Implementation:Scikit learn Scikit learn FactorAnalysis

Knowledge Sources	Scikit_learn Scikit-learn Docs
Domains	Dimensionality Reduction, Latent Variable Models
Last Updated	2026-02-08 15:00 GMT

Overview

Concrete tool for factor analysis with heteroscedastic noise modeling provided by scikit-learn.

Description

FactorAnalysis is 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 with added Gaussian noise, where each feature has its own noise variance. This distinguishes it from probabilistic PCA, which assumes isotropic noise. FactorAnalysis performs maximum likelihood estimation of the loading matrix using an SVD-based approach.

Usage

Use FactorAnalysis when you need a latent variable model that accounts for different noise variances across features. It is well suited for exploratory factor analysis in social sciences, psychology, and any domain where you want to uncover latent structure while modeling feature-specific noise.

Code Reference

Source Location

• Repository: scikit-learn
• File: sklearn/decomposition/_factor_analysis.py

Signature

class FactorAnalysis(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator):
    def __init__(
        self,
        n_components=None,
        *,
        tol=1e-2,
        copy=True,
        max_iter=1000,
        noise_variance_init=None,
        svd_method="randomized",
        iterated_power=3,
        rotation=None,
        random_state=0,
    ):

Import

from sklearn.decomposition import FactorAnalysis

I/O Contract

Inputs

Name	Type	Required	Description
n_components	int	No	Dimensionality of latent space. Defaults to n_features if None.
tol	float	No	Stopping tolerance for log-likelihood increase (default=1e-2).
copy	bool	No	Whether to make a copy of X (default=True).
max_iter	int	No	Maximum number of iterations (default=1000).
noise_variance_init	array-like	No	Initial guess of noise variance for each feature.
svd_method	str	No	SVD method: 'lapack' or 'randomized' (default='randomized').
iterated_power	int	No	Number of iterations for power method in randomized SVD (default=3).
rotation	str	No	Factor rotation method: None, 'varimax', or 'quartimax'.
random_state	int or RandomState	No	Random state for reproducibility (default=0).

Outputs

Name	Type	Description
components_	ndarray of shape (n_components, n_features)	The loading matrix representing the latent factors.
loglike_	list of float	Log-likelihood at each iteration.
noise_variance_	ndarray of shape (n_features,)	Estimated noise variance for each feature.
n_iter_	int	Number of iterations run.
mean_	ndarray of shape (n_features,)	Per-feature empirical mean.

Usage Examples

Basic Usage

import numpy as np
from sklearn.decomposition import FactorAnalysis

X = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]], dtype=float)
fa = FactorAnalysis(n_components=2, random_state=0)
X_transformed = fa.fit_transform(X)
print(X_transformed.shape)  # (4, 2)
print(fa.components_.shape)  # (2, 3)
print(fa.noise_variance_)

Related Pages

• Principle:Scikit_learn_Scikit_learn_Dimensionality_Reduction