两类统计模型中方差和协方差分量的Bayes估计
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摘要
回归模型中的方差和协方差分量的估计在工业、化学、生物和行为科学等领域中有广泛应用,本文中的两类统计模型分别为非线性模型和混合整数线性模型。通过采用Bayes方法,从无先验信息和有逆Gamma先验信息出发,研究了这两类统计模型中的方差和协方差分量的估计。
本文中所研究的非线性模型中方差和协方差分量的Bayes估计包含相关系数ρ,而其他学者提出的线性模型中方差和协方差分量的Bayes估计只是本文的特殊情况。通过实例说明了Bayes方法是可行的,并且有逆Gamma先验信息要比无先验信息得出的结果要好。
对混合整数线性模型的研究。始于GPS定位测量(XuPeiliang,2006),这个问题在统计学中还没有研究。GPS定位测量中研究的只是基线向量β∈R~t和双差整周模糊度θ∈Z~m,没有考虑方差和协方差分量。从统计学的角度来看,方差和协方差分量的估计与其他未知参数的估计同样重要,从而本文所研究的另一个问题是混合整数线性模型中方差因子的Bayes估计,并通过卫星在不同的历元时所得到的实际数据得出混合整数线性模型中方差因子的Bayes估计。同样,结果表明有逆Gamma先验信息要比无先验信息得出的结果要好。
In the regression model, the estimation of variance and covariance components is wider applied in the industrial, chemical, biological, behavioral sciences, and other fields. In this paper, the two kinds of statistical models are the nonlinear model and the mixed-integer linear model. The estimations of variance and covariance components which come from non-informative prior and inverted gamma prior are researched with the Bayesian approach in the two kinds of statistical models.
In this paper, the Bayesian estimation of the variance and covariance components include the correlation coefficientρin the nonlinear model, but the Bayesian estimation of the variance and covariance components in the linear model is provided by other scholars just only a special situation. By an example , the feasibility of the Bayesian approach is demonstrated, and the result of inverted gamma prior is better than non-informative prior.
The research of the mixed-integer linear model began with the GPS measurement (Xu peiliang, 2006), this issue is not studied in the statistics. The baseline vectorβ∈R~l and the difference-two whole ambiguityθ∈Z~m are studied in the GPS measurement, in which the variance and covariance components are not considered. From the view of statistics, the estimation of variance and covariance components is the same important as other estimations of unknown parameters. So another issue of research in this paper is the Bayesian estimation of variance in the mixed-integer linear model by satellites. Similarly, the result indicates that the result of inverted gamma prior is better than non-informative prior.
本文中所研究的非线性模型中方差和协方差分量的Bayes估计包含相关系数ρ,而其他学者提出的线性模型中方差和协方差分量的Bayes估计只是本文的特殊情况。通过实例说明了Bayes方法是可行的,并且有逆Gamma先验信息要比无先验信息得出的结果要好。
对混合整数线性模型的研究。始于GPS定位测量(XuPeiliang,2006),这个问题在统计学中还没有研究。GPS定位测量中研究的只是基线向量β∈R~t和双差整周模糊度θ∈Z~m,没有考虑方差和协方差分量。从统计学的角度来看,方差和协方差分量的估计与其他未知参数的估计同样重要,从而本文所研究的另一个问题是混合整数线性模型中方差因子的Bayes估计,并通过卫星在不同的历元时所得到的实际数据得出混合整数线性模型中方差因子的Bayes估计。同样,结果表明有逆Gamma先验信息要比无先验信息得出的结果要好。
In the regression model, the estimation of variance and covariance components is wider applied in the industrial, chemical, biological, behavioral sciences, and other fields. In this paper, the two kinds of statistical models are the nonlinear model and the mixed-integer linear model. The estimations of variance and covariance components which come from non-informative prior and inverted gamma prior are researched with the Bayesian approach in the two kinds of statistical models.
In this paper, the Bayesian estimation of the variance and covariance components include the correlation coefficientρin the nonlinear model, but the Bayesian estimation of the variance and covariance components in the linear model is provided by other scholars just only a special situation. By an example , the feasibility of the Bayesian approach is demonstrated, and the result of inverted gamma prior is better than non-informative prior.
The research of the mixed-integer linear model began with the GPS measurement (Xu peiliang, 2006), this issue is not studied in the statistics. The baseline vectorβ∈R~l and the difference-two whole ambiguityθ∈Z~m are studied in the GPS measurement, in which the variance and covariance components are not considered. From the view of statistics, the estimation of variance and covariance components is the same important as other estimations of unknown parameters. So another issue of research in this paper is the Bayesian estimation of variance in the mixed-integer linear model by satellites. Similarly, the result indicates that the result of inverted gamma prior is better than non-informative prior.
引文
[1]欧自强.广义方差估计理论及其在测量中的应用[D].长沙:中南工业大学,1991
[2]Grodecki J.On estimation of variance-covariance components for geodetic observation and implications on deformation trend analysis[D].Tech rep 186,Department of Geodesy and Geomatics Engineering,University of New Brunswick,Fredericton,1997
[3]KOCH K R.Maximum Likelihood Estimation of Variance Components[J].Bull Geod,1986,60:329-338
[4]OU Zi-qiang.Estimation of Variance and Covariance Components[J].Bull Geod,1989,63:139-148
[5]YU Zong-chou.An Universal Formula of Maximum Likelihood Estimation of Variance-covariance Compoents[J].Journal of Geodesy,1996,70:233-240
[6]Wang Zhi-zhong,ZHU jian-jun,Estimation and Accucracy Evaluation of Variance and covariance Components Based on More General Functional Model[J].Trans Nonferrous Met Soc China,2000,10(1):123-126
[7]Searle SR,Casella G,McCulloch CE.Variance Component[M].New York:John Wiley,1992
[8]Wang Zhizhong,ZhuJianjun,Estimation of variance and covariance components in the nonlinear Model[J].Acta Geodaetica et Cartoraphica Sinica,2005,12(4):288-293.(王志忠,朱建军.非线性模型中方差和协方差分量的估计[J].测绘学报,2005,12(4):288-293)
[9]KOCH K R.Bayesian inference for variance components[J].Manuscripta Geodaetica,1987,12:309-313
[10]KOCH K R.Bayesian statistics for variance components with informative and non-informative priors[J].Manuscripta Geodaetica,1988,13:370-373
[11]Koch KR.Bayesian inference with geodetic applications[A].Lecture notes in earth science,vol31.New York:Springer,Berlin Heidelberg,1990
[12]Ou Z.Approximative Bayes estimation for Bayes estimates of variance components[J].Manuscr Geod,1991,16:168-172
[13]Ou Z,KOCH K R.Analytical expressions for Bayes estimates of Variance Components[J].Manuscripta Geodaetica,1994,19:284-293
[14]Grodecki J.Generalized maximum-likelihood estimation of variance components with inverted gamma prior[J].Journal of Geodesy,1999,73:367-373
[15]Grodecki J.Generalized maximum-likelihood estimation of variance components with non-informative prior[J].Journal of Geodesy,2001,75:157-163
[16]Xu Peiliang.Voronoi Cells,Probabilistic Bounds,and Hypothesis Testing in Mixed Integer Linear Modes[J].IEEE Transactions on Information Theory,2006,52(7):3122-3138
[17]Hatch R.Instantaneous Ambiguity Resolution[A].Proc IAG Int Symp 107 on Kinematic Systems inGeodey,Surveying and Remote Sensing,10-13 Sept 1990[C].New York:Springer Berlin Heidelberg,1991:299-308
[18]Frei E,Beutler G.Rapid Static Positioning based on the Fast Ambiguity Resolution Approach "FARA":Theory and First Result[J].Manuscr Geod,1990,15:325-356
[19]Teunissen PJG.The Least-squares Ambiguity Decorrelation Ajustment:a Method for Fast GPS Integer Ambiguity Estimation[J].Journal of Geodesy,1995,70:65-82
[20]Teunissen PJG.The Probability Distribution of the Ambiguity bootstrapped GNSS Baselines[J].Journal of Geodesy,2001,75:399-408
[21]Teunissen PJG.The Parameter Distributions of the Integer GPS Model [J].Journal of Geodesy,2002,76:41-48
[22]Betti B,Crespi M,Sanso F.A Geometric Illustration of Ambiguity Resolution in GPS Theory and a Bayesian Approach[J].Manuscr Geod,1993,18:317-330
[23]Zhu J,Ding X,Chen Y.Maximum-likelihood ambiguity resolution based on Bayesian principal[J].Journal of Geodesy,2001,75:175-187
[24]欧自强.广义方差估计理论及起在测量中的应用[D].长沙:中南工业大 学,1991.
[25]王志忠,朱建军.非线性模型中方差和协方差分量的估计[J].测绘学报,2005,12(4):288-293.
[26]KOCH K R.Bayesian inference for variance components[J].Manuscripta Geodaetica,1987,12:309-313
[27]KOCH K R.Bayesian statistics for variance components with informative and non-informative priors[J].Manuscripta Geodaetica,1988,13:370-373
[28]Rice,J.Bandwith choice for nonparametric regression.Ann.Statist,1984,12;1215-1230
[29]Tao Huaxue and Guo Jinyun.A new solution model of nonlinear integral adjustment including different kinds of observing data with different precision[J].AVN,2003,3:98-102
[30]Teunissen PJG.The success rate and precision of GPS ambiguities[J].J Geod,2000,74:321-326
[31]Teunissen PJG.Least-squares estimation of the integer GPS ambiguities[R].Invited lecture,Sect Ⅳ Theory and methodology,IAG General Meeting,Beijing,1993.8
[32]陈希孺,王松桂.近代回归分析[M].合肥:安徽教育出版社,1987
[33]Teunissen PJG.Success probability of integer GPS ambiguity rounding and bootstrapping.Journal of Geodesy,1998,72:606-612
[34]茆诗松编著.贝叶斯统计[M].北京:中国统计出版社,1999
[35]茆诗松等编著.高等数理统计[M].北京:高等教育出版社,1998
[36]中国科学院数学所概率统计室.方差分析[M].北京:科学出版社,1997
[37]Bernardo J M,Smith A F M.Bayesian theory[M].New York Wiley,1994
[38]C.R.Rao.Estimation of parameters in a linear model.Ann.Statist.,1976,4:1023-1037
[39]成平.贝叶斯统计的几点看法[J].数理统计与应用概率.1990,5(4):387-388
[40]陈希孺.数理统计中的两个学派——频率学派和贝叶斯学派[J].数理统计与应用概率.1990,5(4):389-400
[41]L.M.Lye,K.P.Hapuarachchi,S.Ryan.Bayes Estimation of Entreme-Value Reliability Function[J].IEEE Truns on Realibity.1993,42(4):641-644
[42]S.E.Ahmed,N.Reid.Empirical Bayes and Likelihood Inference[M].New York,Springer-Verger,2001
[43]Berger J.O.Statistical Decision Theory and Bayesian Analysis[M].New York,1985
[44]K.E.ajmad,M.E.fakhry,Z.f.Jahean.Empirical Bayes estimation of Burr-type X model[J].Journal of Statistical Planning and Inference,1997(64):297-308
[45]刘基余.GPS导航定位原理与方法[M].北京:科学出版社,2003
[46]ZHANG S X.The theory and application of GPS measurement and positioning[M].National University of Defense Technology Publication,1996
[47]Teunissen P J G.The Probability Distribution of the GPS Baseline for Integer Ambiguity Estimators[J].Journal of Geodesy,1999,73:275-284
[483 FREI E G B.Rapaid Static Positioning Based on the Fast Ambiguity Function Algorithm[J].Journal of Geodesy,1990,15(4):325-356
[49]CHEN D,LACHAPHLLE G.A Comparison of the FAST and Lesst-Square Searche Algorithm for On the Fly Ambiguity Resolution[J].Journal of Institute of Navigation,1995,42(2)
[50]Teunissen P J G.A New Method for Fast Carrier Phase Ambiguity Estimation[A].Proceedings of IEEE PLANS'94[C].[s.l.]:[s.n.],1994.562-573
[51]王松桂等.线性模型引论[M].科学出版社,2004
[52]张尧庭,陈汉峰.贝叶斯统计推断[M].科学出版社,1991
[53]Berger,J.O.,Statistical decision theory and Bayesian analysis,Second edition,Springer-Verlag,New York,1985
[54]成平,吴启光,李国英,二阶原点矩的二次型估计的可容许性,应用数学学报,1983,6(1):18-28
[55]傅惠民.相关系数平稳过程法,机械强度.2002,24(3):400-404
[56]胡跃清,韦博成.非线性模型中误差的相关性检验.中国统计学报,1994,32,499-506
[57]James W.Hardin,,Joesph M.Hilbe.Generalized estimating equations.Chapman&Hall/CRC,2003
[58]Jorgensen,B.Proper dispersion models.Braz.J.Probab.Statist.,1997
[59]刘锋.部分线性模型的序列相关检验和异方差检验.中南大学博士论文,2006
[60]鹿长余,方差分量组合的非负二次估计的可容许,数学年刊,1991,12A (6):699-707
[61]林金官,韦博成。随机权函数非线性回归模型的异方差统计分析。应用概率统计,2002,18(4):431-437
[62]Nelder,J.A.,Wedderburn,R.W.M.Generalized linear models.J.R.Statist.Soc.A,1972,135:370-384
[63]韦博成.加权非线性回归的Score检验及其局部影响分析.应用概率统计,1995,11(4):147-156
[64]徐兴忠,线性模型中误差方差的二次型估计是D—可容许的充要条件,应用数学学报,1992,15(3):410-427
[65]Fisher,R.A.Dispersion on a sphere.Proc.Roy.Soc.Ser.A,1953,217:295-305
[66]王志忠,宋允全.非线性模型方差的贝叶斯估计.云南大学学报,2005,27(4):289-293
[2]Grodecki J.On estimation of variance-covariance components for geodetic observation and implications on deformation trend analysis[D].Tech rep 186,Department of Geodesy and Geomatics Engineering,University of New Brunswick,Fredericton,1997
[3]KOCH K R.Maximum Likelihood Estimation of Variance Components[J].Bull Geod,1986,60:329-338
[4]OU Zi-qiang.Estimation of Variance and Covariance Components[J].Bull Geod,1989,63:139-148
[5]YU Zong-chou.An Universal Formula of Maximum Likelihood Estimation of Variance-covariance Compoents[J].Journal of Geodesy,1996,70:233-240
[6]Wang Zhi-zhong,ZHU jian-jun,Estimation and Accucracy Evaluation of Variance and covariance Components Based on More General Functional Model[J].Trans Nonferrous Met Soc China,2000,10(1):123-126
[7]Searle SR,Casella G,McCulloch CE.Variance Component[M].New York:John Wiley,1992
[8]Wang Zhizhong,ZhuJianjun,Estimation of variance and covariance components in the nonlinear Model[J].Acta Geodaetica et Cartoraphica Sinica,2005,12(4):288-293.(王志忠,朱建军.非线性模型中方差和协方差分量的估计[J].测绘学报,2005,12(4):288-293)
[9]KOCH K R.Bayesian inference for variance components[J].Manuscripta Geodaetica,1987,12:309-313
[10]KOCH K R.Bayesian statistics for variance components with informative and non-informative priors[J].Manuscripta Geodaetica,1988,13:370-373
[11]Koch KR.Bayesian inference with geodetic applications[A].Lecture notes in earth science,vol31.New York:Springer,Berlin Heidelberg,1990
[12]Ou Z.Approximative Bayes estimation for Bayes estimates of variance components[J].Manuscr Geod,1991,16:168-172
[13]Ou Z,KOCH K R.Analytical expressions for Bayes estimates of Variance Components[J].Manuscripta Geodaetica,1994,19:284-293
[14]Grodecki J.Generalized maximum-likelihood estimation of variance components with inverted gamma prior[J].Journal of Geodesy,1999,73:367-373
[15]Grodecki J.Generalized maximum-likelihood estimation of variance components with non-informative prior[J].Journal of Geodesy,2001,75:157-163
[16]Xu Peiliang.Voronoi Cells,Probabilistic Bounds,and Hypothesis Testing in Mixed Integer Linear Modes[J].IEEE Transactions on Information Theory,2006,52(7):3122-3138
[17]Hatch R.Instantaneous Ambiguity Resolution[A].Proc IAG Int Symp 107 on Kinematic Systems inGeodey,Surveying and Remote Sensing,10-13 Sept 1990[C].New York:Springer Berlin Heidelberg,1991:299-308
[18]Frei E,Beutler G.Rapid Static Positioning based on the Fast Ambiguity Resolution Approach "FARA":Theory and First Result[J].Manuscr Geod,1990,15:325-356
[19]Teunissen PJG.The Least-squares Ambiguity Decorrelation Ajustment:a Method for Fast GPS Integer Ambiguity Estimation[J].Journal of Geodesy,1995,70:65-82
[20]Teunissen PJG.The Probability Distribution of the Ambiguity bootstrapped GNSS Baselines[J].Journal of Geodesy,2001,75:399-408
[21]Teunissen PJG.The Parameter Distributions of the Integer GPS Model [J].Journal of Geodesy,2002,76:41-48
[22]Betti B,Crespi M,Sanso F.A Geometric Illustration of Ambiguity Resolution in GPS Theory and a Bayesian Approach[J].Manuscr Geod,1993,18:317-330
[23]Zhu J,Ding X,Chen Y.Maximum-likelihood ambiguity resolution based on Bayesian principal[J].Journal of Geodesy,2001,75:175-187
[24]欧自强.广义方差估计理论及起在测量中的应用[D].长沙:中南工业大 学,1991.
[25]王志忠,朱建军.非线性模型中方差和协方差分量的估计[J].测绘学报,2005,12(4):288-293.
[26]KOCH K R.Bayesian inference for variance components[J].Manuscripta Geodaetica,1987,12:309-313
[27]KOCH K R.Bayesian statistics for variance components with informative and non-informative priors[J].Manuscripta Geodaetica,1988,13:370-373
[28]Rice,J.Bandwith choice for nonparametric regression.Ann.Statist,1984,12;1215-1230
[29]Tao Huaxue and Guo Jinyun.A new solution model of nonlinear integral adjustment including different kinds of observing data with different precision[J].AVN,2003,3:98-102
[30]Teunissen PJG.The success rate and precision of GPS ambiguities[J].J Geod,2000,74:321-326
[31]Teunissen PJG.Least-squares estimation of the integer GPS ambiguities[R].Invited lecture,Sect Ⅳ Theory and methodology,IAG General Meeting,Beijing,1993.8
[32]陈希孺,王松桂.近代回归分析[M].合肥:安徽教育出版社,1987
[33]Teunissen PJG.Success probability of integer GPS ambiguity rounding and bootstrapping.Journal of Geodesy,1998,72:606-612
[34]茆诗松编著.贝叶斯统计[M].北京:中国统计出版社,1999
[35]茆诗松等编著.高等数理统计[M].北京:高等教育出版社,1998
[36]中国科学院数学所概率统计室.方差分析[M].北京:科学出版社,1997
[37]Bernardo J M,Smith A F M.Bayesian theory[M].New York Wiley,1994
[38]C.R.Rao.Estimation of parameters in a linear model.Ann.Statist.,1976,4:1023-1037
[39]成平.贝叶斯统计的几点看法[J].数理统计与应用概率.1990,5(4):387-388
[40]陈希孺.数理统计中的两个学派——频率学派和贝叶斯学派[J].数理统计与应用概率.1990,5(4):389-400
[41]L.M.Lye,K.P.Hapuarachchi,S.Ryan.Bayes Estimation of Entreme-Value Reliability Function[J].IEEE Truns on Realibity.1993,42(4):641-644
[42]S.E.Ahmed,N.Reid.Empirical Bayes and Likelihood Inference[M].New York,Springer-Verger,2001
[43]Berger J.O.Statistical Decision Theory and Bayesian Analysis[M].New York,1985
[44]K.E.ajmad,M.E.fakhry,Z.f.Jahean.Empirical Bayes estimation of Burr-type X model[J].Journal of Statistical Planning and Inference,1997(64):297-308
[45]刘基余.GPS导航定位原理与方法[M].北京:科学出版社,2003
[46]ZHANG S X.The theory and application of GPS measurement and positioning[M].National University of Defense Technology Publication,1996
[47]Teunissen P J G.The Probability Distribution of the GPS Baseline for Integer Ambiguity Estimators[J].Journal of Geodesy,1999,73:275-284
[483 FREI E G B.Rapaid Static Positioning Based on the Fast Ambiguity Function Algorithm[J].Journal of Geodesy,1990,15(4):325-356
[49]CHEN D,LACHAPHLLE G.A Comparison of the FAST and Lesst-Square Searche Algorithm for On the Fly Ambiguity Resolution[J].Journal of Institute of Navigation,1995,42(2)
[50]Teunissen P J G.A New Method for Fast Carrier Phase Ambiguity Estimation[A].Proceedings of IEEE PLANS'94[C].[s.l.]:[s.n.],1994.562-573
[51]王松桂等.线性模型引论[M].科学出版社,2004
[52]张尧庭,陈汉峰.贝叶斯统计推断[M].科学出版社,1991
[53]Berger,J.O.,Statistical decision theory and Bayesian analysis,Second edition,Springer-Verlag,New York,1985
[54]成平,吴启光,李国英,二阶原点矩的二次型估计的可容许性,应用数学学报,1983,6(1):18-28
[55]傅惠民.相关系数平稳过程法,机械强度.2002,24(3):400-404
[56]胡跃清,韦博成.非线性模型中误差的相关性检验.中国统计学报,1994,32,499-506
[57]James W.Hardin,,Joesph M.Hilbe.Generalized estimating equations.Chapman&Hall/CRC,2003
[58]Jorgensen,B.Proper dispersion models.Braz.J.Probab.Statist.,1997
[59]刘锋.部分线性模型的序列相关检验和异方差检验.中南大学博士论文,2006
[60]鹿长余,方差分量组合的非负二次估计的可容许,数学年刊,1991,12A (6):699-707
[61]林金官,韦博成。随机权函数非线性回归模型的异方差统计分析。应用概率统计,2002,18(4):431-437
[62]Nelder,J.A.,Wedderburn,R.W.M.Generalized linear models.J.R.Statist.Soc.A,1972,135:370-384
[63]韦博成.加权非线性回归的Score检验及其局部影响分析.应用概率统计,1995,11(4):147-156
[64]徐兴忠,线性模型中误差方差的二次型估计是D—可容许的充要条件,应用数学学报,1992,15(3):410-427
[65]Fisher,R.A.Dispersion on a sphere.Proc.Roy.Soc.Ser.A,1953,217:295-305
[66]王志忠,宋允全.非线性模型方差的贝叶斯估计.云南大学学报,2005,27(4):289-293