Note
Click here to download the full example code or to run this example in your browser via Binder
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.
# 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
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])
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
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()
Out:
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 0.493 seconds)
