decision-making. A non-additive fuzzy integral based multiple attribute risk allocation decision approach is proposed to effectively aggregate each stakeholder's risk management capability assessment on accepted risk allocation principles that are derived from qualitative judgements and experience based knowledge of experts. Data collected from privately financed and developed power and transport infrastructure projects in Pakistan are used to demonstrate and validate the model for key risk factors that exhibit variable risk allocation preferences. Comparison of results with an additive aggregation approach confirms suitability of the adopted methodology as it performs better when modelling risk allocation preferences of experts due to its ability to handle interdependencies in the risk allocation criteria. Apparently, the allocation and sharing of key risks is significantly influenced by market, sector and project contexts. Résumé : La répartition et le partage appropriés des risques sont des facteurs de réussite essentiels pour les projets de partenariat public-privé, mais les données indiquent que de mauvaises pratiques de répartition des risques prévalent. Cela signifie qu'il faut élaborer un modèle solide pour aider les intervenants à prendre des décisions en matière de répartition des risques. Une approche fondée sur l'intégrale floue non additive à multiples caractéristiques, pour la prise de décisions en matière de répartition des risques est proposée afin de regrouper efficacement l'évaluation de la capacité de gestion des risques de chaque intervenant en fonction de principes acceptés de répartition des risques qui sont dérivés de jugements qualitatifs et de connaissances fondés sur l'expérience d'experts. Les données recueillies dans le cadre de projets d'infrastructure d'électricité et de transport privés au Pakistan sont utilisées pour démontrer et valider le modèle par rapport aux principaux facteurs de risque qui présentent des préférences variables en matière de répartition des risques. La comparaison des résultats avec une approche de regroupement additif confirme l'adéquation de la méthodologie adoptée, car elle permet de mieux modéliser les préférences des experts en matière de répartition des risques en raison de sa capacité à gérer les interdépendances au niveau des critères de répartition des risques. Apparemment, les contextes du marché, du secteur et des projets influent grandement sur la répartition et le partage des principaux risques. [Traduit par la Rédaction] Mots-clés : modèle décisionnel, théorie des ensembles flous, intégrale floue, partenariats public-privé en matière d'infrastructure, répartition des risques.

Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is the most difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms and neural networks, it is hard to say which one is more appropriate and more feasible. Each method has its advantages and limitations. Therefore it is necessary to develop new methods or techniques to learn distinct fuzzy measures. In this paper, we make the first attempt to design a special particle swarm algorithm to determine a type of general fuzzy measures from data, and demonstrate that the algorithm is effective and efficient. Furthermore we extend this algorithm to identify and revise other types of fuzzy measures. To test our algorithms, we compare them with the basic particle swarm algorithms, gradient descent algorithms and genetic algorithms in literatures. In addition, for verifying whether our algorithms are robust in noisy-situations, a number of numerical experiments are conducted. Theoretical analysis and experimental results show that, for determining fuzzy measures, the particle swarm optimization is feasible and has a better performance than the existing genetic algorithms and gradient descent algorithms.

In this paper, we extend the domains of armation and negation operators, and more important, of triangular (semi)norms and (semi)conorms from the unit interval to bounded partially ordered sets. The fundamental properties of the original operators are proven to be conserved under this extension. This clearly shows that they are essentially based upon order-theoretic notions. Consequently, a rather general order-

In this paper, we present the theoretical framework for the generalization of classical hidden Markov models using fuzzy measures and fuzzy integrals. The main characteristic of the generalization is the relaxation of the usual additivity constraint of probability measures. Fuzzy integrals are defined with respect to fuzzy measures, whose key property is monotonicity with respect to set inclusion. This property is far weaker than the usual additivity property of probability measures. As a result of the new formulation, the statistical independence assumption of the classical hidden Markov models is relaxed. Two attractive properties of this generalization are: the generalized hidden Markov model reduces to the classical hidden Markov model if we used the Choquet fuzzy integral and probability measures; and the establishment of a relation between the generalized hidden Markov model and the classical nonstationary hidden Markov model in which the transitional parameters vary with time

This is a subsequent paper of [9]. By using the concepts of fuzzy number fuzzy measures [9] and fuzzy-valued functions [10], a theory of fuzzy integrals of fuzzy-valued functions with respect to fuzzy number fuzzy measures is built up. So far, it is a more general one following Sugeno's [5].

Using parametric form of fuzzy numbers we convert a linear fuzzy Fredholm integral equation of the second kind to a linear system of integral equations of the second kind in crisp case. We use Adomian method and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equation of the second kind. We apply the method to some examples. Ó 2004 Elsevier Inc. All rights reserved.

This paper, based on the fuzzy measures and fuzzy integrals given by Sugeno, first defines interval number fuzzy measures (INF-measures) and fuzzy number fuzzy measures (FNF-measures), and the fuzzy integral of function with respect to INF-measures are given. Then it defines the fuzzy integral of functions with respect to FNF-measures, and the properties and convergence theorems are obtained. All these

A single feature extractor-classifier is not usually able to deal with the diversity of multiple image scenarios. Therefore, integration of features and classifiers can bring benefits to cope with this problem, particularly when the parts are carefully chosen and synergistically combined. In this paper, we address the problem of pedestrian detection by a novel ensemble method. Initially, histograms of oriented gradients (HOGs) and local receptive fields (LRFs), which are provided by a convolutional neural network, have been both classified by multilayer perceptrons (MLPs) and support vector machines (SVMs). A diversity measure is used to refine the initial set of feature extractors and classifiers. A final classifier ensemble was then structured by an HOG and an LRF as features, classified by two SVMs and one MLP. We have analyzed the following two classes of fusion methods of combining the outputs of the component classifiers: 1) majority vote and 2) fuzzy integral. The first part of the performance evaluation consisted of running the final proposed ensemble over the DaimlerChrysler cropwise data set, which was also artificially modified to simulate sunny and shadowy illumination conditions, which is typical of outdoor scenarios. Then, a window-wise study has been performed over a collected video sequence. Experiments have highlighted a state-of-the-art classification system, performing consistently better than the component classifiers and other methods.

In this paper, we have proposed an approach for face recognition by composing Symlet decomposition, Fisherface algorithm, and Sugeno and Choquet Fuzzy Integral. This approach consists of four main sections: the first section uses Symlet, one of the Wavelet families, to transform an image into four sub-images which are called approximate, horizontal, vertical and diagonal partial images respectively. The aim of this work is to extract intrinsic facial features. The second section of this approach uses Fisherface method which is composed of PCA and LDA. The reason for using this was that it is not sensitive to intensive light variations and facial expression and gesture. The third and forth section of this approach, are related to the aggregation of the individual classifiers by means of the fuzzy integral. Both Sugeno and Choquet fuzzy integral are considered as methods for classifier aggregation. In this paper, Olivetti Research Labs face database is used for acquiring experimental results. The approach presented in this paper, will lead to better classification performance compared to other classification methods.

This paper presents four schemes for soft fusion of the outputs of multiple classi®ers. In the ®rst three approaches, the weights assigned to the classi®ers or groups of them are data dependent. The ®rst approach involves the calculation of fuzzy integrals. The second scheme performs weighted averaging with data-dependent weights. The third approach performs linear combination of the outputs of classi®ers via the BADD defuzzi®cation strategy. In the last scheme, the outputs of multiple classi®ers are combined using Zimmermann's compensatory operator. An empirical evaluation using widely accessible data sets substantiates the validity of the approaches with data-dependent weights, compared to various existing combination schemes of multiple classi®ers. Ó 0167-8655/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 -8 6 5 5 ( 9 9 ) 0 0 0 1 2 -4

Acoustic events produced in meeting environments may contain useful information for perceptually aware interfaces and multimodal behavior analysis. In this paper, a system to detect and recognize these events from a multimodal perspective is presented combining information from multiple cameras and microphones. First, spectral and temporal features are extracted from a single audio channel and spatial localization is achieved by exploiting crosscorrelation among microphone arrays. Second, several video cues obtained from multi-person tracking, motion analysis, face recognition, and object detection provide the visual counterpart of the acoustic events to be detected. A multimodal data fusion at score level is carried out using two approaches: weighted mean average and fuzzy integral. Finally, a multimodal database containing a rich variety of acoustic events has been recorded including manual annotations of the data. A set of metrics allow assessing the performance of the presented algorithms. This dataset is made publicly available for research purposes.

The huge number of voxels in fMRI over time poses a major challenge to for effective analysis. Fast, accurate, and reliable classifiers are required for estimating the decoding accuracy of brain activities. Although machine-learning classifiers seem promising, individual classifiers have their own limitations. To address this limitation, the present paper proposes a method based on the ensemble of neural networks to analyze fMRI data for cognitive state classification for application across multiple subjects. Similarly, the fuzzy integral (FI) approach has been employed as an efficient tool for combining different classifiers. The FI approach led to

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.

This is the second paper in a series of two papers describing a novel approach for generalizing classical hidden Markov models using fuzzy measures and fuzzy integrals and their application to the problem of handwritten word recognition. This paper presents an application of the generalized hidden Markov models to handwritten word recognition. The system represents a word image as an ordered list of observation vectors by encoding features computed from each column in the given word image. Word models are formed by concatenating the state chains of the constituent character hidden Markov models. The novel work presented includes the preprocessing, feature extraction, and the application of the generalized hidden Markov models to handwritten word recognition. Methods for training the classical and generalized (fuzzy) models are described. Experiments were performed on a standard data set of handwritten word images obtained from the U. S. Post Office mail stream, which contains real-word samples of different styles and qualities.

Due to limitations of classical weighted average aggregation operators, there is an increase usage of fuzzy integrals, like the Sugeno and Choquet integrals, as alternative aggregation operators. However, their applicability has been threatened by the crux of determining the fuzzy measures in real problems. One way to determine these measures is by using learning data and optimizing the parameters. In this paper we made a comparative study of two well known optimization algorithms, Genetic Algorithm and Gradient Descent to determine fuzzy measures. Two illustrative cases are used to compare the algorithms and assess their performance.

This paper proposes a novel method for aligning multiple genomic or proteomic sequences using a fuzzyfied Hidden Markov Model (HMM). HMMs are known to provide compelling performance among multiple sequence alignment (MSA) algorithms, yet their stochastic nature does not help them cope with the existing dependence among the sequence elements. Fuzzy HMMs are a novel type of HMMs based on fuzzy sets and fuzzy integrals which generalizes the classical stochastic HMM, by relaxing its independence assumptions. In this paper, the fuzzy HMM model for MSA is mathematically defined. New fuzzy algorithms are described for building and training fuzzy HMMs, as well as for their use in aligning multiple sequences. Fuzzy HMMs can also increase the model capability of aligning multiple sequences mainly in terms of computation time. Modeling the multiple sequence alignment procedure with fuzzy HMMs can yield a robust and time-effective solution that can be widely used in bioinformatics in various ap...

Protein function is related to its chemical reaction to the surrounding environment including other proteins. On the other hand, this depends on the spatial shape and tertiary structure of protein and folding of its constituent components in space. The correct identification of protein domain fold solely using extracted information from protein sequence is a complicated and controversial task in the current computational biology. In this article a combined classifier based on the information content of extracted features from the primary structure of protein has been introduced to face this challenging problem. In the first stage of our proposed two-tier architecture, there are several classifiers each of which is trained with a different sequence based feature vector. Apart from the application of the predicted secondary structure, hydrophobicity, van der Waals volume, polarity, polarizability, and different dimensions of pseudo-amino acid composition vectors in similar studies, the position specific scoring matrix (PSSM) has also been used to improve the correct classification rate (CCR) in this study. Using K-fold cross validation on training dataset related to 27 famous folds of SCOP, the 28 dimensional probability output vector from each evidence theoretic K-NN classifier is used to determine the information content or expertness of corresponding feature for discrimination in each fold class. In the second stage, the outputs of classifiers for test dataset are fused using Sugeno fuzzy integral operator to make better decision for target fold class. The expertness factor of each classifier in each fold class has been used to calculate the fuzzy integral operator weights. Results make it possible to provide deeper interpretation about the effectiveness of each feature for discrimination in target classes for query proteins.

In this paper, the concept of fuzzy-valued fuzzy measures is introduced at first and then, based on the generalized fuzzy integral given by Wu et al. [Fuzzy Sets and Systems 57 (1993) 219], the generalized fuzzy integral of fuzzy-valued functions with respect to fuzzy-valued fuzzy measures is defined and the properties and convergence theorems are given. All these are extensions

A problem of decision making under uncertainty in which the choice must be made between two sets of alternatives instead of two single ones is considered. A number of choice rules are proposed and their main properties are investigated, focusing particularly on the generalizations of stochastic dominance and statistical preference. The particular cases where imprecision is present in the utilities or in the beliefs associated to two alternatives are considered.

The systematic aggregation of information is crucial in decision making. If the information is derived from various sources, then the information fusion problem can be transformed into a multi-criteria decision making model. The information fusion problem, thus, is reduced to (a) determining the structure of the hierarchy to be used; (b) computing the degree of satisfaction of criteria based on

In this paper we give representation results for nonlinear functionals in arbitrary spaces, as fuzzy integrals. The main assumption is that of comonotonic maxitivity (i.e. I(f V g) = If V Ig for comonotonic f, g). Under the more restrictive condition of stochastic dominance, we prove a more specific representation as a fuzzy integral with respect to a distorted probability measure. We discuss the particular cases of possibility measures and upper fuzzy expectations. Some partial Randon-Nikodym results are proved for both the Choquet and the fuzzy integrals. Finally an example is shown where the fuzzy integral representation is useful.

To improve recognition results, decisions of multiple neural networks can be aggregated into a committee decision. Aggregation weights assigned to neural networks or groups of networks can be the same in the entire data space or can be different (data dependent) in various regions of the space. In this paper, we propose a method for obtaining data dependent aggregation weights. The proposed approach is tested in two aggregation schemes, namely aggregation through neural network selection, and aggregation by the Choquet integral with respect to the l-fuzzy measure. The effectiveness of the approach is demonstrated on two artificial and three real data sets.

This paper presents an efficient fuzzy neural system which consists of modular neural networks combined by the fuzzy integral with ordered weighted averaging (OWA) operators. The ability of the fuzzy integral to combine the results of multiple sources of information has been established in seL'eral preL'ious works. The key point of this paper is to formalize modular neural networks as information sources, and show the feasibility of the fuzzy integral extended by OWA operators in the problem of combining neural outputs, especially in the case that the networks differ substantially from each other in accuracy. The experimental results with the recognition problem for on-line handwritten characters show that the performance of indiL, idual networks is improced significantly.

Starting from the concept of 'equiordered functions' we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure: Sugeno and Choquet integrals. The characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals, pointing out their formal similarities and their conceptual differences.

This paper describes a fuzzy hierarchical analytic approach to determine the weighting of subjective judgments. In addition, it presents a nonadditive fuzzy integral technique to evaluate a mutual fund case as a fuzzy multi-criteria decision-making (FMCDM) problem. When the investment strategies are evaluated from various aspects, such as market timing, stock selection ability, fund size and team work, it can be regarded as an FMCDM problem. Since investors can not clearly estimate each considered criterion in terms of numerical values for the anticipated alternatives/strategies, fuzziness is considered to be applicable. Consequently, this paper uses triangular fuzzy numbers to establish weights and anticipated achievement values. By ranking fuzzy weights and fuzzy synthetic performance values, we can determine the relative importance of criteria and decide the best strategies.

In this paper, an approach has been proposed for face recognition by composing symlet decomposition, Subclass Discriminant Analysis (SDA), and Sugeno and Choquet Fuzzy Integral. This approach consists of four main sections: the first section uses Symlet, one of the Wavelet families, to transform an image into four sub-images which are called approximate, horizontal, vertical and diagonal partial images respectively. The aim of this work is to extract intrinsic facial features. The second section is composed of PCA and SDA. The reason for using this method was the fact that it is not sensitive to intensive light variations. The third and forth section of this paper are related to the aggregation of the individual classifiers by means of the fuzzy integral. Both Sugeno and Choquet fuzzy integral are considered as methods for classifier aggregation. In this paper, Olivetti Research Labs face database is used for acquiring experimental results. The combination of this method with sym decomposition has yielded a recognition rate which is equal to 97.5 %. The approach presented in this paper, will lead to better classification performance compared to other classification methods.

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.

In this paper the integration formulas of Newton CotÕs methods with positive coefficient for fuzzy integrations in [T. Allahviranloo, Newton CotÕs methods for integration of fuzzy functions, in press] are considered and then are followed by Romberg integration to obtain improvements of the approximations of fuzzy integrations. The proposed algorithm is illustrated by solving some numerical examples.

This paper presents four schemes for soft fusion of the outputs of multiple classi®ers. In the ®rst three approaches, the weights assigned to the classi®ers or groups of them are data dependent. The ®rst approach involves the calculation of fuzzy integrals. The second scheme performs weighted averaging with data-dependent weights. The third approach performs linear combination of the outputs of classi®ers via the BADD defuzzi®cation strategy. In the last scheme, the outputs of multiple classi®ers are combined using Zimmermann's compensatory operator. An empirical evaluation using widely accessible data sets substantiates the validity of the approaches with data-dependent weights, compared to various existing combination schemes of multiple classi®ers. Ó 0167-8655/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 -8 6 5 5 ( 9 9 ) 0 0 0 1 2 -4

A single feature extractor-classifier is not usually able to deal with the diversity of multiple image scenarios. Therefore, integration of features and classifiers can bring benefits to cope with this problem, particularly when the parts are carefully chosen and synergistically combined. In this paper, we address the problem of pedestrian detection by a novel ensemble method. Initially, histograms of oriented gradients (HOGs) and local receptive fields (LRFs), which are provided by a convolutional neural network, have been both classified by multilayer perceptrons (MLPs) and support vector machines (SVMs). A diversity measure is used to refine the initial set of feature extractors and classifiers. A final classifier ensemble was then structured by an HOG and an LRF as features, classified by two SVMs and one MLP. We have analyzed the following two classes of fusion methods of combining the outputs of the component classifiers: 1) majority vote and 2) fuzzy integral. The first part of the performance evaluation consisted of running the final proposed ensemble over the DaimlerChrysler cropwise data set, which was also artificially modified to simulate sunny and shadowy illumination conditions, which is typical of outdoor scenarios. Then, a window-wise study has been performed over a collected video sequence. Experiments have highlighted a state-of-the-art classification system, performing consistently better than the component classifiers and other methods.

Case-Based Reasoning (CBR) simulates human way of solving problems as it solves a new problem using a successful past experience applied to a similar problem. In this paper we describe a CBR system that performs forecasts for cash flow accounts. Forecasting cash flows to a certain degree of accuracy, is an important aspect of a Working Capital decision support system. Working Capital (WC) management decisions reflect a choice among different options on how to arrange the cash flow. The decision establishes an actual event in the cash flow and that means that one needs to envision the consequences of such a decision. Hence, forecasting cash flows accurately can minimize losses caused by usually unpredictable events. Cash flows are usually forecasted by a combination of different techniques enhanced by human experts' feelings about the future, which are grounded in past experience. That is what makes the use of the CBR paradigm the proper choice. Advantages of a CBR system over other Artificial Intelligence techniques are associated to knowledge acquisition, knowledge representation, reuse, updating and justification. An important step in developing a CBR system is the retrieval of similar cases. The proposed system makes use of fuzzy integrals to calculate the synthetic evaluations of similarities between cases instead of the usual weighted mean. * Ackowledgements. I would like to aknowledge the financial support of the Brazilian agency CNPq and the University of South Florida.

In this paper, we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction of Legesgue integrals is explored and exploited. It is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalization of Sugeno's fuzzy integrals.