PCA 和 SVD

先講結論

在統計上我們有 n 個變數構成的矩陣 B = [v1, v2. .... vn]^T
對 B 做 PCA 得到 eigenvector X
就等於直接對 B 做 SVD 得到的 right singular vectors V
PCA = principal value decomposition
SVD = singular value decomposition

[Proof]

PCA on B = eigen value decomposition on S = B^T*B => find X, Λ , s.t. S = XΛX^-1
X is the eigen vector; diagonal elements of Λ is the eigen values.
SVD on B => find U, Σ, V, s.t. B = UΣV^T
U is the left singular matrix; Σ is the singular values; V is the right singular matrix

B^T*B =  VΣU^T*UΣV^T

since U^T*U=I
BB^T =  VΣV^T
因此
X = V
另外, 主成份(Principal vectors)
P = B*

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每次都要從頭想一次，乾脆寫起來～
【參考資料】
線代啟示錄對SVD本身有很好的解釋, 有圖解!!!
https://ccjou.wordpress.com/2009/09/01/%E5%A5%87%E7%95%B0%E5%80%BC%E5%88%86%E8%A7%A3-svd/

對於EVD (eigenvalue decomposition)、PCA、SVD 三者的關係的說明
https://molecular-service-science.com/2014/07/16/eigen-value-singular-value-decomposition-principal-component-analysis/

下面是抄來的

雖然兩者形式十分類似，但從空間運用來看卻有很大不同，前 University of Michigan / Stanford University 的數學教授 Cleve Moler 在他的教科書上是這麼說的：

＂Eigenvalues play an important role in situations where the matrix is a transformation from one vector space onto itself.＂

＂Singular values play an important role where the matrix is a transformation from one vector space to a different vector space, possibly with a different dimension.＂

============== 補充: Matlab 驗證 ===============

假設 mean12L_p 是一個 500*8 的矩陣
    [U,S,V] = svd(mean12L_p(:,[7:12 1 2]));
    ECGv = U*S;
    [~,V_order] = sort(diag(V),'descend');
    V = V(:,V_order);
    mean8L = mean12L_p(:,[7:12 1 2]);
    corV = mean8L'*mean8L;
    [vec,val] = eig(corV);
    [~,val_order] = sort(diag(val),'descend');
    vec = vec(:,val_order);

vec - V_order;
就會是一個零矩陣