Z1=w1·X PCA 云=∑2=∑wx=w∑x=wx Vara)=∑a-z42 (a.b)2=(aTb)2 aTbaTb =∑wx-w刘 aTb(aTb)T aTbbTa =∑如x- Find w1 maximizing =∑w2yrx-06x-刘rw (w1)TSw1 =(wy服∑x-x- w lw2l2=(w1)Tw1=1 =(w1)TCov(x)w1 S=Cov(x)PCA 𝑤1 2 = 𝑤1 𝑇𝑤1 = 1 𝑧ഥ1 = 1 𝑁 ෍𝑧1 = 1 𝑁 ෍𝑤1 ∙ 𝑥 = 𝑤1 ∙ 1 𝑁 ෍𝑥= 𝑤1 ∙ 𝑥ҧ = 1 𝑁 ෍ 𝑥 𝑤1 ∙ 𝑥 − 𝑤1 ∙ 𝑥ҧ 2 = 1 𝑁 ෍ 𝑤1 ∙ 𝑥 − 𝑥ҧ 2 𝑎 ∙ 𝑏 2 = 𝑎 𝑇𝑏 2 = 𝑎 𝑇𝑏𝑎 𝑇𝑏 = 𝑎 𝑇𝑏 𝑎 𝑇𝑏 𝑇 = 𝑎 𝑇𝑏𝑏 𝑇𝑎 = 1 𝑁 ෍ 𝑤1 𝑇 𝑥 − 𝑥ҧ 𝑥 − 𝑥ҧ 𝑇𝑤1 = 𝑤1 𝑇 1 𝑁 ෍ 𝑥 − 𝑥ҧ 𝑥 − 𝑥ҧ 𝑇 𝑤1 = 𝑤1 𝑇𝐶𝑜𝑣 𝑥 𝑤1 Find 𝑤1 maximizing 𝑤1 𝑇𝑆𝑤1 𝑆 = 𝐶𝑜𝑣 𝑥 𝑉𝑎𝑟 𝑧1 = 1 𝑁 ෍ 𝑧1 𝑧1 − 𝑧ഥ1 2 𝑧1 = 𝑤1 ∙ 𝑥