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因子分析（带旋转）以可视化模式#

研究鸢尾花数据集，我们发现花萼长度、花瓣长度和花瓣宽度高度相关。花萼宽度的冗余性较低。矩阵分解技术可以揭示这些潜在模式。对结果组件应用旋转并不会本质上提高衍生潜在空间的预测价值，但可以帮助可视化它们的结构；例如，这里的方差最大化旋转，通过最大化权重的平方方差找到一种结构，其中第二个组件仅在花萼宽度上正加载。

# 作者：scikit-learn 开发者
# SPDX 许可证标识符：BSD-3-Clause

import matplotlib.pyplot as plt
import numpy as np

from sklearn.datasets import load_iris
from sklearn.decomposition import PCA, FactorAnalysis
from sklearn.preprocessing import StandardScaler

%% 加载鸢尾花数据

data = load_iris()
X = StandardScaler().fit_transform(data["data"])
feature_names = data["feature_names"]

绘制鸢尾花特征的协方差

ax = plt.axes()

im = ax.imshow(np.corrcoef(X.T), cmap="RdBu_r", vmin=-1, vmax=1)

ax.set_xticks([0, 1, 2, 3])
ax.set_xticklabels(list(feature_names), rotation=90)
ax.set_yticks([0, 1, 2, 3])
ax.set_yticklabels(list(feature_names))

plt.colorbar(im).ax.set_ylabel("$r$", rotation=0)
ax.set_title("Iris feature correlation matrix")
plt.tight_layout()

进行带有方差最大化旋转的因子分析

n_comps = 2

methods = [
    ("PCA", PCA()),
    ("Unrotated FA", FactorAnalysis()),
    ("Varimax FA", FactorAnalysis(rotation="varimax")),
]
fig, axes = plt.subplots(ncols=len(methods), figsize=(10, 8), sharey=True)

for ax, (method, fa) in zip(axes, methods):
    fa.set_params(n_components=n_comps)
    fa.fit(X)

    components = fa.components_.T
    print("\n\n %s :\n" % method)
    print(components)

    vmax = np.abs(components).max()
    ax.imshow(components, cmap="RdBu_r", vmax=vmax, vmin=-vmax)
    ax.set_yticks(np.arange(len(feature_names)))
    ax.set_yticklabels(feature_names)
    ax.set_title(str(method))
    ax.set_xticks([0, 1])
    ax.set_xticklabels(["Comp. 1", "Comp. 2"])
fig.suptitle("Factors")
plt.tight_layout()
plt.show()

 PCA :

[[ 0.52106591  0.37741762]
 [-0.26934744  0.92329566]
 [ 0.5804131   0.02449161]
 [ 0.56485654  0.06694199]]


 Unrotated FA :

[[ 0.88096009 -0.4472869 ]
 [-0.41691605 -0.55390036]
 [ 0.99918858  0.01915283]
 [ 0.96228895  0.05840206]]


 Varimax FA :

[[ 0.98633022 -0.05752333]
 [-0.16052385 -0.67443065]
 [ 0.90809432  0.41726413]
 [ 0.85857475  0.43847489]]

Total running time of the script: (0 minutes 0.239 seconds)