最小二乘法及其相关方法的数学原理与类比分析
Mathematical Theory and Analogy Analysis of Least Square Method and Its Related Methods
摘要: 在对各种问题的统计分析和建模处理时，经常用到最小二乘法或其相关的方法进行误差分析处理。本文详细叙述了最小二乘法、偏最小二乘法与主成分分析法的数学原理，简述了这些方法的应用场合，对于它们不适用的情况作了说明。描述了它们之间的关联性，概述了它们的相异性。最后，简单说明了回归方程中的参数检验。
Abstract: In the case of statistical analysis and modeling of various problems, the error analysis is often used by least square method or its related methods. In this paper, these mathematical theories are described in detail on least squares, partial least squares and principal component analysis. We sketch the applications of these methods. In the meantime, the case for which they are not appli-cable is explained. We outline the correlation among them and their different phases. At last, the parameter test in the regression equations is simply explained.
文章引用：幸冬梅. 最小二乘法及其相关方法的数学原理与类比分析[J]. 理论数学, 2018, 8(3): 230-238. https://doi.org/10.12677/PM.2018.83029

参考文献

[1] Walpole, R.E., Myers, R.H., Myers, S.L. and Ye, K.Y. 理工科概率统计[M]. 周勇, 马昀蓓, 谢尚宇, 王晓婧, 译.北京: 机械工业出版社, 2010.
[2] Hotelling, H. (1933) Analysis of a Complex of Statistical Variables into Principal Components. Education Psychology, 24, 417-444. [Google Scholar] [CrossRef] 
[3] Massy, W.F. (1965) Principal Components Regression in Exploratory Statistical Research. Journal of the American Statistical Association, 60, 234-256. [Google Scholar] [CrossRef] 
[4] Wold, S., Albano, C. and Dun, M. (1983) Pattern Regression Finding and Using Regularities in Multivariate Data. Analysis Applied Science Publication, London.
[5] Rosipal, R. and Krämer, N. (2006) Overview and Recent Advances in Partial Least Squares. Subspace, Latent Structure and Feature Selection, Bohinj, Slovenia, 23-25 February 2005, 34-51. [Google Scholar] [CrossRef] 
[6] Wold, H. (1982) Soft Modeling: The Basic Design and Some Extensions. In: Jöreskog, J.-K. and Wold, H., Eds., Systems under Indirect Observation, Volume 2, North Holland, Amsterdam, 1-53.
[7] Wold, H. (1985) Partial Least Squares. In: Kotz, S. and Johnson, N.L., Eds., Encyclopedia of the Statistical Sciences, Vol. 6, John Wiley, New York, 581-591.
[8] Wold, S., Ruhe, H., Wold, H. and Dunn III, W.J. (1984) The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverse. SIAM Journal of Scientific and Statistical Computations, 5, 735-743. [Google Scholar] [CrossRef] 
[9] 张红, 谢娜. 基于主成分分析与谱分析的房地产市场周期研究[J]. 清华大学学报(自然科学版), 2008, 48(9): 24-27.

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