因子分析（帶旋轉）以視覺化模式#

研究 Iris 數據集，我們發現萼片長度、花瓣長度和花瓣寬度高度相關。萼片寬度的冗餘較少。矩陣分解技術可以揭示這些潛在模式。將旋轉應用於產生的成分並不會固有地提高導出潛在空間的預測價值，但可以幫助視覺化它們的結構；例如，此處，變異數最大化旋轉（通過最大化權重的平方變異數找到）找到一種結構，其中第二個成分僅在萼片寬度上呈正負載。

# Authors: The scikit-learn developers
# SPDX-License-Identifier: 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

載入 Iris 數據

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

繪製 Iris 特徵的共變異數

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()

運行帶有 Varimax 旋轉的因子分析

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]]

腳本的總運行時間：（0 分鐘 0.433 秒）