在统计学中用标准差描述样本数据的 “散布度” 公式中之所以除以 n-1 而不是 n, 是因为这样使我们以较少的样本集更好的逼近总体标准差。即统计学上所谓的 “无偏估计”。 关于 协方差 与 散度 ：https://blog.csdn.net/wsp_1138886114/article/details/80967843

方差： v a r ( X ) = ∑ i = 1 n ( X i − X ˉ ) ( X i − X ˉ ) n − 1 var(X) = \frac{\sum_{i=1}^n(X_i-\bar{X})(X_i-\bar{X})}{n-1} var(X)=n−1∑i=1n​(Xi​−Xˉ)(Xi​−Xˉ)​

各个维度偏离其均值的程度，协方差： cov ( X , Y ) = ∑ i = 1 n ( X i − X ˉ ) ( Y i − Y ˉ ) n − 1 \text{cov}(X,Y) = \frac{\sum_{i=1}^n(X_i-\bar{X})(Y_i-\bar{Y})}{n-1} cov(X,Y)=n−1∑i=1n​(Xi​−Xˉ)(Yi​−Yˉ)​

协方差矩阵的计算： c o v ( z ) = ( 1 2 3 4 3 4 1 2 2 3 1 4 ) j cov(z) = \begin{pmatrix} 1 & 2 &3 &4 \\ 3&4 &1 & 2\\ 2& 3& 1& 4 \end{pmatrix}j cov(z)=⎝⎛​132​243​311​424​⎠⎞​j

Hessian矩阵定义： 若一元函数 f ( x ) f(x) f(x) 在 x = x ( 0 ) x = x^{(0)} x=x(0) 点的某个领域内具有任意阶导数，则 f ( x ) f(x) f(x) 在 x ( 0 ) x^{(0)} x(0) 点的泰勒展开式为： f ( x ) = f ( x ( 0 ) ) + f ′ ( x ( 0 ) ) Δ x + 1 2 f ′ ′ ( x ( 0 ) ) ( Δ x 2 ) + ⋯ (1) f(x) = f(x^{(0)}) + f'(x^{(0)})\Delta x + \frac{1}{2} f''(x^{(0)})(\Delta x^2)+\cdots \tag{1} f(x)=f(x(0))+f′(x(0))Δx+21​f′′(x(0))(Δx2)+⋯(1)

其中： Δ x = x − x ( 0 ) , Δ x 2 = ( x − x ( 0 ) ) 2 \Delta x = x-x^{(0)},\Delta x^2 = (x-x^{(0)})^2 Δx=x−x(0),Δx2=(x−x(0))2

二元函数 f ( x 1 , x 2 ) f(x_1,x_2) f(x1​,x2​)在 X ( 0 ) ( x 1 ( 0 ) , x 2 ( 0 ) ) X^{(0)}(x^{(0)}_1,x^{(0)}_2) X(0)(x1(0)​,x2(0)​)点处的泰勒展开式为： 1 2 [ ∂ 2 f ∂ 2 x 1 2 ∣ x ( 0 ) Δ x 1 2 + 2 ∂ 2 f ∂ x 1 ∂ x 2 ∣ x ( 0 ) Δ x 1 Δ x 2 + ∂ 2 f ∂ 2 x 2 2 ∣ x ( 0 ) Δ x 2 2 ] + ⋯ (2) \frac{1}{2}\left [ \frac{\partial^2f}{\partial^2x_1^2}|_{x^{(0)}} \Delta x_1^2 + 2\frac{\partial^2f}{\partial x_1\partial x_2}|_{x^{(0)}}\Delta x_1\Delta x_2+\frac{\partial^2f}{\partial^2x_2^2}|_{x^{(0)}} \Delta x_2^2\right ]+\cdots \tag{2} 21​[∂2x12​∂2f​∣x(0)​Δx12​+2∂x1​∂x2​∂2f​∣x(0)​Δx1​Δx2​+∂2x22​∂2f​∣x(0)​Δx22​]+⋯(2)

其中： Δ x 1 = x 1 − x 1 ( 0 ) , Δ x 2 = x 2 − x 2 ( 0 ) \Delta x_1 = x_1-x^{(0)}_1,\Delta x_2 = x_2-x_2^{(0)} Δx1​=x1​−x1(0)​,Δx2​=x2​−x2(0)​

将上述(2)展开式写成矩阵形式，则有： f ( X ) = f ( X ( 0 ) ) + ( ∂ f ∂ x 1 , ∂ f ∂ x 2 ) x ( 0 ) ( Δ x 1 Δ x 2 ) + 1 2 ( Δ x 1 , Δ x 2 ) { ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 } ∣ x ( 0 ) ( Δ x 1 Δ x 2 ) + ⋯ (3) f(X) = f(X^{(0)})+\left ( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2} \right )_{x^{(0)}}\begin{pmatrix} \Delta x_1\\ \Delta x_2 \end{pmatrix}+\frac{1}{2}(\Delta x_1,\Delta x_2)\begin{Bmatrix} \frac{\partial^2f}{\partial x_1^2} & \frac{\partial^2f}{\partial x_1 \partial x_2}\\ \frac{\partial^2f}{\partial x_2 \partial x_1}& \frac{\partial^2f}{\partial x_2^2} \end{Bmatrix}|_{x^{(0)}} \begin{pmatrix} \Delta x_1\\ \Delta x_2 \end{pmatrix} +\cdots \tag{3} f(X)=f(X(0))+(∂x1​∂f​,∂x2​∂f​)x(0)​(Δx1​Δx2​​)+21​(Δx1​,Δx2​){∂x12​∂2f​∂x2​∂x1​∂2f​​∂x1​∂x2​∂2f​∂x22​∂2f​​}∣x(0)​(Δx1​Δx2​​)+⋯(3)

即为： f ( X ) = f ( X ( 0 ) ) + ∇ f ( X ( 0 ) ) T + 1 2 Δ x T G ( X ( 0 ) ) Δ X + ⋯ (4) f(X) = f(X^{(0)})+\nabla f(X^{(0)})^T + \frac{1}{2} \Delta x^T G(X^{(0)}) \Delta X +\cdots \tag{4} f(X)=f(X(0))+∇f(X(0))T+21​ΔxTG(X(0))ΔX+⋯(4)

其中： G ( X ( 0 ) ) = { ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 } ∣ x ( 0 ) ,    Δ X = ( Δ x 1 Δ x 2 ) G(X^{(0)}) = \begin{Bmatrix} \frac{\partial^2f}{\partial x_1^2} & \frac{\partial^2f}{\partial x_1 \partial x_2}\\ \frac{\partial^2f}{\partial x_2 \partial x_1}& \frac{\partial^2f}{\partial x_2^2} \end{Bmatrix}|_{x^{(0)}}, ~~\Delta X = \begin{pmatrix} \Delta x_1\\ \Delta x_2 \end{pmatrix} G(X(0))={∂x12​∂2f​∂x2​∂x1​∂2f​​∂x1​∂x2​∂2f​∂x22​∂2f​​}∣x(0)​,  ΔX=(Δx1​Δx2​​)

G ( X ( 0 ) ) G(X^{(0)}) G(X(0))是 f ( x 1 , x 2 ) f(x_1,x_2) f(x1​,x2​) 在 X ( 0 ) X^{(0)} X(0) 点处的Hessian矩阵。它是由函数 f ( x 1 , x 2 ) f(x_1,x_2) f(x1​,x2​) 在 X ( 0 ) X^{(0)} X(0)点处的二阶偏导数所组成的方阵。我们一般将其表示为:

H ( f ) = [ ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ⋯ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x n ∂ x 2 ⋯ ∂ 2 f ∂ x n 2 ] H(f) = \begin{bmatrix} \frac{\partial^2f}{\partial x_1^2} & \frac{\partial^2f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2f}{\partial x_1 \partial x_n} \\ \frac{\partial^2f}{\partial x_2 \partial x_1} & \frac{\partial^2f}{\partial x_2^2} & \cdots & \frac{\partial^2f}{\partial x_2 \partial x_n}\\ \vdots & \vdots & \ddots &\vdots \\ \frac{\partial^2f}{\partial x_n \partial x_1} & \frac{\partial^2f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2f}{\partial x_n^2} \end{bmatrix} H(f)=⎣⎢⎢⎢⎢⎢⎡​∂x12​∂2f​∂x2​∂x1​∂2f​⋮∂xn​∂x1​∂2f​​∂x1​∂x2​∂2f​∂x22​∂2f​⋮∂xn​∂x2​∂2f​​⋯⋯⋱⋯​∂x1​∂xn​∂2f​∂x2​∂xn​∂2f​⋮∂xn2​∂2f​​⎦⎥⎥⎥⎥⎥⎤​

简写成： Q H e s s i a n = [ I x x I x y I y x I y y ] \mathbf{Q_{Hessian}} = \begin{bmatrix} I_{xx} & I_{xy}\\ I_{yx} & I_{yy} \end{bmatrix} QHessian​=[Ixx​Iyx​​Ixy​Iyy​​]

在线性代数中，正定矩阵（positive definite matrix）简称正定阵。 定义：A是n阶方阵，如果对于任何非零向量x都有 x T A x > 0 x^TAx>0 xTAx>0就称A正定矩阵。