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"""Factor Analysis.
A latent linear variable model.
FactorAnalysis is similar to probabilistic PCA implemented by PCA.score
While PCA assumes Gaussian noise with the same variance for each
feature, the FactorAnalysis model assumes different variances for
each of them.
This implementation is based on David Barber's Book,
Bayesian Reasoning and Machine Learning,
http://www.cs.ucl.ac.uk/staff/d.barber/brml,
Algorithm 21.1
"""
# Author: Christian Osendorfer <osendorf@gmail.com>
# Alexandre Gramfort <alexandre.gramfort@inria.fr>
# Denis A. Engemann <denis-alexander.engemann@inria.fr>
# License: BSD3
import warnings
from math import sqrt, log
import numpy as np
from scipy import linalg
from ..base import BaseEstimator, TransformerMixin
from ..utils import check_array, check_random_state
from ..utils.extmath import fast_logdet, randomized_svd, squared_norm
from ..utils.validation import check_is_fitted, _deprecate_positional_args
from ..exceptions import ConvergenceWarning
class FactorAnalysis(TransformerMixin, BaseEstimator):
"""Factor Analysis (FA)
A simple linear generative model with Gaussian latent variables.
The observations are assumed to be caused by a linear transformation of
lower dimensional latent factors and added Gaussian noise.
Without loss of generality the factors are distributed according to a
Gaussian with zero mean and unit covariance. The noise is also zero mean
and has an arbitrary diagonal covariance matrix.
If we would restrict the model further, by assuming that the Gaussian
noise is even isotropic (all diagonal entries are the same) we would obtain
:class:`PPCA`.
FactorAnalysis performs a maximum likelihood estimate of the so-called
`loading` matrix, the transformation of the latent variables to the
observed ones, using SVD based approach.
Read more in the :ref:`User Guide <FA>`.
.. versionadded:: 0.13
Parameters
----------
n_components : int | None
Dimensionality of latent space, the number of components
of ``X`` that are obtained after ``transform``.
If None, n_components is set to the number of features.
tol : float
Stopping tolerance for log-likelihood increase.
copy : bool
Whether to make a copy of X. If ``False``, the input X gets overwritten
during fitting.
max_iter : int
Maximum number of iterations.
noise_variance_init : None | array, shape=(n_features,)
The initial guess of the noise variance for each feature.
If None, it defaults to np.ones(n_features)
svd_method : {'lapack', 'randomized'}
Which SVD method to use. If 'lapack' use standard SVD from
scipy.linalg, if 'randomized' use fast ``randomized_svd`` function.
Defaults to 'randomized'. For most applications 'randomized' will
be sufficiently precise while providing significant speed gains.
Accuracy can also be improved by setting higher values for
`iterated_power`. If this is not sufficient, for maximum precision
you should choose 'lapack'.
iterated_power : int, optional
Number of iterations for the power method. 3 by default. Only used
if ``svd_method`` equals 'randomized'
random_state : int, RandomState instance, default=0
Only used when ``svd_method`` equals 'randomized'. Pass an int for
reproducible results across multiple function calls.
See :term:`Glossary <random_state>`.
Attributes
----------
components_ : array, [n_components, n_features]
Components with maximum variance.
loglike_ : list, [n_iterations]
The log likelihood at each iteration.
noise_variance_ : array, shape=(n_features,)
The estimated noise variance for each feature.
n_iter_ : int
Number of iterations run.
mean_ : array, shape (n_features,)
Per-feature empirical mean, estimated from the training set.
Examples
--------
>>> from sklearn.datasets import load_digits
>>> from sklearn.decomposition import FactorAnalysis
>>> X, _ = load_digits(return_X_y=True)
>>> transformer = FactorAnalysis(n_components=7, random_state=0)
>>> X_transformed = transformer.fit_transform(X)
>>> X_transformed.shape
(1797, 7)
References
----------
.. David Barber, Bayesian Reasoning and Machine Learning,
Algorithm 21.1
.. Christopher M. Bishop: Pattern Recognition and Machine Learning,
Chapter 12.2.4
See also
--------
PCA: Principal component analysis is also a latent linear variable model
which however assumes equal noise variance for each feature.
This extra assumption makes probabilistic PCA faster as it can be
computed in closed form.
FastICA: Independent component analysis, a latent variable model with
non-Gaussian latent variables.
"""
@_deprecate_positional_args
def __init__(self, n_components=None, *, tol=1e-2, copy=True,
max_iter=1000,
noise_variance_init=None, svd_method='randomized',
iterated_power=3, random_state=0):
self.n_components = n_components
self.copy = copy
self.tol = tol
self.max_iter = max_iter
if svd_method not in ['lapack', 'randomized']:
raise ValueError('SVD method %s is not supported. Please consider'
' the documentation' % svd_method)
self.svd_method = svd_method
self.noise_variance_init = noise_variance_init
self.iterated_power = iterated_power
self.random_state = random_state
def fit(self, X, y=None):
"""Fit the FactorAnalysis model to X using SVD based approach
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data.
y : Ignored
Returns
-------
self
"""
X = self._validate_data(X, copy=self.copy, dtype=np.float64)
n_samples, n_features = X.shape
n_components = self.n_components
if n_components is None:
n_components = n_features
self.mean_ = np.mean(X, axis=0)
X -= self.mean_
# some constant terms
nsqrt = sqrt(n_samples)
llconst = n_features * log(2. * np.pi) + n_components
var = np.var(X, axis=0)
if self.noise_variance_init is None:
psi = np.ones(n_features, dtype=X.dtype)
else:
if len(self.noise_variance_init) != n_features:
raise ValueError("noise_variance_init dimension does not "
"with number of features : %d != %d" %
(len(self.noise_variance_init), n_features))
psi = np.array(self.noise_variance_init)
loglike = []
old_ll = -np.inf
SMALL = 1e-12
# we'll modify svd outputs to return unexplained variance
# to allow for unified computation of loglikelihood
if self.svd_method == 'lapack':
def my_svd(X):
_, s, V = linalg.svd(X, full_matrices=False)
return (s[:n_components], V[:n_components],
squared_norm(s[n_components:]))
elif self.svd_method == 'randomized':
random_state = check_random_state(self.random_state)
def my_svd(X):
_, s, V = randomized_svd(X, n_components,
random_state=random_state,
n_iter=self.iterated_power)
return s, V, squared_norm(X) - squared_norm(s)
else:
raise ValueError('SVD method %s is not supported. Please consider'
' the documentation' % self.svd_method)
for i in range(self.max_iter):
# SMALL helps numerics
sqrt_psi = np.sqrt(psi) + SMALL
s, V, unexp_var = my_svd(X / (sqrt_psi * nsqrt))
s **= 2
# Use 'maximum' here to avoid sqrt problems.
W = np.sqrt(np.maximum(s - 1., 0.))[:, np.newaxis] * V
del V
W *= sqrt_psi
# loglikelihood
ll = llconst + np.sum(np.log(s))
ll += unexp_var + np.sum(np.log(psi))
ll *= -n_samples / 2.
loglike.append(ll)
if (ll - old_ll) < self.tol:
break
old_ll = ll
psi = np.maximum(var - np.sum(W ** 2, axis=0), SMALL)
else:
warnings.warn('FactorAnalysis did not converge.' +
' You might want' +
' to increase the number of iterations.',
ConvergenceWarning)
self.components_ = W
self.noise_variance_ = psi
self.loglike_ = loglike
self.n_iter_ = i + 1
return self
def transform(self, X):
"""Apply dimensionality reduction to X using the model.
Compute the expected mean of the latent variables.
See Barber, 21.2.33 (or Bishop, 12.66).
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
The latent variables of X.
"""
check_is_fitted(self)
X = check_array(X)
Ih = np.eye(len(self.components_))
X_transformed = X - self.mean_
Wpsi = self.components_ / self.noise_variance_
cov_z = linalg.inv(Ih + np.dot(Wpsi, self.components_.T))
tmp = np.dot(X_transformed, Wpsi.T)
X_transformed = np.dot(tmp, cov_z)
return X_transformed
def get_covariance(self):
"""Compute data covariance with the FactorAnalysis model.
``cov = components_.T * components_ + diag(noise_variance)``
Returns
-------
cov : array, shape (n_features, n_features)
Estimated covariance of data.
"""
check_is_fitted(self)
cov = np.dot(self.components_.T, self.components_)
cov.flat[::len(cov) + 1] += self.noise_variance_ # modify diag inplace
return cov
def get_precision(self):
"""Compute data precision matrix with the FactorAnalysis model.
Returns
-------
precision : array, shape (n_features, n_features)
Estimated precision of data.
"""
check_is_fitted(self)
n_features = self.components_.shape[1]
# handle corner cases first
if self.n_components == 0:
return np.diag(1. / self.noise_variance_)
if self.n_components == n_features:
return linalg.inv(self.get_covariance())
# Get precision using matrix inversion lemma
components_ = self.components_
precision = np.dot(components_ / self.noise_variance_, components_.T)
precision.flat[::len(precision) + 1] += 1.
precision = np.dot(components_.T,
np.dot(linalg.inv(precision), components_))
precision /= self.noise_variance_[:, np.newaxis]
precision /= -self.noise_variance_[np.newaxis, :]
precision.flat[::len(precision) + 1] += 1. / self.noise_variance_
return precision
def score_samples(self, X):
"""Compute the log-likelihood of each sample
Parameters
----------
X : array, shape (n_samples, n_features)
The data
Returns
-------
ll : array, shape (n_samples,)
Log-likelihood of each sample under the current model
"""
check_is_fitted(self)
Xr = X - self.mean_
precision = self.get_precision()
n_features = X.shape[1]
log_like = -.5 * (Xr * (np.dot(Xr, precision))).sum(axis=1)
log_like -= .5 * (n_features * log(2. * np.pi)
- fast_logdet(precision))
return log_like
def score(self, X, y=None):
"""Compute the average log-likelihood of the samples
Parameters
----------
X : array, shape (n_samples, n_features)
The data
y : Ignored
Returns
-------
ll : float
Average log-likelihood of the samples under the current model
"""
return np.mean(self.score_samples(X))