一种新的线性回归模型及其应用示例
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摘要
回归分析是统计分析中常用的方法之一。传统的回归模型不具备全域分析能力,而变量场之间的关系多采用SVD(Singular Value Decomposition)进行分析,与传统的回归分析有所脱节。更为广义的线性回归模型是传统线性回归模型的延拓,在标量情况下,该模型可转化为传统线性回归模型。该模型的基本特征包含乘法不可互易性、等价于传统线性回归(因子项为标量时)、可分析性、延拓性、降维特征及容错性等。该模型解决了传统的线性回归模型不具备全域分析能力及模型表达能力受限于模型维数的现实问题。本文采用了NCEP(National Centers for Environmental Prediction)降水、高度场、风场月平均资料及国家气候中心西太平洋副热带高压指数资料,利用该模型和传统回归方案进行对比分析,分析结果表明,该模型具有一定的实用参考价值。
Regression analysis is one of the commonly used methods in statistical analysis. However, traditional regression models have less ability for global analysis, and the relationship between variables is often analyzed by methods like the SVD(Singular Value Decomposition), which lack connections with traditional regression analysis. A MGLRM(more generalized linear regression model) is a continuation of traditional linear regression model. In the case that both the predictand and the predictors are scalars, the MGLRM can be transformed into the traditional linear regression model. The MGLRM's basic features include non-commutative multiplication, equivalence to traditional linear regression as predictors in the model are scalars, analysis, extension, dimension-reduction, and robustness, etc. The MGLRM solves problems in traditional linear regression models that have less ability for global analysis and limited expressive ability due to the dimensions of the regression equation. In this paper, the MGLRM and the traditional regression model are applied for statistical analysis of monthly average data of precipitation, height, and wind fields from the NCEP(National Centers for Environmental Prediction) and the western Pacific subtropical high index data from the National Climate Center. Comparison of the results show that the MGLRM has practical implications.
Regression analysis is one of the commonly used methods in statistical analysis. However, traditional regression models have less ability for global analysis, and the relationship between variables is often analyzed by methods like the SVD(Singular Value Decomposition), which lack connections with traditional regression analysis. A MGLRM(more generalized linear regression model) is a continuation of traditional linear regression model. In the case that both the predictand and the predictors are scalars, the MGLRM can be transformed into the traditional linear regression model. The MGLRM's basic features include non-commutative multiplication, equivalence to traditional linear regression as predictors in the model are scalars, analysis, extension, dimension-reduction, and robustness, etc. The MGLRM solves problems in traditional linear regression models that have less ability for global analysis and limited expressive ability due to the dimensions of the regression equation. In this paper, the MGLRM and the traditional regression model are applied for statistical analysis of monthly average data of precipitation, height, and wind fields from the NCEP(National Centers for Environmental Prediction) and the western Pacific subtropical high index data from the National Climate Center. Comparison of the results show that the MGLRM has practical implications.
引文
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黄菲,王宏,戴平.2007.南海季风型海-气耦合模态的时空特征及其与中国夏季降水的关系[J].中国海洋大学学报(自然科学版),37(3):351-356.Huang Fei,Wang Hong,Dai Ping.2007.Spatial-temporal characters of the monsoon-ocean coupled mode over the South China Sea and its relation with summer precipitation of China[J].Periodical of Ocean University of China(in Chinese),37(3):351-356,doi:10.3969/j.issn.1672-5174.2007.03.002.
黄嘉佑,刘舸,赵昕奕.2004.副高、极涡因子对我国夏季降水的影响[J].大气科学,28(4):517-526.Huang Jiayou,Liu Ge,Zhao Xinyi.2004.The influence of subtropical high indexes and polar vortex indexes on the summertime precipitation in China[J].Chinese Journal of Atmospheric Sciences(in Chinese),28(4):517-526,doi:10.3878/j.issn.1006-9895.2004.04.04.
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靳莉君,赵平.2012.夏季南海季风对长江中下游季风降水影响的观测分析和数值模拟[J].气象学报,70(4):670-680.Jin Lijun,Zhao Ping.2012.Observationally analyzing and numerically simulating the impacts of the South China Sea summer monsoon on summer monsoon rainfall in the middle-lower reaches of the Yangtze River[J].Acta Meteorologica Sinica(in Chinese),70(4):670-680,doi:10.11676/qxxb2012.054.
王会军,陈丽娟,李维京,等.2007.中国区域月平均温度和降水的模式可预报性分析[J].气象学报,65(5):725-732.Wang Huijun,Chen Lijuan,Li Weijing,et al.2007.Predictability of DERF on monthly mean temperature and precipitation over China[J].Acta Meteorologica Sinica(in Chinese),65(5):725-732,doi:10.11676/qxxb2007.068.
魏凤英.2007.现代气候统计诊断与预测技术(第二版)[M].北京:气象出版社,182-184.Wei Fengying.2007.Modern Climate Statistical Diagnosis and Prediction Technology(second edition)(in Chinese)[M].Beijing:China Meteorological Press,182-184.
魏凤英,黄嘉佑.2010.我国东部夏季降水量统计降尺度的可预测性研究[J].热带气象学报,26(4):483-488.Wei Fengying,Huang Jiayou.2010.A study of predictability for summer precipitation on East China using downscaling techniques[J].Journal of Tropical Meteorology(in Chinese),26(4):483-488,doi:10.3969/j.issn.1004-4965.2010.04.013.
吴洪宝,吴蕾.2010.气候变率诊断和预测方法(第2版)[M].北京:气象出版社,112-116.Wu Hongbao,Wu Lei.2010.Methods for Diagnosing and Forecasting Climate Variability(second edition)(in Chinese)[M].Beijing:China Meteorological Press,112-116.
吴尚森,梁建茵,李春晖.2003.南海夏季风强度与我国汛期降水的关系[J].热带气象学报,19(Z1):25-36.Wu Shangsen,Liang Jianyin,Li Chunhui.2003.Relationship between the intensity of South China Sea summer monsoon and the precipitation in raining seasons in China[J].Journal of Tropical Meteorology(in Chinese),19(Z1):25-36,doi:10.3969/j.issn.1004-4965.2003.z1.003.
严华生,严小冬,王会军.2003.用500 h Pa月高度场作月降水预报所提供的可预报性时空分布[J].热带气象学报,19(4):389-396.Yan Huasheng,Yan Xiaodong,Wang Huijun.2003.The predictability spatial and temporal distribution of making monthly rainfall prediction with 500hPa monthly height field[J].Journal of Tropical Meteorology(in Chinese),19(4):389-396,doi:10.3969/j.issn.1004-4965.2003.04.006.
杨林,魏应植,丁一汇,等.2007.东亚地区夏季旱涝分布与南海夏季风爆发时间关系[J].自然灾害学报,16(5):52-58.Yang Lin,Wei Yingzhi,Ding Yihui,et al.2007.Relationship between summer droughtflood distribution in East Asia region and breaking out time of South China Sea summer monsoon[J].Journal of Natural Disasters(in Chinese),16(5):52-58.
姚愚,严华生,程建刚.2004.主汛期(6~8月)副高各指数与中国160站降雨的关系[J].热带气象学报,20(6):651-661.Yao Yu,Yan Huasheng,Cheng Jiangang.2004.Relationship between subtropical high indexes at the main raining period with the rainfall of 160 sampling stations in China[J].Journal of Tropical Meteorology(in Chinese),20(6):651-661,doi:10.3969/j.issn.1004-4965.2004.06.005.
姚永红,钱永甫.2001.用湿位涡定义的南海西南季风指数及其与我国区域降水的关系研究[J].南京大学学报(自然科学),37(6):781-788.Yao Yonghong,Qian Yongfu.2001.A study on the South China Sea monsoon index and the relationship between the index and regional rainfalls of China[J].Journal of Nanjing University(Natural Sciences)(in Chinese),37(6):781-788,doi:10.3321/j.issn:0469-5097.2001.06.020.
俞亚勋,王式功,钱正安,等.2013.夏半年西太副高位置与东亚季风雨带(区)的气候联系[J].高原气象,32(5):1510-1525.Yu Yachu,Wang Shigong,Qian Zheng’an,et al.2013.Climatic linkages between SHWP position and EASM rainy-belts and-areas in east part of China in summer half year[J].Plateau Meteorology(in Chinese),32(5):1510-1525,doi:10.7522/j.issn.1000-0534.2013.00033.
袁媛,李崇银.2009.热带印度洋海温异常不同模态对南海夏季风爆发的可能影响[J].大气科学,33(2):325-336.Yuan Yuan,Li Chongyin.2009.Possible impacts of the tropical Indian Ocean SST anomaly modes on the South China Sea summer monsoon onset[J].Chinese Journal of Atmospheric Sciences(in Chinese),33(2):325-336,doi:10.3878/j.issn.1006-9895.2009.02.11.
赵俊虎,封国林,王启光,等.2011.2010年我国夏季降水异常气候成因分析及预测[J].大气科学,35(6):1069-1078.Zhao Junhu,Feng Guolin,Wang Qiguang,et al.2011.Cause and prediction of summer rainfall anomaly distribution in China in 2010[J].Chinese Journal of Atmospheric Sciences(in Chinese),35(6):1069-1078,doi:10.3878/j.issn.1006-9895.2011.06.07.
赵俊虎,支蓉,申茜,等.2014.2012年我国夏季降水预测与异常成因分析[J].大气科学,38(2):237-250.Zhao Junhu,Zhi Rong,Shen Qian,et al.2014.Prediction of the distribution of the 2012 summer rainfall in China and analysis of the cause for anomaly[J].Chinese Journal of Atmospheric Sciences(in Chinese),38(2):237-250,doi:10.3878/j.issn.1006-9895.2013.12215.
郑彬,施能.2006.南海夏季风对华南夏季降水年代际变化的影响[J].南京气象学院学报,29(4):477-483.Zheng Bin,Shi Neng.2006.Effects of South China Sea summer monsoon on the interdecadal variability of summer rainfall over South China[J].Journal of Nanjing Institute of Meteorology(in Chinese),29(4):477-483,doi:10.3969/j.issn.1674-7097.2006.04.007.
郑崇伟,高悦,陈璇.2017.巴基斯坦瓜达尔港风能资源的历史变化趋势及预测[J].北京大学学报(自然科学版),53(4):617-626.Zheng Chongwei,Gao Yue,Chen Xuan.2017.Climatic long term trend and prediction of the wind energy resource in the Gwadar Port[J].Acta Scientiarum Naturalium Universitatis Pekinensis(in Chinese),53(4):617-626,doi:10.13209/j.0479-8023.2017.012.
郑崇伟.2018a.21世纪海上丝绸之路:风能的长期变化趋势[J].哈尔滨工程大学学报,39(3):399-405.Zheng Chongwei.2018a.Wind energy trend in the 21st Century Maritime Silk Road[J].Journal of Harbin Engineering University(in Chinese),39(3):399-405,doi:10.11990/jheu.201704019.
郑崇伟.2018b.21世纪海上丝绸之路:斯里兰卡海域的波浪能评估及决策建议[J].哈尔滨工程大学学报,39(4):614-621.Zheng Chongwei.2018b.21st Century Maritime Silk Road:Wave energy evaluation and decision and proposal of the Sri Lankan waters[J].Journal of Harbin Engineering University(in Chinese),39(4):614-621,doi:10.11990/jheu.201704020.
郑崇伟.2018c.21世纪海上丝绸之路:关键节点的能源困境及应对[J].太平洋学报,26(7):71-78.Zheng Chongwei.2018c.Energy predicament and countermeasures on key junctions of the 21st Century Maritime Silk Road[J].Pacific Journal(in Chinese),26(7):71-78.
Chen Xuan.2018.A preliminary study on dimension-reduction algorithm for variational methods in three dimensions[OL].Preprints,doi:10.20944/preprints201801.0293.v1.
陈璇,郑崇伟,张伟涛,等.2017.更为广义的线性回归模型及其气象应用[J].解放军理工大学学报(自然科学版),18(2):144-149.Chen Xuan,Zheng Chongwei,Zhang Weitao,et al.2017.More generalized linear regression model and its meteorological application[J].Journal of PLA University of Science and Technology(Natural Science Edition)(in Chinese),18(2):144-149,doi:10.12018/j.issn.1009-3443.20161215002.
Haykin S O.2009.Neural Networks and Learning Machines[M].New Jersey:Prentice Hall,50-55.
黄菲,王宏,戴平.2007.南海季风型海-气耦合模态的时空特征及其与中国夏季降水的关系[J].中国海洋大学学报(自然科学版),37(3):351-356.Huang Fei,Wang Hong,Dai Ping.2007.Spatial-temporal characters of the monsoon-ocean coupled mode over the South China Sea and its relation with summer precipitation of China[J].Periodical of Ocean University of China(in Chinese),37(3):351-356,doi:10.3969/j.issn.1672-5174.2007.03.002.
黄嘉佑,刘舸,赵昕奕.2004.副高、极涡因子对我国夏季降水的影响[J].大气科学,28(4):517-526.Huang Jiayou,Liu Ge,Zhao Xinyi.2004.The influence of subtropical high indexes and polar vortex indexes on the summertime precipitation in China[J].Chinese Journal of Atmospheric Sciences(in Chinese),28(4):517-526,doi:10.3878/j.issn.1006-9895.2004.04.04.
黄玮,程建刚,王学锋,等.2013.基于气候模式的动力-统计降尺度技术在云南降水预测中的应用[J].云南大学学报(自然科学版),35(6):782-790.Huang Wei,Cheng Jiangang,Wang Xuefeng,et al.2013.An application of a dynamical-statistical downscaling method based on climate model to precipitation forecast over Yunnan[J].Journal of Yunnan University(Natural Sciences)(in Chinese),35(6):782-790,doi:10.7540/j.ynu.20120078.
靳莉君,赵平.2012.夏季南海季风对长江中下游季风降水影响的观测分析和数值模拟[J].气象学报,70(4):670-680.Jin Lijun,Zhao Ping.2012.Observationally analyzing and numerically simulating the impacts of the South China Sea summer monsoon on summer monsoon rainfall in the middle-lower reaches of the Yangtze River[J].Acta Meteorologica Sinica(in Chinese),70(4):670-680,doi:10.11676/qxxb2012.054.
王会军,陈丽娟,李维京,等.2007.中国区域月平均温度和降水的模式可预报性分析[J].气象学报,65(5):725-732.Wang Huijun,Chen Lijuan,Li Weijing,et al.2007.Predictability of DERF on monthly mean temperature and precipitation over China[J].Acta Meteorologica Sinica(in Chinese),65(5):725-732,doi:10.11676/qxxb2007.068.
魏凤英.2007.现代气候统计诊断与预测技术(第二版)[M].北京:气象出版社,182-184.Wei Fengying.2007.Modern Climate Statistical Diagnosis and Prediction Technology(second edition)(in Chinese)[M].Beijing:China Meteorological Press,182-184.
魏凤英,黄嘉佑.2010.我国东部夏季降水量统计降尺度的可预测性研究[J].热带气象学报,26(4):483-488.Wei Fengying,Huang Jiayou.2010.A study of predictability for summer precipitation on East China using downscaling techniques[J].Journal of Tropical Meteorology(in Chinese),26(4):483-488,doi:10.3969/j.issn.1004-4965.2010.04.013.
吴洪宝,吴蕾.2010.气候变率诊断和预测方法(第2版)[M].北京:气象出版社,112-116.Wu Hongbao,Wu Lei.2010.Methods for Diagnosing and Forecasting Climate Variability(second edition)(in Chinese)[M].Beijing:China Meteorological Press,112-116.
吴尚森,梁建茵,李春晖.2003.南海夏季风强度与我国汛期降水的关系[J].热带气象学报,19(Z1):25-36.Wu Shangsen,Liang Jianyin,Li Chunhui.2003.Relationship between the intensity of South China Sea summer monsoon and the precipitation in raining seasons in China[J].Journal of Tropical Meteorology(in Chinese),19(Z1):25-36,doi:10.3969/j.issn.1004-4965.2003.z1.003.
严华生,严小冬,王会军.2003.用500 h Pa月高度场作月降水预报所提供的可预报性时空分布[J].热带气象学报,19(4):389-396.Yan Huasheng,Yan Xiaodong,Wang Huijun.2003.The predictability spatial and temporal distribution of making monthly rainfall prediction with 500hPa monthly height field[J].Journal of Tropical Meteorology(in Chinese),19(4):389-396,doi:10.3969/j.issn.1004-4965.2003.04.006.
杨林,魏应植,丁一汇,等.2007.东亚地区夏季旱涝分布与南海夏季风爆发时间关系[J].自然灾害学报,16(5):52-58.Yang Lin,Wei Yingzhi,Ding Yihui,et al.2007.Relationship between summer droughtflood distribution in East Asia region and breaking out time of South China Sea summer monsoon[J].Journal of Natural Disasters(in Chinese),16(5):52-58.
姚愚,严华生,程建刚.2004.主汛期(6~8月)副高各指数与中国160站降雨的关系[J].热带气象学报,20(6):651-661.Yao Yu,Yan Huasheng,Cheng Jiangang.2004.Relationship between subtropical high indexes at the main raining period with the rainfall of 160 sampling stations in China[J].Journal of Tropical Meteorology(in Chinese),20(6):651-661,doi:10.3969/j.issn.1004-4965.2004.06.005.
姚永红,钱永甫.2001.用湿位涡定义的南海西南季风指数及其与我国区域降水的关系研究[J].南京大学学报(自然科学),37(6):781-788.Yao Yonghong,Qian Yongfu.2001.A study on the South China Sea monsoon index and the relationship between the index and regional rainfalls of China[J].Journal of Nanjing University(Natural Sciences)(in Chinese),37(6):781-788,doi:10.3321/j.issn:0469-5097.2001.06.020.
俞亚勋,王式功,钱正安,等.2013.夏半年西太副高位置与东亚季风雨带(区)的气候联系[J].高原气象,32(5):1510-1525.Yu Yachu,Wang Shigong,Qian Zheng’an,et al.2013.Climatic linkages between SHWP position and EASM rainy-belts and-areas in east part of China in summer half year[J].Plateau Meteorology(in Chinese),32(5):1510-1525,doi:10.7522/j.issn.1000-0534.2013.00033.
袁媛,李崇银.2009.热带印度洋海温异常不同模态对南海夏季风爆发的可能影响[J].大气科学,33(2):325-336.Yuan Yuan,Li Chongyin.2009.Possible impacts of the tropical Indian Ocean SST anomaly modes on the South China Sea summer monsoon onset[J].Chinese Journal of Atmospheric Sciences(in Chinese),33(2):325-336,doi:10.3878/j.issn.1006-9895.2009.02.11.
赵俊虎,封国林,王启光,等.2011.2010年我国夏季降水异常气候成因分析及预测[J].大气科学,35(6):1069-1078.Zhao Junhu,Feng Guolin,Wang Qiguang,et al.2011.Cause and prediction of summer rainfall anomaly distribution in China in 2010[J].Chinese Journal of Atmospheric Sciences(in Chinese),35(6):1069-1078,doi:10.3878/j.issn.1006-9895.2011.06.07.
赵俊虎,支蓉,申茜,等.2014.2012年我国夏季降水预测与异常成因分析[J].大气科学,38(2):237-250.Zhao Junhu,Zhi Rong,Shen Qian,et al.2014.Prediction of the distribution of the 2012 summer rainfall in China and analysis of the cause for anomaly[J].Chinese Journal of Atmospheric Sciences(in Chinese),38(2):237-250,doi:10.3878/j.issn.1006-9895.2013.12215.
郑彬,施能.2006.南海夏季风对华南夏季降水年代际变化的影响[J].南京气象学院学报,29(4):477-483.Zheng Bin,Shi Neng.2006.Effects of South China Sea summer monsoon on the interdecadal variability of summer rainfall over South China[J].Journal of Nanjing Institute of Meteorology(in Chinese),29(4):477-483,doi:10.3969/j.issn.1674-7097.2006.04.007.
郑崇伟,高悦,陈璇.2017.巴基斯坦瓜达尔港风能资源的历史变化趋势及预测[J].北京大学学报(自然科学版),53(4):617-626.Zheng Chongwei,Gao Yue,Chen Xuan.2017.Climatic long term trend and prediction of the wind energy resource in the Gwadar Port[J].Acta Scientiarum Naturalium Universitatis Pekinensis(in Chinese),53(4):617-626,doi:10.13209/j.0479-8023.2017.012.
郑崇伟.2018a.21世纪海上丝绸之路:风能的长期变化趋势[J].哈尔滨工程大学学报,39(3):399-405.Zheng Chongwei.2018a.Wind energy trend in the 21st Century Maritime Silk Road[J].Journal of Harbin Engineering University(in Chinese),39(3):399-405,doi:10.11990/jheu.201704019.
郑崇伟.2018b.21世纪海上丝绸之路:斯里兰卡海域的波浪能评估及决策建议[J].哈尔滨工程大学学报,39(4):614-621.Zheng Chongwei.2018b.21st Century Maritime Silk Road:Wave energy evaluation and decision and proposal of the Sri Lankan waters[J].Journal of Harbin Engineering University(in Chinese),39(4):614-621,doi:10.11990/jheu.201704020.
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