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# `sklearn.decomposition`.FactorAnalysis

class sklearn.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)

  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 `PCA`.

  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 User Guide.

  New in version 0.13.

  Parameters:
    n_components int, default=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, default=1e-2

      Stopping tolerance for log-likelihood increase.

    copy bool, default=True

      Whether to make a copy of X. If `False`, the input X gets overwritten during fitting.

    max_iter int, default=1000

      Maximum number of iterations.

    noise_variance_init array-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 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, default=3

      Number of iterations for the power method. 3 by default. Only used if `svd_method` equals ‘randomized’.

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

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

      New in version 0.24.

    random_state int or RandomState instance, default=0

      Only used when `svd_method` equals ‘randomized’. Pass an int for reproducible results across multiple function calls. See Glossary.

  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 during fit.

      New in version 0.24.

    feature_names_in_ndarray of shape (`n_features_in_`,)

      Names of features seen during fit. Defined only when `X` has feature names that are all strings.

      New 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

      >>> 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)

  Methods

  `fit`(X[, y]) | Fit the FactorAnalysis model to X using SVD based approach.
  `fit_transform`(X[, y]) | Fit to data, then transform it.
  `get_covariance`() | Compute data covariance with the FactorAnalysis model.
  `get_feature_names_out`([input_features]) | Get output feature names for transformation.
  `get_metadata_routing`() | Get metadata routing of this object.
  `get_params`([deep]) | Get parameters for this estimator.
  `get_precision`() | Compute data precision matrix with the FactorAnalysis model.
  `score`(X[, y]) | Compute the average log-likelihood of the samples.
  `score_samples`(X) | Compute the log-likelihood of each sample.
  `set_output`(*[, transform]) | Set output container.
  `set_params`(**params) | Set the parameters of this estimator.
  `transform`(X) | Apply dimensionality reduction to X using the model.

  fit(X, y=None)

    Fit the FactorAnalysis model to X using SVD based approach.

    Parameters:
      X array-like of shape (n_samples, n_features)

        Training data.

      y Ignored

        Ignored parameter.

    Returns:
      self object

        FactorAnalysis class instance.

  fit_transform(X, y=None, **fit_params)

    Fit to data, then transform it.

    Fits transformer to `X` and `y` with optional parameters `fit_params` and returns a transformed version of `X`.

    Parameters:
      X array-like of shape (n_samples, n_features)

        Input samples.

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

        Target values (None for unsupervised transformations).

      **fit_params dict

        Additional fit parameters.

    Returns:
      X_new ndarray array of shape (n_samples, n_features_new)

        Transformed array.

  get_covariance()

    Compute data covariance with the FactorAnalysis model.

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

    Returns:
      cov ndarray of shape (n_features, n_features)

        Estimated covariance of data.

  get_feature_names_out(input_features=None)

    Get output feature names for transformation.

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

    Parameters:
      input_features array-like of str or None, default=None

        Only used to validate feature names with the names seen in `fit`.

    Returns:
      feature_names_out ndarray of str objects

        Transformed feature names.

  get_metadata_routing()

    Get metadata routing of this object.

    Please check User Guide on how the routing mechanism works.

    Returns:
      routing MetadataRequest

        A `MetadataRequest` encapsulating routing information.

  get_params(deep=True)

    Get parameters for this estimator.

    Parameters:
      deep bool, default=True

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

    Returns:
      params dict

        Parameter names mapped to their values.

  get_precision()

    Compute data precision matrix with the FactorAnalysis model.

    Returns:
      precision ndarray of shape (n_features, n_features)

        Estimated precision of data.

  score(X, y=None)

    Compute the average log-likelihood of the samples.

    Parameters:
      X ndarray of shape (n_samples, n_features)

        The data.

      y Ignored

        Ignored parameter.

    Returns:
      ll float

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

  score_samples(X)

    Compute the log-likelihood of each sample.

    Parameters:
      X ndarray of shape (n_samples, n_features)

        The data.

    Returns:
      ll ndarray of shape (n_samples,)

        Log-likelihood of each sample under the current model.

  set_output(*, transform=None)

    Set output container.

    See Introducing the set_output API for an example on how to use the API.

    Parameters:
      transform{“default”, “pandas”}, default=None

        Configure output of `transform` and `fit_transform`.

        - `"default"`: Default output format of a transformer

        - `"pandas"`: DataFrame output

        - `None`: Transform configuration is unchanged

    Returns:
      self estimator instance

        Estimator instance.

  set_params(**params)

    Set the parameters of this estimator.

    The method works on simple estimators as well as on nested objects (such as `Pipeline`). The latter have parameters of the form `<component>__<parameter>` so that it’s possible to update each component of a nested object.

    Parameters:
      **params dict

        Estimator parameters.

    Returns:
      self estimator instance

        Estimator instance.

  transform(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 of shape (n_samples, n_features)

        Training data.

    Returns:
      X_new ndarray of shape (n_samples, n_components)

        The latent variables of X.

## Examples using `sklearn.decomposition.FactorAnalysis`

Faces dataset decompositions

Faces dataset decompositions

Factor Analysis (with rotation) to visualize patterns

Factor Analysis (with rotation) to visualize patterns

Model selection with Probabilistic PCA and Factor Analysis (FA)

Model selection with Probabilistic PCA and Factor Analysis (FA)