%matplotlib inline

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA

from sklearn.datasets import load_iris

iris = load_iris()
y = iris.target
x = iris.data
print(x.shape)

(150, 4)

PCA数据可视化

pca = PCA(n_components=2)
pca = pca.fit(x)
x_dr = pca.transform(x)
print(x_dr.shape)

(150, 2)

colors = ["red", "black", "orange"]
plt.figure()

for i in range(3):
    plt.scatter(x_dr[y == i, 0]
                ,x_dr[y == i, 1]
                ,alpha=0.7
                ,c=colors[i]
                ,label=iris.target_names[i]
               )
plt.legend()
plt.title("PCA of IRIS datasets")
plt.show()

选择 n_components

(1) 累积可解释性方差贡献率曲线

pca_line = PCA().fit(x)

plt.plot([1, 2, 3, 4], np.cumsum(pca_line.explained_variance_ratio_))
plt.xticks([1, 2, 3, 4])
plt.xlabel("number of components after dimension reduction")
plt.ylabel("cumulative explained variance ratio")
plt.show()

pca_line.explained_variance_ratio_

array([0.92461872, 0.05306648, 0.01710261, 0.00521218])

(2) 最大似然估计自选超参数

pca_mle = PCA(n_components="mle")
pca_mle = pca_mle.fit(x)
x_mle = pca_mle.transform(x)
print(x_mle.shape)

(150, 3)

pca_mle.explained_variance_ratio_.sum()

0.9947878161267246

(3) 按信息量占比选超参

pca_f = PCA(n_components=0.97, svd_solver="full")
pca_f = pca_f.fit(x)
x_f = pca_f.transform(x)

pca_f.explained_variance_ratio_

array([0.92461872, 0.05306648])

pca_f.components_

array([[ 0.36138659, -0.08452251,  0.85667061,  0.3582892 ],
       [ 0.65658877,  0.73016143, -0.17337266, -0.07548102]])

x.shape

(150, 4)

当 x 为图像时，components_ 可以用于可视化，探索新特征空间从原始数据提取了什么重要信息

from sklearn.datasets import fetch_lfw_people

faces = fetch_lfw_people(data_home=".", min_faces_per_person=60)

print("数据: {}".format(faces.images.shape))
print("特征矩阵: {}".format(faces.data.shape))

数据: (1348, 62, 47)
特征矩阵: (1348, 2914)

fig, axes = plt.subplots(4, 5
                         ,figsize=(8, 4)
                         ,subplot_kw={"xticks":[], "yticks":[]} # 不显示坐标轴
                        )

for i, ax in enumerate(axes.flat):
    ax.imshow(faces.images[i,:,:]
              ,cmap="gray" # 色彩模式
             )

fig

X = faces.data
pca = PCA(n_components=150).fit(X)

pca.components_[:3]

array([[-0.00579721, -0.00595371, -0.00615771, ..., -0.01000111,
        -0.00901092, -0.00813917],
       [ 0.01708354,  0.01623676,  0.01622024, ..., -0.03474263,
        -0.03416986, -0.03298327],
       [-0.01833663, -0.01670169, -0.01557038, ..., -0.03540288,
        -0.03147691, -0.02929779]], dtype=float32)

V = pca.components_
print(V.shape)

(150, 2914)

fig, axes = plt.subplots(3, 8
                         ,figsize=(8, 4)
                         ,subplot_kw={"xticks":[], "yticks":[]}
                        )

for i, ax in enumerate(axes.flat):
    ax.imshow(V[i,:].reshape(62, 47), cmap="gray")

inverse_transform 无法完全还原数据

X_dr = pca.transform(X)
X_dr.shape

(1348, 150)

X_inverse = pca.inverse_transform(X_dr)

X_inverse.shape

(1348, 2914)

fig, ax = plt.subplots(2, 10
                       ,figsize=(10, 2.5)
                       ,subplot_kw={"xticks":[], "yticks":[]}
                      )

for i in range(10):
    ax[0, i].imshow(faces.images[i, :, :], cmap="binary_r")
    ax[1, i].imshow(X_inverse[i].reshape(62, 47), cmap="binary_r")

fig