



@book {2890,
	title = {Minimum error entropy classification},
	series = {Studies in Computational Intelligence},
	volume = {420},
	year = {2013},
	note = {Export Date: 12 December 2013Source: Scopus},
	month = {2013///},
	pages = {1 - 278},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-84867489006\&partnerID=40\&md5=45e5679e51ca7e192b098b2320ec561a},
	author = {Marques De S{\'a}, J.P. and Silva, L.M.A. and Santos, J.M.F. and Alexandre, L.A.}
}
@article {2253,
	title = {The MEE principle in data classification: A perceptron-based analysis},
	journal = {Neural ComputationNeural Comp.},
	volume = {22},
	year = {2010},
	note = {Cited By (since 1996): 3Export Date: 21 November 2012Source: Scopusdoi: 10.1162/NECO_a_00013PubMed ID: 20569178Language of Original Document: EnglishCorrespondence Address: Silva, L.M.; Instituto de Engenharia Biom{\'e}dica, Divis{\~a}o de Sinal e Imagem, Porto 4200-465, Portugal; email: lmsilva@fe.up.ptReferences: Alexandre, L.A., Marques de S{\'a}, J., Error entropy minimization for LSTM training (2006) Proceedings of the 16th International Conference on Artificial Neural Networks, pp. 244-253. , Berlin: Springer;Asuncion, A., Newman, D., (2007) UCI machine learning repository. Irvine University of California, Irvine, , http://www.ics.uci.edu/~mlearn/MLRepository.html, School of Information and Computer Sciences. Available online at; 
Bishop, C., (1995) Neural networks for pattern recognition, , New York: OxfordUniversity Press; 
Erdogmus, D., Hild K., II., Pr{\'\i}ncipe, J.C., Blind source separation using R{\'e}nyi{\textquoteright}s α-marginal entropies (2002) Neurocomputing, 49, pp. 25-38; 
Erdogmus, D., Pr{\'\i}ncipe, J.C., Comparison of entropy and mean square error criteria in adaptive system training using higher order statistics (2000) Proceedings of the Intl. Conf. on ICA and Signal Separation, pp. 75-80. , Berlin: Springer-Verlag; 
Erdogmus, D., Pr{\'\i}ncipe, J.C., An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems (2002) IEEE Transactions on Signal Processing, 50 (7), pp. 1780-1786; 
Erdogmus, D., Pr{\'\i}ncipe, J.C., Convergence properties and data efficiency of the minimum error entropy criterion in adaline training (2003) IEEE Transactions on Signal Processing, 51 (7), pp. 1966-1978; 
Gokcay, E., Pr{\'\i}ncipe, J.C., Information theoretic clustering (2002) IEEE Trans. on Pattern Analysis and Machine Learning, 24 (2), pp. 158-171; 
Han, L., Kernel partial least squares (K-PLS) for scientific data mining. Unpublished doctoral dissertation (2007) Rensselaer Polytechnic Institute; 
Hild, I.I., Erdogmus, K., Pr{\'\i}ncipe D., J.C., Blind source separation using R{\'e}nyi{\textquoteright}s mutual information (2001) IEEE Signal Processing Letters, 8, pp. 174-176; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., Adaptive mixtures of local experts (1991) Neural Computation, 3, pp. 79-87; 
Jenssen, R., Hild, K., Erdogmus, D., Pr{\'\i}ncipe, J.-C., Eltoft, T., Clustering using R{\'e}nyis entropy (2003) Proceedings of the Int. Joint Conference on Neural Networks, pp. 523-528. , Piscataway, NJ: IEEE Press; 
Johnson, N., Kotz, S., (1970) Continuous univariate distributions, , Hoboken, NJ:Wiley; 
Kapur, J., Maximum-entropy models in science and engineering (1993), (rev. ed.). Hoboken, NJ: WileyLagarias, J., Reeds, J.A., Wright, M.H., Wright, P.E., Convergence properties of the nelder-mead simplex method in low dimensions (1998) SIAM Journal of Optimization, 9 (1), pp. 112-147; 
Linsker, R., Self-organization in a perceptual network (1988) IEEE Computer, 21, pp. 105-117; 
Liu, W., Pokharel, P., Pr{\'\i}ncipe, J., Correntropy: Properties and applications in non-gaussian signal processing (2007) IEEE Transactions on Signal Processing, 55 (11), pp. 5286-5298; 
Parzen, E., On the estimation of a probability density function and the mode (1962) Annals of Mathematical Statistics, 33, pp. 1065-1076; 
Pr{\'\i}ncipe, J.C., Xu, D., Fisher, J., Information theoretic learning (2000) Unsupervised adaptive filtering, 1, pp. 265-319. , S. Haykin (Ed.), Blind source separation, Hoboken, NJ: Wiley; 
R{\'e}nyi, A., (1970) Probability theory, , Dordrecht: North-Holland; 
Richard, M., Lippmann, R., Neural network classifiers estimate Bayesian a posteriori probabilities (1991) Neural Computation, 3, pp. 461-483; 
Santos, J., (2007) Data classification with neural networks and entropic criteria, , Unpublished doctoral dissertation, University of Porto; 
Santos, J.M., Marques de S{\'a}, J., Alexandre, L.A., Batch-sequential algorithm for neural networks trained with entropic criteria (2005) Proceedings of the International Conference on Artificial Neural Networks, pp. 91-96. , Berlin: Springer-Verlag; 
Santos, J.M., Marques de S{\'a}, J., Alexandre, L.A., Legclust-a clustering algorithm based on layered entropic subgraphs (2007) IEEE Transactions on Pattern Analysis and Machine Intelligence, 30 (1), pp. 62-75; 
Santos, J.M., Marques de S{\'a}, J., Alexandre, L.A., Sereno, F., Optimization of the error entropy minimization algorithm for neural network classification (2004) Intelligent engineering systems through artificial neural networks, pp. 81-86. , C. Dagli, A. Buczak, D. Enke, M. Embrechts, \& O. Ersoy (Eds.), Washington, DC: ASME; 
Silva, L.M., (2008) Neural networks with error-density risk functionals for data classification, , Unpublished doctoral dissertation, University of Porto; 
Silva, L.M., Embrechts, M., Santos, J.M., Marques de S{\'a}, J., The influence of the risk functional in data classification with MLPs (2008) Proceedings of the International Conference on Artificial Neural Networks, pp. 185-194. , Berlin: Springer-Verlag; 
Silva, L.M., Felgueiras, C., Alexandre, L.A., Marques de S{\'a}, J., Error entropy in classification problems: A univariate data analysis (2006) Neural Computation, 18 (9), pp. 2036-2061; 
Silva, L.M., Marques de S{\'a}, J., Alexandre, L.A., Neural network classification using Shannon{\textquoteright}s entropy (2005) Proceedings of the European Symposium on Artificial Neural Networks, pp. 217-222. , Bruges: d-side; 
Stoller, D., Univariate two-population distribution free discrimination (1954) Journal of the American Statistical Association, 49, pp. 770-777; 
Thompson, J., Tapia, R., (1978) Nonparametric probability density estimation, , Baltimore, MD: Johns Hopkins University Press; 
Watanabe, S., Pattern recognition as a quest for minimum entropy (1981) Pattern Recognition, 13 (5), pp. 381-387
},
	month = {2010///},
	pages = {2698 - 2728},
	abstract = {This letter focuses on the issue of whether risk functionals derived from information-theoretic principles, such as Shannon or R{\'e}nyi{\textquoteright}s entropies, are able to cope with the data classification problem in both the sense of attaining the risk functional minimum and implying the minimum probability of error allowed by the family of functions implemented by the classifier, here denoted by min Pe. The analysis of this so-called minimization of error entropy (MEE) principle is carried out in a single perceptron with continuous activation functions, yielding continuous error distributions. In spite of the fact that the analysis is restricted to single perceptrons, it reveals a large spectrum of behaviors that MEE can be expected to exhibit in both theory and practice. In what concerns the theoretical MEE, our study clarifies the role of the parameters controlling the perceptron activation function (of the squashing type) in often reaching the minimum probability of error. Our study also clarifies the role of the kernel density estimator of the error density in achieving the minimum probability of error in practice. {\textcopyright} 2010 Massachusetts Institute of Technology.},
	keywords = {algorithm, Algorithms, artificial intelligence, artificial neural network, automated pattern recognition, Computer simulation, entropy, letter, Mathematical Concepts, mathematical phenomena, Models, Statistical, Neural Networks (Computer), Pattern Recognition, Automated, standard, statistical model},
	isbn = {08997667 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-78149334887\&partnerID=40\&md5=97c7445cfa4693d406d979afc4b42319},
	author = {Silva, L.M. and Marques de S{\'a}, J. and Alexandre, L.A.}
}
@book {2299,
	title = {Decision trees using the minimum Entropy-of-error principle},
	series = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Lect. Notes Comput. Sci.},
	volume = {5702 LNCS},
	year = {2009},
	note = {Conference code: 77256Export Date: 21 November 2012Source: Scopusdoi: 10.1007/978-3-642-03767-2_97Language of Original Document: EnglishCorrespondence Address: De S{\'a}, J. P. M.; INEB-Instituto de Engenharia Biom{\'e}dica, Porto, Portugal; email: jmsa@fe.up.ptReferences: Asuncion, A., Newman, D.J., (2007) UCI Machine Learning Repository, , http://www.ics.uci.edu/̃mlearn/MLRepository.html, Univ. of California, SICS, Irvine, CA;Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J., (1993) Classification and Regression Trees, , Chapman \& Hall/CRC, Boca Raton; 
Devroye, L., Gi{\"o}rfi, L., Lugosi, G., (1996) A Probabilistic Theory of Pattern Recognition, , Springer, Heidelberg; 
Forina, M., Armanino, C., Eigenvector Projection and Simplified Nonlinear Mapping of Fatty Acid Content of Italian Olive Oils (1981) Ann. Chim, 72, pp. 127-155; 
Loh, W.-Y., Shih, Y.-S., Split Selection Methods for Classification Trees (1997) Statistica Sinica, 7, pp. 815-840; 
Marques de S{\'a}, J.P., (2007) Applied Statistics Using SPSS, STATISTICA, MATLAB and R, , 2nd edn. Springer, Heidelberg; 
Quinlan, J.R., Induction of Decision Trees (1986) Machine Learning, 1, pp. 81-106; 
Quinlan, J.R.: C4.5 Programs for Machine Learning. Morgan Kaufmann, San Francisco (1993)Silva, L.M., Felgueiras, C.S., Alexandre, L.A., Marques de S{\'a}, J., Error Entropy in Classification Problems: A Univariate Data Analysis (2006) Neural Comp, 18, pp. 2036-2061; 
Silva, L.M., Embrechts, M.J., Santos, J.M., de S{\'a}, J.M., The influence of the risk functional in data classification with mLPs (2008) LNCS, 5163, pp. 185-194. , K{\r u}rkov{\'a}, V, Neruda, R, Koutn{\'\i}k, J, eds, ICANN 2008, Part I, Springer, Heidelberg
Sponsors: University of Munster; International Association for Pattern Recognition; Olympus Soft Imaging Solutions GmbH; Philips},
	month = {2009///},
	pages = {799 - 807},
	address = {Munster},
	abstract = {Binary decision trees based on univariate splits have traditionally employed so-called impurity functions as a means of searching for the best node splits. Such functions use estimates of the class distributions. In the present paper we introduce a new concept to binary tree design: instead of working with the class distributions of the data we work directly with the distribution of the errors originated by the node splits. Concretely, we search for the best splits using a minimum entropy-of-error (MEE) strategy. This strategy has recently been applied in other areas (e.g. regression, clustering, blind source separation, neural network training) with success. We show that MEE trees are capable of producing good results with often simpler trees, have interesting generalization properties and in the many experiments we have performed they could be used without pruning. {\textcopyright} 2009 Springer Berlin Heidelberg.},
	keywords = {Binary decision trees, Binary trees, Blind source separation, Class distributions, decision trees, Electronic medical equipment, entropy, entropy-of-error, Generalization properties, Image analysis, Minimum entropy, Neural network training, Neural networks, New concept, node split criteria, Univariate},
	isbn = {03029743 (ISSN); 3642037666 (ISBN); 9783642037665 (ISBN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-70349306674\&partnerID=40\&md5=39d60bd9d77f6ad61720d8ddd3f68e4d},
	author = {De S{\'a}, J.P.M. and Gama, J. and Sebasti{\~a}o, R. and Alexandre, L.A.}
}
@article {2335,
	title = {Data classification with multilayer perceptrons using a generalized error function},
	journal = {Neural NetworksNeural Netw.},
	volume = {21},
	year = {2008},
	note = {Cited By (since 1996): 6Export Date: 21 November 2012Source: ScopusCODEN: NNETEdoi: 10.1016/j.neunet.2008.04.004PubMed ID: 18572384Language of Original Document: EnglishCorrespondence Address: Silva, L.M.; INEB-Instituto de Engenharia Biom{\'e}dica, Porto, Portugal; email: lmsilva@fe.up.ptReferences: Baum, E.B., Wilczek, F., Supervised learning of probability distributions by neural networks (1987) NIPS, pp. 52-61. , Anderson D.Z. (Ed), American Institute of Physics;Bishop, C., (1995) Neural networks for pattern recognition, , Oxford University Press; 
Blake, C., \& Merz, C. (1998). UCI repository of machine learning databases. University of California, Irvine, Dept. of Information and Computer SciencesForina, M., Armanino, C., Eigenvector projection and simplified non-linear mapping of fatty acid content of italian olive oils (1981) Annali di Chimica (Rome), 50, pp. 127-155; 
Gish, H. (1990). A probabilistic approach to the understanding and training of neural network classifiers. In Proceedings of the 1990 IEEE international conference on acoustics, speech, and signal processingHampshire, J., Waibel, A., A novel objective function for improved phoneme recognition using time-delay neural networks (1990) IEEE Transactions on Neural Networks, 1 (2), pp. 216-228; 
Hinton, G.E., Connectionist learning procedures (1989) Artificial Intelligence, 40 (1-3), pp. 185-234; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., Adaptive mixtures of local experts (1991) Neural Computation, 3, pp. 79-87; 
M{\o}ller, M. (1993). Efficient training of feed-forward neural networks. Ph.D. thesis. Computer Science Department - Aarhus UniversityMarques de S{\'a}, J., (2001) Pattern recognition: Concepts, methods and applications, , Springer Verlag; 
Matsuoka, K., \& Yi, J. (1991). Backpropagation based on the logarithmic error function and elimination of local minima. In Proceedings of the 1990 IEEE international joint conference on neural networksRichard, M., Lippmann, R., Neural network classifiers estimate bayesian a posteriori probabilities (1991) Neural Computation, 3, pp. 461-483; 
Santos, J., Alexandre, L., \& Marques~de S{\'a}, J. (2004). The error entropy minimization algorithm for neural network classification. In int. conf. on recent advances in soft computingSilva, L., Alexandre, L., Marques de S{\'a}, J., Neural network classification: Maximizing zero-error density (2005) LNCS, 3686, pp. 127-135. , ICAPR 2005; 
Silva, L., Alexandre, L.A., \& Marques~de S{\'a}, J. (2006). New developments of the Z-EDM algorithm. In Proceedings of the sixth international conference intelligent systems design and applications: Vol.~1 (pp. 1067-1072)Silva, L., Marques~de S{\'a}, J., \& Alexandre, L. (2005). Neural network classification using Shannon{\textquoteright}s entropy. In European symposium on artificial neural networksSilverman, B., (1986) Density estimation for statistics and data analysis, , Chapman \& Hall; 
Solla, S.A., Levin, E., Fleisher, M., Accelerated learning in layered neural networks (1988) Complex Systems, 2 (6), pp. 625-639
},
	month = {2008///},
	pages = {1302 - 1310},
	abstract = {The learning process of a multilayer perceptron requires the optimization of an error function E (y, t) comparing the predicted output, y, and the observed target, t. We review some usual error functions, analyze their mathematical properties for data classification purposes, and introduce a new one, E                        Exp, inspired by the Z-EDM algorithm that we have recently proposed. An important property of E
                        Exp is its ability to emulate the behavior of other error functions by the sole adjustment of a real-valued parameter. In other words, E
                        Exp is a sort of generalized error function embodying complementary features of other functions. The experimental results show that the flexibility of the new, generalized, error function allows one to obtain the best results achievable with the other functions with a performance improvement in some cases. {\textcopyright} 2008 Elsevier Ltd. All rights reserved.
},
	keywords = {algorithm, Algorithms, analytical error, article, calculation, Classification, Complementary features, Cybernetics, data analysis, Data classification, Data classifications, Data Interpretation, Statistical, Electric loads, entropy, Error functions, learning, Learning processes, mathematical analysis, Mathematical properties, Models, Statistical, Multilayer neural networks, Multilayer perceptron, Multilayer perceptrons, Multilayers, Neural networks, Neural Networks (Computer), Pattern recognition systems, perceptron, Performance improvements, priority journal, Probability density function},
	isbn = {08936080 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-54449100428\&partnerID=40\&md5=30f8de18c0272f4a440395045bd3c164},
	author = {Silva, L.M. and Marques de S{\'a}, J. and Alexandre, L.A.}
}
@article {2365,
	title = {LEGClust - A clustering algorithm based on layered entropic subgraphs},
	journal = {IEEE Transactions on Pattern Analysis and Machine IntelligenceIEEE Trans Pattern Anal Mach Intell},
	volume = {30},
	year = {2008},
	note = {Cited By (since 1996): 8Export Date: 21 November 2012Source: ScopusCODEN: ITPIDdoi: 10.1109/TPAMI.2007.1142Language of Original Document: EnglishCorrespondence Address: Santos, J.M.; Department of Mathematics, ISEP - Polytechnic School of Engineering, R. Dr. A. Bernardino de Almeida 431, 4200-072 Porto, Portugal; email: jms@isep.ipp.ptReferences: Jain, A.K., Dubes, R.C., (1988) Algorithms for Clustering Data, , Prentice Hall International;Jain, A.K., Murty, M.N., Flynn, P.J., Data clustering: A review (1999) ACM Computing Surveys, 31 (3), pp. 264-323; 
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},
	month = {2008///},
	pages = {62 - 75},
	abstract = {Hierarchical clustering is a stepwise clustering method usually based on proximity measures between objects or sets of objects from a given data set. The most common proximity measures are distance measures. The derived proximity matrices can be used to build graphs, which provide the basic structure for some clustering methods. We present here a new proximity matrix based on an entropic measure and also a clustering algorithm (LEGClust) that builds layers of subgraphs based on this matrix, and uses them and a hierarchical agglomerative clustering technique to form the clusters. Our approach capitalizes on both a graph structure and a hierarchical construction. Moreover, by using entropy as a proximity measure we are able, with no assumption about the cluster shapes, to capture the local structure of the data, forcing the clustering method to reflect this structure. We present several experiments on artificial and real data sets that provide evidence on the superior performance of this new algorithm when compared with competing ones. {\textcopyright} 2007 IEEE.},
	keywords = {algorithm, Algorithms, article, artificial intelligence, automated pattern recognition, Cluster Analysis, Clustering, Clustering algorithms, computer assisted diagnosis, Distance measurement, Entropic subgraphs, entropy, Graph theory, Graphs, Hierarchical clustering, Hierarchical systems, Image enhancement, Image Interpretation, Computer-Assisted, Imaging, Three-Dimensional, methodology, Pattern Recognition, Automated, reproducibility, Reproducibility of Results, sensitivity and specificity, three dimensional imaging},
	isbn = {01628828 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-36448984894\&partnerID=40\&md5=f549049e4ae5bff1f3a95aaae22cefb8},
	author = {Santos, J.M. and De S{\'a}, J.M. and Alexandre, L.A.}
}
@article {2405,
	title = {Error entropy in classification problems: A univariate data analysis},
	journal = {Neural ComputationNeural Comp.},
	volume = {18},
	year = {2006},
	note = {Cited By (since 1996): 7Export Date: 21 November 2012Source: Scopusdoi: 10.1162/neco.2006.18.9.2036PubMed ID: 16846386Language of Original Document: EnglishCorrespondence Address: Silva, L.M.; Institute de Engenharia Biom{\'e}dica, Laborat{\'o}rio Sinal e Imagem Biom{\'e}dica, 4200-465, Porto, Portugal; email: lmsilva@fe.up.ptReferences: Devroye, L., Gyorfi, L., Lugosi, G., (1996) A Probabilistic Theory of Pattern Recognition, , Berlin: Springer-Verlag;Erdogmus, D., Principe, J., Comparison of entropy and mean square error criteria in adaptive system training using higher order statistics (2000) Proceedings of the Intl. Conf., on ICA and Signal Separation, pp. 75-80. , Helsinki, Finland; 
Erdogmus, D., Principe, J.C., An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems (2002) IEEE Transactions on Signal Processing, 50 (7), pp. 1780-1786; 
Kapur, J., (1993) Maximum-entropy Models in Science and Engineering (Rev. Ed), , New York: Wiley; 
Kullback, S., (1959) Statistics and Information Theory, , New York: Wiley; 
Linsker, R., Self-organization in a perceptual network (1988) IEEE Computer, 21, pp. 105-117; 
Marques De Sa, J., (2001) Pattern Recognition: Concepts, Methods and Applications, , Berlin: Springer-Verlag; 
Newman, D., Hettich, S., Blake, C., Merz, C., (1998) UCI Repository of Machine Learning Databases, , http://www.ics.uci.edu/~mlearn/MLRepository.html, Irvine: University of California, Irvine, Department of Information and Computer Sciences; 
Parzen, E., On the estimation of a probability density function and the mode (1962) Annals of Mathematical Statistics, 33, pp. 1065-1076; 
Press, W., Teukolsky, S., Vetterling, W., Flannery, B., (1992) Numerical Recipes in C: The Art of Scientific Computing, , Cambridge: Cambridge University Press; 
Principe, J.C., Xu, D., Fisher, J., Information theoretic learning (2000) Unsupervised Adaptive Filtering, Vol. 1: Blind Source Separation, 1, pp. 265-319. , S. Haykin (Ed.), New York: Wiley; 
Santos, J., Alexandre, L., Marques De S{\'a}, J., The error entropy minimization algorithm for neural network classification (2004) Int. Conf. on Recent Advances in Soft Computing, , Nottingham, U.K.: Nottingham Trent University; 
Santos, J., Marques De Sa, J., Alexandre, L., Sereno, F., Optimization of the error entropy minimization algorithm for neural network classification (2004) Intelligent Engineering Systems Through Artificial Neural Networks, 14, pp. 81-86. , C. Dagli, A. Buczak, D. Enke, M. Embrechts, \& O. Ersoy (Eds.), New York: American Society of Mechanical Engineers; 
Shannon, C.E., A mathematical theory of communication (1948) Bell System Technical Journal, 27, pp. 379-423; 
Silva, L., Marques De Sa, J., Alexandre, L., Neural network classification using Shannon{\textquoteright}s entropy (2005) European Symposium on Artificial Neural Networks, , Bruges, Belgium: d-side publications; 
Silverman, B., (1986) Density Estimation for Statistics and Data Analysis, , London: Chapman \& Hall; 
Stoller, D., Univariate two-population distribution free discrimination (1954) Journal of the American Statistical Association, 49, pp. 770-777
},
	month = {2006///},
	pages = {2036 - 2061},
	abstract = {Entropy-based cost functions are enjoying a growing attractiveness in unsupervised and supervised classification tasks. Better performances in terms both of error rate and speed of convergence have been reported. In this letter, we study the principle of error entropy minimization (EEM) from a theoretical point of view. We use Shannon{\textquoteright}s entropy and study univariate data splitting in two-class problems. In this setting, the error variable is a discrete random variable, leading to a not too complicated mathematical analysis of the error entropy. We start by showing that for uniformly distributed data, there is equivalence between the EEM split and the optimal classifier. In a more general setting, we prove the necessary conditions for this equivalence and show the existence of class configurations where the optimal classifier corresponds to maximum error entropy. The presented theoretical results provide practical guidelines that are illustrated with a set of experiments with both real and simulated data sets, where the effectiveness of EEM is compared with the usual mean square error minimization. {\textcopyright} 2006 Massachusetts Institute of Technology.},
	keywords = {article, Classification, entropy, methodology, Models, Theoretical, normal distribution, theoretical model},
	isbn = {08997667 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-33748288090\&partnerID=40\&md5=47b51b3d90bc273df156889a92fef01d},
	author = {Silva, L.M. and Felgueiras, C.S. and Alexandre, L.A. and Marques de S{\'a}, J.}
}
@book {2395,
	title = {Error entropy minimization for LSTM training},
	series = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Lect. Notes Comput. Sci.},
	volume = {4131 LNCS - I},
	year = {2006},
	note = {Conference code: 68317Cited By (since 1996): 3Export Date: 21 November 2012Source: ScopusLanguage of Original Document: EnglishCorrespondence Address: Alexandre, L.A.; Department of Informatics, IT-Networks and Multimedia Group, University of Beira Interior, Covilh{\~a}, Portugal; email: lfbaa@di.ubi.ptReferences: Hochreiter, S., Schmidhuber, J., Long short-term memory (1997) Neural Computation, 9 (8), pp. 1735-1780;Gers, F., Schmidhuber, J., Cummins, F., Learning to forget: Continual prediction with LSTM (2000) Neural Computation, 12 (10), pp. 2451-2471; 
Gers, F., Schmidhuber, J., Recurrent nets that time and count (2000) Proc. IJCNN{\textquoteright}2000, Int. Joint Conf. on Neural Networks, , Como, Italy; 
P{\'e}rez-Ortiz, J., Gers, F., Eck, D., Schmidhuber, J., Kalman filters improve LSTM network performance in problems unsolvable by traditional recurrent nets (2003) Neural Networks, 16 (2), pp. 241-250; 
Haykin, S., (1999) Neural Networks: A Comprehensive Foundation, 2nd Edition, , Prentice Hall; 
Erdogmus, D., Principe, J., An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems (2002) IEEE Trans. Signal Processing, 50 (7), pp. 1780-1786; 
Santos, J., Alexandre, L., Sereno, F., Marques De S{\'a}, J., Optimization of the error entropy minimization algorithm for neural network classification (2004) ANNIE 2004, Intelligent Engineering Systems Through Artificial Neural Networks, 14, pp. 81-86. , St.Louis, USA, ASME Press Series; 
Santos, J., Alexandre, L., Marques De S{\'a}, J., The error entropy minimization algorithm for neural network classification (2004) Proceedings of the 5th International Conference on Recent Advances in Soft Computing, pp. 92-97. , Lofti, A., ed.: Nottingham, United Kingdom; 
Silva, L., Marques De S{\'a}, J., Alexandre, L., Neural network classification using Shannon{\textquoteright}s entropy (2005) 13th European Symposium on Artificial Neural Networks - ESANN 2005, pp. 217-222. , Bruges, Belgium
},
	month = {2006///},
	pages = {244 - 253},
	address = {Athens},
	abstract = {In this paper we present a new training algorithm for the Long Short-Term Memory (LSTM) recurrent neural network. This algorithm uses entropy instead of the usual mean squared error as the cost function for the weight update. More precisely we use the Error Entropy Minimization approach, were the entropy of the error is minimized after each symbol is present to the network. Our experiments show that this approach enables the convergence of the LSTM more frequently than with the traditional learning algorithm. This in turn relaxes the burden of parameter tuning since learning is achieved for a wider range of parameter values. The use of EEM also reduces, in some cases, the number of epochs needed for convergence. {\textcopyright} Springer-Verlag Berlin Heidelberg 2006.},
	keywords = {Algorithms, Convergence of numerical methods, Cost function, entropy, Error Entropy Minimization, Errors, Long Short-Term Memory (LSTM), Neural networks, Optimization},
	isbn = {03029743 (ISSN); 3540386254 (ISBN); 9783540386254 (ISBN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-33749837658\&partnerID=40\&md5=5f40159a3f1faf4760405dd581e79bf7},
	author = {Alexandre, L.A. and Marques De S{\'a}, J.P.}
}
@conference {2390,
	title = {Modular neural network task decomposition via entropic clustering},
	booktitle = {Proceedings - ISDA 2006: Sixth International Conference on Intelligent Systems Design and ApplicationsProc. ISDA Sixth Int. Conf. Intelligent Syst. Design Applic.},
	series = {ISDA 2006: Sixth International Conference on Intelligent Systems Design and Applications},
	volume = {1},
	year = {2006},
	note = {Conference code: 69992Cited By (since 1996): 1Export Date: 21 November 2012Source: ScopusArt. No.: 4021410doi: 10.1109/ISDA.2006.198Language of Original Document: EnglishCorrespondence Address: Santos, J.M.; Instituto Superior de Engenharia do Porto, Instituto de Engenharia Biom{\'e}dica, Porto, Portugal; email: jms@isep.ipp.ptReferences: Santos, J.M., Marques de S{\'a}, J., Alexandre, L.A., LEGClust - a clustering algorithm based on layered entropie subgraphs (2005) IEEE Transactions on Pattern Analysis and Machine Intelligence, , submitted;Bauer, E., Kohavi, R., An empirical comparison of voting classification algorithms: Bagging, boosting, and variants (1999) Machine Learning, 36 (1-2), pp. 105-139; 
Dietterich, T.G., An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting, and randomization (2000) Machine Learning, 40 (2), pp. 139-157; 
Kittler, J., Hatef, M., Duin, R.P.W., Matas, J., On combining classifiers (1998) IEEE Trans. Pattern Anal. Mach. Intell, 20 (3), pp. 226-239; 
Allwein, E.L., Schapire, R.E., Singer, Y., Reducing multiclass to binary: A unifying approach for margin classifiers (2000) Proc. 17th International Conf on Machine Learning, pp. 9-16. , Morgan Kaufmann, San Francisco, CA; 
Alexandre, L.A., Campilho, A.C., Kamel, M., Bounds for the average generalization error of the mixture of experts neural network (2004) LNCS, 3138, pp. 618-625. , 5th International Workshop on Statistical Techniques in Pattern Recognition; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., Adaptive mixtures of local experts (1991) Neural Computation, 3, pp. 79-87; 
Auda, G., Kamel, M., Modular neural network classifiers: A comparative study (1998) Intel. Robotic Systems, 21, pp. 117-129; 
Jordan, M., Jacobs, R., Hierarchical mixture of experts and the EM algorithm (1994) Neural Computation, 6, pp. 181-214; 
Jacobs, R., Peng, F., Tanner, M., A bayesian approach to model selection in hierarchical mixtures-of-experts architectures (1997) Neural Networks, 10 (2), pp. 231-241; 
Vilalta, R., Achari, M.K., Eick, C.F., Class decomposition via clustering: A new framework for lowvariance classifiers (2003) Third IEEE International Conference on Data Mining, pp. 673-676; 
Alexandre, L.A., Campilho, A.C., Kamel, M., A probabilistic model for the cooperative modular neural network (2003) LNCS, 2652, pp. 11-18. , Proceedings of the IbPRIA 2003; 
Ennaji, A., Ribert, A., Lecourtier, Y., From data topology to a modular classifier (2003) International Journal on Document Analysis and Recognition, 6 (1), pp. 1-9; 
Ng, A.Y., Jordan, M.I., Weiss, Y., On spectral clustering: Analysis and an algorithm (2001) Advances in Neural Information Processing Systems, 14; 
Bowman, A.W., Azzalini, A., (1997) Applied Smooting Techniques for Data Analysis, , London:Oxford University Press; 
Marques de S{\'a}, J., (2003) Applied statistics using SPSS, STATISTICA andMATLAB, , Springer; 
Blake, C., Keogh, E., Merz, C., (1998) UCI repository of machine learning databases; 
Forina, M., Armanino, C., Eigenvector projection and simplified non-linear mapping of fatty acid content of Italian olive oils (1981) Ann. Chim. (Rome), 72, pp. 127-155
},
	month = {2006///},
	pages = {62 - 67},
	address = {Jinan},
	abstract = {The use of monolithic neural networks (such as a multilayer perceptron) has some drawbacks: e.g. slow learning, weight coupling, the black box effect. These can be alleviated by the use of a modular neural network. The creation of a MNN has three steps: task decomposition, module creation and decision integration. In this paper we propose the use of an entropie clustering algorithm as a way of performing task decomposition. We present experiments on several real world classification problems that show the performance of this approach. {\textcopyright} 2006 IEEE.},
	keywords = {Classification (of information), Clustering algorithms, Decision Theory, Entropic clustering, Learning systems, Modular neural network task decomposition, Monolithic neural networks, Neural networks, Weight coupling},
	isbn = {0769525288 (ISBN); 9780769525280 (ISBN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-34547502901\&partnerID=40\&md5=c784fc5fb020181d30a63f65ed112c41},
	author = {Santos, J.M. and Alexandre, L.A. and De S{\'a}, J.M.}
}
@book {2397,
	title = {A multiclassifier approach for lung nodule classification},
	series = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Lect. Notes Comput. Sci.},
	volume = {4142 LNCS},
	year = {2006},
	note = {Conference code: 68310Cited By (since 1996): 3Export Date: 21 November 2012Source: ScopusLanguage of Original Document: EnglishCorrespondence Address: Pereira, C.S.; Instituto de Engenharia Biom{\'e}dica, Rua Roberto Frias, 4200-465 Porto, Portugal; email: cpereira@estgl.ipv.ptReferences: Jemal, A., Murray, T., Ward, E., Samuels, A., Tiwari, R.C., Ghafoor, A., Feuer, E.J., Thun, M.J., (2005) Cancer Statistics, 2005, 55, pp. 10-30. , CA Cancer J Clin;Keserci, B., Yoshida, H., Computerized detection of pulmonary nodules in chest radiographs based on morphological features and wavelet snake model (2002) Medical Image Analysis, 6, pp. 431-447; 
Van Ginneken, B., Romeny, B.M.H., Viergever, M.A., Computer-aided diagnosis in chest radiography: A survey (2001) IEEE Transactions on Medical Imaging, 20 (12), pp. 1228-1241; 
Wei, J., Hagihara, Y., Shimizu, A., Kobatake, H., Optimal image feature set for detecting lung nodules on chest X-ray images (2002) CARS 2002; 
Van Ginneken, B., Katsuragawa, S., Ter Haar Romeny, B.M., Kunio, D., Viergever, M.A., Automatic detection of abnormalities in chest radiographs using local texture analysis (2002) IEEE Transactions on Medical Imaging, 21, pp. 139-149; 
Schilham, A.M.R., Van Ginneken, B., Loog, M., Multi-scale nodule detection in chest radiographs (2003) Lecture Notes in Computer Science, LNCS 2878, pp. 602-609; 
Shiraishi, J., Katsuragawa, S., Ikezoe, J., Matsumoto, T., Kobayashi, T., Matsu, K.-I.K., Matsui, M., Doi, K., Development of a digital image database for chest radiographs with and without a lung nodule: Receiver operating characteristic analysis of radiologists{\textquoteright} detection of pulmonary nodules (2000) American Journal of Roentgenology, 174, pp. 71-74; 
Yoshida, H., Local contralateral subtraction based on bilateral symmetry of lung for reduction of false positives in computerized detection of pulmonary nodules (2004) IEEE Transactions on Biomedical Engineering, 51, pp. 778-789; 
Suzuki, K., Shiraishi, J., Abe, H., MacMahon, H., Doi, K., False-positive reduction in computer-aided diagnostic scheme for detecting nodules in chest radiographs by means of massive training artificial neural network (2005) Academic Radiology, 12, pp. 191-201; 
Randen, T., Husoy, J.H., Filtering for texture classification: A comparative study (1999) IEEE Transactions on Pattern Analysis and Machine Intelligence, 21 (4); 
Grigorescu, S.E., Petkov, N., Kruizinga, P., Comparison of texture features based on Gabor filters (2002) IEEE Transactions on Image Processing, 11 (10), pp. 195-203; 
Kumar, A., Pang, G.K.H., Defect detection in textured materials using Gabor filters (2002) IEEE Transactions on Industry Applications, 38 (2); 
Varma, M., Zisserman, A., Unifying statistical texture classification frameworks (2004) Image and Vision Computing, 22, pp. 1175-1183; 
Leung, T., Malik, J., Representing and recognizing the visual appearance of materials using three-dimensional textons (2001) International Journal of Computer Vision, 43, pp. 29-44; 
Alexandre, L.A., Campilho, A.C., Kamel, M., On combining classifiers using sum and product rules (2001) Pattern Recognition Letters, 22, pp. 1283-1289
},
	month = {2006///},
	pages = {612 - 623},
	address = {Povoa de Varzim},
	abstract = {The aim of this paper is to examine a multiclassifier approach to the classification of the lung nodules in X-ray chest radiographs. The approach investigated here is based on an image region-based classification whose output is the information of the presence or absence of a nodule in an image region. The classification was made, essentially, in two steps: firstly, a set of rotation invariant features was extracted from the responses of a multi-scale and multi-orientation filter bank; secondly, different classifiers (multi-layer perceptrons) are designed using different features sets and trained in different data. These classifiers are further combined in order to improve the classification performance. The obtained results are promising and can be used for reducing the false-positives nodules detected in a computer-aided diagnosis system. {\textcopyright} Springer-Verlag Berlin Heidelberg 2006.},
	keywords = {Biological organs, Classification (of information), Computer aided diagnosis, False-positives nodules, Medical imaging, Multi-orientation filter bank, Radiography, X rays, X-ray chest radiographs},
	isbn = {03029743 (ISSN); 3540448942 (ISBN); 9783540448945 (ISBN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-33749658606\&partnerID=40\&md5=f6e83e868f5b1ee08b400ee9191138cd},
	author = {Pereira, C.S. and Alexandre, L.A. and Mendon{\c c}a, A.M. and Campilho, A.}
}
@conference {2389,
	title = {New developments of the Z-EDM algorithm},
	booktitle = {Proceedings - ISDA 2006: Sixth International Conference on Intelligent Systems Design and ApplicationsProc. ISDA Sixth Int. Conf. Intelligent Syst. Design Applic.},
	series = {ISDA 2006: Sixth International Conference on Intelligent Systems Design and Applications},
	volume = {1},
	year = {2006},
	note = {Conference code: 69992Cited By (since 1996): 2Export Date: 21 November 2012Source: ScopusArt. No.: 4021587doi: 10.1109/ISDA.2006.204Language of Original Document: EnglishCorrespondence Address: Silva, L.M.; INEB - Instituto de Engenharia Biom{\'e}dica, Porto, Portugal; email: lmsilva@fe.up.ptReferences: Bishop, C., (1995) Neural Networks for Pattern Recognition, , Oxford University Press;Blake, C., Merz, C., (1998) UCI Repository of machine learning databases University of California, , http://www.ics.uci.edu/~mlearn/MLRepository.html, Irvine, Dept. of Information and Computer Sciences; 
Erdogmus, D., Principe, J.C., An Error-Entropy Minimization Algorithm for Supervised Training of Nonlinear Adaptive Systems (2002) IEEE Transactions on Signal Processing, 50 (7), pp. 1780-1786; 
Forina, M., Armanino, C., Eigenvector projection and simplified non-linear mapping of fatty acid content of Italian olive oils (1981) Ann. Chim. (Rome), 50, pp. 127-155; 
Marques de S{\'a}, J., (2001) Pattern Recognition: Concepts, Methods and Applications, , Springer Verlag; 
J. Santos, L. Alexandre, and J. Marques de S{\'a}. The Error Entropy Minimization Algorithm for Neural Network Classification. In Int. Conf. on Recent Advances in Soft Computing, Nottingham, United Kingdom, 2004Silva, L., Alexandre, L., Marques de S{\'a}, J., Neural network classification: Maximizing zero-error density (2005) LNCS, 3686, pp. 127-135. , ICAPR 2005; 
Silva, L., Marques de S{\'a}, J., Alexandre, L., Neural Network Classification using Shannon{\textquoteright}s Entropy (2005) European Symposium on Artificial Neural Networks; 
Silverman, B., (1986) Density Estimation for Statistics and Data Analysis, , Chapman \& Hall
},
	month = {2006///},
	pages = {1067 - 1072},
	address = {Jinan},
	abstract = {In this paper we address some open questions on the recently proposed Zero-Error Density Maximization algorithm for MLP training. We propose a new version of the cost function that solves a training problem encountered in previous work and prove that the use of a nonparametric density estimator preserves the optimal solution. Some experiments are reported comparing this cost function to the usual mean-square error and cross entropy cost junctions. {\textcopyright} 2006 IEEE.},
	keywords = {Algorithms, Cost functions, Mean square error, MLP training, Nonparametric density estimators, Optimization, Personnel training, Problem solving, Zero-Error Density Maximization algorithm},
	isbn = {0769525288 (ISBN); 9780769525280 (ISBN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-34547496988\&partnerID=40\&md5=65cadef393b0843f5b21f6534bc122ff},
	author = {Silva, L.M. and Marques de S{\'a}, J. and Alexandre, L.A.}
}
@conference {2442,
	title = {Neural network classification: Maximizing zero-error density},
	booktitle = {Lecture Notes in Computer ScienceLect. Notes Comput. Sci.},
	series = {Third International Conference on Advances in Patten Recognition, ICAPR 2005},
	volume = {3686},
	year = {2005},
	note = {Conference code: 65875Cited By (since 1996): 2Export Date: 21 November 2012Source: ScopusLanguage of Original Document: EnglishCorrespondence Address: Silva, L.M.; Instituto de Engenharia Biom{\'e}dica, Lab. de Sinai e Imagem Biom{\'e}dica, Campus da FEUP, Rua Dr. Roberto Frias, 4200 - 465 Porto, PortugalReferences: Erdogmus, D., Principe, J.C., An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems (2002) IEEE Transactions on Signal Processing, 50 (7), pp. 1780-1786;Erdogmus, D., Principe, J., Generalized information potential criterion for adaptive system training (2002) IEEE Transactions on Neural Networks, 13 (5), pp. 1035~1044; 
Santos, J.M., Alexandre, L.A., Marques De S{\'a}, J., The error entropy minimization algorithm for neural network classification (2004) Int. Conf. on Recent Advances in Soft Computing, , Nottingham, United Kingdom; 
Santos, J.M., Alexandre, L.A., De Sa, J.M., Optimization of the error entropy minimization algorithm for neural network classification (2004) Intelligent Engineering Systems Through Artificial Neural Networks, 14, pp. 81-86. , C. Dagli, A. Buczak, D. Enke, M. Embrechts, and O. Ersoy, editors, ASME Press Series; 
Silva, L.M., De Sa, J.M., Alexandre, L.A., Neural network classification using shannon{\textquoteright}s entropy (2005) European Symposium on Artificial Neural Networks; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., Adaptive mixtures of local experts (1991) Neural Computation, 3, pp. 79-87; 
Blake, C.L., Merz, C.J., (1998) UCI Repository of Machine Learning Databases, , http://www.ics.uci.edu/~mlearn/MLRepository.html, University of California, Irvine, Dept. of Information and Computer Sciences
},
	month = {2005///},
	pages = {127 - 135},
	address = {Bath},
	abstract = {We propose a new cost function for neural network classification: the error density at the origin. This method provides a simple objective function that can be easily plugged in the usual backpropagation algorithm, giving a simple and efficient learning scheme. Experimental work shows the effectiveness and superiority of the proposed method when compared to the usual mean square error criteria in four well known datasets. {\textcopyright} Springer-Verlag Berlin Heidelberg 2005.},
	keywords = {Algorithms, Backpropagation, Backpropagation algorithm, Classification (of information), Costs, Error analysis, Function evaluation, Learning systems, Mean square error, Neural network classification, Neural networks, Zero-error density},
	isbn = {03029743 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-27244450344\&partnerID=40\&md5=73ee3c0c073fb3bc7bc85013bf7dcce6},
	author = {Silva, L.M. and Alexandre, L.A. and De S{\'a}, J.M.},
	editor = {Singh S. and Singh M. and Apte C. and Perner P.}
}
@article {2457,
	title = {Neural networks trained with the EEM algorithm: Tuning the smoothing parameter},
	journal = {WSEAS Transactions on SystemsWSEAS Trans. Syst.},
	volume = {4},
	year = {2005},
	note = {Cited By (since 1996): 3Export Date: 21 November 2012Source: ScopusLanguage of Original Document: EnglishCorrespondence Address: Santos, J.M.; Intituto de Engenharia Biom{\'e}dica, Porto, Portugal; email: jms@isep.ipp.ptReferences: Erdogmus, D., Principe, J., "An Error-Entropy Minimization Algorithm for Supervised Training of Nonlinear Adaptive Systems" (2002) Trans. On Signal Processing, 50 (7), pp. 1780-1786;Santos, J.M., Alexandre, L.A., Marques de S{\'a}, J., "The error entropy minimization algorithm for neural network classification" (2004) Proceedings of the 5th International Conference on Recent Advances in Soft Computing, pp. 92-97. , A. Lofti, Ed. Nottingham Trent University; 
Santos, J.M., Marques de S{\'a}, J., Alexandre, L.A., Sereno, F., "Optimization of the error entropy minimization algorithm for neural network classification" (2004) Intelligent Engineering Systems Through Artificial Neural Networks, 14, pp. 81-86. , C.H. Dagli, A.L. Buczak, D.L. Enke, M.J. Embrechts, and O. Ersoy, Eds., ASME Press; 
Erdogmus, D., Principe, J., "Entropy Minimization Algorithm for Neural Networks" (2001) Proceedings of the Intl. Joint Conf. on Neural Networks, pp. 3003-3008; 
Xu, D., Princpe, J., "Training mlps layer-by-layer with the information potential" (1999) Intl. Joint Conf. on Neural Networks, pp. 1716-1720; 
Silverman, B.W., (1986) Density Estimation for Statistics and Data Analisys, 26. , Chapman \& Hall; 
Bowman, A.W., Azzalini, A., (1997) Applied Smooting Techniques for Data Analysis, , London: Oxford University Press; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., "Adaptive mixtures of local experts" (1991) Neural Computation, pp. 79-87
},
	month = {2005///},
	pages = {295 - 299},
	abstract = {The training of Neural Networks and particularly Multi-Layer Perceptrons (MLP{\textquoteright}s) is made by minimizing an error function usually known as "cost function". In our previous works, we apply the Error Entropy Minimization (EEM) algorithm in classification and its optimized version using, as cost function, the entropy of the errors between the outputs and the desired targets of the neural network. One of the difficulties in implementing the EEM algorithm is the choice of the smoothing parameter, also known as window size, in the Parzen Window probability density function estimation for the computation of the entropy and its gradient. We present here a formula yielding the value of the smoothing parameter, depending on the number of data samples and on the neural network output dimension. Several experiments with real data sets were made in order to show the validity of the proposed formula.},
	keywords = {Cost function, entropy, Error entropy minimization (EEM) algorithm, Learning algorithms, Learning systems, Neural networks, Optimization, Parzen, Probability density function, Smoothing parameter},
	isbn = {11092777 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-21844450579\&partnerID=40\&md5=31c652daee111a68194cd4f006d7ca7b},
	author = {Santos, J.M. and Marques de S{\'a}, J. and Alexandre, L.A.}
}
@book {2462,
	title = {Bounds for the Average Generalization Error of the Mixture of Experts Neural Network},
	series = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Lect. Notes Comput. Sci.},
	volume = {3138},
	year = {2004},
	note = {Cited By (since 1996): 2Export Date: 21 November 2012Source: ScopusLanguage of Original Document: EnglishCorrespondence Address: Alexandre, L.A.; Networks and Multimedia Group, IT, Covilh{\~a}, Portugal; email: lfbaa@di.ubi.ptReferences: Gu, H., Takahashi, H., Towards more practical average bounds on supervised learning (1996) IEEE Trans. Neural Networks, 7, pp. 953-968;Vapnik, V., (1999) The Nature of Statistical Learning Theory., , Springer; 
Haykin, S., (1999) Neural Networks: A Comprehensive Foundation, 2nd Edition., , Prentice Hall; 
Alexandre, L., Campilho, A., Kamel, M., Average error bound for the mixture of experts MNN architecture (2002) Proceedings of the 12th Portuguese Conference on Pattern Recognition, , Aveiro, Portugal; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., Adaptive mixtures of local experts (1991) Neural Computation, pp. 79-87; 
Auda, G., Kamel, M., Modular neural network classifiers: A comparative study (1998) J. Intel. Robotic Systems, pp. 117-129; 
Jordan, M., Jacobs, R., Hierarchical mixture of experts and the em algorithm (1994) Neural Computation, pp. 181-214; 
Jacobs, R., Peng, F., Tanner, M., A bayesian approach to model selection in hierarchical mixtures-of-experts architectures (1997) Neural Networks, 10, pp. 231-241; 
De Marques Sa, J., Pattern Recognition: Concepts (2001) Methods and Applications, , Springer; 
Blake, C., Keogh, E., Merz, C., (1998) UCI repository of machine learning databases, , http://www.ics.uci.edu/~mlearn/MLRepository.html; 
Gu, H., Takahashi, H., Estimating learning curves of concept learning (1997) Neural Networks, 10, pp. 1089-1102; 
Bishop, C., (1995) Neural Networks for Pattern Recognition., , Oxford University Press; 
Riedmiller, M., Braun, H., A direct adaptive method for faster backpropagation learning: The RPROP algorithm (1993) IEEE International Conference on Neural Networks, pp. 586-591. , San Francisco
},
	month = {2004///},
	pages = {618 - 625},
	abstract = {In this paper we derive an upper bound for the average-case generalization error of the mixture of experts modular neural network, based on an average-case generalization error bound for an isolated neural network. By doing this we also generalize a previous bound for this architecture that was restricted to special problems. We also present a correction factor for the original average generalization error, that was empirically obtained, that yields more accurate error bounds for the 6 data sets used in the experiments. These experiments illustrate the validity of the derived error bound for the mixture of experts modular neural network and show how it can be used in practice. {\textcopyright} Springer-Verlag 2004.},
	keywords = {Generalization error bounds, Mixture of experts, Modular neural networks},
	isbn = {03029743 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-35048839786\&partnerID=40\&md5=10183183d399e4055c331416faef909e},
	author = {Alexandre, L.A. and Campilho, A. and Kamel, M.}
}
@book {2504,
	title = {A probabilistic model for the cooperative modular neural network},
	series = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Lect. Notes Comput. Sci.},
	volume = {2652},
	year = {2003},
	note = {Cited By (since 1996): 2Export Date: 21 November 2012Source: ScopusLanguage of Original Document: EnglishCorrespondence Address: Alexandre, L.A.; Departamento de Inform{\'a}tica, Universidade da Beira Interior, Covilh{\~a}, Portugal; email: lfbaa@di.ubi.ptReferences: Auda, G., Kamel, M., Modular neural network classifiers: A comparative study (1998) J. Intel. Robotic Systems, pp. 117-129;De Bollivier, M., Gallinari, P., Thiria, S., Cooperation of neural nets and task decomposition (1991) Int. Joint Conf. on Neural Networks, 2, pp. 573-576. , Seattle, USA; 
Jacobs, R., Jordan, M., Nowlan, S., Hinton, G., Adaptive mixtures of local experts (1991) Neural Computation, pp. 79-87; 
Jacobs, R., Peng, F., Tanner, M., A bayesian approach to model selection in hierarchical mixtures-of-experts architectures (1997) Neural Networks, 10, pp. 231-241; 
Wanas, N., Kamel, M., Auda, G., Karray, F., Feature-based decision aggregation in modular neural network classifiers (1999) Pattern Recognition Letters, 20, pp. 1353-1359; 
Alexandre, L., Campilho, A., Kamel, M., Combining independent and unbiased classifiers using weighted average (2000) Proceedings of the 15th International Conference on Pattern Recognition, 2, pp. 495-498. , Barcelona, Spain, IEEE Press; 
Alexandre, L., Campilho, A., Kamel, M., On combining classifiers using sum and product rules (2001) Pattern Recognition Letters, 22, pp. 1283-1289; 
Kittler, J., Hatef, M., Duin, R., Matas, J., On combining classifiers (1998) IEEE Trans. PAMI, 20, pp. 226-239; 
Auda, G., Kamel, M., CMNN: Cooperative modular neural networks for pattern recognition (1997) Pattern Recognition Letters, 18, pp. 1391-1398; 
Auda, G., Kamel, M., Raafat, H., Voting schemes for cooperative neural network classifiers (1995) IEEE Int. Conference on Neural Networks, 3, pp. 1240-1243. , Australia
},
	month = {2003///},
	pages = {11 - 18},
	abstract = {This paper presents a model for the probability of correct classification for the Cooperative Modular Neural Network (CMNN). The model enables the estimation of the performance of the CMNN using parameters obtained from the data set. The performance estimates for the experiments presented are quite accurate (less than 1\% relative difference). We compare the CMNN with a multi-layer perceptron with equal number of weights and conclude that the CMNN is preferred for complex problems. We also investigate the error introduced by one of the CMNN voting strategies. {\textcopyright} Springer-Verlag Berlin Heidelberg 2003.},
	isbn = {03029743 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-34547529574\&partnerID=40\&md5=fe30a6406b8bc4deb9b0cc36ab961d59},
	author = {Alexandre, L.A. and Campilho, A. and Kamel, M.}
}
@article {2542,
	title = {On combining classifiers using sum and product rules},
	journal = {Pattern Recognition LettersPattern Recogn. Lett.},
	volume = {22},
	year = {2001},
	note = {Cited By (since 1996): 51Export Date: 21 November 2012Source: Scopusdoi: 10.1016/S0167-8655(01)00073-3Language of Original Document: EnglishCorrespondence Address: Alexandre, L.A.; INEB-Inst. de Engenharia Biomedica, Rua Dr. Roberto Frias s/n, Porto 4200-465, Portugal; email: lfbaa@noe.ubi.ptReferences: Alexandre, L., Campilho, A., Kamel, M., Combining independent and unbiased classifiers using weighted average (2000) Proc. 15th Internat. Conf. on Pattern Recognition, 2. , IEEE Press, Barcelona, Spain;Bishop, C., (1995) Neural Networks for Pattern Recognition, , Oxford University Press, Oxford; 
Blake, C., Keogh, E., Merz, C., (1998) UCI Repository of Machine Learning Databases, , http://www.ics.uci.edu/~mlearn/MLRepository.html; 
Demuth, H., Beale, M., (1998) Neural Network Toolbox User{\textquoteright}s Guide, , The MathWorks, Inc; 
Hansen, L., Salamon, P., Neural network ensembles (1990) IEEE Trans. PAMI, 12 (10), pp. 993-1001; 
Hashem, S., Optimal linear combinations of neural networks (1997) Neural Networks, 10 (4), pp. 599-614; 
Kittler, J., Hatef, M., Duin, R., Combining classifiers (1996) Proc. Internat. Conf. on Pattern Recognition{\textquoteright}96; 
Kittler, J., Hatef, M., Duin, R., Matas, J., On combining classifiers (1998) IEEE Trans. PAMI, 20 (3), pp. 226-239; 
Opitz, D., Maclin, R., Popular ensemble methods: An empirical study (1999) J. Artif. Intell. Res, 11, pp. 169-198; 
Richard, M., Lippmann, R., Neural network classifiers estimate Bayesian a posteriori probabilities (1991) Neural Comput, 3, pp. 461-483; 
Rogova, G., Combining the results of several neural network classifiers (1994) Neural Networks, 7 (5), pp. 777-781; 
Tax, D., Duin, R., Breukelen, M., Comparison between product and mean classifier combination rules (1997) Workshop on Statistical Techniques in Pattern Recognition, , Prague Czech Republic; 
Ueda, N., Optimal linear combination of neural networks for improving classification performance (2000) IEEE Trans. PAMI, 22 (2), pp. 207-215
},
	month = {2001///},
	pages = {1283 - 1289},
	abstract = {This paper presents a comparative study of the performance of arithmetic and geometric means as rules to combine multiple classifiers. For problems with two classes, we prove that these combination rules are equivalent when using two classifiers and the sum of the estimates of the a posteriori probabilities is equal to one. We also prove that the case of a two class problem and a combination of two classifiers is the only one where such equivalence occurs. We present experiments illustrating the equivalence of the rules under the above mentioned assumptions. {\textcopyright} 2001 Elsevier Science B.V. All rights reserved.},
	keywords = {artificial neural network, Classification, Classifier fusion, Combinatorial mathematics, Combining classifiers, Equivalence classes, Estimation, K nearest-neighbours, Multiple classifiers, Neural networks, Pattern recognition, probability},
	isbn = {01678655 (ISSN)},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0034875843\&partnerID=40\&md5=becfe9c0c4a7d97f7bbaf33f3059ef8f},
	author = {Alexandre, L.A. and Campilho, A.C. and Kamel, M.}
}
