# 高斯过程

## 定義

${\displaystyle X_{t_{1},\ldots ,t_{k}}=(X_{t_{1}},\ldots ,X_{t_{k}})}$

## 协方差函数

### 常見的协方差函數

• 常值：${\displaystyle K_{\operatorname {C} }(x,x')=C}$
• 線性：${\displaystyle K_{\operatorname {L} }(x,x')=x^{T}x'}$
• 高斯噪聲: ${\displaystyle K_{\operatorname {GN} }(x,x')=\sigma ^{2}\delta _{x,x'}}$
• 平方指數: ${\displaystyle K_{\operatorname {SE} }(x,x')=\exp {\Big (}-{\frac {\|d\|^{2}}{2\ell ^{2}}}{\Big )}}$
• Ornstein–Uhlenbeck : ${\displaystyle K_{\operatorname {OU} }(x,x')=\exp \left(-{\frac {|d|}{\ell }}\right)}$
• Matérn: ${\displaystyle K_{\operatorname {Matern} }(x,x')={\frac {2^{1-\nu }}{\Gamma (\nu )}}{\Big (}{\frac {{\sqrt {2\nu }}|d|}{\ell }}{\Big )}^{\nu }K_{\nu }{\Big (}{\frac {{\sqrt {2\nu }}|d|}{\ell }}{\Big )}}$
• 定期: ${\displaystyle K_{\operatorname {P} }(x,x')=\exp \left(-{\frac {2\sin ^{2}\left({\frac {d}{2}}\right)}{\ell ^{2}}}\right)}$
• 有理二次方: ${\displaystyle K_{\operatorname {RQ} }(x,x')=(1+|d|^{2})^{-\alpha },\quad \alpha \geq 0}$

## 註譯

1. ^ Platypus Innovation: A Simple Intro to Gaussian Processes (a great data modelling tool). [2016-11-02]. （原始内容存档于2018-05-01）.
2. ^ Chen, Zexun; Wang, Bo; Gorban, Alexander N. Multivariate Gaussian and Student-t process regression for multi-output prediction. Neural Computing and Applications. 2019-12-31. ISSN 0941-0643. doi:10.1007/s00521-019-04687-8 （英语）.
3. ^ Rasmussen, C. E. Gaussian Processes in Machine Learning. Advanced Lectures on Machine Learning. Lecture Notes in Computer Science 3176. 2004: 63–71. ISBN 978-3-540-23122-6. doi:10.1007/978-3-540-28650-9_4.
4. Bishop, C.M. Pattern Recognition and Machine Learning. Springer. 2006. ISBN 0-387-31073-8.
5. ^ Simon, Barry. Functional Integration and Quantum Physics. Academic Press. 1979.
6. ^ Seeger, Matthias. Gaussian Processes for Machine Learning. International Journal of Neural Systems. 2004, 14 (2): 69–104. doi:10.1142/s0129065704001899.
7. ^ Barber, David. Bayesian Reasoning and Machine Learning. Cambridge University Press. 2012 [2018-06-26]. ISBN 978-0-521-51814-7. （原始内容存档于2020-11-11）.
8. Rasmussen, C.E.; Williams, C.K.I. Gaussian Processes for Machine Learning. MIT Press. 2006 [2018-06-26]. ISBN 0-262-18253-X. （原始内容存档于2021-05-22）.
9. ^ Grimmett, Geoffrey; David Stirzaker. Probability and Random Processes. Oxford University Press. 2001. ISBN 0198572220.