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 or None, optional (default=0) If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. Only used when ``svd_method`` equals 'randomized'.
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.