查看“︁散布矩阵”︁的源代码

[多元变量统计]]和[[概率论]]中，'''散布矩阵'''是一种[[统计量]]，用于估计[[协方差矩阵]]，例如[[多元正态分布]]的协方差矩阵。

==定义==
给定m维数据的n个样本，写作m×n矩阵<math>X=[\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n]</math>，则[[样本均值]]为

:<math>\overline{\mathbf{x}} = \frac{1}{n}\sum_{j=1}^n \mathbf{x}_j</math>

其中<math>\mathbf{x}_j</math>是<math>X</math>的第j列。<ref>{{Cite web |last=Raghavan |date=2018-08-16 |title=Scatter matrix , Covariance and Correlation Explained |url=https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56 |access-date=2022-12-28 |website=Medium |language=en |archive-date=2022-12-28 |archive-url=https://web.archive.org/web/20221228171745/https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56 |dead-url=no }}</ref>

'''散布矩阵'''是m×m正半定矩阵

:<math>S = \sum_{j=1}^n (\mathbf{x}_j-\overline{\mathbf{x}})(\mathbf{x}_j-\overline{\mathbf{x}})^T = \sum_{j=1}^n (\mathbf{x}_j-\overline{\mathbf{x}})\otimes(\mathbf{x}_j-\overline{\mathbf{x}}) = \left( \sum_{j=1}^n \mathbf{x}_j \mathbf{x}_j^T \right) - n \overline{\mathbf{x}} \overline{\mathbf{x}}^T </math>

其中<math>(\cdot)^T</math>表示[[矩阵转置]]，<ref>{{Cite web |last=Raghavan |date=2018-08-16 |title=Scatter matrix , Covariance and Correlation Explained |url=https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56 |access-date=2022-12-28 |website=Medium |language=en |archive-date=2022-12-28 |archive-url=https://web.archive.org/web/20221228171745/https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56 |dead-url=no }}</ref>乘法为[[外积]]。散布矩阵可更简洁地表为

:<math>S = X\,C_n\,X^T</math>

其中<math>\,C_n</math>是n×n[[中心化矩阵]]。

==应用==
给定n个样本的多元正态分布协方差矩阵的[[最大似然估计]]值可表为归一化散布矩阵
:<math>C_{ML}=\frac{1}{n}S.</math><ref>{{cite thesis |last=Liu |first=Zhedong |date=April 2019 |title=Robust Estimation of Scatter Matrix, Random Matrix Theory and an Application to Spectrum Sensing |url=https://repository.kaust.edu.sa/bitstream/handle/10754/652444/Thesis.pdf?sequence=1&isAllowed=y |type=Master of Science |publisher=King Abdullah University of Science and Technology |access-date=2023-10-01 |archive-date=2022-12-28 |archive-url=https://web.archive.org/web/20221228171753/https://repository.kaust.edu.sa/bitstream/handle/10754/652444/Thesis.pdf?sequence=1&isAllowed=y |dead-url=no }}</ref>

当<math>X</math>的列从多元正态分布中独立采样时，<math>S</math>遵循[[威沙特分布]]。

==另见==
* [[威沙特分布]]
* [[外积]]—<math>XX^\top</math>或X⊗X是X与自身的外积。
* [[格拉姆矩阵]]

== 参考文献 ==
{{reflist}}

[[Category:矩阵]]


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