



@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 Unpublished doctoral dissertation (2007) Rensselaer Polytechnic Institute; 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.}
}
@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.}
}
@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.}
}
