协方差矩阵推导

协方差定义： $cov({\bf{x}},{\bf{y}}) = E[({\bf{x}} - {u_x}){({\bf{y}} - {u_y})^T}] = E({\bf{x}}{{\bf{y}}^T}) - E({\bf{x}})E({\bf{y}})$ ，其中 ${u_x},{u_y}$ 分别为向量 ${\rm{x}},{\bf{y}}$ 的均值。

设已知矩阵 $% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgatCvAUfeBSn0BKvguHDwzZbqegSSZmxoasaacH8srps % 0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr % 0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeqabaqaci % GacaGaaeqabaWaaeaaeaqbbOqaaiaadogacaWGVbGaamODaiaacIca % caWH4bGaaiilaiaahMhacaGGPaGaeyypa0JaamyraiaacUfacaGGOa % GaaCiEaiabgkHiTiaadwhadaWgaaWcbaGaamiEaaqabaGccaGGPaGa % aiikaiaahMhacqGHsislcaWG1bWaaSbaaSqaaiaadMhaaeqaaOGaai % ykamaaCaaaleqabaGaamivaaaakiaac2facqGH9aqpcaWGfbGaaiik % aiaahIhacaWH5bWaaWbaaSqabeaacaWGubaaaOGaaiykaiaam2caca % WGfbGaaiikaiaahIhacaGGPaGaamyraiaacIcacaWH5bGaaiykaaaa % !6485! \[cov({\bf{x}},{\bf{y}}) = E[({\bf{x}} - {u_x}){({\bf{y}} - {u_y})^T}] = E({\bf{x}}{{\bf{y}}^T}) - E({\bf{x}})E({\bf{y}})\]$

$X = \left[ {\begin{array}{*{20}{c}} {{x_{11}}}&{{x_{12}}}& \cdots &{{x_{1n}}}\\ {{x_{21}}}&{{x_{22}}}& \cdots &{{x_{2n}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{x_{m1}}}&{{x_{m2}}}& \cdots &{{x_{mn}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\bf{x}}_{\bf{1}}}}&{{{\bf{x}}_{\bf{2}}}}& \cdots &{{{\bf{x}}_{\bf{n}}}} \end{array}} \right]$

则

$cov(X) = cov(\left[ {\begin{array}{*{20}{c}} {{{\bf{x}}_{\bf{1}}}}&{{{\bf{x}}_{\bf{2}}}}& \cdots &{{{\bf{x}}_{\bf{n}}}} \end{array}} \right])$

$= \left[ {\begin{array}{*{20}{c}} {cov({{\bf{x}}_1},{{\bf{x}}_1})}&{cov({{\bf{x}}_1},{{\bf{x}}_2})}& \cdots &{cov({{\bf{x}}_1},{{\bf{x}}_n})}\\ {cov({{\bf{x}}_2},{{\bf{x}}_1})}&{cov({{\bf{x}}_2},{{\bf{x}}_2})}& \cdots &{cov({{\bf{x}}_2},{{\bf{x}}_n})}\\ \vdots & \vdots & \ddots & \vdots \\ {cov({{\bf{x}}_n},{{\bf{x}}_1})}&{cov({{\bf{x}}_n},{{\bf{x}}_2})}& \cdots &{cov({{\bf{x}}_n},{{\bf{x}}_n})} \end{array}} \right]$

$= \left[ {\begin{array}{*{20}{c}} {E[{{({{\bf{x}}_1} - {u_{{x_1}}})}^T}({{\bf{x}}_1} - {u_{{x_1}}})]}&{E[{{({{\bf{x}}_1} - {u_{{x_1}}})}^T}({{\bf{x}}_2} - {u_{{x_2}}})]}& \cdots &{E[{{({{\bf{x}}_1} - {u_{{x_1}}})}^T}({{\bf{x}}_n} - {u_{{x_n}}})]}\\ {E[{{({{\bf{x}}_2} - {u_{{x_2}}})}^T}({{\bf{x}}_1} - {u_{{x_1}}})]}&{E[{{({{\bf{x}}_2} - {u_{{x_2}}})}^T}({{\bf{x}}_2} - {u_{{x_2}}})]}& \cdots &{E[{{({{\bf{x}}_2} - {u_{{x_2}}})}^T}({{\bf{x}}_n} - {u_{{x_n}}})]}\\ \vdots & \vdots & \ddots & \vdots \\ {E[{{({{\bf{x}}_n} - {u_{{x_n}}})}^T}({{\bf{x}}_1} - {u_{{x_1}}})]}&{E[{{({{\bf{x}}_n} - {u_{{x_n}}})}^T}({{\bf{x}}_2} - {u_{{x_2}}})]}& \cdots &{E[{{({{\bf{x}}_n} - {u_{{x_n}}})}^T}({{\bf{x}}_n} - {u_{{x_n}}})]} \end{array}} \right]$

$= \frac{1}{{m - 1}}\left[ {\begin{array}{*{20}{c}} {{{({{\bf{x}}_1} - {u_{{x_1}}})}^T}({{\bf{x}}_1} - {u_{{x_1}}})}&{{{({{\bf{x}}_1} - {u_{{x_1}}})}^T}({{\bf{x}}_2} - {u_{{x_2}}})}& \cdots &{{{({{\bf{x}}_1} - {u_{{x_1}}})}^T}({{\bf{x}}_n} - {u_{{x_n}}})}\\ {{{({{\bf{x}}_2} - {u_{{x_2}}})}^T}({{\bf{x}}_1} - {u_{{x_1}}})}&{{{({{\bf{x}}_2} - {u_{{x_2}}})}^T}({{\bf{x}}_2} - {u_{{x_2}}})}& \cdots &{{{({{\bf{x}}_2} - {u_{{x_2}}})}^T}({{\bf{x}}_n} - {u_{{x_n}}})}\\ \vdots & \vdots & \ddots & \vdots \\ {{{({{\bf{x}}_n} - {u_{{x_n}}})}^T}({{\bf{x}}_1} - {u_{{x_1}}})}&{{{({{\bf{x}}_n} - {u_{{x_n}}})}^T}({{\bf{x}}_2} - {u_{{x_2}}})}& \cdots &{{{({{\bf{x}}_n} - {u_{{x_n}}})}^T}({{\bf{x}}_n} - {u_{{x_n}}})} \end{array}} \right]$

样本自由度m-1，设 ${{\bf{\tilde x}}_i} = {{\bf{x}}_i} - {u_{{x_i}}}$ ， $\tilde X = [\begin{array}{*{20}{c}} {{{{\bf{\tilde x}}}_1}}&{{{{\bf{\tilde x}}}_2}}& \cdots &{{{{\bf{\tilde x}}}_n}} \end{array}]$ ，则 $cov(X) = \frac{1}{{m - 1}}{\tilde X^T}\tilde X$