FactorAnalysis#

classsklearn.decomposition.FactorAnalysis(n_components=None,*,tol=0.01,copy=True,max_iter=1000,noise_variance_init=None,svd_method='randomized',iterated_power=3,rotation=None,random_state=0)[source]#

Factor Analysis (FA).

A simple linear generative model with Gaussian latent variables.

The observations are assumed to be caused by a linear transformation oflower dimensional latent factors and added Gaussian noise.Without loss of generality the factors are distributed according to aGaussian with zero mean and unit covariance. The noise is also zero meanand has an arbitrary diagonal covariance matrix.

If we would restrict the model further, by assuming that the Gaussiannoise is even isotropic (all diagonal entries are the same) we would obtainPCA.

FactorAnalysis performs a maximum likelihood estimate of the so-calledloading matrix, the transformation of the latent variables to theobserved ones, using SVD based approach.

Read more in theUser Guide.

Added in version 0.13.

Parameters:

n_componentsint, default=None

Dimensionality of latent space, the number of componentsofX that are obtained aftertransform.If None, n_components is set to the number of features.

tolfloat, default=1e-2

Stopping tolerance for log-likelihood increase.

copybool, default=True

Whether to make a copy of X. IfFalse, the input X gets overwrittenduring fitting.

max_iterint, default=1000

Maximum number of iterations.

noise_variance_initarray-like of shape (n_features,), default=None

The initial guess of the noise variance for each feature.If None, it defaults to np.ones(n_features).

svd_method{‘lapack’, ‘randomized’}, default=’randomized’

Which SVD method to use. If ‘lapack’ use standard SVD fromscipy.linalg, if ‘randomized’ use fastrandomized_svd function.Defaults to ‘randomized’. For most applications ‘randomized’ willbe sufficiently precise while providing significant speed gains.Accuracy can also be improved by setting higher values foriterated_power. If this is not sufficient, for maximum precisionyou should choose ‘lapack’.

iterated_powerint, default=3

Number of iterations for the power method. 3 by default. Only usedifsvd_method equals ‘randomized’.

rotation{‘varimax’, ‘quartimax’}, default=None

If not None, apply the indicated rotation. Currently, varimax andquartimax are implemented. See“The varimax criterion for analytic rotation in factor analysis”H. F. Kaiser, 1958.

Added in version 0.24.

random_stateint or RandomState instance, default=0

Only used whensvd_method equals ‘randomized’. Pass an int forreproducible results across multiple function calls.SeeGlossary.

Attributes:

components_ndarray of shape (n_components, n_features)

Components with maximum variance.

loglike_list of shape (n_iterations,)

The log likelihood at each iteration.

noise_variance_ndarray of shape (n_features,)

The estimated noise variance for each feature.

n_iter_int

Number of iterations run.

mean_ndarray of shape (n_features,)

Per-feature empirical mean, estimated from the training set.

n_features_in_int

Number of features seen duringfit.

Added in version 0.24.

feature_names_in_ndarray of shape (n_features_in_,)

Names of features seen duringfit. Defined only whenXhas feature names that are all strings.

Added in version 1.0.

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.

References

• David Barber, Bayesian Reasoning and Machine Learning,Algorithm 21.1.

• Christopher M. Bishop: Pattern Recognition and Machine Learning,Chapter 12.2.4.

Examples

>>>fromsklearn.datasetsimportload_digits>>>fromsklearn.decompositionimportFactorAnalysis>>>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)

fit(X,y=None)[source]#

Fit the FactorAnalysis model to X using SVD based approach.

Parameters:

Xarray-like of shape (n_samples, n_features)

Training data.

yIgnored

Ignored parameter.

Returns:

selfobject

FactorAnalysis class instance.

fit_transform(X,y=None,**fit_params)[source]#

Fit to data, then transform it.

Fits transformer toX andy with optional parametersfit_paramsand returns a transformed version ofX.

Parameters:

Xarray-like of shape (n_samples, n_features)

Input samples.

yarray-like of shape (n_samples,) or (n_samples, n_outputs), default=None

Target values (None for unsupervised transformations).

**fit_paramsdict

Additional fit parameters.

Returns:

X_newndarray array of shape (n_samples, n_features_new)

Transformed array.

get_covariance()[source]#

Compute data covariance with the FactorAnalysis model.

cov=components_.T*components_+diag(noise_variance)

Returns:

covndarray of shape (n_features, n_features)

Estimated covariance of data.

get_feature_names_out(input_features=None)[source]#

Get output feature names for transformation.

The feature names out will prefixed by the lowercased class name. Forexample, if the transformer outputs 3 features, then the feature namesout are:["class_name0","class_name1","class_name2"].

Parameters:

input_featuresarray-like of str or None, default=None

Only used to validate feature names with the names seen infit.

Returns:

feature_names_outndarray of str objects

Transformed feature names.

get_metadata_routing()[source]#

Get metadata routing of this object.

Please checkUser Guide on how the routingmechanism works.

Returns:

routingMetadataRequest

AMetadataRequest encapsulatingrouting information.

get_params(deep=True)[source]#

Get parameters for this estimator.

Parameters:

deepbool, default=True

If True, will return the parameters for this estimator andcontained subobjects that are estimators.

Returns:

paramsdict

Parameter names mapped to their values.

get_precision()[source]#

Compute data precision matrix with the FactorAnalysis model.

Returns:

precisionndarray of shape (n_features, n_features)

Estimated precision of data.

score(X,y=None)[source]#

Compute the average log-likelihood of the samples.

Parameters:

Xndarray of shape (n_samples, n_features)

The data.

yIgnored

Ignored parameter.

Returns:

llfloat

Average log-likelihood of the samples under the current model.

score_samples(X)[source]#

Compute the log-likelihood of each sample.

Parameters:

Xndarray of shape (n_samples, n_features)

The data.

Returns:

llndarray of shape (n_samples,)

Log-likelihood of each sample under the current model.

set_output(*,transform=None)[source]#

Set output container.

SeeIntroducing the set_output APIfor an example on how to use the API.

Parameters:

transform{“default”, “pandas”, “polars”}, default=None

Configure output oftransform andfit_transform.

• "default": Default output format of a transformer

• "pandas": DataFrame output

• "polars": Polars output

• None: Transform configuration is unchanged

Added in version 1.4:"polars" option was added.

Returns:

selfestimator instance

Estimator instance.

set_params(**params)[source]#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects(such asPipeline). The latter haveparameters of the form<component>__<parameter> so that it’spossible to update each component of a nested object.

Parameters:

**paramsdict

Estimator parameters.

Returns:

selfestimator instance

Estimator instance.

transform(X)[source]#

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:

Xarray-like of shape (n_samples, n_features)

Training data.

Returns:

X_newndarray of shape (n_samples, n_components)

The latent variables of X.

Gallery examples#

Faces dataset decompositions

Faces dataset decompositions

Model selection with Probabilistic PCA and Factor Analysis (FA)

Model selection with Probabilistic PCA and Factor Analysis (FA)

Factor Analysis (with rotation) to visualize patterns

Factor Analysis (with rotation) to visualize patterns