import numpy as np # 適当なデータを用意する 5次元6件のデータを用意した data = np.random.random((6, 5)) # 平均を引く d2 = data - data.mean(axis=0) # 共分散行列を作る cov = np.einsum("ij,ik->jk", d2, d2) # 共分散行列はデータの次元の正方行列 >>> cov.shape (5, 5) # 共分散行列を固有値分解する >>> np.linalg.eig(cov) (array([1.34012189, 0.77663231, 0.58201555, 0.08694709, 0.03991043]), array([[ 0.67869952, 0.35340645, -0.37436676, 0.50602025, 0.13514392], [ 0.45792096, 0.20519401, 0.77490418, -0.08403411, -0.37505412], [-0.46695028, 0.17178326, 0.04681577, 0.68316922, -0.53248104], [-0.26671759, 0.89310368, -0.08178549, -0.34559589, 0.07142937], [-0.20123254, 0.07652205, 0.50049221, 0.38823327, 0.74325791]])) # 比較対象のPCA実装 from sklearn.decomposition import PCA # データにフィットさせる m = PCA().fit(data) # 主成分軸を表示 >>> m.components_ array([[ 0.67869952, 0.45792096, -0.46695028, -0.26671759, -0.20123254], [ 0.35340645, 0.20519401, 0.17178326, 0.89310368, 0.07652205], [-0.37436676, 0.77490418, 0.04681577, -0.08178549, 0.50049221], [ 0.50602025, -0.08403411, 0.68316922, -0.34559589, 0.38823327], [ 0.13514392, -0.37505412, -0.53248104, 0.07142937, 0.74325791]]) # 共分散行列の固有値分解の結果と、PCAの主成分軸を比較すると、差がほぼゼロ >>> np.linalg.eig(cov)[1] - m.components_.T array([[-4.44089210e-16, 1.11022302e-16, 4.99600361e-16, 1.11022302e-16, 1.66533454e-16], [-2.22044605e-16, -6.66133815e-16, 4.44089210e-16, -3.88578059e-16, -1.11022302e-16], [ 1.11022302e-16, -3.05311332e-16, -2.56739074e-16, -3.33066907e-16, 3.33066907e-16], [-2.22044605e-16, 0.00000000e+00, 2.08166817e-15, 3.33066907e-16, -3.60822483e-16], [ 2.22044605e-16, -1.02695630e-15, -1.11022302e-16, 6.66133815e-16, 0.00000000e+00]]) # 特異値分類(SVD)との関係(軸の一部の符号が反転していることを除けば同じもの) >>> -np.linalg.svd(d2)[2] array([[ 0.67869952, 0.45792096, -0.46695028, -0.26671759, -0.20123254], [ 0.35340645, 0.20519401, 0.17178326, 0.89310368, 0.07652205], [-0.37436676, 0.77490418, 0.04681577, -0.08178549, 0.50049221], [-0.50602025, 0.08403411, -0.68316922, 0.34559589, -0.38823327], [-0.13514392, 0.37505412, 0.53248104, -0.07142937, -0.74325791]]) >>> m.components_ array([[ 0.67869952, 0.45792096, -0.46695028, -0.26671759, -0.20123254], [ 0.35340645, 0.20519401, 0.17178326, 0.89310368, 0.07652205], [-0.37436676, 0.77490418, 0.04681577, -0.08178549, 0.50049221], [ 0.50602025, -0.08403411, 0.68316922, -0.34559589, 0.38823327], [ 0.13514392, -0.37505412, -0.53248104, 0.07142937, 0.74325791]])