高斯分布

x ∈ R p x\in \mathbb{R}^p x∈Rp
x ∼ N ( μ , ∑ ) = 1 ( 2 π ) p 2 ∣ Σ ∣ 1 2 e x p ( − 1 2 ( x − μ ) T Σ − 1 ( x − μ ) ) x\sim N(\mu,\sum)=\frac{1}{(2\pi)^{\frac{p}{2}}|\Sigma|^{\frac{1}{2}}}exp(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)) x∼N(μ,∑)=(2π)2p​∣Σ∣21​1​exp(−21​(x−μ)TΣ−1(x−μ))
( X − μ ) T Σ − 1 ( X − μ ) (X-\mu)^T\Sigma^{-1}(X-\mu) (X−μ)TΣ−1(X−μ)为x与 μ \mu μ之间的马氏距离
当 Σ = I \Sigma=I Σ=I时，马氏距离=欧式距离
将 Σ \Sigma Σ进行特征值分解
Σ = U Λ U T , U U T = U T U = I \Sigma=U\Lambda U^T,UU^T=U^TU=I Σ=UΛUT,UUT=UTU=I
Σ − 1 = ∑ i = 1 p u i 1 λ i u i T \Sigma^{-1}=\sum_{i=1}^{p}u_i\frac{1}{\lambda_i}u_i^T Σ−1=∑i=1p​ui​λi​1​uiT​
令 y i = ( x − μ ) T u i y_i=(x-\mu)^Tu_i yi​=(x−μ)Tui​
马氏距离 Δ = ( x − μ ) T Σ − 1 ( x − μ ) = ∑ i = 1 p ( x − μ ) T u i 1 λ i u i T ( x − μ ) = ∑ i = 1 p y i 2 λ i \Delta=(x-\mu)^T\Sigma^{-1}(x-\mu)=\sum_{i=1}^{p}(x-\mu)^Tu_i\frac{1}{\lambda_i}u_i^T(x-\mu)=\sum_{i=1}^{p}\frac{y_i^2}{\lambda_i} Δ=(x−μ)TΣ−1(x−μ)=∑i=1p​(x−μ)Tui​λi​1​uiT​(x−μ)=∑i=1p​λi​yi2​​