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Principal Component Analysis Explained
1.
2.
3.
𝜆1 𝒆1 𝜆2 𝒆2 𝒆1 𝒆2 𝜆1>
𝜆2
4.
𝑌 = 𝑃𝑋 𝐶
𝑌 = 1 𝑛 𝑌𝑌 𝑇 = 1 𝑛 𝑃𝑋 𝑃𝑋 𝑇 = 1 𝑛 𝑃𝑋𝑋 𝑇 𝑃 𝑇 = 𝑃𝐶 𝑋 𝑃 𝑇 𝐴𝒙 = 𝜆𝒙 𝐴𝐸 = [𝐴𝒆1, 𝐴𝒆2, … , 𝐴𝒆 𝑛] 𝐸𝐷 = [𝜆1 𝒆1, 𝜆2 𝒆2, … ,𝜆 𝑛 𝒆 𝑛] 𝐴 = 𝐸𝐷𝐸−1 = 𝐸𝐷𝐸 𝑇 𝐴𝐸 = 𝐸𝐷 𝐶 𝑋 = 𝐸𝐶 𝑌 𝐸 𝑇 𝒆1 … , 𝒆 𝑚 ∴ 𝐸 = 𝑃 𝑇
5.
𝑋={Γ1, Γ2,…, Γ
𝑁} 𝑤 x ℎ = 𝑑 Γ = 𝑑 x 1 Ψ= 1 𝑀 𝑖=1 𝑁 Γ𝑖 Γ𝑖= Γ 𝑖−𝐸(Γ) 𝜎Γ Φ𝑖=Γ𝑖- Ψ 𝐶 = 1 𝑀 𝑖=1 𝑁 Φ𝑖 𝑇 Φ𝑖 = 𝐴𝐴 𝑇 𝐶 = 𝐸Λ𝐸 𝑇 𝑢𝑙= 1 𝑀 𝑘=1 𝑀 𝜆𝑙𝑘Φ 𝑘 , 𝑙 = 1, … , 𝑀 𝜔𝑗=𝑢𝑖 𝑇 Φ𝑖 Φ𝑖-mean= 𝑗=1 𝑀 𝜔𝑗 𝑢𝑗 𝜔 𝑘=𝑢 𝑘 𝑇 Γ − Ψ Ω 𝑇 =[𝜔1, . . , 𝜔 𝑀] Φ = 𝑗=1 𝐾 𝜔𝑗 𝑢𝑗 , 𝜀 𝑘=||Ω − Ω 𝑘||2 Thank you.
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