# 【机器学习中的数学】基函数与函数空间

## 说说径向基函数

<a href="http://www.codecogs.com/eqnedit.php?latex=p(t|X,\delta2)=\frac{1}{(2\pi){N/2}|K|{1/2}}exp(-\frac{1}{2}tTK^{-1}t)" target="_blank">

$p(t|X,\delta^2)=\frac{1}{(2\pi)^{N/2}|K|^{1/2}}exp(-\frac{1}{2}t^TK^{-1}t)$
p(t|X,\delta^2)=\frac{1}{(2\pi)^{N/2}|K|^{1/2}}exp(-\frac{1}{2}t^TK^{-1}t)
</a>

<a href="http://www.codecogs.com/eqnedit.php?latex=K=\alpha&space;XXT&plus;\delta2I&space;\Rightarrow&space;K=\alpha&space;\Phi&space;\PhiT&space;&plus;&space;\delta2I" target="_blank">

$K=\alpha&space;XX^T+\delta^2I&space;\Rightarrow&space;K=\alpha&space;\Phi&space;\Phi^T&space;+&space;\delta^2I$
K=\alpha XX^T+\delta^2I \Rightarrow K=\alpha \Phi \Phi^T + \delta^2I
</a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\Phi&space;=&space;\begin{bmatrix}&space;1&space;&&space;x_1&space;&&space;x_12&space;\&space;1&space;&&space;x_2&space;&&space;x_22&space;\&space;:&space;&&space;:&space;&&space;:&space;\&space;1&space;&&space;x_n&space;&&space;x_n^2&space;\end{bmatrix}" target="_blank">

$\Phi&space;=&space;\begin{bmatrix}&space;1&space;&&space;x_1&space;&&space;x_1^2&space;\\&space;1&space;&&space;x_2&space;&&space;x_2^2&space;\\&space;:&space;&&space;:&space;&&space;:&space;\\&space;1&space;&&space;x_n&space;&&space;x_n^2&space;\end{bmatrix}$
\Phi = \begin{bmatrix} 1 & x_1 & x_1^2 \ 1 & x_2 & x_2^2 \ : & : & : \ 1 & x_n & x_n^2 \end{bmatrix}
</a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\Phi&space;=&space;\begin{bmatrix}&space;exp(-2(x_1-1)2)&space;&&space;...&space;\&space;exp(-2(x_2-1)2)&space;&&space;...&space;\&space;:&space;&&space;:&space;\&space;exp(-2(x_n-1)^2)&space;&&space;...&space;\end{bmatrix}" target="_blank">

$\Phi&space;=&space;\begin{bmatrix}&space;exp(-2(x_1-1)^2)&space;&&space;...&space;\\&space;exp(-2(x_2-1)^2)&space;&&space;...&space;\\&space;:&space;&&space;:&space;\\&space;exp(-2(x_n-1)^2)&space;&&space;...&space;\end{bmatrix}$
\Phi = \begin{bmatrix} exp(-2(x_1-1)^2) & ... \ exp(-2(x_2-1)^2) & ... \ : & : \ exp(-2(x_n-1)^2) & ... \end{bmatrix}
</a>

## 径向基核函数

<a href="http://www.codecogs.com/eqnedit.php?latex=k(x,x')=exp(-(x-x')2)=exp(-x2)exp(-x'2)exp(2xx')&space;=exp(-x2)exp(-x'2)[\sum_{i=0}{\infty&space;}\frac{(2xx')i}{i!}]&space;=\sum_{i=0}{\infty&space;}[exp(-x2)exp(-x'2)\sqrt{\frac{2i}{i!}}\sqrt{\frac{2i}{i!}}xix'i]&space;=\Phi(x)^T\Phi(x')" target="_blank">

$k(x,x')=exp(-(x-x')^2)=exp(-x^2)exp(-x'^2)exp(2xx')&space;=exp(-x^2)exp(-x'^2)[\sum_{i=0}^{\infty&space;}\frac{(2xx')^i}{i!}]&space;=\sum_{i=0}^{\infty&space;}[exp(-x^2)exp(-x'^2)\sqrt{\frac{2^i}{i!}}\sqrt{\frac{2^i}{i!}}x^ix'^i]&space;=\Phi(x)^T\Phi(x')$
k(x,x')=exp(-(x-x')^2)=exp(-x^2)exp(-x'^2)exp(2xx') =exp(-x^2)exp(-x'^2)[\sum_{i=0}^{\infty }\frac{(2xx')^i}{i!}] =\sum_{i=0}^{\infty }[exp(-x^2)exp(-x'^2)\sqrt{\frac{2^i}{i!}}\sqrt{\frac{2^i}{i!}}x^ix'^i] =\Phi(x)^T\Phi(x')
</a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\Phi(x)=exp(-x2)(1,\sqrt{\frac{2}{1!}}x,\sqrt{\frac{22}{2!}}x^2,...)" target="_blank">

$\Phi(x)=exp(-x^2)(1,\sqrt{\frac{2}{1!}}x,\sqrt{\frac{2^2}{2!}}x^2,...)$
\Phi(x)=exp(-x^2)(1,\sqrt{\frac{2}{1!}}x,\sqrt{\frac{2^2}{2!}}x^2,...)
</a>

## 函数空间浅显解释

Github博客主页(http://jasonding1354.github.io/)
GitCafe博客主页(http://jasonding1354.gitcafe.io/)
CSDN博客(http://blog.csdn.net/jasonding1354)