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mathematics_basic_for_ML
  • README
  • README
    • Summary
    • Geometry
      • EulerAngle
      • Gimbal lock
      • Quaternion
      • RiemannianManifolds
      • RotationMatrix
      • SphericalHarmonics
    • Information
      • Divergence
      • 信息熵 entropy
    • LinearAlgebra
      • 2D仿射变换(2D Affine Transformation)
      • 2DTransformation
      • 3D变换(3D Transformation)
      • ComplexTransformation
      • Conjugate
      • Hessian
      • IllConditioning
      • 逆变换(Inverse transform)
      • SVD
      • det
      • eigendecomposition
      • 矩阵
      • norm
      • orthogonal
      • special_matrix
      • trace
      • vector
    • Mathematics
      • Complex
      • ExponentialDecay
      • average
      • calculus
      • convex
      • derivative
      • 距离
      • function
      • space
      • Formula
        • euler
        • jensen
        • taylor
        • trigonometric
    • Numbers
      • 几何级数
      • SpecialNumbers
    • NumericalComputation
      • ConstrainedOptimization
      • GradientDescent
      • Newton
      • Nominal
      • ODE_SDE
      • Preprocessing
    • Probability
      • bayes
      • distribution
      • expectation_variance
      • 贝叶斯公式
      • functions
      • likelihood
      • mixture_distribution
      • 一些术语
      • probability_distribution
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  • 期望
  • 方差
  • 协方差
  • 协方差矩阵

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  1. README
  2. Probability

expectation_variance

期望

离散型变量期望:

Ex∼P[f(x)]=∑xP(x)f(x)E_{x\sim P}[f(x)] = \sum_x P(x)f(x)Ex∼P​[f(x)]=x∑​P(x)f(x)

连续型变量期望:Ex∼P[f(x)]=∫p(x)f(x)dxE_{x\sim P}[f(x)] = \int p(x)f(x)dxEx∼P​[f(x)]=∫p(x)f(x)dx

方差

Var(f(x))=E[(f(x)−E[f(x)])2]Var(f(x)) = E[(f(x) - E[f(x)])^2]Var(f(x))=E[(f(x)−E[f(x)])2]

标准差=$\sqrt \text{方差}$

协方差

两个变量线性相关性的强度以及这些变量的尺度

Cov(f(x),g(x))=E[(f(x)−E[f(x)])(g(x)−E[g(x)])]Cov(f(x),g(x)) = E[(f(x)-E[f(x)])(g(x)-E[g(x)])]Cov(f(x),g(x))=E[(f(x)−E[f(x)])(g(x)−E[g(x)])]

意义(没看懂):

  1. 绝对值很大:变量值变化很大,距离各自均值很远

  2. 负的:一个变量倾向于取得相对较大的值,另一个变量倾向于取得相对较小的值。反之亦然。

  3. 为0:没有线性关系,但不一定独立

协方差矩阵

x∈RnCov(X)i,j=Cov(Xi,Xj)方阵Cov(X)i,i=Var(Xi)\begin{aligned} x \in R^n\\ Cov(X)_{i,j} = Cov(X_i, X_j) \text{方阵}\\ Cov(X)_{i,i} = Var(X_i) \end{aligned}x∈RnCov(X)i,j​=Cov(Xi​,Xj​)方阵Cov(X)i,i​=Var(Xi​)​
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Last updated 2 years ago

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