# 多项式回归

## 定义和实例

${\displaystyle y=\beta _{0}+\beta _{1}x+\varepsilon ,\,}$

${\displaystyle y=\beta _{0}+\beta _{1}x+\beta _{2}x^{2}+\varepsilon }$

${\displaystyle y=\beta _{0}+\beta _{1}x+\beta _{2}x^{2}+\beta _{3}x^{3}+\cdots +\beta _{n}x^{n}+\varepsilon }$

## 矩阵形式和估计计算

${\displaystyle y_{i}\,=\,\beta _{0}+\beta _{1}x_{i}+\beta _{2}x_{i}^{2}+\cdots +\beta _{m}x_{i}^{m}+\varepsilon _{i}\ (i=1,2,\dots ,n)}$

${\displaystyle {\begin{bmatrix}y_{1}\\y_{2}\\y_{3}\\\vdots \\y_{n}\end{bmatrix}}={\begin{bmatrix}1&x_{1}&x_{1}^{2}&\dots &x_{1}^{m}\\1&x_{2}&x_{2}^{2}&\dots &x_{2}^{m}\\1&x_{3}&x_{3}^{2}&\dots &x_{3}^{m}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{n}&x_{n}^{2}&\dots &x_{n}^{m}\end{bmatrix}}{\begin{bmatrix}\beta _{0}\\\beta _{1}\\\beta _{2}\\\vdots \\\beta _{m}\end{bmatrix}}+{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\vdots \\\varepsilon _{n}\end{bmatrix}},}$

${\displaystyle {\vec {y}}=\mathbf {X} {\vec {\beta }}+{\vec {\varepsilon }}}$

${\displaystyle {\widehat {\vec {\beta }}}=(\mathbf {X} ^{\mathsf {T}}\mathbf {X} )^{-1}\;\mathbf {X} ^{\mathsf {T}}{\vec {y}},\,}$

## 参考文献

1. ^ Shaw, P; 等. Intellectual ability and cortical development in children and adolescents. Nature. 2006, 440 (7084): 676–679. PMID 16572172. doi:10.1038/nature04513.
2. ^ Barker, PA; Street-Perrott, FA; Leng, MJ; Greenwood, PB; Swain, DL; Perrott, RA; Telford, RJ; Ficken, KJ. A 14,000-Year Oxygen Isotope Record from Diatom Silica in Two Alpine Lakes on Mt. Kenya. Science. 2001, 292 (5525): 2307–2310. PMID 11423656. doi:10.1126/science.1059612.
3. ^ Greenland, Sander. Dose-Response and Trend Analysis in Epidemiology: Alternatives to Categorical Analysis. Epidemiology. 1995, 6 (4): 356–365. JSTOR 3702080. PMID 7548341. doi:10.1097/00001648-199507000-00005.
4. ^ Yin-Wen Chang; Cho-Jui Hsieh; Kai-Wei Chang; Michael Ringgaard; Chih-Jen Lin. Training and testing low-degree polynomial data mappings via linear SVM. Journal of Machine Learning Research. 2010, 11: 1471–1490 [2019-03-18]. （原始内容存档于2020-11-21）.
5. ^ Gergonne, J. D. The application of the method of least squares to the interpolation of sequences. Historia Mathematica Translated by Ralph St. John and S. M. Stigler from the 1815 French. November 1974, 1 (4): 439–447 [1815]. doi:10.1016/0315-0860(74)90034-2.
6. ^ Stigler, Stephen M. Gergonne's 1815 paper on the design and analysis of polynomial regression experiments. Historia Mathematica. November 1974, 1 (4): 431–439. doi:10.1016/0315-0860(74)90033-0.
7. ^ Smith, Kirstine. On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of the Observations. Biometrika. 1918, 12 (1/2): 1–85. JSTOR 2331929. doi:10.2307/2331929.
8. ^ Such "non-local" behavior is a property of analytic functions that are not constant (everywhere). Such "non-local" behavior has been widely discussed in statistics:
9. ^ Fan, Jianqing. Local Polynomial Modelling and Its Applications: From linear regression to nonlinear regression. Monographs on Statistics and Applied Probability. Chapman & Hall/CRC. 1996. ISBN 978-0-412-98321-4.