时间序列分析的早期发展
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摘要
在当前信息化时代,数据的处理和分析至关重要,时间序列分析作为一套独具特色、自成体系的数据分析和处理方法被广泛应用,逐渐引起科学家和社会学家的高度重视。对于这样备受关注的学科,尽管国内外的教材较为丰富,有关历史发展研究的文献却不多见,而且多是一些概括性、粗略性的介绍,零散地分布于各种专著和论文之中,因此,系统地研究时间序列分析的历史演化过程具有重要的理论价值和现实意义。
本文在已有研究基础上,以“为什么数学”作为方法论指导,主要运用了以下研究方法：以原始文献为依据,内史与外史相结合；全面进行纵向和横向比较,注重不同时期关键人物和核心工作之间的传承关系；注重不同学科间的交叉和融合。主要研究成果为：
1.首次以时间序列分析的起源、基本概念对学科发展的推动、以及时间序列分析两大主体方法的初步发展为主线,系统地梳理了时间序列分析早期发展的历史脉络,为时间序列分析学科史的研究提供了一条清晰的线索。
2.对于时间序列分析的起源问题,众说纷纭、观点不一,本文通过对原始文献的剖析,首次提出了平稳时间序列产生的背景正是Graunt的关键性创新思想——统计比率对于时间和空间的稳定性,并根据Graunt各种现代时问序列思想的萌芽,以及对Pearson的学术影响,肯定了Graunt在时间序列分析上的先驱地位。
3.探讨了时间序列分析基本概念——差分、指数与滑动平均等从金融算术到政治算术,到科学算术,最后进入到现代化数学领域的发展历程,展现了这些最初只是金融家进行贸易猜测和掩盖真相的技术工具,如何被统计学家逐步借鉴、运用到实证研究中,并推动着现代时间序列分析形成的发展历程。
4.概括了时间序列分析两大方法——频域分析和时域分析的早期发展,详细考察了Schuster在Fourier级数理论的基础上,创建周期图方法的背景、过程及应用。对周期图方法缺陷的讨论和修订,不仅引导着频域分析方法的提升,而且刺激了时域分析方法的初步发展,文章细致分析了时域分析中平稳时间序列三大模型——线性自回归AR模型、移动平均MA模型和自回归移动平均ARM A模型的创建过程及其相互之间的传承关系,并特别强调,Wold对于离散平稳时间序列的综合研究,不仅概括了AR模型和MA模型,更结合了Khinchin、Kolmogorov研究平稳随机过程的概率理论,以及Schuster的周期图方法。最后,以Wold分解定理的基础、诞生、发展及影响为主线,梳理了时间序列分解的发展历程。
5.分析了经济学、统计学思想与时间序列分析学科的交叉、融合：以价格平衡问题、稳定和扰动概念、经济学家和统计学家对贸易循环的共同关注以及Lexis的实证研究等典型问题为例,阐述了经济学对时间序列分析的影响；以回归与相关引发“时间相关问题”,促使回归概念发展到自回归概念为主线,阐述了统计学对时间序列分析的影响。
In present information age, it is essential to process and analyse data. Due to its unique, self-contained methods for data analysis and processing, time series analysis has been widely applied to many areas and is drawing scientist's great attention. In spite of the rich teaching material, research literature on its historical development for such a frontier subject is still wanting, and most of historical introduction in existence are briefly and concisely scattering in various monographs and papers. Systematic studies on the historical evolution of time series analysis are therefore necessary and of both theoretical and practical significance.
Based on the existing research work on the considered field and regarding "why mathematics?" as its methodological guide, the thesis takes the following main research approaches:deep investigation into the original literature, combination of the internal and external historical view; comparitive studies comprehensively from the vertical and horizontal perspective, in particular, in the inheritance relationship between the key figures in different periods and their core theory; emphasizing the interaction and integration between different disciplines. The main results are as follows:
l.The historical context of the early development of time series analysis is systematically sorted for the first time by major consideration of the origin of the basic concepts which impulse the growth of the subject and the initial development of its two principal methods. The thesis therefore provides a clear clue for the study of the discipline history of time series analysis.
2. Facing diverse opinions for the origin of time series analysis, the thesis first clearly points out by analyzing the original literature that the background of arising of stationary time series was right the concept "stable statistical ratios over time and space", which was Graunt's key innovative idea. Combined with the exposition of Graunt's all germinating ideas of modern time series and his academic influence on Pearson, it establishes firmly the historical place of Graunt as a pioneer of time series analysis.
3. The development process of the basic concepts including the difference, the index number and the moving average is explored, from monetary arithmetic to political arithmetic, then to scientific arithmetic, and finally into the modern mathematical areas. It displays how the statisticians gradually applied these technical tools, which were initially used to speculate and cover up the truth by financiers, into empirical investigation and laid theoretical foundation for the modern time series analysis.
4. The early development of the two principle methods——the frequency domain method and the time domain method of time series is summarized. The background, evolution and application of periodogram method created by Schuster on bases of the theory of Fourier series are discussed in detail. The improvment of periodogram method not only led to the advancement of frequency domain method but also stimulated the early development of time domain method. The thesis analyses carefully the establishing process and the inheritance relationship of three models of stationary time series (auto regression model, moving average model and auto regression moving average model), with emphasis on that Wold's comprehensive study in the discrete stationary time series not only integrated the AR model and MA model, but also combined Khinchin and Kolmogorov's probability theory which had been used to study the stationary random process as well as Schuster's periodogram method. At last, the development of time series decomposition is described by observing mainly the birth, progress and influence of Wold decomposition theorem.
5. The interaction and integration of economics and statistics with time series analysis are investigated. The impact of economics on time series analysis is described by taking the typical cases such as:the price equilibrium problem, the concept of stationary and disturbance, economists and statisticians'common concern about the trade cycle, Lexis' empirical investigation; The main line of elaborating the impact of statistics on time series analysis is the fact that regression and correlation led to time-correlation problem and promoted the concept evolution from regression to autoregression.
本文在已有研究基础上,以“为什么数学”作为方法论指导,主要运用了以下研究方法：以原始文献为依据,内史与外史相结合；全面进行纵向和横向比较,注重不同时期关键人物和核心工作之间的传承关系；注重不同学科间的交叉和融合。主要研究成果为：
1.首次以时间序列分析的起源、基本概念对学科发展的推动、以及时间序列分析两大主体方法的初步发展为主线,系统地梳理了时间序列分析早期发展的历史脉络,为时间序列分析学科史的研究提供了一条清晰的线索。
2.对于时间序列分析的起源问题,众说纷纭、观点不一,本文通过对原始文献的剖析,首次提出了平稳时间序列产生的背景正是Graunt的关键性创新思想——统计比率对于时间和空间的稳定性,并根据Graunt各种现代时问序列思想的萌芽,以及对Pearson的学术影响,肯定了Graunt在时间序列分析上的先驱地位。
3.探讨了时间序列分析基本概念——差分、指数与滑动平均等从金融算术到政治算术,到科学算术,最后进入到现代化数学领域的发展历程,展现了这些最初只是金融家进行贸易猜测和掩盖真相的技术工具,如何被统计学家逐步借鉴、运用到实证研究中,并推动着现代时间序列分析形成的发展历程。
4.概括了时间序列分析两大方法——频域分析和时域分析的早期发展,详细考察了Schuster在Fourier级数理论的基础上,创建周期图方法的背景、过程及应用。对周期图方法缺陷的讨论和修订,不仅引导着频域分析方法的提升,而且刺激了时域分析方法的初步发展,文章细致分析了时域分析中平稳时间序列三大模型——线性自回归AR模型、移动平均MA模型和自回归移动平均ARM A模型的创建过程及其相互之间的传承关系,并特别强调,Wold对于离散平稳时间序列的综合研究,不仅概括了AR模型和MA模型,更结合了Khinchin、Kolmogorov研究平稳随机过程的概率理论,以及Schuster的周期图方法。最后,以Wold分解定理的基础、诞生、发展及影响为主线,梳理了时间序列分解的发展历程。
5.分析了经济学、统计学思想与时间序列分析学科的交叉、融合：以价格平衡问题、稳定和扰动概念、经济学家和统计学家对贸易循环的共同关注以及Lexis的实证研究等典型问题为例,阐述了经济学对时间序列分析的影响；以回归与相关引发“时间相关问题”,促使回归概念发展到自回归概念为主线,阐述了统计学对时间序列分析的影响。
In present information age, it is essential to process and analyse data. Due to its unique, self-contained methods for data analysis and processing, time series analysis has been widely applied to many areas and is drawing scientist's great attention. In spite of the rich teaching material, research literature on its historical development for such a frontier subject is still wanting, and most of historical introduction in existence are briefly and concisely scattering in various monographs and papers. Systematic studies on the historical evolution of time series analysis are therefore necessary and of both theoretical and practical significance.
Based on the existing research work on the considered field and regarding "why mathematics?" as its methodological guide, the thesis takes the following main research approaches:deep investigation into the original literature, combination of the internal and external historical view; comparitive studies comprehensively from the vertical and horizontal perspective, in particular, in the inheritance relationship between the key figures in different periods and their core theory; emphasizing the interaction and integration between different disciplines. The main results are as follows:
l.The historical context of the early development of time series analysis is systematically sorted for the first time by major consideration of the origin of the basic concepts which impulse the growth of the subject and the initial development of its two principal methods. The thesis therefore provides a clear clue for the study of the discipline history of time series analysis.
2. Facing diverse opinions for the origin of time series analysis, the thesis first clearly points out by analyzing the original literature that the background of arising of stationary time series was right the concept "stable statistical ratios over time and space", which was Graunt's key innovative idea. Combined with the exposition of Graunt's all germinating ideas of modern time series and his academic influence on Pearson, it establishes firmly the historical place of Graunt as a pioneer of time series analysis.
3. The development process of the basic concepts including the difference, the index number and the moving average is explored, from monetary arithmetic to political arithmetic, then to scientific arithmetic, and finally into the modern mathematical areas. It displays how the statisticians gradually applied these technical tools, which were initially used to speculate and cover up the truth by financiers, into empirical investigation and laid theoretical foundation for the modern time series analysis.
4. The early development of the two principle methods——the frequency domain method and the time domain method of time series is summarized. The background, evolution and application of periodogram method created by Schuster on bases of the theory of Fourier series are discussed in detail. The improvment of periodogram method not only led to the advancement of frequency domain method but also stimulated the early development of time domain method. The thesis analyses carefully the establishing process and the inheritance relationship of three models of stationary time series (auto regression model, moving average model and auto regression moving average model), with emphasis on that Wold's comprehensive study in the discrete stationary time series not only integrated the AR model and MA model, but also combined Khinchin and Kolmogorov's probability theory which had been used to study the stationary random process as well as Schuster's periodogram method. At last, the development of time series decomposition is described by observing mainly the birth, progress and influence of Wold decomposition theorem.
5. The interaction and integration of economics and statistics with time series analysis are investigated. The impact of economics on time series analysis is described by taking the typical cases such as:the price equilibrium problem, the concept of stationary and disturbance, economists and statisticians'common concern about the trade cycle, Lexis' empirical investigation; The main line of elaborating the impact of statistics on time series analysis is the fact that regression and correlation led to time-correlation problem and promoted the concept evolution from regression to autoregression.
引文
①Anderson T. W., The Statistical Analysis of Time Series[M]. New York:John Wiley & Sons,1971:1-3.
②为了发展农业生产,古埃及人特别关注尼罗河的泛滥规律,对尼罗河涨落情况的长期观察、记录,使他们发现,尼罗河的泛滥开始于天狼星第一次和太阳同时升起之后的200天左右,洪水持续七、八十天之后,土地肥沃,适于农业种植。
③王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：1.
④罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117.
②Kirchgassner G., Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007:2.⑤王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：2.
① Kirchgassner G, Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007:2-5.
②Wold H., A Study in the Analysis of Stationary Time Series[M]. Stockholm:Almqvist and Wiksell, [1938]1954.
③Box G.E. P., Jenkins G.M., Time-Series Analysis:Forecasting and Control[M]. San Francisco:Holden Day,1970.
④这些时间单位包括年、季、月、日,甚至也可能是时、分、秒等。
①Engle R. F., Auto Regressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation [J]. Econometrics,1982,50(4):987-1007.
②ARCH族计量模型包括GARCH模型(G)ARCH-M模型FGARCH模型EGARCH模型JGARCH模型和TGARCH模型等等,当前研究ARCH族模型的论文、论著非常丰富,此处不再赘述。
③聂淑媛.时间序列分析的历史发展[J].广西民族大学学报(自然科学版),2012,18(1)：24-28.
④比如,有限域的历史、博弈论前史、组合学史、生物数学史、行列式理论、傅里叶级数的理论、偏微分方程理论等。
⑤Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997.
① Kirchgassner G, Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007.
②张尧庭.时间序列分析[J].统计教育,1996,23(4)：23-27.
③吴正英.浅谈时间序列发展史[J].统计与预测,1999,5：55-56.
④程振源.时间序列分析：历史回顾与未来展望[J].统计与决策,2002,9：45-46.
⑤罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117.
⑥汤岩.时间序列分析的研究与应用[D].哈尔滨：东北农业大学,2007.
①曲安京.中国数学史研究的两次运动[J].科学,2004,56(2)：27-30.
②曲安京.中国数学史研究范式的转换[J].中国科技史杂志,2005,26(1)：50-58.
①Graunt J., Natural and Political Observations Made Upon the Bills of Mortality[M]. London:John Martin and James Allestry, [1662] 1665.
①陈希孺.数理统计学简史[M].长沙：湖南教育出版社,2005：131-135.
③所谓三分法,是指在一个比例关系a/b=c/d中,用三个已知数a、6、c去求解第四个未知数d的简单算术技巧。
在欧洲,该法则由于其开创性的思想而受到极高的评价。
④Graunt的《观察》一书,共包括12章,用8个表格细致整理了死亡公报中的庞大数据,内容涉及到死亡原因(尤其是当时比较多见的黑死病)、男女差异、伦敦人口数及增长情况等诸多问题,Graunt的结论正是通过对数据的分析而推断得出。
⑤所谓统计比率的稳定性,是指某个事件或某种特性出现的频率,随着观察次数的逐渐增加而趋向于一个固定值。
⑥慢性病如结核病等,偶然因素如淹死等。
①聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7.
②汤岩.时间序列分析的研究与应用[D].哈尔滨：东北农业大学,2007.
③张尧庭.时间序列分析[J].统计教育,1996,23(4)：23-27.
④王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：1.
⑥罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117.
⑥Kirchgassner G, Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007:2.
①张树京,齐立心.时间序列分析简明教程[M].北京：清华大学出版社,北方交通大学出版社,2003：19-20.
②评语为：对任何缺少实际测量和三分法的事情,我都没有信心。
①聂淑媛.格朗特与时间序列分析[J].自然辩证法通讯,2011,33(3)：50-52.
①Hald A., A History of Probability and Statistics and their Applications before 1750[M]. New York:Wiley,1990:55.
①如莱布尼兹(Gottfried Wilhelm Leibniz,1646-1716)1666年在其《组合艺术》中考察平方数序列时,也引入了阶差分和二阶差分,而且也正是由此开始,他注意到序列求和与求差之间的互逆关系,并进一步发展了微积分。
②Box G. E. P., Jenkins G. M., Time-Series Analysis:Forecasting and Control[M]. San Francisco:Holden Day,1970:60-98.
③Perman R., Cointegration:An Introduction to The Literature[J]. Journal of Economic Studies,1991,18(3):3-30.
④Rao B. B., Cointegration for the Applied Economist[M]. New York:St.Martin's Press,1994:1-3.
①所谓“政治人”,指小依靠抽象的语言和看似巧妙的推理,一切都依靠数字说明,并最终讨论和决定政治政策。
②Statistics, ed. Henry Higgs and George Udny Yule[M]. London:Macmillan,1913:25-46.
①Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997:66.
①Norton J., Statistical Studies in the New York Money Market[M]. New York:Macmillan,1902.
①Cave-Browne-Cave F. E., Pearson K., On the Influence of the Time Factor on the Correlation between the Barometric Heights at Stations More Than 1000 Miles Apart[J]. Proceedings of the Royal Statistical Society of London,1904,74: 403-413.
①Hooker R., The Suspension of the Berlin Produce Exchange and its Effect upon Corn Prices[J]. Journal of the Royal Statistical Society,1901,64(3):574-613.
②Hooker R., On the Correlation of Successive Observations Illustrated by Corn Prices [J]. Journal of the Royal Statistical Society,1905,68(4):696-703.
①Student, The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(1):179-180.
②Anderson O., The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(2-3): 269-279.
①聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7.
②1938年,Wold把这种危险序列定义为非稳定过程或发展过程。
①Yule差分思想的发展历程,可参见第五章5.1节。
②聂淑媛.经济时间序列中差分的历史研究[J].科学技术哲学研究,2012,29(1)：70-75.
①Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997:78-85.
①Poynting J. H., The Drunkenness Statistics of the Large Towns in England and Wales. In Collected Scientific Papers[C]. Cambridge:Cambridge University Press, [187711920:497-503.
②Poynting J. H., A Comparison of the Fluctuations in the Price of Wheat and in Cotton and Silk Imports into Great Britain[J]. Journal of the Royal Statistical Society,1884,47(1):34-74.
①Poynting J. H., The Drunkenness Statistics of the Large Towns in England and Wales. In Collected Scientific Papers[C]. Cambridge:Cambridge University Press, [1877]1920:497-503.
②谷物计量单位,即1/4吨。
②需要说明,经济学家勃雷(Arthur Lyon Bowley,1869—1957)1895年研究工资问题时,也发展完善了比较的概念,他根据不同地区、不同职业和不同时间的大量工资数据,从杂乱元章的情形中建构了连续指数序列,联系这一链条的关键止是变化率。
②Jevons W. S., The Periodicity of Commercial Crises and Its Physical Expianation. In Investigations in Currency and Finance [M]. London:Macmillan, [1878b]1884:206-219.
①Poynting J. H., A Comparison of the Fluctuations in the Price of Wheat and in Cotton and Silk Imports into Great Britain[J]. Journal of the Royal Statistical Society,1884,47(1):34-74.
②尽管Poynting从未在论文中提到Jevons,但一个不容忽视的事实是,当Poynting向皇家统计学会提交论文之后,Jevons的工作才逐渐出现在皇家统计学会的讨论中。
①Hooker R., Correlation of the Marriage Rate with Trade[J]. Journal of the Royal Statistical Society,1901,64(3):485-492.
①聂淑媛,梁铁旺.指数和滑动平均的历史发展探究[J].统计与决策,2011,24：4-7.
①统计学家把这种规律性很强的贸易循环称为平稳过程。
② The Theory of the Stability of Statistical Series. WPA No.165-1-71-124. [M]. Minneapolis:University of Minnesota, U.S.Works Projects Administration[1879]1942:20-80.
①贾随军.傅里叶级数理论的起源[D].西安：西北大学,2010.
①贾随军.傅里叶级数理论的起源[D].西安：西北大学,2010.
②[法]约瑟夫·傅立叶著,桂质亮译.热的解析理论[M].北京：北京大学出版社,2008.
①Schuster A., On Sun-Spot Periodicities. Preliminary Notice[J]. Proceedings of the Royal Society of London. Series A, 1906,77(515):141-145.
②Schuster A., On the Periodicities of Sunspots [J]. Philosophical Transactions of the Royal Society of London. Series A, 1906,206:69-100.
③Schuster A., On the Periodicity of Sun-Spots[J]. Proceedings of the Royal Society of London, Series A,1911,85(575): 50-53.
④Simpson G.C., Obituary:Sir Arthur Schuster, F.R.S. [J]. Nature,1934,134:595-597.
⑤Simpson G. C., Sir Arthur Schuster.1851-1934. [J]. Obituary Notices of Fellows of the Royal Society,1935,1(4):408-423.
①日冕是太阳大气的最外层,厚度达几百万公里以上,温度有100万摄氏度,可人为地分为内冕、中冕和外冕3层。
②Simpson G. C., Obituary:Sir Arthur Schuster, F.R.S. [J]. Nature,1934,134:595-597.
③Simpson G. C., Sir Arthur Schuster.1851-1934. [J]. Obituary Notices of Fellows of the Royal Society,1935,1(4):408-423.
④Schuster A., On the Investigation of Hidden Periodicities with Application to a Supposed 26 Day Period of Meteorological Phenomena[J]. Terrestrial Magnetism,1898,3(1):13-41.
①Schuster A., On Sun-Spot Periodicities. Preliminary Notice[J]. Proceedings of the Royal Society of London. Series A, 1906,77(515):141-145.
①Schuster A., On the Periodicities of Sunspots [J]. Philosophical Transactions of the Royal Society of London. Series A, 1906,206:69-100.
②Schuster使用的是沃尔夫观察得到的太阳黑子数,而把由平滑过程得到的数据放在单独的表中作为“补偿数”,因为他认为平滑过程削弱了数字的精确程度,不利于科学地检验周期。
①Schuster A., On the Periodicity of Sun-Spots[J]. Proceedings of the Royal Society of London, Series A,1911,85(575): 50-53.
②Schuster指出Oppenheim通过对拉蒙特(Lamont)地区1836—1886年的观察值取连续两年的平均,使用与Schuster完全不同的方法,在磁下降记录中也独立地发现了相唰的周期4.92年,Schuster强调,因为Oppenheim以两年平均 为基础,且只有12个完整周期,故4.92年和4.78年的差别可以忽略。
①中国大百科全书.天文学[M].北京：中国大百科全书出版社,1980,
①Yule G.U., On the Time-Correlation Problem, with Especial Reference to the Variate- Difference Correlation Method[J]. Journal of the Royal Statistical,1921,84(4):497-537.
②Yule GU., Why do we Sometimes get Nonsense-Correlations between Time-Series?--A Study in Sampling and the Nature of Time-Series[J]. Journal of the Royal Statistical,1926,89(1):1-63.
③Yule G U., On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers[J]. Philosophical Transactions of the Royal Society of London. Series A,1927,226:267-298.
④Yule G.U., On the Theory of Correlation[J]. Journal of the Royal Statistical,1897,60(4):812-854.
⑤Yule G.U. Kendall M. G, An introduction to the theory of statistics[M]. London:Charles Griffin,1950:88-90.
①Kendall M. G, Obituaries:Mr. GUdny Yule, C.B.E, F.R.S. [J]. Nature,1951,168(4274):542-543.
②Kendall M. G, George Udny Yule,1871-1951 [J]. Review of the International Statistical Institute,1952,20(1):92-93.
③Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997.
④这些思想已经在第三章3.2节中进行了细致分析。
④Hooker R., Correlation of the Marriage Rate with Trade[J]. Journal of the Royal Statistical Society,1901,64(3):485-492.
①聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7.
②聂淑媛.经济时间序列中差分的历史研究[J].科学技术哲学研究,2012,29(1)：70-75.
③目前通常采用ε或εt表示扰动,Yule原文中用的也是ε,但Walker原文中vx表示扰动,此处未作更改,下同。第六章中Wold使用的一些符号,与当前的常用符号也略有差异,本文在分析这些数学家的工作时,尽量忠实于原著,不做符号的改动。
①Walker G.,On Periodicity in Series of Related Terms[J].Proceedings of the Royal Society,Series A,1931,131(818): 518.532.
②Walker强调,虽然与周期相联系的相关系数法不是一种新方法,但却被广泛使用。
①Slutzky E. E., The Summation of Random Causes As the Source of Cyclic Processes[J].Econometrica,1937,5(2):105-146.
①Slutzky在原文中用严格的数学语言表达和证明了正弦极限定理,本文对他的数学语言进行了转换。
②Slutzky认为,经济调查者凭借自己的敏锐和直觉,对商业循环的周期至少有一个大概的正确判断,他们是可靠的。
③他的同事Kondratieff在1928年出版了著作《The Long Wave Cycle》之后,不仅被迫关闭了研究所,后来甚至被送进了监狱。
④Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997:276-279.
①Pearson K., The Problem of the Random Walk [J]. Nature,1905(72):294,342.
①王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：150.
②Bachelier L., Theory of Speculation. Trans. James Boness. In the Random Character of Stock Market Pricesr[M]. Cambridge, MA:MIT Press, [1900]1964:17-78.
①关于布朗运动的资料非常丰富,对其历史发展的研究可以参阅中文文献[58]、[59]、[60]等,此处不再赘述。
②Kendall M. G.,The Analysis of Economic Time Series—Part 1:Prices [J]. Journal of the Royal Statistical Society, SeriesA, 1953,116(1):11-25.
①Wold H., A Study in the Analysis of Stationary Time Series[M]. Stockholm:Almqvist and Wiksell, [1938]1954:84-92.
①Persons W. M., Correlation of Time Series. In Handbook of Mathematical Statisticas[M]. Boston:Houghton Mifflin,1924: 150-165.
②Yulec对 Anderson高阶差分方法的质疑可以参见第二章2.1.3节和第五章5.1.2节。
①王振龙.应用时间序列分析[M].北京：中国统计出版社,2010：93-94
②王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：112.
[1]Akaike H., A New Look at the Statistical Model Identification[J]. IEEE Transactions on Automatic Control,1974,19(6):716-723
[2]Aldrich J., Correlations Genuine and Spurious in Pearson and Yule[J]. Statistical Science,1995,10(4):364-376
[3]Anderson O., The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(2-3):269-279
[4]Anderson O., On the Logic of the Decomposition of Statistical Series into Separate Components[J]. Journal of the Royal Statistical Society,1927,90(3):548-569
[5]Anderson T. W., The Statistical Analysis of Time Series[M]. New York:John Wiley & Sons,1971
[6]Bachelier L., Theory of Speculation. In the Random Character of Stock Market Pricesr[M]. Cambridge, MA:MIT Press, [1900]1964:17-78
[7]Bartlett M. S., On the Theoretical Specification of Sampling Properties of Autocorrelated Time Series[J], Journal of the Royal Statistical Society B,1946,8(1): 27-41
[8]Beveridge W. H., Wheat Prices and Rainfall in Western Europe[J]. Journal of the Royal Statistical Society,1922,85(3):412-475
[9]Bollerslev T., Generalized Auto Regressive Conditional Heteroskedasticity[J]. Journal of Econometrics,1986,31(3):307-327
[10]Box G. E. P., Jenkins G. M., Time-Series Analysis:Forecasting and Control[M]. San Francisco:Holden Day,1970
[11]Brillinger D. R., Time Series:Data Analysis and Theory[D]. Philadelphia, SIAM, 2001
[12]Brockwell P. J., Davis R. A., Time Series:Theory and Methods(2nd ed)[M]. New York:Springer,1991
[13]Brockwell P. J., Davis R. A., Introduction to Time Series and Forecasting (2nd ed) [M]. New York:Springer,2002
[14]Brown R. G., Smoothing, Forecasting and Prediction of Discrete Time Series[M]. Englewood Cliffs, NJ: Prentice-Hall, 1962
[15]Cave-Browne-Cave F.E., Pearson K., On the Influence of the Time Factor on the Correlation between the Barometric Heights at Stations More Than 1000 Miles Apart[J]. Proceedings of the Royal Statistical Society of London, 1904, 74: 403-413
[16]Cave B., Pearson K., Numerical Illustrations of the Variate Difference Correlation Method[J].Biometrika, 1914, 10(2-3): 340-355
[17]Chatfield C, The Analysis of Time Series: An Introduction(3nd ed) [M]. London: Chapman and Hall, 1984
[18]Cooley J. W., Tukey J. W., An Algorithm for the Machine Calculation of Complex Fourier Series[J]. Mathematics of Computation, 1965, 19(90): 297-301
[19]Cramer H., Leadbetter M. R., Stationary and Related Random Processes[M]. New York: John Wiley&Sons, 1967
[20]Draper N. R., Smith H., Applied Regression Analysis(2nd ed)[M]. New York: John Wiley & Sons, 1981
[21]Durbin J., Koopman S. J., Time Series Analysis by State Space Methods[M].Oxford: Oxford University Press, 2001
[22]Engle R. F., Auto Regressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation[J]. Econometrica, 1982, 50(4): 987-1007
[23]Fisher I., Our Unstable Dollar and the So-called Business Cycle[J]. Journal of the American Statistical Association, 1925, 20(150): 179-202
[24]Frisch R., Propagation Problems and Impulse Problems in Dynamic Economics. In Economic Essays in Honour of Gustav Cassel [M]. London: George Allen & Unwin, 1933: 171-205
[25]Fuller W. A., Introduction to Statistical Time Series(2nd ed)[M]. New York: John Wiley & Sons, 1996
[26]Galton F., Regression towards Mediocrity in Hereditary Stature[J]. The Journal of the Anthropological Institute of Great Britain and Ireland, 1886, 15: 246-263
[27]Galton F., Essays on Eugenics[M]. London: Eugenics Eduation Society, 1909
[28]GiffenR., The Excess of Imports [J]. Journal of the Royal Statistical Society, 1899, 62(1):1-82
[29]Giffen R., Economic Inquiries and Studies[M]. London:George Bell and Sons,1904
[30]Giffen R., Statistics, ed. Henry Higgs and George Udny Yule[M]. London: Macmillan,1913
[31]Graunt J., Natural and Political Observations Made Upon the Bills of Mortality[M]. London:John Martin and James Allestry, [1662] 1665
[32]Great Britain, Parliament. Report of the Committee on the Bank of England Charter 1832[C]. House of Commons Paper 722,1832
[33]Hacking I., The Emergence of Probability[M]. Cambridge:Cambridge University Press,1975
[34]Hald A., A History of Probability and Statistics and their Applications before 1750 [M]. New York:Wiley,1990
[35]Hald A., A History of Mathematical Statistics from 1750 to 1930[M]. New York: Wiley,1998
[36]Hamilton J. D., Time Series Analysis[M]. Princeton:Princeton University press,1994
[37]Harvey A. C., The Econometric Analysis of Time Series(2nd ed)[M]. Boston:MIT press,1990
[38]Hooker R., The Suspension of the Berlin Produce Exchange and its Effect upon Corn Prices[J]. Journal of the Royal Statistical Society,1901,64(3):574-613
[39]Hooker R., Correlation of the Marriage Rate with Trade[J]. Journal of the Royal Statistical Society,1901,64(3):485-492
[40]Hooker R., On the Correlation of Successive Observations Illustrated by Corn Prices [J]. Journal of the Royal Statistical Society,1905,68(4):696-703
[41]Hotelling H., Differential Equations Subject to Error and Population Estimates[J]. Journal of the American Statistical Association,1927,22(159):283-314
[42]Hotelling H., The Impact of R. A. Fisher on Statistics[J].Journal of the American Statistical Association,1951,46(253):35-46
[43]Hurvich C. M., Tsai C. L., Regression and Time Series Model Selection in Small Samples[J]. Biometrika,1989,76(2):297-307
[44]Jenkins G. M., General Considerations in Estimation of Spectra[J]. Technometrics, 1961,3(2):133-166
[45]Jenkins G. M., Watts D. G., Spectral Analysis and Its Applications[M]. San Francisco:Holden-Day,1968
[46]Jevons W. S., The Periodicity of Commercial Crises and Its Physical Expianation. In Investigations in Currency and Finance [M]. London:Macmillan, [1878b] 1884: 206-219
[47]Kendall M. G., Obituaries:Mr. G.Udny Yule, C.B.E, F.R.S. [J]. Nature,1951, 168(4274):542-543
[48]Kendall M. G., George Udny Yule,1871-1951[J]. Review of the International Statistical Institute,1952,20(1):92-93
[49]Kendall M. G., Hill A.B., The Analysis of Economic Time Series—Part 1:Prices [J]. Journal of the Royal Statistical Society, Series A,1953,116(1):11-34
[50]Kirchgassner G., Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007
[51]Klein J. L., Statistical Visions in Time:A History of Time Series Analysis, 1662-1938[M]. Cambridge:Cambridge University Press,1997
[52]Lexis W., The Theory of the Stability of Statistical Series. WPA No.165-1-71-124 [M].Minneapolis:University of Minnesota, U.S.Works Projects Administration [1879]1942
[53]Makridakis S., A Survey of Time Series[J].International Statistical Review,1976, 44(1):29-70
[54]Moore H. L., Economic Cycles:Their Law and Cause[M]. New York:Macmillan, 1914
[55]Moore H. L., Generating Economic Cycles[M]. New York:Macmillan,1923
[56]Nelson C. R., Applied Time Series Analysis for Managerial Forecasting[M]. San Francisco:Holden-Day,1973
[57]Norton J., Statistical Studies in the New York Money Market[M]. New York: Macmillan,1902
[58]Parker E. H., A Note on some Statistics Regarding China[J]. Journal of the Royal Statistical Society,1899,62(1):150-156
[59]Parzen E., Mathematical Considerations in Estimation of Spectra[J]. Technometrics, 1961,3(2):167-190
[60]Parzen E., An Approach to Time Series Analysis[J]. The Annals of Mathematical Statistics,1961,32(4):951-989
[61]Pearson K., Mathematical Contributions to the Theory of Evolution-Ⅲ. Regression, Heredity and Panmixia[J]. Philosophical Transactions of the Royal Society of London. Series A,1896,187:253-318
[62]Pearson K., The Chance of Death. In the Chances of Death and Other Studies in Evolution [M]. London:Edward Arnold,1897
[63]Pearson K., Lee A., On the Distribrution of Frequency(Variation and Correlation) of the Barometric Height of Divers Stations[J]. Philosophical Transactions of the Royal Society of London. Series A,1897,190:423-469
[64]Pearson K., The Problem of the Random Walk [J]. Nature,1905,72:294,342
[65]Pearson K., Notes on the History of Correlation[J]. Biometrika,1920,13(1):25-45
[66]Pearson K., Elderton E., On the Variate Difference Method[J]. Biometrika,1923, 14(3-4):281-310
[67]Pearson K., The Grammer of Science[M]. London:Walter Scott, [1892]1969
[68]Persons W. M., Correlation of Time Series[J]. Journal of the American Statistical Association,1923,18(142):713-726
[69]Persons W. M., Correlation of Time Series. In Handbook of Mathematical Statisticas[M]. Boston:Houghton Mifflin,1924:150-165
[70]Percival D. B., Walden A. T., Spectral Analysis for Physical Applications [M]. Cambridge:Cambridge University Press,1993
[71]Perman R., Cointegration:An Introduction to the Literature[J]. Journal of Economic Studies,1991,18(3):3-30
[72]Phillips P. C. B., New Tools for Understanding Spurious Regressions[J]. Econometrica,1998,66(6):1299-1325
[73]Poynting J. H., A Comparison of the Fluctuations in the Price of Wheat and in Cotton and Silk Imports into Great Britain[J]. Journal of the Royal Statistical Society,1884, 47(1):34-74
[74]Poynting J. H., The Drunkenness Statistics of the Large Towns in England and Wales In Collected Scientific Papers [C]. Cambridge:Cambridge University Press, [1877]1920:497-503
[75]Rao B. B., Cointegration for the Applied Economist[M]. New York:St.Martin's Press,1994
[76]Schuster A., Lunar and Solar Periodicities of Earthquakes[J]. Proceedings of the Royal Society of London,1897,61:455-465
[77]Schuster A., On the Investigation of Hidden Periodicities with Application to a Supposed 26 Day Period of Meteorological Phenomena[J]. Terrestrial Magnetism, 1898,3(1):13-41
[78]Schuster A., On Sun-Spot Periodicities. Preliminary Notice[J]. Proceedings of the Royal Society of London. Series A,1906,77(515):141-145
[79]Schuster A., On the Periodicities of Sunspots [J]. Philosophical Transactions of the Royal Society of London. Series A,1906,206:69-100
[80]Schuster A., On the Periodicity of Sun-Spots[J]. Proceedings of the Royal Society of London, Series A,1911,85(575):50-53
[81]Simpson G. C., Obituary:Sir Arthur Schuster, F.R.S. [J]. Nature,1934,134:595-597
[82]Simpson G.C., Sir Arthur Schuster.1851-1934. [J]. Obituary Notices of Fellows of the Royal Society,1935,1(4):408-423
[83]Slutzky E. E., The Summation of Random Causes As the Source of Cyclic Processes[J]. Econometrica,1937,5(2):105-146
[84]Solo V, The End of Time Series[J]. Journal of the American Statistical Association, 2000,95(452):1346-1349
[85]Stewart B., Meteorology:Terrestrial Magnetism. In Encyclopedia Britannica(9nd ed) [M]. New York:Charles Scribner's Sons, [1883] 1889:159-184
[86]Stigler S. M., The History of Statistics:The Measurement of Uncertainty before 1900 [M]. Cambridge, MA:Belknap Press of Harvard University Press,1986
[87]Stigler S., Fisher in 1921 [J]. Statistical Science,2005,20(1):32-49
[88]Student., The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(1):179-180
[89]Taylor S. J., Modeling Financial Time Series[M]. Chichester:John Wiley & Sons, 1986
[90]Turner H. H., Preliminary Report of the Expedition to Aswan to Observe the Total Solar Eclipse of August 30,1905[J]. Proceedings of the Royal Society of London, Series A,1906,77(514):77-96
[91]Walker G., On Periodicity in Series of Related Terms [J]. Proceedings of the Royal Society, Series A,1931,131(818):518-532
[92]Weigend A. S., Time Series Analysis and Prediction[D]. Colorado:Univerisity of Colorado,1994
[93]Whittle P., Obituary:Professor Herman Wold [J]. Journal of the Royal Statistical Society, Series A,1992,155(3):466-469
[94]Wichern D. W., The Behavior of the Sample Autocorrelation Function for an Intrgrated Moving Average Process[J]. Biometrika,1973,60(2):235-239
[95]Wilson J., Fluctuations of Currency, Commerce and Manufactures Referable to the Corn Laws[M]. London:Longman, Orme, Brown, Greenard & Longmans, 1840
[96]Wishart J., Tests of Significance in Analysis of Covariance[J]. Supplement to the Journal of the Royal Statistical Society,1936,3(1):79-82
[97]Wold H., A Study in the Analysis of Stationary Time Series[M]. Stockholm: Almqvist and Wiksell, [1938] 1954
[98]Wold H., On Prediction in Stationary Time Series[J]. The Annals of Mathematical Statistics,1948,19(4):558-567
[99]Working H., A Random-Difference Series for Use in the Analysis Time Series[J]. Journal of the American Statistical Association,1934,29(185):11-24
[100]Yule G.U., On the Theory of Correlation[J]. Journal of the Royal Statistical,1897, 60(4):812-854
[101]Yule G.U., On the Time-Correlation Problem, with Especial Reference to the Variate-Difference Correlation Method[J]. Journal of the Royal Statistical,1921, 84(4):497-537
[102]Yule G. U., Why do we Sometimes get Nonsense-Correlations between Time-Series?--A Study in Sampling and the Nature of Time-Series[J]. Journal of the Royal Statistical,1926,89(1):1-63
[103]Yule G. U., On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers[J]. Philosophical Transactions of the Royal Society of London. Series A,1927,226:267-298
[104]Yule G. U., Kendall M. G., An Introduction to the Theory of Statistics[M]. London:Charles Griffin,1950
[1]安鸿志,陈敏.非线性时间序列分析[M].上海：上海科学技术出版社,1998
[2][美]A·吉特尔曼著,欧阳绛译.数学史[M].北京：科学普及出版社,1987
[3][英]彼得·迈克尔·哈曼著,龚少明译.19世纪物理学概念的发展[M].上海：复旦大学出版社,2000
[4][美]布洛克威尔等著,田铮译.时间序列的理论与方法[M].北京：高等教育出版社,1991
[5]陈善林,张浙.统计发展史[M].上海：立信会计图书用品社,1987
[6]陈希孺.数理统计学简史[M].长沙：湖南教育出版社,2005
[7]陈兆国.时间序列及其谱分析[M].北京：科学出版社,1988
[8]程振源.时间序列分析：历史回顾与未来展望[J].统计与决策,2002,9：45-46
[9]范福仁.生物统计学(修订本)[M].南京：江苏科学技术出版社,1980
[10]复旦大学.概率论基础[M].北京：人民教育出版社,1983
[11]复旦大学.数理统计[M].北京：人民教育出版社,1983
[12]复旦大学.随机过程[M].北京：人民教育出版社,1983
[13]何书元.应用时间序列分析[M].北京：北京大学出版社,2003
[14]贾随军.傅里叶级数理论的起源[D].西安：西北大学,2010
[15]贾随军,贾小勇.傅里叶与热传导理论数学化[J].自然辩证法通讯,2009,31(5)：66-70
[16]蒋长荣,王骁勇,刘树勇.爱因斯坦与布朗运动[J].首都师范大学学报(自然科学版),2005,26(3)：28-32
[17][美]卡茨著,李文林,邹建成,胥鸣伟等译.数学史通论(第二版)[M].北京：高等教育出版社,2004
[18]李爱云.太阳黑子周其与行星的引潮力[J].枣庄学院学报,2010,27(2)：16-18
[19]李迪.中外数学史教程[M].福州：福建教育出版社,1993
[20]李家良译.数学在科学起源中的作用[M].湖南教育出版社,1992
[21]李锐,向书坚.我国时间序列分析研究工作综述[J].统计教育,2006,7：6-8
[22]李文林.数学珍宝—历史文献精选[M].北京：科学出版社,2000
[23]李文林.数学史概论[M].北京：高等教育出版社,2002
[24]李文林.数学的进化[M].北京：科学出版社,2005
[25]梁宗巨,王青建,孙宏安.世界数学通史(下册)[M].沈阳：辽宁教育出版社,2001
[26]林德光.生物统计的数学原理[M].沈阳：辽宁人民出版社,1982
[27]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京：高等教育出版社,2001
[28]罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117
[29]明道绪.生物统计[M].北京：中国农业科技出版社,1998
[30][美]莫里斯·克莱因著,北京大学数学系数学史翻译组译.古今数学思想(第四册)[M].上海：上海科学技术出版社,1979
[31][美]莫里斯·克莱因著,张祖贵译.西方文化中的数学[M].上海：复旦大学出版社,2005
[32]聂清香,张海霞,左庆林等.太阳活动11年周期的形成原因新探[J].自然杂志,2007,29(4)：225-228
[33]聂淑媛.格朗特与时间序列分析[J].自然辩证法通讯,2011,33(3)：50-52
[34]聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7
[35]聂淑媛,梁铁旺.指数和滑动平均的历史发展探究[J].统计与决策,2011,24：4-7
[36]聂淑媛.经济时间序列中差分的历史研究[J].科学技术哲学研究,2012,29(1)： 70-75
[37]聂淑媛.时间序列分析的历史发展[J].广西民族大学学报(自然科学版),2012,18(1)：24-28
[38]聂淑媛.沃尔德与离散平稳时间序列[J].咸阳师范学院学报(自然科学版),2012,27(2)：72-75
[39]聂淑媛.数学史的职业教育价值[J].高等理科教育,2011,6：115-117
[40][美]乔纳森·克莱尔等著,潘红宇等译.时间序列分析及应用：R语言(原书第二版)[M].北京：机械工业出版社,2011
[41][美]乔治·E.P.博克斯等著,王成璋等译.时间序列分析：预测与控制(原书第四版)[M].北京：机械工业出版社,2011
[42]曲安京.中国数学史研究的两次运动[J].科学,2004,56(2)：27-30
[43]曲安京.中国数学史研究范式的转换[J].中国科技史杂志,2005,26(1)：50-58
[44][美]塞路蒙·波克纳著,李家良译.数学在科学起源中的作用[M].湖南教育出版社,1992
[45]《数学辞海》编辑委员会.数学辞海(第三卷)[M].南京：东南大学出版社,北京：中国科学技术出版社,太原：山西教育出版社,2002
[46]汤岩.时间序列分析的研究与应用[D].哈尔滨：东北农业大学,2007
[47]王黎明,王连,杨楠.应用时间序列分析[M].上海：复旦大学出版社,2009
[48]王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008
[49]王振龙.应用时间序列分析[M].北京：中国统计出版社,2010
[50][美]魏武雄著,易丹辉,刘超,贺学强等译.时间序列分析—单变量和多变量方法(第二版)[M].北京：中国人民大学出版社,2009
[51]魏宗舒.概率论与数理统计教程[M].北京：高等教育出版社,2005
[52]武娜.傅里叶级数的起源和发展[D].石家庄：河北师范大学,2008
[53]吴文俊主编.世界著名数学家传记(上集)[M].北京：科学出版社,2003
[54]吴文俊主编.世界著名数学家传记(下集)[M].北京：科学出版社,2003
[55]吴正英.浅谈时间序列发展史[J].统计与预测,1999,5：55-56.
[56]徐传胜.概率论简史[J].数学通报,2004(9)：36-39
[57]徐传胜.彼得堡数学学派的概率思想研究[D].西安：西北大学,2007
[58]杨静.布朗运动的数学理论的历史研究[D].北京：中国科学院研究生院,2006
[59]杨静,唐泉.维纳和布朗运动[J].数学的实践和认识,2008,38(10)：162-169
[60]杨静,王丽霞.爱因斯坦与布朗运动的数学理论[J].西北大学学报(自然科学版),2006,36(1)：169-172
[61]杨静,徐传胜,王朝旺.试析巴夏里埃的《投机理论》对数学的影响[J].自然科学史研究,2008,27(1)：94-104
[62]叶中行,王蓉华,徐晓玲等.概率论与数理统计[M].北京：北京大学出版社,2009
[63][美]约翰·塔巴克著杨静译.概率论和统计学：不明确的科学(数学之旅)[M].北京：商务印书馆,2007
[64][法]约瑟夫·傅立叶著,桂质亮译.热的解析理论[M].北京：北京大学出版社,2008
[65]张美英,何杰.时间序列预测模型研究简介[J].江西科学,2009,27(5)：697-701
[66]张树京,齐立心.时间序列分析简明教程[M].北京：清华大学出版社,北方交通大学出版社,2003
[67]张尧庭.时间序列分析[J].统计教育,1996,23(4)：23-27
[68]赵晨阳,翟少丹.高尔顿与相关理论的产生[J].西北大学学报(自然科学版),2008,38(4)：680-684
[69]中国大百科全书.天文学[M].北京：中国大百科全书出版社,1980
②为了发展农业生产,古埃及人特别关注尼罗河的泛滥规律,对尼罗河涨落情况的长期观察、记录,使他们发现,尼罗河的泛滥开始于天狼星第一次和太阳同时升起之后的200天左右,洪水持续七、八十天之后,土地肥沃,适于农业种植。
③王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：1.
④罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117.
②Kirchgassner G., Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007:2.⑤王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：2.
① Kirchgassner G, Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007:2-5.
②Wold H., A Study in the Analysis of Stationary Time Series[M]. Stockholm:Almqvist and Wiksell, [1938]1954.
③Box G.E. P., Jenkins G.M., Time-Series Analysis:Forecasting and Control[M]. San Francisco:Holden Day,1970.
④这些时间单位包括年、季、月、日,甚至也可能是时、分、秒等。
①Engle R. F., Auto Regressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation [J]. Econometrics,1982,50(4):987-1007.
②ARCH族计量模型包括GARCH模型(G)ARCH-M模型FGARCH模型EGARCH模型JGARCH模型和TGARCH模型等等,当前研究ARCH族模型的论文、论著非常丰富,此处不再赘述。
③聂淑媛.时间序列分析的历史发展[J].广西民族大学学报(自然科学版),2012,18(1)：24-28.
④比如,有限域的历史、博弈论前史、组合学史、生物数学史、行列式理论、傅里叶级数的理论、偏微分方程理论等。
⑤Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997.
① Kirchgassner G, Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007.
②张尧庭.时间序列分析[J].统计教育,1996,23(4)：23-27.
③吴正英.浅谈时间序列发展史[J].统计与预测,1999,5：55-56.
④程振源.时间序列分析：历史回顾与未来展望[J].统计与决策,2002,9：45-46.
⑤罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117.
⑥汤岩.时间序列分析的研究与应用[D].哈尔滨：东北农业大学,2007.
①曲安京.中国数学史研究的两次运动[J].科学,2004,56(2)：27-30.
②曲安京.中国数学史研究范式的转换[J].中国科技史杂志,2005,26(1)：50-58.
①Graunt J., Natural and Political Observations Made Upon the Bills of Mortality[M]. London:John Martin and James Allestry, [1662] 1665.
①陈希孺.数理统计学简史[M].长沙：湖南教育出版社,2005：131-135.
③所谓三分法,是指在一个比例关系a/b=c/d中,用三个已知数a、6、c去求解第四个未知数d的简单算术技巧。
在欧洲,该法则由于其开创性的思想而受到极高的评价。
④Graunt的《观察》一书,共包括12章,用8个表格细致整理了死亡公报中的庞大数据,内容涉及到死亡原因(尤其是当时比较多见的黑死病)、男女差异、伦敦人口数及增长情况等诸多问题,Graunt的结论正是通过对数据的分析而推断得出。
⑤所谓统计比率的稳定性,是指某个事件或某种特性出现的频率,随着观察次数的逐渐增加而趋向于一个固定值。
⑥慢性病如结核病等,偶然因素如淹死等。
①聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7.
②汤岩.时间序列分析的研究与应用[D].哈尔滨：东北农业大学,2007.
③张尧庭.时间序列分析[J].统计教育,1996,23(4)：23-27.
④王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：1.
⑥罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117.
⑥Kirchgassner G, Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007:2.
①张树京,齐立心.时间序列分析简明教程[M].北京：清华大学出版社,北方交通大学出版社,2003：19-20.
②评语为：对任何缺少实际测量和三分法的事情,我都没有信心。
①聂淑媛.格朗特与时间序列分析[J].自然辩证法通讯,2011,33(3)：50-52.
①Hald A., A History of Probability and Statistics and their Applications before 1750[M]. New York:Wiley,1990:55.
①如莱布尼兹(Gottfried Wilhelm Leibniz,1646-1716)1666年在其《组合艺术》中考察平方数序列时,也引入了阶差分和二阶差分,而且也正是由此开始,他注意到序列求和与求差之间的互逆关系,并进一步发展了微积分。
②Box G. E. P., Jenkins G. M., Time-Series Analysis:Forecasting and Control[M]. San Francisco:Holden Day,1970:60-98.
③Perman R., Cointegration:An Introduction to The Literature[J]. Journal of Economic Studies,1991,18(3):3-30.
④Rao B. B., Cointegration for the Applied Economist[M]. New York:St.Martin's Press,1994:1-3.
①所谓“政治人”,指小依靠抽象的语言和看似巧妙的推理,一切都依靠数字说明,并最终讨论和决定政治政策。
②Statistics, ed. Henry Higgs and George Udny Yule[M]. London:Macmillan,1913:25-46.
①Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997:66.
①Norton J., Statistical Studies in the New York Money Market[M]. New York:Macmillan,1902.
①Cave-Browne-Cave F. E., Pearson K., On the Influence of the Time Factor on the Correlation between the Barometric Heights at Stations More Than 1000 Miles Apart[J]. Proceedings of the Royal Statistical Society of London,1904,74: 403-413.
①Hooker R., The Suspension of the Berlin Produce Exchange and its Effect upon Corn Prices[J]. Journal of the Royal Statistical Society,1901,64(3):574-613.
②Hooker R., On the Correlation of Successive Observations Illustrated by Corn Prices [J]. Journal of the Royal Statistical Society,1905,68(4):696-703.
①Student, The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(1):179-180.
②Anderson O., The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(2-3): 269-279.
①聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7.
②1938年,Wold把这种危险序列定义为非稳定过程或发展过程。
①Yule差分思想的发展历程,可参见第五章5.1节。
②聂淑媛.经济时间序列中差分的历史研究[J].科学技术哲学研究,2012,29(1)：70-75.
①Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997:78-85.
①Poynting J. H., The Drunkenness Statistics of the Large Towns in England and Wales. In Collected Scientific Papers[C]. Cambridge:Cambridge University Press, [187711920:497-503.
②Poynting J. H., A Comparison of the Fluctuations in the Price of Wheat and in Cotton and Silk Imports into Great Britain[J]. Journal of the Royal Statistical Society,1884,47(1):34-74.
①Poynting J. H., The Drunkenness Statistics of the Large Towns in England and Wales. In Collected Scientific Papers[C]. Cambridge:Cambridge University Press, [1877]1920:497-503.
②谷物计量单位,即1/4吨。
②需要说明,经济学家勃雷(Arthur Lyon Bowley,1869—1957)1895年研究工资问题时,也发展完善了比较的概念,他根据不同地区、不同职业和不同时间的大量工资数据,从杂乱元章的情形中建构了连续指数序列,联系这一链条的关键止是变化率。
②Jevons W. S., The Periodicity of Commercial Crises and Its Physical Expianation. In Investigations in Currency and Finance [M]. London:Macmillan, [1878b]1884:206-219.
①Poynting J. H., A Comparison of the Fluctuations in the Price of Wheat and in Cotton and Silk Imports into Great Britain[J]. Journal of the Royal Statistical Society,1884,47(1):34-74.
②尽管Poynting从未在论文中提到Jevons,但一个不容忽视的事实是,当Poynting向皇家统计学会提交论文之后,Jevons的工作才逐渐出现在皇家统计学会的讨论中。
①Hooker R., Correlation of the Marriage Rate with Trade[J]. Journal of the Royal Statistical Society,1901,64(3):485-492.
①聂淑媛,梁铁旺.指数和滑动平均的历史发展探究[J].统计与决策,2011,24：4-7.
①统计学家把这种规律性很强的贸易循环称为平稳过程。
② The Theory of the Stability of Statistical Series. WPA No.165-1-71-124. [M]. Minneapolis:University of Minnesota, U.S.Works Projects Administration[1879]1942:20-80.
①贾随军.傅里叶级数理论的起源[D].西安：西北大学,2010.
①贾随军.傅里叶级数理论的起源[D].西安：西北大学,2010.
②[法]约瑟夫·傅立叶著,桂质亮译.热的解析理论[M].北京：北京大学出版社,2008.
①Schuster A., On Sun-Spot Periodicities. Preliminary Notice[J]. Proceedings of the Royal Society of London. Series A, 1906,77(515):141-145.
②Schuster A., On the Periodicities of Sunspots [J]. Philosophical Transactions of the Royal Society of London. Series A, 1906,206:69-100.
③Schuster A., On the Periodicity of Sun-Spots[J]. Proceedings of the Royal Society of London, Series A,1911,85(575): 50-53.
④Simpson G.C., Obituary:Sir Arthur Schuster, F.R.S. [J]. Nature,1934,134:595-597.
⑤Simpson G. C., Sir Arthur Schuster.1851-1934. [J]. Obituary Notices of Fellows of the Royal Society,1935,1(4):408-423.
①日冕是太阳大气的最外层,厚度达几百万公里以上,温度有100万摄氏度,可人为地分为内冕、中冕和外冕3层。
②Simpson G. C., Obituary:Sir Arthur Schuster, F.R.S. [J]. Nature,1934,134:595-597.
③Simpson G. C., Sir Arthur Schuster.1851-1934. [J]. Obituary Notices of Fellows of the Royal Society,1935,1(4):408-423.
④Schuster A., On the Investigation of Hidden Periodicities with Application to a Supposed 26 Day Period of Meteorological Phenomena[J]. Terrestrial Magnetism,1898,3(1):13-41.
①Schuster A., On Sun-Spot Periodicities. Preliminary Notice[J]. Proceedings of the Royal Society of London. Series A, 1906,77(515):141-145.
①Schuster A., On the Periodicities of Sunspots [J]. Philosophical Transactions of the Royal Society of London. Series A, 1906,206:69-100.
②Schuster使用的是沃尔夫观察得到的太阳黑子数,而把由平滑过程得到的数据放在单独的表中作为“补偿数”,因为他认为平滑过程削弱了数字的精确程度,不利于科学地检验周期。
①Schuster A., On the Periodicity of Sun-Spots[J]. Proceedings of the Royal Society of London, Series A,1911,85(575): 50-53.
②Schuster指出Oppenheim通过对拉蒙特(Lamont)地区1836—1886年的观察值取连续两年的平均,使用与Schuster完全不同的方法,在磁下降记录中也独立地发现了相唰的周期4.92年,Schuster强调,因为Oppenheim以两年平均 为基础,且只有12个完整周期,故4.92年和4.78年的差别可以忽略。
①中国大百科全书.天文学[M].北京：中国大百科全书出版社,1980,
①Yule G.U., On the Time-Correlation Problem, with Especial Reference to the Variate- Difference Correlation Method[J]. Journal of the Royal Statistical,1921,84(4):497-537.
②Yule GU., Why do we Sometimes get Nonsense-Correlations between Time-Series?--A Study in Sampling and the Nature of Time-Series[J]. Journal of the Royal Statistical,1926,89(1):1-63.
③Yule G U., On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers[J]. Philosophical Transactions of the Royal Society of London. Series A,1927,226:267-298.
④Yule G.U., On the Theory of Correlation[J]. Journal of the Royal Statistical,1897,60(4):812-854.
⑤Yule G.U. Kendall M. G, An introduction to the theory of statistics[M]. London:Charles Griffin,1950:88-90.
①Kendall M. G, Obituaries:Mr. GUdny Yule, C.B.E, F.R.S. [J]. Nature,1951,168(4274):542-543.
②Kendall M. G, George Udny Yule,1871-1951 [J]. Review of the International Statistical Institute,1952,20(1):92-93.
③Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997.
④这些思想已经在第三章3.2节中进行了细致分析。
④Hooker R., Correlation of the Marriage Rate with Trade[J]. Journal of the Royal Statistical Society,1901,64(3):485-492.
①聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7.
②聂淑媛.经济时间序列中差分的历史研究[J].科学技术哲学研究,2012,29(1)：70-75.
③目前通常采用ε或εt表示扰动,Yule原文中用的也是ε,但Walker原文中vx表示扰动,此处未作更改,下同。第六章中Wold使用的一些符号,与当前的常用符号也略有差异,本文在分析这些数学家的工作时,尽量忠实于原著,不做符号的改动。
①Walker G.,On Periodicity in Series of Related Terms[J].Proceedings of the Royal Society,Series A,1931,131(818): 518.532.
②Walker强调,虽然与周期相联系的相关系数法不是一种新方法,但却被广泛使用。
①Slutzky E. E., The Summation of Random Causes As the Source of Cyclic Processes[J].Econometrica,1937,5(2):105-146.
①Slutzky在原文中用严格的数学语言表达和证明了正弦极限定理,本文对他的数学语言进行了转换。
②Slutzky认为,经济调查者凭借自己的敏锐和直觉,对商业循环的周期至少有一个大概的正确判断,他们是可靠的。
③他的同事Kondratieff在1928年出版了著作《The Long Wave Cycle》之后,不仅被迫关闭了研究所,后来甚至被送进了监狱。
④Klein J. L., Statistical Visions in Time:A History of Time Series Analysis,1662-1938[M]. Cambridge:Cambridge University Press,1997:276-279.
①Pearson K., The Problem of the Random Walk [J]. Nature,1905(72):294,342.
①王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：150.
②Bachelier L., Theory of Speculation. Trans. James Boness. In the Random Character of Stock Market Pricesr[M]. Cambridge, MA:MIT Press, [1900]1964:17-78.
①关于布朗运动的资料非常丰富,对其历史发展的研究可以参阅中文文献[58]、[59]、[60]等,此处不再赘述。
②Kendall M. G.,The Analysis of Economic Time Series—Part 1:Prices [J]. Journal of the Royal Statistical Society, SeriesA, 1953,116(1):11-25.
①Wold H., A Study in the Analysis of Stationary Time Series[M]. Stockholm:Almqvist and Wiksell, [1938]1954:84-92.
①Persons W. M., Correlation of Time Series. In Handbook of Mathematical Statisticas[M]. Boston:Houghton Mifflin,1924: 150-165.
②Yulec对 Anderson高阶差分方法的质疑可以参见第二章2.1.3节和第五章5.1.2节。
①王振龙.应用时间序列分析[M].北京：中国统计出版社,2010：93-94
②王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008：112.
[1]Akaike H., A New Look at the Statistical Model Identification[J]. IEEE Transactions on Automatic Control,1974,19(6):716-723
[2]Aldrich J., Correlations Genuine and Spurious in Pearson and Yule[J]. Statistical Science,1995,10(4):364-376
[3]Anderson O., The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(2-3):269-279
[4]Anderson O., On the Logic of the Decomposition of Statistical Series into Separate Components[J]. Journal of the Royal Statistical Society,1927,90(3):548-569
[5]Anderson T. W., The Statistical Analysis of Time Series[M]. New York:John Wiley & Sons,1971
[6]Bachelier L., Theory of Speculation. In the Random Character of Stock Market Pricesr[M]. Cambridge, MA:MIT Press, [1900]1964:17-78
[7]Bartlett M. S., On the Theoretical Specification of Sampling Properties of Autocorrelated Time Series[J], Journal of the Royal Statistical Society B,1946,8(1): 27-41
[8]Beveridge W. H., Wheat Prices and Rainfall in Western Europe[J]. Journal of the Royal Statistical Society,1922,85(3):412-475
[9]Bollerslev T., Generalized Auto Regressive Conditional Heteroskedasticity[J]. Journal of Econometrics,1986,31(3):307-327
[10]Box G. E. P., Jenkins G. M., Time-Series Analysis:Forecasting and Control[M]. San Francisco:Holden Day,1970
[11]Brillinger D. R., Time Series:Data Analysis and Theory[D]. Philadelphia, SIAM, 2001
[12]Brockwell P. J., Davis R. A., Time Series:Theory and Methods(2nd ed)[M]. New York:Springer,1991
[13]Brockwell P. J., Davis R. A., Introduction to Time Series and Forecasting (2nd ed) [M]. New York:Springer,2002
[14]Brown R. G., Smoothing, Forecasting and Prediction of Discrete Time Series[M]. Englewood Cliffs, NJ: Prentice-Hall, 1962
[15]Cave-Browne-Cave F.E., Pearson K., On the Influence of the Time Factor on the Correlation between the Barometric Heights at Stations More Than 1000 Miles Apart[J]. Proceedings of the Royal Statistical Society of London, 1904, 74: 403-413
[16]Cave B., Pearson K., Numerical Illustrations of the Variate Difference Correlation Method[J].Biometrika, 1914, 10(2-3): 340-355
[17]Chatfield C, The Analysis of Time Series: An Introduction(3nd ed) [M]. London: Chapman and Hall, 1984
[18]Cooley J. W., Tukey J. W., An Algorithm for the Machine Calculation of Complex Fourier Series[J]. Mathematics of Computation, 1965, 19(90): 297-301
[19]Cramer H., Leadbetter M. R., Stationary and Related Random Processes[M]. New York: John Wiley&Sons, 1967
[20]Draper N. R., Smith H., Applied Regression Analysis(2nd ed)[M]. New York: John Wiley & Sons, 1981
[21]Durbin J., Koopman S. J., Time Series Analysis by State Space Methods[M].Oxford: Oxford University Press, 2001
[22]Engle R. F., Auto Regressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation[J]. Econometrica, 1982, 50(4): 987-1007
[23]Fisher I., Our Unstable Dollar and the So-called Business Cycle[J]. Journal of the American Statistical Association, 1925, 20(150): 179-202
[24]Frisch R., Propagation Problems and Impulse Problems in Dynamic Economics. In Economic Essays in Honour of Gustav Cassel [M]. London: George Allen & Unwin, 1933: 171-205
[25]Fuller W. A., Introduction to Statistical Time Series(2nd ed)[M]. New York: John Wiley & Sons, 1996
[26]Galton F., Regression towards Mediocrity in Hereditary Stature[J]. The Journal of the Anthropological Institute of Great Britain and Ireland, 1886, 15: 246-263
[27]Galton F., Essays on Eugenics[M]. London: Eugenics Eduation Society, 1909
[28]GiffenR., The Excess of Imports [J]. Journal of the Royal Statistical Society, 1899, 62(1):1-82
[29]Giffen R., Economic Inquiries and Studies[M]. London:George Bell and Sons,1904
[30]Giffen R., Statistics, ed. Henry Higgs and George Udny Yule[M]. London: Macmillan,1913
[31]Graunt J., Natural and Political Observations Made Upon the Bills of Mortality[M]. London:John Martin and James Allestry, [1662] 1665
[32]Great Britain, Parliament. Report of the Committee on the Bank of England Charter 1832[C]. House of Commons Paper 722,1832
[33]Hacking I., The Emergence of Probability[M]. Cambridge:Cambridge University Press,1975
[34]Hald A., A History of Probability and Statistics and their Applications before 1750 [M]. New York:Wiley,1990
[35]Hald A., A History of Mathematical Statistics from 1750 to 1930[M]. New York: Wiley,1998
[36]Hamilton J. D., Time Series Analysis[M]. Princeton:Princeton University press,1994
[37]Harvey A. C., The Econometric Analysis of Time Series(2nd ed)[M]. Boston:MIT press,1990
[38]Hooker R., The Suspension of the Berlin Produce Exchange and its Effect upon Corn Prices[J]. Journal of the Royal Statistical Society,1901,64(3):574-613
[39]Hooker R., Correlation of the Marriage Rate with Trade[J]. Journal of the Royal Statistical Society,1901,64(3):485-492
[40]Hooker R., On the Correlation of Successive Observations Illustrated by Corn Prices [J]. Journal of the Royal Statistical Society,1905,68(4):696-703
[41]Hotelling H., Differential Equations Subject to Error and Population Estimates[J]. Journal of the American Statistical Association,1927,22(159):283-314
[42]Hotelling H., The Impact of R. A. Fisher on Statistics[J].Journal of the American Statistical Association,1951,46(253):35-46
[43]Hurvich C. M., Tsai C. L., Regression and Time Series Model Selection in Small Samples[J]. Biometrika,1989,76(2):297-307
[44]Jenkins G. M., General Considerations in Estimation of Spectra[J]. Technometrics, 1961,3(2):133-166
[45]Jenkins G. M., Watts D. G., Spectral Analysis and Its Applications[M]. San Francisco:Holden-Day,1968
[46]Jevons W. S., The Periodicity of Commercial Crises and Its Physical Expianation. In Investigations in Currency and Finance [M]. London:Macmillan, [1878b] 1884: 206-219
[47]Kendall M. G., Obituaries:Mr. G.Udny Yule, C.B.E, F.R.S. [J]. Nature,1951, 168(4274):542-543
[48]Kendall M. G., George Udny Yule,1871-1951[J]. Review of the International Statistical Institute,1952,20(1):92-93
[49]Kendall M. G., Hill A.B., The Analysis of Economic Time Series—Part 1:Prices [J]. Journal of the Royal Statistical Society, Series A,1953,116(1):11-34
[50]Kirchgassner G., Wolters J., Introduction to Modern Time Series Analysis[M]. New York:Springer-Verlag,2007
[51]Klein J. L., Statistical Visions in Time:A History of Time Series Analysis, 1662-1938[M]. Cambridge:Cambridge University Press,1997
[52]Lexis W., The Theory of the Stability of Statistical Series. WPA No.165-1-71-124 [M].Minneapolis:University of Minnesota, U.S.Works Projects Administration [1879]1942
[53]Makridakis S., A Survey of Time Series[J].International Statistical Review,1976, 44(1):29-70
[54]Moore H. L., Economic Cycles:Their Law and Cause[M]. New York:Macmillan, 1914
[55]Moore H. L., Generating Economic Cycles[M]. New York:Macmillan,1923
[56]Nelson C. R., Applied Time Series Analysis for Managerial Forecasting[M]. San Francisco:Holden-Day,1973
[57]Norton J., Statistical Studies in the New York Money Market[M]. New York: Macmillan,1902
[58]Parker E. H., A Note on some Statistics Regarding China[J]. Journal of the Royal Statistical Society,1899,62(1):150-156
[59]Parzen E., Mathematical Considerations in Estimation of Spectra[J]. Technometrics, 1961,3(2):167-190
[60]Parzen E., An Approach to Time Series Analysis[J]. The Annals of Mathematical Statistics,1961,32(4):951-989
[61]Pearson K., Mathematical Contributions to the Theory of Evolution-Ⅲ. Regression, Heredity and Panmixia[J]. Philosophical Transactions of the Royal Society of London. Series A,1896,187:253-318
[62]Pearson K., The Chance of Death. In the Chances of Death and Other Studies in Evolution [M]. London:Edward Arnold,1897
[63]Pearson K., Lee A., On the Distribrution of Frequency(Variation and Correlation) of the Barometric Height of Divers Stations[J]. Philosophical Transactions of the Royal Society of London. Series A,1897,190:423-469
[64]Pearson K., The Problem of the Random Walk [J]. Nature,1905,72:294,342
[65]Pearson K., Notes on the History of Correlation[J]. Biometrika,1920,13(1):25-45
[66]Pearson K., Elderton E., On the Variate Difference Method[J]. Biometrika,1923, 14(3-4):281-310
[67]Pearson K., The Grammer of Science[M]. London:Walter Scott, [1892]1969
[68]Persons W. M., Correlation of Time Series[J]. Journal of the American Statistical Association,1923,18(142):713-726
[69]Persons W. M., Correlation of Time Series. In Handbook of Mathematical Statisticas[M]. Boston:Houghton Mifflin,1924:150-165
[70]Percival D. B., Walden A. T., Spectral Analysis for Physical Applications [M]. Cambridge:Cambridge University Press,1993
[71]Perman R., Cointegration:An Introduction to the Literature[J]. Journal of Economic Studies,1991,18(3):3-30
[72]Phillips P. C. B., New Tools for Understanding Spurious Regressions[J]. Econometrica,1998,66(6):1299-1325
[73]Poynting J. H., A Comparison of the Fluctuations in the Price of Wheat and in Cotton and Silk Imports into Great Britain[J]. Journal of the Royal Statistical Society,1884, 47(1):34-74
[74]Poynting J. H., The Drunkenness Statistics of the Large Towns in England and Wales In Collected Scientific Papers [C]. Cambridge:Cambridge University Press, [1877]1920:497-503
[75]Rao B. B., Cointegration for the Applied Economist[M]. New York:St.Martin's Press,1994
[76]Schuster A., Lunar and Solar Periodicities of Earthquakes[J]. Proceedings of the Royal Society of London,1897,61:455-465
[77]Schuster A., On the Investigation of Hidden Periodicities with Application to a Supposed 26 Day Period of Meteorological Phenomena[J]. Terrestrial Magnetism, 1898,3(1):13-41
[78]Schuster A., On Sun-Spot Periodicities. Preliminary Notice[J]. Proceedings of the Royal Society of London. Series A,1906,77(515):141-145
[79]Schuster A., On the Periodicities of Sunspots [J]. Philosophical Transactions of the Royal Society of London. Series A,1906,206:69-100
[80]Schuster A., On the Periodicity of Sun-Spots[J]. Proceedings of the Royal Society of London, Series A,1911,85(575):50-53
[81]Simpson G. C., Obituary:Sir Arthur Schuster, F.R.S. [J]. Nature,1934,134:595-597
[82]Simpson G.C., Sir Arthur Schuster.1851-1934. [J]. Obituary Notices of Fellows of the Royal Society,1935,1(4):408-423
[83]Slutzky E. E., The Summation of Random Causes As the Source of Cyclic Processes[J]. Econometrica,1937,5(2):105-146
[84]Solo V, The End of Time Series[J]. Journal of the American Statistical Association, 2000,95(452):1346-1349
[85]Stewart B., Meteorology:Terrestrial Magnetism. In Encyclopedia Britannica(9nd ed) [M]. New York:Charles Scribner's Sons, [1883] 1889:159-184
[86]Stigler S. M., The History of Statistics:The Measurement of Uncertainty before 1900 [M]. Cambridge, MA:Belknap Press of Harvard University Press,1986
[87]Stigler S., Fisher in 1921 [J]. Statistical Science,2005,20(1):32-49
[88]Student., The Elimination of Spurious Correlation Due to Position in Time or Space[J]. Biometrika,1914,10(1):179-180
[89]Taylor S. J., Modeling Financial Time Series[M]. Chichester:John Wiley & Sons, 1986
[90]Turner H. H., Preliminary Report of the Expedition to Aswan to Observe the Total Solar Eclipse of August 30,1905[J]. Proceedings of the Royal Society of London, Series A,1906,77(514):77-96
[91]Walker G., On Periodicity in Series of Related Terms [J]. Proceedings of the Royal Society, Series A,1931,131(818):518-532
[92]Weigend A. S., Time Series Analysis and Prediction[D]. Colorado:Univerisity of Colorado,1994
[93]Whittle P., Obituary:Professor Herman Wold [J]. Journal of the Royal Statistical Society, Series A,1992,155(3):466-469
[94]Wichern D. W., The Behavior of the Sample Autocorrelation Function for an Intrgrated Moving Average Process[J]. Biometrika,1973,60(2):235-239
[95]Wilson J., Fluctuations of Currency, Commerce and Manufactures Referable to the Corn Laws[M]. London:Longman, Orme, Brown, Greenard & Longmans, 1840
[96]Wishart J., Tests of Significance in Analysis of Covariance[J]. Supplement to the Journal of the Royal Statistical Society,1936,3(1):79-82
[97]Wold H., A Study in the Analysis of Stationary Time Series[M]. Stockholm: Almqvist and Wiksell, [1938] 1954
[98]Wold H., On Prediction in Stationary Time Series[J]. The Annals of Mathematical Statistics,1948,19(4):558-567
[99]Working H., A Random-Difference Series for Use in the Analysis Time Series[J]. Journal of the American Statistical Association,1934,29(185):11-24
[100]Yule G.U., On the Theory of Correlation[J]. Journal of the Royal Statistical,1897, 60(4):812-854
[101]Yule G.U., On the Time-Correlation Problem, with Especial Reference to the Variate-Difference Correlation Method[J]. Journal of the Royal Statistical,1921, 84(4):497-537
[102]Yule G. U., Why do we Sometimes get Nonsense-Correlations between Time-Series?--A Study in Sampling and the Nature of Time-Series[J]. Journal of the Royal Statistical,1926,89(1):1-63
[103]Yule G. U., On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers[J]. Philosophical Transactions of the Royal Society of London. Series A,1927,226:267-298
[104]Yule G. U., Kendall M. G., An Introduction to the Theory of Statistics[M]. London:Charles Griffin,1950
[1]安鸿志,陈敏.非线性时间序列分析[M].上海：上海科学技术出版社,1998
[2][美]A·吉特尔曼著,欧阳绛译.数学史[M].北京：科学普及出版社,1987
[3][英]彼得·迈克尔·哈曼著,龚少明译.19世纪物理学概念的发展[M].上海：复旦大学出版社,2000
[4][美]布洛克威尔等著,田铮译.时间序列的理论与方法[M].北京：高等教育出版社,1991
[5]陈善林,张浙.统计发展史[M].上海：立信会计图书用品社,1987
[6]陈希孺.数理统计学简史[M].长沙：湖南教育出版社,2005
[7]陈兆国.时间序列及其谱分析[M].北京：科学出版社,1988
[8]程振源.时间序列分析：历史回顾与未来展望[J].统计与决策,2002,9：45-46
[9]范福仁.生物统计学(修订本)[M].南京：江苏科学技术出版社,1980
[10]复旦大学.概率论基础[M].北京：人民教育出版社,1983
[11]复旦大学.数理统计[M].北京：人民教育出版社,1983
[12]复旦大学.随机过程[M].北京：人民教育出版社,1983
[13]何书元.应用时间序列分析[M].北京：北京大学出版社,2003
[14]贾随军.傅里叶级数理论的起源[D].西安：西北大学,2010
[15]贾随军,贾小勇.傅里叶与热传导理论数学化[J].自然辩证法通讯,2009,31(5)：66-70
[16]蒋长荣,王骁勇,刘树勇.爱因斯坦与布朗运动[J].首都师范大学学报(自然科学版),2005,26(3)：28-32
[17][美]卡茨著,李文林,邹建成,胥鸣伟等译.数学史通论(第二版)[M].北京：高等教育出版社,2004
[18]李爱云.太阳黑子周其与行星的引潮力[J].枣庄学院学报,2010,27(2)：16-18
[19]李迪.中外数学史教程[M].福州：福建教育出版社,1993
[20]李家良译.数学在科学起源中的作用[M].湖南教育出版社,1992
[21]李锐,向书坚.我国时间序列分析研究工作综述[J].统计教育,2006,7：6-8
[22]李文林.数学珍宝—历史文献精选[M].北京：科学出版社,2000
[23]李文林.数学史概论[M].北京：高等教育出版社,2002
[24]李文林.数学的进化[M].北京：科学出版社,2005
[25]梁宗巨,王青建,孙宏安.世界数学通史(下册)[M].沈阳：辽宁教育出版社,2001
[26]林德光.生物统计的数学原理[M].沈阳：辽宁人民出版社,1982
[27]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京：高等教育出版社,2001
[28]罗芳琼,吴春梅.时间序列分析的理论与应用综述[J].柳州师专学报,2009,24(3)：113-117
[29]明道绪.生物统计[M].北京：中国农业科技出版社,1998
[30][美]莫里斯·克莱因著,北京大学数学系数学史翻译组译.古今数学思想(第四册)[M].上海：上海科学技术出版社,1979
[31][美]莫里斯·克莱因著,张祖贵译.西方文化中的数学[M].上海：复旦大学出版社,2005
[32]聂清香,张海霞,左庆林等.太阳活动11年周期的形成原因新探[J].自然杂志,2007,29(4)：225-228
[33]聂淑媛.格朗特与时间序列分析[J].自然辩证法通讯,2011,33(3)：50-52
[34]聂淑媛.尤尔建立时间序列线性自回归AR(P)模型的历史过程探析[J].统计与决策,2011,3：4-7
[35]聂淑媛,梁铁旺.指数和滑动平均的历史发展探究[J].统计与决策,2011,24：4-7
[36]聂淑媛.经济时间序列中差分的历史研究[J].科学技术哲学研究,2012,29(1)： 70-75
[37]聂淑媛.时间序列分析的历史发展[J].广西民族大学学报(自然科学版),2012,18(1)：24-28
[38]聂淑媛.沃尔德与离散平稳时间序列[J].咸阳师范学院学报(自然科学版),2012,27(2)：72-75
[39]聂淑媛.数学史的职业教育价值[J].高等理科教育,2011,6：115-117
[40][美]乔纳森·克莱尔等著,潘红宇等译.时间序列分析及应用：R语言(原书第二版)[M].北京：机械工业出版社,2011
[41][美]乔治·E.P.博克斯等著,王成璋等译.时间序列分析：预测与控制(原书第四版)[M].北京：机械工业出版社,2011
[42]曲安京.中国数学史研究的两次运动[J].科学,2004,56(2)：27-30
[43]曲安京.中国数学史研究范式的转换[J].中国科技史杂志,2005,26(1)：50-58
[44][美]塞路蒙·波克纳著,李家良译.数学在科学起源中的作用[M].湖南教育出版社,1992
[45]《数学辞海》编辑委员会.数学辞海(第三卷)[M].南京：东南大学出版社,北京：中国科学技术出版社,太原：山西教育出版社,2002
[46]汤岩.时间序列分析的研究与应用[D].哈尔滨：东北农业大学,2007
[47]王黎明,王连,杨楠.应用时间序列分析[M].上海：复旦大学出版社,2009
[48]王燕.应用时间序列分析[M].北京：中国人民大学出版社,2008
[49]王振龙.应用时间序列分析[M].北京：中国统计出版社,2010
[50][美]魏武雄著,易丹辉,刘超,贺学强等译.时间序列分析—单变量和多变量方法(第二版)[M].北京：中国人民大学出版社,2009
[51]魏宗舒.概率论与数理统计教程[M].北京：高等教育出版社,2005
[52]武娜.傅里叶级数的起源和发展[D].石家庄：河北师范大学,2008
[53]吴文俊主编.世界著名数学家传记(上集)[M].北京：科学出版社,2003
[54]吴文俊主编.世界著名数学家传记(下集)[M].北京：科学出版社,2003
[55]吴正英.浅谈时间序列发展史[J].统计与预测,1999,5：55-56.
[56]徐传胜.概率论简史[J].数学通报,2004(9)：36-39
[57]徐传胜.彼得堡数学学派的概率思想研究[D].西安：西北大学,2007
[58]杨静.布朗运动的数学理论的历史研究[D].北京：中国科学院研究生院,2006
[59]杨静,唐泉.维纳和布朗运动[J].数学的实践和认识,2008,38(10)：162-169
[60]杨静,王丽霞.爱因斯坦与布朗运动的数学理论[J].西北大学学报(自然科学版),2006,36(1)：169-172
[61]杨静,徐传胜,王朝旺.试析巴夏里埃的《投机理论》对数学的影响[J].自然科学史研究,2008,27(1)：94-104
[62]叶中行,王蓉华,徐晓玲等.概率论与数理统计[M].北京：北京大学出版社,2009
[63][美]约翰·塔巴克著杨静译.概率论和统计学：不明确的科学(数学之旅)[M].北京：商务印书馆,2007
[64][法]约瑟夫·傅立叶著,桂质亮译.热的解析理论[M].北京：北京大学出版社,2008
[65]张美英,何杰.时间序列预测模型研究简介[J].江西科学,2009,27(5)：697-701
[66]张树京,齐立心.时间序列分析简明教程[M].北京：清华大学出版社,北方交通大学出版社,2003
[67]张尧庭.时间序列分析[J].统计教育,1996,23(4)：23-27
[68]赵晨阳,翟少丹.高尔顿与相关理论的产生[J].西北大学学报(自然科学版),2008,38(4)：680-684
[69]中国大百科全书.天文学[M].北京：中国大百科全书出版社,1980