Polynomial Expansion Of The Probability Density Function About Gaussian Mixtures

A polynomial expansion to probability density function (pdf ) approximation about Gaussian mixture densities is proposed in this paper. Using known polynomial series expansions we apply the Parzen estimator to derive an orthonormal basis that is able to represent the characteristics of probability distributions that are not concentrated in the vicinity of the mean point such as the Gaussian pdf. The blind source separation problem is used to illustrate the applicability of the proposal in practical analysis of the dynamics of the recovered data pdf estimation. Simulations are carried out to illustrate the analysis. © 2004 IEEE.163172Cavalcante, C.C., (2004) On Blind Source Separation: Proposals and Analysis of Multi-user Processing Strategies, , Ph.D. thesis, State University of Campinas (UNICAMP) - DECOM, Campinas, SP - Brazil, AprilCavalcante, C.C., Cavalcanti, F.R.P., Mota, J.C.M., Romano, J.M.T., A Constrained version of fitting PDF algorithm for blind source separation (2003) Proceeding of IEEE Signal Processing Advances for Wireless Communications (SPAWC 2003), , Rome, Italy, June, 15-18Haykin, S., (1998) Neural Networks: A Comprehensive Foundation, , Prentice Hall, 2nd ednHyvärinen, A., Oja, E., Karhunen, J., (2001) Independent Component Analysis, , John Wiley & SonsLacoume, J.-L., Amblard, P.-O., Comon, P., (1997) Statistiques D'Ordre Supérieur Pour le Traitement du Signal, , (Traitement du Signal), Paris: MassonLaster, J.D., (1997) Robust GMSK Demodulation Using Demodulator Diversity and BER Estimation, , Ph.D. thesis, Faculty of the Virginia Polytechnic Institute and State University, Blacksburg, Virginia, MarchNikias, C.L., Petropulu, A.P., (1993) Higher-order Spectra Analysis, , Prentice HallPapadias, C.B., (2000) Blind Separation of Independent Sources Based on Multiuser Kurtosis Optimization Criteria, 2, pp. 147-179. , John-Wiley & Sons, chap. 4Papoulis, A., (1991) Probability, Random Variables and Stochastic Processes, , (Electrical & Electronic Engineering Series), McGraw-Hill International, 3rd ednParzen, E., On estimation of a probability density function and mode (1962) The Annals of Mathematical Statistics, 33 (3), pp. 1066-1076. , SeptemberSilverman, B.W., (1986) Density Estimation for Statistics and Data Analysis, , (Monographs on Statistics and Applied Probability), Bristol, Great Britain: Chapman and HallTherrien, C.W., (1992) Discrete Random Signals and Statistical Signal Processing, , (Prentice-Hall Signal Processing Series), Prentice-Hall InternationalWegman, E.J., Nonparametric probability density estimation: I. A summary of available methods (1972) Technometrics, 14 (3), pp. 533-546. , Augus

Modeling Of Concepts And Mathematical Processes By Petri Nets: The Case Of Integrability Of Real Functions [modelagem De Conceitos E Processos Matemáticos Por Redes De Petri Coloridas: O Caso Da Integrabilidade De Funções Reais]

Petri Nets (PNs) are a mathematical and graphical tool for general use. In this paper, the use of PNs is promoted as a modeling tool to organize the teaching of mathematical notions. The modeling presented in this work relies on the strong mathematical basis of PNs. As a case study, a model is constructed for teaching integrability of real functions, based on the evolution of the integral concept. With this aim, it is considered a real bounded function, defined on a closed and bounded interval, that is analyzed according to conditions which ensure or exclude its integration by Cauchy, by Riemann, by Lebesgue or by none of these integrals. The idea of modeling integrals using PN was originated in the context of mathematical classes, as the integral was being taught to first year engineering students.27457595Aalst, W., The application of Petri nets to workflow management (1998) The Journal of Circuits, Systems and Computers, 8 (1), pp. 1-53. , LondonArtigue, M., Ingénierie Didactique (1989) Recherches en Didactique des Mathématiques, 9 (3), pp. 281-308. , ParisBachelard, G., (1999) Le Nouvel Sprit Scientifique, , 6. ed. Paris: Quadrige/PUFBarroso, N., (2009) Um Modelo de Ensino Dos Conceitos de Cálculo Para Os Cursos de Engenharia Fundamentado em Uma Epistemologia Histórica e Baseado Na Metodologia da Engenharia Didática: Validação Por Meio Do Conceito de Integral, p. 187. , Tese (Doutorado em Engenharia de Teleinformática) - Universidade Federal do Ceará, Fortaleza, 2009Barroso, N., Soares, J.M., Mota, J.C.M., Neto, H.B., Uma Seqüência de Ensino para a Introdução do Conceito de Integral de Riemann (2007) ENCONTRO NACIONAL de EDUCAÇÃO MATEMÁTICA, pp. 1-17. , http://www.sbem.com.br/files/ix_enem/Html/comunicacaoCientifica.html, 9., 2007, Belo Horizonte. Anais... Belo Horizonte: Sociedade Brasileira de Educação Matemática (SBEM), Acesso em: 21 mar. 2013Brousseau, G., Fondements et méthodes de la Didactique des Mathématiques (1986) Recherches en Didactique des Matematiques, 7 (2), pp. 33-116. , ParisChen, J., Smith, S., Cardinality of the set of real functions with a given continuity set (2008) Pi Mu Epsilon Journal, Worcester, 12 (8), pp. 449-454. , http://www.sju.edu/~smith/Current_Courses/continuitysetsrev.pdf, Massachusetts, Acesso em: 21 mar. 2013Dahan-Dalmedico, A., Peiffer, J., (1986) Une Histoire des Mathématiques: Routes et Dédales, , Paris: Le SeuilD'Amore, B., (2005) Epistemologia e Didática da Matemática, , São Paulo: EscriturasFinney, R., Weir, M., Giordano, F., (2004) Cálculo George B. Thomas, 1. , 10. ed. São Paulo: Pearson-Addison WesleyJensen, K., Coloured petri nets (1997) Basic Concepts, Analysis Methods and Practical Use, 1. , 2. ed. Berlin: Springer-VerlagLeithold, L., (1994) Cálculo Com Geometria Analítica, , 3. ed. São Paulo: HarbraMoreira, M., (1999) Teorias da Aprendizagem, , São Paulo: EPUNovak, J.D., Cañas, A.J., Theoretical origins of concept maps, how to contruct them, and uses in education (2007) Refleting Education, 3 (1), pp. 29-42. , Londres, NovPádua, S., Silva, A., Porto, A., Inamasu, R., O potencial das redes de Petri em modelagem e análise de processos de negócios (2004) Gestão & Produção, 11 (1), pp. 109-119. , São Carlos, jan./abrPais, L.C., (2001) Didática da Matemática: Uma Análise da Influência Francesa, , 2. ed. Belo Horizonte: Autêntica, (Coleção Tendências em Educação Matemática)Park, S., Kim, Y., Applying petri nets to model customized learning and cooperative learning with competence (2008) International Journal of Computer Science and Network Security, 8 (2), pp. 127-132. , Seul, FebPeterson, J.L., (1981) Petri Net Theory and the Modeling of Systems, , New Jersey: Prentice HallPontes, J., Silva, J., Teoria da Aprendizagem Significativa de David Ausubel: Perguntas e Respostas (2006) Série-Estudos: Periódicos Do Mestrado em Educação da UCDB, 21, pp. 117-130. , Campo Grande, jan./junRoyden, H., (1971) Real Analysis, , 3. ed. New Jersey: Prentice HallRudin, W., (1976) Principles of Mathematical Analysis, , 3.ed. New York: McGraw-HillRussel, B., (2007) Introdução À Filosofia Matemática, , 1. ed. Rio de Janeiro: ZaharSingh, S., (1999) O Último Teorema de Fermat, , 5. ed. Rio de Janeiro: RecordStewart, I., (2004) Galois Theory, , 3. ed. Flórida: Chapman & Hall/CR