一种局部多项式时空地理加权回归方法
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摘要
基于加权最小二乘估计的时空地理加权回归方法,在随机项方差相同且最小的假设条件下估计回归参数和拟合值,由于没有考虑时空分析中异方差影响而导致估计结果存在一定偏差。局部多项式估计是一种消除异方差影响的非参数估计方法。本文在局部多项式估计原理基础上,提出了局部多项式时空地理加权回归方法。它是采用三元一阶泰勒级数展开式重构时空回归系数和自变量矩阵,进而建立满足高斯-马尔可夫独立同分布假定要求的新模型,利用新模型回归系数估计值、拟合值以及新模型与原模型的关系,可得到原模型回归系数估计值和拟合值。本文采用模拟数据和真实数据进行试验,以GTWR与局部线性地理加权回归作为对比方法,从方法适用性、整体估计效果、回归系数估计偏差和拟合优度、整体估计偏差等方面分析了LPGTWR方法性能,有效证明了LPGTWR方法能消除异方差影响提升估计精度。
Geographically and temporally weight regression(GTWR)estimates regression coefficients and fitted value by weighted least squares(WLS),which under the assumption of the same minimum random variance.As without considering the spatio-temporal heteroscedasticity,it may reduce the accuracy of estimation.Local polynomial estimation is a nonparametric estimation method to eliminate heteroscedasticity in statistics.On the basis of the local polynomial estimation,the local polynomial geographically and weight regression temporally(LPGTWR)approach is proposed in this paper.It reconstructs the spatio-temporal coefficients using threedimensional Taylor Series in order to satisfy the Gauss-Markov assumption of independent identical distribution.Then estimate the regression coefficients and fitting value using weighted least squares.The experiments use both simulated data and real data to compare LPGTWR,GTWR and local linear-fitting-based geographically weight regression(LGWR).Experiments using simulated data showed that LPGTWR can significantly improve the accuracy of estimation not only in goodness-of-fit of the fitted value,but also in reducing bias of the coefficient estimation and the estimation.It is useful by adopting LPGTWR to eliminate heteroscedasticity effect and improve estimation accuracy.
Geographically and temporally weight regression(GTWR)estimates regression coefficients and fitted value by weighted least squares(WLS),which under the assumption of the same minimum random variance.As without considering the spatio-temporal heteroscedasticity,it may reduce the accuracy of estimation.Local polynomial estimation is a nonparametric estimation method to eliminate heteroscedasticity in statistics.On the basis of the local polynomial estimation,the local polynomial geographically and weight regression temporally(LPGTWR)approach is proposed in this paper.It reconstructs the spatio-temporal coefficients using threedimensional Taylor Series in order to satisfy the Gauss-Markov assumption of independent identical distribution.Then estimate the regression coefficients and fitting value using weighted least squares.The experiments use both simulated data and real data to compare LPGTWR,GTWR and local linear-fitting-based geographically weight regression(LGWR).Experiments using simulated data showed that LPGTWR can significantly improve the accuracy of estimation not only in goodness-of-fit of the fitted value,but also in reducing bias of the coefficient estimation and the estimation.It is useful by adopting LPGTWR to eliminate heteroscedasticity effect and improve estimation accuracy.
引文
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[14]黄砚玲.地理加权空间经济计量模型的GMM估计及区域金融发展收敛性实证研究[D].广州:华南理工大学,2012.HUANG Yanling.GMM Estimate for SGWR Model and a Spatial Analysis of Regional Financial Convergence in China[D].Guangzhou:South China University of Technology,2012.
[15]杨毅.顾及时空非平稳性的地理加权回归方法研究[D].武汉:武汉大学,2016.YANG Yi.Research on Geographically and Temporally Weighted Regression for Spatial and Temporal Nonstationarity[D].Wuhan:Wuhan University,2016.
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[19]柏中强,王卷乐,杨雅萍,等.基于乡镇尺度的中国25省区人口分布特征及影响因素[J].地理学报,2015,70(8):1229-1242.BAI Zhongqiang,WANG Juanle,YANG Yaping,et al.Characterizing Spatial Patterns of Population Distribution at Township Level across the 25Provinces in China[J].Acta Geographica Sinica,2015,70(8):1229-1242.
[20]戚伟,李颖,刘盛和,等.城市昼夜人口空间分布的估算及其特征——以北京市海淀区为例[J].地理学报,2013,68(10):1344-1356.QI Wei,LI Ying,LIU Shenghe,et al.Estimation of Urban Population at Daytime and Nighttime and Analyses of Their Spatial Pattern:A Case Study of Haidian District,Beijing[J].Acta Geographica Sinica,2013,68(10):1344-1356.
[21]鲁楠,张委伟,陈利军,等.顾及城乡差异的大区域人口密度估算——以山东省为例[J].测绘学报,2015,44(12):1384-1391.DOI:10.11947/j.AGCS.2015.20150005.LU Nan,ZHANG Weiwei,CHEN Lijun,et al.Estimation of Large Regional Urban and Rural Population Density Based on the Differences of Population Distribution between Urban and Rural:Take Shandong Province as Example[J].Acta Geodaetica et Cartographica Sinica,2015,44(12):1384-1391.DOI:10.11947/j.AGCS.2015.20150005.
[22]陈彦光.基于Matlab的地理数据分析[M].北京:高等教育出版社,2012:34-45.CHEN Yanguang.Geographical Data Analysis with Matlab[M].Beijing:Higher Education Press,2012:34-45.
[23]SONG Weize,JIA Haifeng,HUANG Jingfeng,et al.A Satellite-based Geographically Weighted Regression Model for Regional PM2.5 Estimation over the Pearl River Delta Region in China[J].Remote Sensing of Environment,2014,154:1-7.
[2]BRUNSDON C,FOTHERINGHAM A S,CHARLTON M E.Geographically Weighted Regression:A Method for Exploring Spatial Nonstationarity[J].Geographical Analysis,1996,28(4):281-298.
[3]覃文忠.地理加权回归基本理论与应用研究[D].上海:同济大学,2007.QIN Wenzhong.The Basic Theoretics and Application Research on Geographically Weighted Regression[D].Shanghai:Tongji University,2007.
[4]BRUNSDON C,AITKIN M,FOTHERINGHAM A S,et al.A Comparison of Random Coefficient Modelling and Geographically Weighted Regression for Spatially Nonstationary Regression Problems[J].Geographical and Environmental Modelling,1999,3(1):47-62.
[5]FOTHERINGHAM A S,BRUNSDON C,CHARLTON M.Geographically Weighted Regression:the Analysis of Spatially Varying Relationships[M].New York:John Wiley&Sons,2002:34-45.
[6]WU Bo,LI Rongrong,HUANG Bo.A Geographically and Temporally Weighted Autoregressive Model with Application to Housing Prices[J].International Journal of Geographical Information Science,2014,28(5):1186-1204.
[7]WRENN D H,SAM A G.Geographically and Temporally Weighted Likelihood Regression:Exploring the Spatiotemporal Determinants of Land Use Change[J].Regional Science and Urban Economics,2014,44:60-74.
[8]BAI Yang,WU Lixin,QIN Kai,et al.A Geographically and Temporally Weighted Regression Model for GroundLevel PM2.5 Estimation from Satellite-derived 500m Resolution AOD[J].Remote Sensing,2016,8(3):262.
[9]CHU H J,HUANG Bo,LIN C Y.Modeling the Spatiotemporal Heterogeneity in the PM10-PM2.5 Relationship[J].Atmospheric Environment,2015,102:176-182.
[10]WANG Ning,MEI Changlin,YAN Xiaodong.Local Linear Estimation of Spatially Varying Coefficient Models:An Improvement on the Geographically Weighted Regression Technique[J].Environment and Planning A:Economy and Space,2008,40(4):986-1005.
[11]ZHANG Huiguo,MEI Changlin.Local Least Absolute Deviation Estimation of Spatially Varying Coefficient Models:Robust Geographically Weighted Regression Approaches[J].International Journal of Geographical Information Science,2011,25(9):1467-1489.
[12]BRUNSDON C,FOTHERINGHAM A S,CHARLTON M.Some Notes on Parametric Significance Tests for Geographically Weighted Regression[J].Journal of Regional Science,1999,39(3):497-524.
[13]刘美玲.时空地理加权回归模型的统计诊断[D].西安:西安建筑科技大学,2013.LIU Meiling.Statistical Diagnostics in Geographically and Temporally Weighted Regression Models[D].Xi'an:Xi'an University of Architecture and Technology,2013.
[14]黄砚玲.地理加权空间经济计量模型的GMM估计及区域金融发展收敛性实证研究[D].广州:华南理工大学,2012.HUANG Yanling.GMM Estimate for SGWR Model and a Spatial Analysis of Regional Financial Convergence in China[D].Guangzhou:South China University of Technology,2012.
[15]杨毅.顾及时空非平稳性的地理加权回归方法研究[D].武汉:武汉大学,2016.YANG Yi.Research on Geographically and Temporally Weighted Regression for Spatial and Temporal Nonstationarity[D].Wuhan:Wuhan University,2016.
[16]HURVICH C M,SIMONOFF J S,TSAI C L.Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion[J].Journal of The Royal Statistical Society:Series B,1998,60(2):271-293.
[17]CLEVELAND W S.Robust Locally Weighted Regression and Smoothing Scatterplots[J].Journal of the American Statistical Association,1979,74(368):829-836.
[18]LU Binbin,CHARLTON M,HARRIS P,et al.Geographically Weighted Regression with a Non-euclidean Distance Metric:A Case Study Using Hedonic House Price Data[J].International Journal of Geographical Information Science,2014,28(4):660-681.
[19]柏中强,王卷乐,杨雅萍,等.基于乡镇尺度的中国25省区人口分布特征及影响因素[J].地理学报,2015,70(8):1229-1242.BAI Zhongqiang,WANG Juanle,YANG Yaping,et al.Characterizing Spatial Patterns of Population Distribution at Township Level across the 25Provinces in China[J].Acta Geographica Sinica,2015,70(8):1229-1242.
[20]戚伟,李颖,刘盛和,等.城市昼夜人口空间分布的估算及其特征——以北京市海淀区为例[J].地理学报,2013,68(10):1344-1356.QI Wei,LI Ying,LIU Shenghe,et al.Estimation of Urban Population at Daytime and Nighttime and Analyses of Their Spatial Pattern:A Case Study of Haidian District,Beijing[J].Acta Geographica Sinica,2013,68(10):1344-1356.
[21]鲁楠,张委伟,陈利军,等.顾及城乡差异的大区域人口密度估算——以山东省为例[J].测绘学报,2015,44(12):1384-1391.DOI:10.11947/j.AGCS.2015.20150005.LU Nan,ZHANG Weiwei,CHEN Lijun,et al.Estimation of Large Regional Urban and Rural Population Density Based on the Differences of Population Distribution between Urban and Rural:Take Shandong Province as Example[J].Acta Geodaetica et Cartographica Sinica,2015,44(12):1384-1391.DOI:10.11947/j.AGCS.2015.20150005.
[22]陈彦光.基于Matlab的地理数据分析[M].北京:高等教育出版社,2012:34-45.CHEN Yanguang.Geographical Data Analysis with Matlab[M].Beijing:Higher Education Press,2012:34-45.
[23]SONG Weize,JIA Haifeng,HUANG Jingfeng,et al.A Satellite-based Geographically Weighted Regression Model for Regional PM2.5 Estimation over the Pearl River Delta Region in China[J].Remote Sensing of Environment,2014,154:1-7.