Model selection with Probabilistic (PCA) and Factor Analysis (FA)

Probabilistic PCA and Factor Analysis are probabilistic models. The consequence is that the likelihood of new data can be used for model selection and covariance estimation. Here we compare PCA and FA with cross-validation on low rank data corrupted with homoscedastic noise (noise variance is the same for each feature) or heteroscedastic noise (noise variance is the different for each feature). In a second step we compare the model likelihood to the likelihoods obtained from shrinkage covariance estimators.

One can observe that with homoscedastic noise both FA and PCA succeed in recovering the size of the low rank subspace. The likelihood with PCA is higher than FA in this case. However PCA fails and overestimates the rank when heteroscedastic noise is present. Under appropriate circumstances the low rank models are more likely than shrinkage models.

The automatic estimation from Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604 by Thomas P. Minka is also compared.

Python source code: plot_pca_vs_fa_model_selection.py

print(__doc__)

# Authors: Alexandre Gramfort
#          Denis A. Engemann
# License: BSD 3 clause

import numpy as np
import pylab as pl
from scipy import linalg

from sklearn.decomposition import PCA, FactorAnalysis
from sklearn.covariance import ShrunkCovariance, LedoitWolf
from sklearn.cross_validation import cross_val_score
from sklearn.grid_search import GridSearchCV

###############################################################################
# Create the data

n_samples, n_features, rank = 1000, 50, 10
sigma = 1.
rng = np.random.RandomState(42)
U, _, _ = linalg.svd(rng.randn(n_features, n_features))
X = np.dot(rng.randn(n_samples, rank), U[:, :rank].T)

# Adding homoscedastic noise
X_homo = X + sigma * rng.randn(n_samples, n_features)

# Adding heteroscedastic noise
sigmas = sigma * rng.rand(n_features) + sigma / 2.
X_hetero = X + rng.randn(n_samples, n_features) * sigmas

###############################################################################
# Fit the models

n_components = np.arange(0, n_features, 5)  # options for n_components


def compute_scores(X):
    pca = PCA()
    fa = FactorAnalysis()

    pca_scores, fa_scores = [], []
    for n in n_components:
        pca.n_components = n
        fa.n_components = n
        pca_scores.append(np.mean(cross_val_score(pca, X)))
        fa_scores.append(np.mean(cross_val_score(fa, X)))

    return pca_scores, fa_scores


def shrunk_cov_score(X):
    shrinkages = np.logspace(-2, 0, 30)
    cv = GridSearchCV(ShrunkCovariance(), {'shrinkage': shrinkages})
    a = cv.fit(X).best_estimator_
    return np.mean(cross_val_score(cv.fit(X).best_estimator_, X))


def lw_score(X):
    a = LedoitWolf().fit(X)
    return np.mean(cross_val_score(LedoitWolf(), X))


for X, title in [(X_homo, 'Homoscedastic Noise'),
                 (X_hetero, 'Heteroscedastic Noise')]:
    pca_scores, fa_scores = compute_scores(X)
    n_components_pca = n_components[np.argmax(pca_scores)]
    n_components_fa = n_components[np.argmax(fa_scores)]

    pca = PCA(n_components='mle')
    pca.fit(X)
    n_components_pca_mle = pca.n_components_

    print("best n_components by PCA CV = %d" % n_components_pca)
    print("best n_components by FactorAnalysis CV = %d" % n_components_fa)
    print("best n_components by PCA MLE = %d" % n_components_pca_mle)

    pl.figure()
    pl.plot(n_components, pca_scores, 'b', label='PCA scores')
    pl.plot(n_components, fa_scores, 'r', label='FA scores')
    pl.axvline(rank, color='g', label='TRUTH: %d' % rank, linestyle='-')
    pl.axvline(n_components_pca, color='b',
               label='PCA CV: %d' % n_components_pca, linestyle='--')
    pl.axvline(n_components_fa, color='r',
               label='FactorAnalysis CV: %d' % n_components_fa, linestyle='--')
    pl.axvline(n_components_pca_mle, color='k',
               label='PCA MLE: %d' % n_components_pca_mle, linestyle='--')

    # compare with other covariance estimators
    pl.axhline(shrunk_cov_score(X), color='violet',
               label='Shrunk Covariance MLE', linestyle='-.')
    pl.axhline(lw_score(X), color='orange',
               label='LedoitWolf MLE' % n_components_pca_mle, linestyle='-.')

    pl.xlabel('nb of components')
    pl.ylabel('CV scores')
    pl.legend(loc='lower right')
    pl.title(title)

pl.show()