Sklearn PCA explained variance and explained variance ratio difference

Question

I'm trying to get the variances from the eigen vectors.

What is the difference between explained_variance_ratio_ and explained_variance_ in PCA?

Answer 1

The percentage of the explained variance is:

explained_variance_ratio_

The variance i.e. the eigenvalues of the covariance matrix is:

explained_variance_

Formula: explained_variance_ratio_ = explained_variance_ / np.sum(explained_variance_)

Example:

import numpy as np
from sklearn.decomposition import PCA
X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
pca = PCA(n_components=2)
pca.fit(X)  

pca.explained_variance_
array([7.93954312, 0.06045688]) # the actual eigenvalues (variance)

pca.explained_variance_ratio_ # the percentage of the variance
array([0.99244289, 0.00755711])

Also based on the above formula:

7.93954312 / (7.93954312+ 0.06045688) = 0.99244289

From the documentation:

explained_variance_ : array, shape (n_components,) The amount of variance explained by each of the selected components.

Equal to n_components largest eigenvalues of the covariance matrix of X.

New in version 0.18.

explained_variance_ratio_ : array, shape (n_components,) Percentage of variance explained by each of the selected components.

If n_components is not set then all components are stored and the sum of the ratios is equal to 1.0.