تحلیل مؤلفه‌های اصلی

${\displaystyle {t_k}_{(i)}={\mathbf{x}}_{(i)}\cdot {\mathbf{w}}_{(k)}\mathrm{f}\mathrm{o}\mathrm{r}i=1,\dots ,nk=1,\dots ,l}$

${\displaystyle {\mathbf{w}}_{(1)}=\arg \underset{\Vert \mathbf{w}\Vert =1}{max}\left\{\sum _i(t_1{)}_{(i)}^2\right\}=\arg \underset{\Vert \mathbf{w}\Vert =1}{max}\left\{\sum _i{\left({\mathbf{x}}_{(i)}\cdot \mathbf{w}\right)}^2\right\}}$

${\displaystyle {\mathbf{w}}_{(1)}=\arg \underset{\Vert \mathbf{w}\Vert =1}{max}\left\{{\Vert \mathbf{X}\mathbf{w}\Vert }^2\right\}=\arg \underset{\Vert \mathbf{w}\Vert =1}{max}\left\{{\mathbf{w}}^{\mathsf{T}}{\mathbf{X}}^{\mathsf{T}}\mathbf{X}\mathbf{w}\right\}}$

${\displaystyle {\mathbf{w}}_{(1)}=\arg max\left\{\frac{{\mathbf{w}}^{\mathsf{T}}{\mathbf{X}}^{\mathsf{T}}\mathbf{X}\mathbf{w}}{{\mathbf{w}}^{\mathsf{T}}\mathbf{w}}\right\}}$

${\displaystyle {\hat{\mathbf{X}}}_k=\mathbf{X}-\sum _{s=1}^{k-1}\mathbf{X}{\mathbf{w}}_{(s)}{\mathbf{w}}_{(s)}^{\mathsf{T}}}$

${\displaystyle {\mathbf{w}}_{(k)}=\underset{\Vert \mathbf{w}\Vert =1}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\Vert {\hat{\mathbf{X}}}_k\mathbf{w}\Vert }^2\right\}=\arg max\left\{\frac{{\mathbf{w}}^{\mathsf{T}}{\hat{\mathbf{X}}}_k^{\mathsf{T}}{\hat{\mathbf{X}}}_k\mathbf{w}}{{\mathbf{w}}^T\mathbf{w}}\right\}}$

${\displaystyle \mathbf{T}=\mathbf{X}\mathbf{W}}$

${\displaystyle \mathbf{X}=\mathbf{U}\mathbf{\Sigma }{\mathbf{W}}^T}$

${\displaystyle u[m]=\frac{1}{N}\sum _{i=1}^{N}X[m,i]}$

${\displaystyle B=X-uh}$

${\displaystyle C=\mathbb{E}[B\otimes B]=\mathbb{E}[B\cdot B^{\ast }]=\frac{1}{N}B\cdot B^{\ast }}$

${\displaystyle V^{-1}CV=D}$

${\displaystyle g[m]=\sum _{q=1}^m{\lambda }_q}$

${\displaystyle g[m=l]\le 90\mathrm{\%}}$

${\displaystyle W[p,q]=V[p,q],p=1\dots M,q=1\dots l}$

${\displaystyle s[i]=\sqrt{C[i,i]}}$

${\displaystyle Z=\frac{B}{s}}$

${\displaystyle Y=W^{\ast }.Z}$