Factor Analysis (with rotation) to visualize patterns

Investigating the Iris dataset, we see that sepal length, petal length and petal width are highly correlated. Sepal width is less redundant. Matrix decomposition techniques can uncover these latent patterns. Applying rotations to the resulting components does not inherently improve the predictive value of the derived latent space, but can help visualise their structure; here, for example, the varimax rotation, which is found by maximizing the squared variances of the weights, finds a structure where the second component only loads positively on sepal width.

# Authors: Jona Sassenhagen
# License: 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

Load Iris data

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

Plot covariance of Iris features

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])

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

Run factor analysis with Varimax rotation

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):

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

    vmax = np.abs(components).max()
    ax.imshow(components, cmap="RdBu_r", vmax=vmax, vmin=-vmax)
    ax.set_xticks([0, 1])
    ax.set_xticklabels(["Comp. 1", "Comp. 2"])
Factors, PCA, Unrotated FA, Varimax FA
 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 0.532 seconds)

Related examples

The Iris Dataset

The Iris Dataset

Model selection with Probabilistic PCA and Factor Analysis (FA)

Model selection with Probabilistic PCA and Factor Analysis (FA)

Comparison of LDA and PCA 2D projection of Iris dataset

Comparison of LDA and PCA 2D projection of Iris dataset

Logistic Regression 3-class Classifier

Logistic Regression 3-class Classifier

Faces dataset decompositions

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