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%matplotlib inline

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 predictve 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.

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# Authors: Jona Sassenhagen
# License: BSD 3 clause

import matplotlib.pyplot as plt
import numpy as np

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

print(__doc__)

Load Iris data

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

Plot covariance of Iris features

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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()

Run factor analysis with Varimax rotation

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n_comps = 2

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

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)))
    if ax.is_first_col():
        ax.set_yticklabels(feature_names)
    else:
        ax.set_yticklabels([])
    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()