Estimation of Probability Distribution and Its Application in Bayesian Classification and Maximum Likelihood Regression

Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challengi...

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Published in:	Interdisciplinary sciences : computational life sciences Vol. 11; no. 3; pp. 559 - 574
Main Authors:	Dai, Hao, Wang, Wei, Xu, Qin, Xiong, Yi, Wei, Dong-Qing
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2019 Springer Nature B.V
Subjects:	Algorithms Bayes Theorem Bayesian analysis Biomedical and Life Sciences Classification Computational Biology - methods Computational Biology/Bioinformatics Computational Science and Engineering Computer Appl. in Life Sciences Computer simulation Continuity (mathematics) Distribution functions Economic models Health Sciences Life Sciences Likelihood Functions Mathematical and Computational Physics Mathematical models Maximum likelihood estimation Medicine Models, Statistical Monte Carlo simulation Numerical methods Original Research Article Probability Probability density functions Probability distribution Probability theory Random variables Regression Analysis Regression models Research Design Spline functions Statistical analysis Statistical methods Statistics for Life Sciences Statistics, Nonparametric Theoretical Theoretical and Computational Chemistry Distribution function estimation Density function estimation Bayesian classification Spline regression Smoothing spline Maximum likelihood regression
ISSN:	1913-2751, 1867-1462, 1867-1462
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Abstract	Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.
AbstractList	Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models. Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.
Author	Wei, Dong-Qing Xiong, Yi Dai, Hao Xu, Qin Wang, Wei
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Cites_doi	10.1002/9780470430583 10.1214/aoms/1177728190 10.1016/0167-7152(92)90003-N 10.1007/978-1-4612-6333-3 10.1007/BF02162161 10.1007/978-1-4613-2279-5 10.1145/1143844.1143865 10.1080/01621459.1990.10476223 10.1109/34.494647 10.1109/LCOMM.2006.1603370 10.1109/ICASSP.2015.7178633 10.1109/CISP.2009.5300813 10.1214/aoms/1177698795 10.1109/TPAMI.2003.1233899 10.1080/00401706.1974.10489142 10.1016/S0167-4730(98)00019-8 10.1214/aoms/1177704472 10.1109/34.310693 10.1007/978-1-4899-3324-9
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References	RosenblattMRemarks on some nonparametric estimates of a density functionAnn Math Stat195627383283710.1214/aoms/1177728190 Rennie JDM, Shih L, Teevan J, Karger D (2003) Tackling the poor assumptions of Naive Bayes text classifiers. In: Proceedings of the twentieth international conference on machine learning, Washington, DC, USA SilvermanBWDensity estimation for statistics and data analysis1986LondonChapman & Hall10.1007/978-1-4899-3324-9 Howard RM (2010) PDF estimation via characteristic function and an orthonormal basis set. In: Wseas international conference on systems PfanzaglJHambökerRParametric statistical theoryJ Am Stat Assoc199691433269287 ParzenEOn estimation of probability density function and modeAnn Math Stat19623331065107610.1214/aoms/1177704472 SchwartzSCEstimation of probability density by an orthogonal seriesAnn Math Stat19673841261126510.1214/aoms/1177698795 MarshLCormierDRSpline regression modelsJ R Stat Soc20025234958 MoodAMGraybillFABoesDCIntroduction to the theory of statistics19743New YorkMcGraw-Hill Education VannucciMNonparametric density estimation using wavelets; Discussion Paper 95–26, ISDS1998DurhamDuke University BowersNLExpansion of probability density functions as a sum of gamma densities with applications in risk theoryTrans Soc Actuar196618 PT.152125147 Van KhuongHKongHYGeneral expression for pdf of a sum of independent exponential random variablesIEEE Commun Lett200610315916110.1109/LCOMM.2006.1603370 De BoorCA practical guide to splines1978New YorkSpringer10.1007/978-1-4612-6333-3 BishopCMNeural networks for pattern recognition1996New YorkOxford University Press CandyJVBayesian signal processing: classical, modern and particle filtering methods2009New YorkWiley-Interscience10.1002/9780470430583 GirolamiMHeCProbability density estimation from optimally condensed data samplesIEEE Trans Pattern Anal Mach Intell200325101253126410.1109/TPAMI.2003.1233899 JeonBLandgrebeDAFast Parzen density estimation using clustering-based branch andboundIEEE Trans Pattern Anal Mach Intell199416995095410.1109/34.310693 Xie J, Wang Z (2009) Probability density function estimation based on windowed fourier transform of characteristic function. In: International congress on image and signal processing EngelJDensity estimation with Haar seriesStat Probab Lett19909211111710.1016/0167-7152(92)90003-N BabichGACampsOIWeighted Parzen windows for pattern classificationIEEE Trans Pattern Anal Mach Intell199618556757010.1109/34.494647 Kitahara D, Yamada I (2015) Probability density function estimation by positive quartic C 2 -spline functions. In: IEEE international conference on acoustics, speech and signal processing AlexandreLAA solve-the-equation approach for unidimensional data kernel bandwidth selection2008CovilhãUniversity of Beira Interior MansourAMeslehRAggouneEHMBlind estimation of statistical properties of non-stationary random variablesJ Adv Signal Process2015511309314 ZongZLamKYEstimation of complicated distributions using B-spline functionsStruct Saf199820434135510.1016/S0167-4730(98)00019-8 TerrellGRThe maximal smoothing principle in density estimationJ Am Stat Assoc19908541047047710.1080/01621459.1990.10476223 MitchellTMCarbonellJGMichalskiRSMachine learning: a guide to current research1986NorwellKluwer Academic Publishers10.1007/978-1-4613-2279-5 Caruana R, Niculescu-Mizil A (2006) An empirical comparison of supervised learning algorithms. In: Proceedings of the 23rd international conference on machine learning, Pittsburgh, PA, USA WoldSSpline functions in data analysisTechnometrics197416111110.1080/00401706.1974.10489142 ReinschCHSmoothing by spline functionsNumer Math196710317718310.1007/BF02162161 Zhang H (2005) The optimality of Naive Bayes. In: Seventeenth international florida artificial intelligence research society conference, Miami Beach, Florida, USA 343_CR18 TM Mitchell (343_CR3) 1986 343_CR19 AM Mood (343_CR4) 1974 LA Alexandre (343_CR9) 2008 E Parzen (343_CR6) 1962; 33 JV Candy (343_CR1) 2009 Z Zong (343_CR23) 1998; 20 A Mansour (343_CR24) 2015; 51 H Van Khuong (343_CR14) 2006; 10 BW Silverman (343_CR7) 1986 CH Reinsch (343_CR21) 1967; 10 B Jeon (343_CR10) 1994; 16 J Engel (343_CR16) 1990; 9 343_CR25 NL Bowers (343_CR13) 1966; 18 PT.1 SC Schwartz (343_CR15) 1967; 38 343_CR27 343_CR29 343_CR28 GA Babich (343_CR11) 1996; 18 M Vannucci (343_CR17) 1998 C De Boor (343_CR26) 1978 J Pfanzagl (343_CR30) 1996; 91 M Rosenblatt (343_CR5) 1956; 27 M Girolami (343_CR12) 2003; 25 L Marsh (343_CR22) 2002; 52 S Wold (343_CR20) 1974; 16 GR Terrell (343_CR8) 1990; 85 CM Bishop (343_CR2) 1996
References_xml	– reference: ParzenEOn estimation of probability density function and modeAnn Math Stat19623331065107610.1214/aoms/1177704472 – reference: MarshLCormierDRSpline regression modelsJ R Stat Soc20025234958 – reference: Van KhuongHKongHYGeneral expression for pdf of a sum of independent exponential random variablesIEEE Commun Lett200610315916110.1109/LCOMM.2006.1603370 – reference: AlexandreLAA solve-the-equation approach for unidimensional data kernel bandwidth selection2008CovilhãUniversity of Beira Interior – reference: SchwartzSCEstimation of probability density by an orthogonal seriesAnn Math Stat19673841261126510.1214/aoms/1177698795 – reference: Zhang H (2005) The optimality of Naive Bayes. In: Seventeenth international florida artificial intelligence research society conference, Miami Beach, Florida, USA – reference: BowersNLExpansion of probability density functions as a sum of gamma densities with applications in risk theoryTrans Soc Actuar196618 PT.152125147 – reference: TerrellGRThe maximal smoothing principle in density estimationJ Am Stat Assoc19908541047047710.1080/01621459.1990.10476223 – reference: Xie J, Wang Z (2009) Probability density function estimation based on windowed fourier transform of characteristic function. In: International congress on image and signal processing – reference: Howard RM (2010) PDF estimation via characteristic function and an orthonormal basis set. In: Wseas international conference on systems – reference: SilvermanBWDensity estimation for statistics and data analysis1986LondonChapman & Hall10.1007/978-1-4899-3324-9 – reference: De BoorCA practical guide to splines1978New YorkSpringer10.1007/978-1-4612-6333-3 – reference: MansourAMeslehRAggouneEHMBlind estimation of statistical properties of non-stationary random variablesJ Adv Signal Process2015511309314 – reference: JeonBLandgrebeDAFast Parzen density estimation using clustering-based branch andboundIEEE Trans Pattern Anal Mach Intell199416995095410.1109/34.310693 – reference: BabichGACampsOIWeighted Parzen windows for pattern classificationIEEE Trans Pattern Anal Mach Intell199618556757010.1109/34.494647 – reference: Rennie JDM, Shih L, Teevan J, Karger D (2003) Tackling the poor assumptions of Naive Bayes text classifiers. 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In: IEEE international conference on acoustics, speech and signal processing – reference: CandyJVBayesian signal processing: classical, modern and particle filtering methods2009New YorkWiley-Interscience10.1002/9780470430583 – reference: GirolamiMHeCProbability density estimation from optimally condensed data samplesIEEE Trans Pattern Anal Mach Intell200325101253126410.1109/TPAMI.2003.1233899 – reference: Caruana R, Niculescu-Mizil A (2006) An empirical comparison of supervised learning algorithms. 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SubjectTerms	Algorithms Bayes Theorem Bayesian analysis Biomedical and Life Sciences Classification Computational Biology - methods Computational Biology/Bioinformatics Computational Science and Engineering Computer Appl. in Life Sciences Computer simulation Continuity (mathematics) Distribution functions Economic models Health Sciences Life Sciences Likelihood Functions Mathematical and Computational Physics Mathematical models Maximum likelihood estimation Medicine Models, Statistical Monte Carlo simulation Numerical methods Original Research Article Probability Probability density functions Probability distribution Probability theory Random variables Regression Analysis Regression models Research Design Spline functions Statistical analysis Statistical methods Statistics for Life Sciences Statistics, Nonparametric Theoretical Theoretical and Computational Chemistry
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Title	Estimation of Probability Distribution and Its Application in Bayesian Classification and Maximum Likelihood Regression
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