数学基础(高等数学)

2018-09-26  本文已影响0人  Li77159

一些在机器学习中可能用到的数学基础以及常用的数学公式

机器学习中常见的函数

y = sgnx= \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \\ \end{cases}

y = [x]

y = D(x)= \begin{cases} 1 & x 为有理数 \\ 0 & x 为无理数 \\ \end{cases}

y = max[{f(x),g(x)}]
y = min[{f(x),g(x)}]

x =\log_a{N}

N = a^x

P(X=k) = \frac{\lambda^k}{k!} e^{-\lambda},k = 0,11...

f(x) = \frac{1}{\sqrt{2\pi}\sigma}exp(-\frac{(x-\mu)^2}{2\sigma^2})

S(x) = \frac{1}{1+e^{-x}}

y = \begin{cases} x \\ x^2 \\ x^3 \\ x^4 \\ \end{cases}

y = ax +b

y = \begin{cases} sinx & 正弦函数\\ cosx & 余弦函数 \\ tanx & 正切函数\\ \end{cases}

y = \begin{cases} arcsinx & 反正弦函数\\ arccosx & 反余弦函数 \\ arctanx & 反正切函数\\ \end{cases}

导数与微积分

某点的导数即为该点的斜率
k = \lim_{x\rightarrow x_0}{\frac{f(x)-f(x_0)}{x - x_0}}

y' = \begin{cases} 0 & (C)' \\ cosx & (sinx)' \\ -sinx & (cosx)' \\ nx^{n-1} & (x^n)'\\ \frac{1}{xlna} & (log_a x)'\\ \frac{1}{x} & (lnx)'\\ a^x lna & (a^x)'\\ e^x & (e^x)'\\ \end{cases}

dy = f'(x_0)\Delta x = f'(x_0)dx

[f(x)\pm g(x)]' = f'(x) \pm g'(x)
[f(x) \ast g(x)]' = f'(x)g(x) + f(x)g'(x)
[\frac{f(x)}{g(x)}]' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}(g(x) \neq 0)

定积分

\int^b_a f(x)dx = \lim_{\lambda \rightarrow 0} \sum^n_{i=1} f(\xi _i)\Delta x_i

\int^b_a f(x)dx = \begin{cases} 0 & a = b\\ -\int^a_b f(x)dx & b < a\\ \end{cases}

\int^b_a[f(x)+g(x)]dx = \int^b_a f(x)dx + \int^b_a g(x)dx
\int^b_a kf(x)dx = k\int^b_af(x)dx
\int^b_a [\alpha f(x) + \beta g(x)]dx = \alpha \int^b_a f(x)dx + \beta \int^b_a g(x)dx
\int^b_a f(x)dx = \int^c_a f(x)dx + \int^b_c f(x)dx
\int^b_a f(x)dx \geq 0
若f(x)在[a,b]上连续,则存在\xi \in [a,b] ,使 \int^b_a f(x)dx = f(\xi)(b-a)
设M,m分别是f(x)在区间[a,b]上的最大值及最小值,则m(b-a) \leq \int ^b_a f(x)dx \leq M(b-a)

向量代数及空间解析几何

定义
既有大小又有方向的量

\vec a + \vec b = \vec b + \vec a
\vec a + \vec b + \vec c = (\vec a + \vec b ) + \vec c = \vec a + (\vec b + \vec c)
\vec a +(- \vec a) = \vec 0
\vec a -\vec b = \vec a +(- \vec b)
\lambda (\mu \vec a) = \mu (\lambda \vec a) = (\mu \lambda)\vec a
(\lambda + \mu) \vec a = \lambda \vec a + \mu \vec a
\lambda(\vec a + \vec b ) = \lambda \vec a + \lambda \vec b

\vec a \cdot \vec b = |\vec a||\vec b |cos \theta

a \cdot b = b \cdot a
(a+b)\cdot c = a \cdot c + b \cdot c
(\lambda a) \cdot b = a \cdot (\lambda b) = \lambda(a \cdot b)
(\lambda a ) \cdot (\mu b) = \lambda \mu (a \cdot b)

cos \theta = \frac{a \cdot b}{|a||b|}=\frac{a_xb_x+a_yb_y+a_zb_z}{\sqrt{a_x^2+a_y^2+a_z^2} \sqrt{b_x^2+b_y^2+b_z^2}}

(x-x_0)^2+(y-y_0)^2+(z-z_0)^2 = R^2

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

梯度

定义

函数z=f(x,y)在平面区域D内具有一阶连续偏导数,则对于点P_0(x_0,y_0) \in D,都可以定出向量 \frac{\alpha f}{\alpha x} \vec i+\frac{\alpha f}{\alpha y} \vec j, 这向量称为函数z = f(x,y) 在点P_0(x_0,y_0)的梯度,记为 gradf(x_0,y_0) = \nabla f(x_0,y_0) = f_x(x_0,y_0) \vec i +f_y(x_0,y_0) \vec j 它的模等于方向导数的最大值 |gradf(x,y)| = \sqrt{(\frac {\alpha f}{\alpha x})^2 + (\frac {\alpha f}{\alpha y})^2}

\Theta^1 = \Theta^0 - \alpha \nabla J(\Theta) evaluated at \Theta^0
假设有一个单变量的函数 J(\theta) = \theta^2,函数的导数是J'(\theta)=2\theta,起点为\theta^0 = 1,学习率\alpha = 0.4,开始计算下降梯度\theta^0=1 \theta^1 = \theta^0-\alpha *J'(\theta^0)=1-0.4*2=0.2 \theta^2 = \theta^1-\alpha *J'(\theta^1)=0.2-0.4*0.4=0.04

二重积分

\int \int_Dkf(x,y)d \sigma = k\int\int_Df(x,y)d \sigma
\int \int_D[f(x,y) \pm g(x,y)]d \sigma = \int\int_Df(x,y)d \sigma + \int\int_Dg(x,y)d \sigma
\int \int_Df(x,y)d \sigma = \int\int_{D_1} f(x,y)d \sigma + \int\int_{D_2} f(x,y)d \sigma

结束语

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