Covariance

EllipticEnvelope¶

Module Sklearn.Covariance.EllipticEnvelope wraps Python class sklearn.covariance.EllipticEnvelope.

type t

create¶

constructor and attributes create

val create :
  ?store_precision:bool ->
  ?assume_centered:bool ->
  ?support_fraction:float ->
  ?contamination:float ->
  ?random_state:int ->
  unit ->
  t

An object for detecting outliers in a Gaussian distributed dataset.

Read more in the :ref:User Guide <outlier_detection>.

Parameters

store_precision : bool, default=True Specify if the estimated precision is stored.
assume_centered : bool, default=False If True, the support of robust location and covariance estimates is computed, and a covariance estimate is recomputed from it, without centering the data. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, the robust location and covariance are directly computed with the FastMCD algorithm without additional treatment.
support_fraction : float, default=None The proportion of points to be included in the support of the raw MCD estimate. If None, the minimum value of support_fraction will be used within the algorithm: [n_sample + n_features + 1] / 2. Range is (0, 1).
contamination : float, default=0.1 The amount of contamination of the data set, i.e. the proportion of outliers in the data set. Range is (0, 0.5).
random_state : int or RandomState instance, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls. See :term: Glossary <random_state>.

Attributes

location_ : ndarray of shape (n_features,) Estimated robust location
covariance_ : ndarray of shape (n_features, n_features) Estimated robust covariance matrix
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True)
support_ : ndarray of shape (n_samples,) A mask of the observations that have been used to compute the robust estimates of location and shape.
offset_ : float Offset used to define the decision function from the raw scores. We have the relation: decision_function = score_samples - offset_. The offset depends on the contamination parameter and is defined in such a way we obtain the expected number of outliers (samples with decision function < 0) in training.

.. versionadded:: 0.20
raw_location_ : ndarray of shape (n_features,) The raw robust estimated location before correction and re-weighting.
raw_covariance_ : ndarray of shape (n_features, n_features) The raw robust estimated covariance before correction and re-weighting.
raw_support_ : ndarray of shape (n_samples,) A mask of the observations that have been used to compute the raw robust estimates of location and shape, before correction and re-weighting.
dist_ : ndarray of shape (n_samples,) Mahalanobis distances of the training set (on which :meth:fit is called) observations.

Examples

>>> import numpy as np
>>> from sklearn.covariance import EllipticEnvelope
>>> true_cov = np.array([[.8, .3],
...                      [.3, .4]])
>>> X = np.random.RandomState(0).multivariate_normal(mean=[0, 0],
...                                                  cov=true_cov,
...                                                  size=500)
>>> cov = EllipticEnvelope(random_state=0).fit(X)
>>> # predict returns 1 for an inlier and -1 for an outlier
>>> cov.predict([[0, 0],
...              [3, 3]])
array([ 1, -1])
>>> cov.covariance_
array([[0.7411..., 0.2535...],
       [0.2535..., 0.3053...]])
>>> cov.location_
array([0.0813... , 0.0427...])

Notes

Outlier detection from covariance estimation may break or not perform well in high-dimensional settings. In particular, one will always take care to work with n_samples > n_features ** 2.

References

.. [1] Rousseeuw, P.J., Van Driessen, K. 'A fast algorithm for the minimum covariance determinant estimator' Technometrics 41(3), 212 (1999)

correct_covariance¶

method correct_covariance

val correct_covariance :
  data:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Apply a correction to raw Minimum Covariance Determinant estimates.

Correction using the empirical correction factor suggested by Rousseeuw and Van Driessen in [RVD]_.

Parameters

data : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. The data set must be the one which was used to compute the raw estimates.

Returns

covariance_corrected : ndarray of shape (n_features, n_features) Corrected robust covariance estimate.

References

.. [RVD] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS

decision_function¶

method decision_function

val decision_function :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Compute the decision function of the given observations.

Parameters

X : array-like of shape (n_samples, n_features) The data matrix.

Returns

decision : ndarray of shape (n_samples, ) Decision function of the samples. It is equal to the shifted Mahalanobis distances. The threshold for being an outlier is 0, which ensures a compatibility with other outlier detection algorithms.

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fit the EllipticEnvelope model.

Parameters

X : {array-like, sparse matrix} of shape (n_samples, n_features) Training data.
y : Ignored Not used, present for API consistency by convention.

fit_predict¶

method fit_predict

val fit_predict :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Perform fit on X and returns labels for X.

Returns -1 for outliers and 1 for inliers.

Parameters

X : {array-like, sparse matrix, dataframe} of shape (n_samples, n_features)
y : Ignored Not used, present for API consistency by convention.

Returns

y : ndarray of shape (n_samples,) 1 for inliers, -1 for outliers.

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

predict¶

method predict

val predict :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Predict the labels (1 inlier, -1 outlier) of X according to the fitted model.

Parameters

X : array-like of shape (n_samples, n_features) The data matrix.

Returns

is_inlier : ndarray of shape (n_samples,) Returns -1 for anomalies/outliers and +1 for inliers.

reweight_covariance¶

method reweight_covariance

val reweight_covariance :
  data:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  ([>`ArrayLike] Np.Obj.t * [>`ArrayLike] Np.Obj.t * Py.Object.t)

Re-weight raw Minimum Covariance Determinant estimates.

Re-weight observations using Rousseeuw's method (equivalent to deleting outlying observations from the data set before computing location and covariance estimates) described in [RVDriessen]_.

Parameters

data : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. The data set must be the one which was used to compute the raw estimates.

Returns

location_reweighted : ndarray of shape (n_features,) Re-weighted robust location estimate.
covariance_reweighted : ndarray of shape (n_features, n_features) Re-weighted robust covariance estimate.
support_reweighted : ndarray of shape (n_samples,), dtype=bool A mask of the observations that have been used to compute the re-weighted robust location and covariance estimates.

References

.. [RVDriessen] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS

score¶

method score

val score :
  ?sample_weight:[>`ArrayLike] Np.Obj.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  y:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Returns the mean accuracy on the given test data and labels.

In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted.

Parameters

X : array-like of shape (n_samples, n_features) Test samples.
y : array-like of shape (n_samples,) or (n_samples, n_outputs) True labels for X.
sample_weight : array-like of shape (n_samples,), default=None Sample weights.

Returns

score : float Mean accuracy of self.predict(X) w.r.t. y.

score_samples¶

method score_samples

val score_samples :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Compute the negative Mahalanobis distances.

Parameters

X : array-like of shape (n_samples, n_features) The data matrix.

Returns

negative_mahal_distances : array-like of shape (n_samples,) Opposite of the Mahalanobis distances.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

support_¶

attribute support_

val support_ : t -> [>`ArrayLike] Np.Obj.t
val support_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

offset_¶

attribute offset_

val offset_ : t -> float
val offset_opt : t -> (float) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

raw_location_¶

attribute raw_location_

val raw_location_ : t -> [>`ArrayLike] Np.Obj.t
val raw_location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

raw_covariance_¶

attribute raw_covariance_

val raw_covariance_ : t -> [>`ArrayLike] Np.Obj.t
val raw_covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

raw_support_¶

attribute raw_support_

val raw_support_ : t -> [>`ArrayLike] Np.Obj.t
val raw_support_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

dist_¶

attribute dist_

val dist_ : t -> [>`ArrayLike] Np.Obj.t
val dist_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

EmpiricalCovariance¶

Module Sklearn.Covariance.EmpiricalCovariance wraps Python class sklearn.covariance.EmpiricalCovariance.

type t

create¶

constructor and attributes create

val create :
  ?store_precision:bool ->
  ?assume_centered:bool ->
  unit ->
  t

Maximum likelihood covariance estimator

Read more in the :ref:User Guide <covariance>.

Parameters

store_precision : bool, default=True Specifies if the estimated precision is stored.
assume_centered : bool, default=False If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False (default), data are centered before computation.

Attributes

location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean.
covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo-inverse matrix. (stored only if store_precision is True)

Examples

>>> import numpy as np
>>> from sklearn.covariance import EmpiricalCovariance
>>> from sklearn.datasets import make_gaussian_quantiles
>>> real_cov = np.array([[.8, .3],
...                      [.3, .4]])
>>> rng = np.random.RandomState(0)
>>> X = rng.multivariate_normal(mean=[0, 0],
...                             cov=real_cov,
...                             size=500)
>>> cov = EmpiricalCovariance().fit(X)
>>> cov.covariance_
array([[0.7569..., 0.2818...],
       [0.2818..., 0.3928...]])
>>> cov.location_
array([0.0622..., 0.0193...])

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fits the Maximum Likelihood Estimator covariance model according to the given training data and parameters.

Parameters

X : array-like of shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features.
y : Ignored Not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

GraphicalLasso¶

Module Sklearn.Covariance.GraphicalLasso wraps Python class sklearn.covariance.GraphicalLasso.

type t

create¶

constructor and attributes create

val create :
  ?alpha:float ->
  ?mode:[`Cd | `Lars] ->
  ?tol:float ->
  ?enet_tol:float ->
  ?max_iter:int ->
  ?verbose:int ->
  ?assume_centered:bool ->
  unit ->
  t

Sparse inverse covariance estimation with an l1-penalized estimator.

Read more in the :ref:User Guide <sparse_inverse_covariance>.

.. versionchanged:: v0.20 GraphLasso has been renamed to GraphicalLasso

Parameters

alpha : float, default=0.01 The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf].
mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where p > n. Elsewhere prefer cd which is more numerically stable.
tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf].
enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. Range is (0, inf].
max_iter : int, default=100 The maximum number of iterations.
verbose : bool, default=False If verbose is True, the objective function and dual gap are plotted at each iteration.
assume_centered : bool, default=False If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation.

Attributes

location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean.
covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix.
n_iter_ : int Number of iterations run.

Examples

>>> import numpy as np
>>> from sklearn.covariance import GraphicalLasso
>>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0],
...                      [0.0, 0.4, 0.0, 0.0],
...                      [0.2, 0.0, 0.3, 0.1],
...                      [0.0, 0.0, 0.1, 0.7]])
>>> np.random.seed(0)
>>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0],
...                                   cov=true_cov,
...                                   size=200)
>>> cov = GraphicalLasso().fit(X)
>>> np.around(cov.covariance_, decimals=3)
array([[0.816, 0.049, 0.218, 0.019],
       [0.049, 0.364, 0.017, 0.034],
       [0.218, 0.017, 0.322, 0.093],
       [0.019, 0.034, 0.093, 0.69 ]])
>>> np.around(cov.location_, decimals=3)
array([0.073, 0.04 , 0.038, 0.143])

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fits the GraphicalLasso model to X.

Parameters

X : array-like of shape (n_samples, n_features) Data from which to compute the covariance estimate
y : Ignored Not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

n_iter_¶

attribute n_iter_

val n_iter_ : t -> int
val n_iter_opt : t -> (int) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

GraphicalLassoCV¶

Module Sklearn.Covariance.GraphicalLassoCV wraps Python class sklearn.covariance.GraphicalLassoCV.

type t

create¶

constructor and attributes create

val create :
  ?alphas:Py.Object.t ->
  ?n_refinements:int ->
  ?cv:[`BaseCrossValidator of [>`BaseCrossValidator] Np.Obj.t | `I of int | `Arr of [>`ArrayLike] Np.Obj.t] ->
  ?tol:float ->
  ?enet_tol:float ->
  ?max_iter:int ->
  ?mode:[`Cd | `Lars] ->
  ?n_jobs:int ->
  ?verbose:int ->
  ?assume_centered:bool ->
  unit ->
  t

Sparse inverse covariance w/ cross-validated choice of the l1 penalty.

See glossary entry for :term:cross-validation estimator.

Read more in the :ref:User Guide <sparse_inverse_covariance>.

.. versionchanged:: v0.20 GraphLassoCV has been renamed to GraphicalLassoCV

Parameters

alphas : int or array-like of shape (n_alphas,), dtype=float, default=4 If an integer is given, it fixes the number of points on the grids of alpha to be used. If a list is given, it gives the grid to be used. See the notes in the class docstring for more details. Range is (0, inf] when floats given.
n_refinements : int, default=4 The number of times the grid is refined. Not used if explicit values of alphas are passed. Range is [1, inf).
cv : int, cross-validation generator or iterable, default=None Determines the cross-validation splitting strategy. Possible inputs for cv are:
- None, to use the default 5-fold cross-validation,
- integer, to specify the number of folds.
- :term:CV splitter,
- An iterable yielding (train, test) splits as arrays of indices.
For integer/None inputs :class:KFold is used.
Refer :ref:User Guide <cross_validation> for the various cross-validation strategies that can be used here.

.. versionchanged:: 0.20 cv default value if None changed from 3-fold to 5-fold.
tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf].
enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. Range is (0, inf].
max_iter : int, default=100 Maximum number of iterations.
mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where number of features is greater than number of samples. Elsewhere prefer cd which is more numerically stable.
n_jobs : int, default=None number of jobs to run in parallel. None means 1 unless in a :obj:joblib.parallel_backend context. -1 means using all processors. See :term:Glossary <n_jobs> for more details.

.. versionchanged:: v0.20 n_jobs default changed from 1 to None
verbose : bool, default=False If verbose is True, the objective function and duality gap are printed at each iteration.
assume_centered : bool, default=False If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation.

Attributes

location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean.
covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix.
precision_ : ndarray of shape (n_features, n_features) Estimated precision matrix (inverse covariance).
alpha_ : float Penalization parameter selected.
cv_alphas_ : list of shape (n_alphas,), dtype=float All penalization parameters explored.
grid_scores_ : ndarray of shape (n_alphas, n_folds) Log-likelihood score on left-out data across folds.
n_iter_ : int Number of iterations run for the optimal alpha.

Examples

>>> import numpy as np
>>> from sklearn.covariance import GraphicalLassoCV
>>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0],
...                      [0.0, 0.4, 0.0, 0.0],
...                      [0.2, 0.0, 0.3, 0.1],
...                      [0.0, 0.0, 0.1, 0.7]])
>>> np.random.seed(0)
>>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0],
...                                   cov=true_cov,
...                                   size=200)
>>> cov = GraphicalLassoCV().fit(X)
>>> np.around(cov.covariance_, decimals=3)
array([[0.816, 0.051, 0.22 , 0.017],
       [0.051, 0.364, 0.018, 0.036],
       [0.22 , 0.018, 0.322, 0.094],
       [0.017, 0.036, 0.094, 0.69 ]])
>>> np.around(cov.location_, decimals=3)
array([0.073, 0.04 , 0.038, 0.143])

Notes

The search for the optimal penalization parameter (alpha) is done on an iteratively refined grid: first the cross-validated scores on a grid are computed, then a new refined grid is centered around the maximum, and so on.

One of the challenges which is faced here is that the solvers can fail to converge to a well-conditioned estimate. The corresponding values of alpha then come out as missing values, but the optimum may be close to these missing values.

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fits the GraphicalLasso covariance model to X.

Parameters

X : array-like of shape (n_samples, n_features) Data from which to compute the covariance estimate
y : Ignored Not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

alpha_¶

attribute alpha_

val alpha_ : t -> float
val alpha_opt : t -> (float) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

cv_alphas_¶

attribute cv_alphas_

val cv_alphas_ : t -> Py.Object.t
val cv_alphas_opt : t -> (Py.Object.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

grid_scores_¶

attribute grid_scores_

val grid_scores_ : t -> [>`ArrayLike] Np.Obj.t
val grid_scores_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

n_iter_¶

attribute n_iter_

val n_iter_ : t -> int
val n_iter_opt : t -> (int) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

LedoitWolf¶

Module Sklearn.Covariance.LedoitWolf wraps Python class sklearn.covariance.LedoitWolf.

type t

create¶

constructor and attributes create

val create :
  ?store_precision:bool ->
  ?assume_centered:bool ->
  ?block_size:int ->
  unit ->
  t

LedoitWolf Estimator

Ledoit-Wolf is a particular form of shrinkage, where the shrinkage coefficient is computed using O. Ledoit and M. Wolf's formula as described in 'A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices', Ledoit and Wolf, Journal of Multivariate Analysis, Volume 88, Issue 2, February 2004, pages 365-411.

Read more in the :ref:User Guide <shrunk_covariance>.

Parameters

store_precision : bool, default=True Specify if the estimated precision is stored.
assume_centered : bool, default=False If True, data will not be centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False (default), data will be centered before computation.
block_size : int, default=1000 Size of the blocks into which the covariance matrix will be split during its Ledoit-Wolf estimation. This is purely a memory optimization and does not affect results.

Attributes

covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix.
location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean.
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True)
shrinkage_ : float Coefficient in the convex combination used for the computation of the shrunk estimate. Range is [0, 1].

Examples

>>> import numpy as np
>>> from sklearn.covariance import LedoitWolf
>>> real_cov = np.array([[.4, .2],
...                      [.2, .8]])
>>> np.random.seed(0)
>>> X = np.random.multivariate_normal(mean=[0, 0],
...                                   cov=real_cov,
...                                   size=50)
>>> cov = LedoitWolf().fit(X)
>>> cov.covariance_
array([[0.4406..., 0.1616...],
       [0.1616..., 0.8022...]])
>>> cov.location_
array([ 0.0595... , -0.0075...])

Notes

The regularised covariance is:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features and shrinkage is given by the Ledoit and Wolf formula (see References)

References

'A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices', Ledoit and Wolf, Journal of Multivariate Analysis, Volume 88, Issue 2, February 2004, pages 365-411.

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fit the Ledoit-Wolf shrunk covariance model according to the given training data and parameters.

Parameters

X : array-like of shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features.
y : Ignored not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

shrinkage_¶

attribute shrinkage_

val shrinkage_ : t -> float
val shrinkage_opt : t -> (float) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

MinCovDet¶

Module Sklearn.Covariance.MinCovDet wraps Python class sklearn.covariance.MinCovDet.

type t

create¶

constructor and attributes create

val create :
  ?store_precision:bool ->
  ?assume_centered:bool ->
  ?support_fraction:float ->
  ?random_state:int ->
  unit ->
  t

Minimum Covariance Determinant (MCD): robust estimator of covariance.

The Minimum Covariance Determinant covariance estimator is to be applied on Gaussian-distributed data, but could still be relevant on data drawn from a unimodal, symmetric distribution. It is not meant to be used with multi-modal data (the algorithm used to fit a MinCovDet object is likely to fail in such a case). One should consider projection pursuit methods to deal with multi-modal datasets.

Read more in the :ref:User Guide <robust_covariance>.

Parameters

store_precision : bool, default=True Specify if the estimated precision is stored.
assume_centered : bool, default=False If True, the support of the robust location and the covariance estimates is computed, and a covariance estimate is recomputed from it, without centering the data. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, the robust location and covariance are directly computed with the FastMCD algorithm without additional treatment.
support_fraction : float, default=None The proportion of points to be included in the support of the raw MCD estimate. Default is None, which implies that the minimum value of support_fraction will be used within the algorithm: (n_sample + n_features + 1) / 2. The parameter must be in the range (0, 1).
random_state : int or RandomState instance, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls.
See :term: Glossary <random_state>.

Attributes

raw_location_ : ndarray of shape (n_features,) The raw robust estimated location before correction and re-weighting.
raw_covariance_ : ndarray of shape (n_features, n_features) The raw robust estimated covariance before correction and re-weighting.
raw_support_ : ndarray of shape (n_samples,) A mask of the observations that have been used to compute the raw robust estimates of location and shape, before correction and re-weighting.
location_ : ndarray of shape (n_features,) Estimated robust location.
covariance_ : ndarray of shape (n_features, n_features) Estimated robust covariance matrix.
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True)
support_ : ndarray of shape (n_samples,) A mask of the observations that have been used to compute the robust estimates of location and shape.
dist_ : ndarray of shape (n_samples,) Mahalanobis distances of the training set (on which :meth:fit is called) observations.

Examples

>>> import numpy as np
>>> from sklearn.covariance import MinCovDet
>>> from sklearn.datasets import make_gaussian_quantiles
>>> real_cov = np.array([[.8, .3],
...                      [.3, .4]])
>>> rng = np.random.RandomState(0)
>>> X = rng.multivariate_normal(mean=[0, 0],
...                                   cov=real_cov,
...                                   size=500)
>>> cov = MinCovDet(random_state=0).fit(X)
>>> cov.covariance_
array([[0.7411..., 0.2535...],
       [0.2535..., 0.3053...]])
>>> cov.location_
array([0.0813... , 0.0427...])

References

.. [Rouseeuw1984] P. J. Rousseeuw. Least median of squares regression. J. Am Stat Ass, 79:871, 1984. .. [Rousseeuw] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS .. [ButlerDavies] R. W. Butler, P. L. Davies and M. Jhun, Asymptotics For The Minimum Covariance Determinant Estimator, The Annals of Statistics, 1993, Vol. 21, No. 3, 1385-1400

correct_covariance¶

method correct_covariance

val correct_covariance :
  data:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Apply a correction to raw Minimum Covariance Determinant estimates.

Correction using the empirical correction factor suggested by Rousseeuw and Van Driessen in [RVD]_.

Parameters

data : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. The data set must be the one which was used to compute the raw estimates.

Returns

covariance_corrected : ndarray of shape (n_features, n_features) Corrected robust covariance estimate.

References

.. [RVD] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fits a Minimum Covariance Determinant with the FastMCD algorithm.

Parameters

X : array-like of shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features.
y: Ignored Not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

reweight_covariance¶

method reweight_covariance

val reweight_covariance :
  data:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  ([>`ArrayLike] Np.Obj.t * [>`ArrayLike] Np.Obj.t * Py.Object.t)

Re-weight raw Minimum Covariance Determinant estimates.

Re-weight observations using Rousseeuw's method (equivalent to deleting outlying observations from the data set before computing location and covariance estimates) described in [RVDriessen]_.

Parameters

data : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. The data set must be the one which was used to compute the raw estimates.

Returns

location_reweighted : ndarray of shape (n_features,) Re-weighted robust location estimate.
covariance_reweighted : ndarray of shape (n_features, n_features) Re-weighted robust covariance estimate.
support_reweighted : ndarray of shape (n_samples,), dtype=bool A mask of the observations that have been used to compute the re-weighted robust location and covariance estimates.

References

.. [RVDriessen] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

raw_location_¶

attribute raw_location_

val raw_location_ : t -> [>`ArrayLike] Np.Obj.t
val raw_location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

raw_covariance_¶

attribute raw_covariance_

val raw_covariance_ : t -> [>`ArrayLike] Np.Obj.t
val raw_covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

raw_support_¶

attribute raw_support_

val raw_support_ : t -> [>`ArrayLike] Np.Obj.t
val raw_support_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

support_¶

attribute support_

val support_ : t -> [>`ArrayLike] Np.Obj.t
val support_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

dist_¶

attribute dist_

val dist_ : t -> [>`ArrayLike] Np.Obj.t
val dist_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

OAS¶

Module Sklearn.Covariance.OAS wraps Python class sklearn.covariance.OAS.

type t

create¶

constructor and attributes create

val create :
  ?store_precision:bool ->
  ?assume_centered:bool ->
  unit ->
  t

Oracle Approximating Shrinkage Estimator

Read more in the :ref:User Guide <shrunk_covariance>.

OAS is a particular form of shrinkage described in 'Shrinkage Algorithms for MMSE Covariance Estimation' Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.

The formula used here does not correspond to the one given in the article. In the original article, formula (23) states that 2/p is multiplied by Trace(cov*cov) in both the numerator and denominator, but this operation is omitted because for a large p, the value of 2/p is so small that it doesn't affect the value of the estimator.

Parameters

store_precision : bool, default=True Specify if the estimated precision is stored.
assume_centered : bool, default=False If True, data will not be centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False (default), data will be centered before computation.

Attributes

covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix.
location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean.
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True)
shrinkage_ : float coefficient in the convex combination used for the computation of the shrunk estimate. Range is [0, 1].

Examples

>>> import numpy as np
>>> from sklearn.covariance import OAS
>>> from sklearn.datasets import make_gaussian_quantiles
>>> real_cov = np.array([[.8, .3],
...                      [.3, .4]])
>>> rng = np.random.RandomState(0)
>>> X = rng.multivariate_normal(mean=[0, 0],
...                             cov=real_cov,
...                             size=500)
>>> oas = OAS().fit(X)
>>> oas.covariance_
array([[0.7533..., 0.2763...],
       [0.2763..., 0.3964...]])
>>> oas.precision_
array([[ 1.7833..., -1.2431... ],
       [-1.2431...,  3.3889...]])
>>> oas.shrinkage_
0.0195...

Notes

The regularised covariance is:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features and shrinkage is given by the OAS formula (see References)

References

'Shrinkage Algorithms for MMSE Covariance Estimation' Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fit the Oracle Approximating Shrinkage covariance model according to the given training data and parameters.

Parameters

X : array-like of shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features.
y : Ignored not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

shrinkage_¶

attribute shrinkage_

val shrinkage_ : t -> float
val shrinkage_opt : t -> (float) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

ShrunkCovariance¶

Module Sklearn.Covariance.ShrunkCovariance wraps Python class sklearn.covariance.ShrunkCovariance.

type t

create¶

constructor and attributes create

val create :
  ?store_precision:bool ->
  ?assume_centered:bool ->
  ?shrinkage:float ->
  unit ->
  t

Covariance estimator with shrinkage

Read more in the :ref:User Guide <shrunk_covariance>.

Parameters

store_precision : bool, default=True Specify if the estimated precision is stored
assume_centered : bool, default=False If True, data will not be centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data will be centered before computation.
shrinkage : float, default=0.1 Coefficient in the convex combination used for the computation of the shrunk estimate. Range is [0, 1].

Attributes

covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix
location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean.
precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True)

Examples

>>> import numpy as np
>>> from sklearn.covariance import ShrunkCovariance
>>> from sklearn.datasets import make_gaussian_quantiles
>>> real_cov = np.array([[.8, .3],
...                      [.3, .4]])
>>> rng = np.random.RandomState(0)
>>> X = rng.multivariate_normal(mean=[0, 0],
...                                   cov=real_cov,
...                                   size=500)
>>> cov = ShrunkCovariance().fit(X)
>>> cov.covariance_
array([[0.7387..., 0.2536...],
       [0.2536..., 0.4110...]])
>>> cov.location_
array([0.0622..., 0.0193...])

Notes

The regularized covariance is given by:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features

error_norm¶

method error_norm

val error_norm :
  ?norm:[`Frobenius | `Spectral] ->
  ?scaling:bool ->
  ?squared:bool ->
  comp_cov:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm).

Parameters

comp_cov : array-like of shape (n_features, n_features) The covariance to compare with.
norm : {'frobenius', 'spectral'}, default='frobenius' The type of norm used to compute the error. Available error types:
- 'frobenius' (default): sqrt(tr(A^t.A))
- 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scaling : bool, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squared : bool, default=True Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns

result : float The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit¶

method fit

val fit :
  ?y:Py.Object.t ->
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  t

Fit the shrunk covariance model according to the given training data and parameters.

Parameters

X : array-like of shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features.
y: Ignored not used, present for API consistence purpose.

Returns

self : object

get_params¶

method get_params

val get_params :
  ?deep:bool ->
  [> tag] Obj.t ->
  Dict.t

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 : mapping of string to any Parameter names mapped to their values.

get_precision¶

method get_precision

val get_precision :
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Getter for the precision matrix.

Returns

precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object.

mahalanobis¶

method mahalanobis

val mahalanobis :
  x:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  [>`ArrayLike] Np.Obj.t

Computes the squared Mahalanobis distances of given observations.

Parameters

X : array-like of shape (n_samples, n_features) The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

Returns

dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations.

score¶

method score

val score :
  ?y:Py.Object.t ->
  x_test:[>`ArrayLike] Np.Obj.t ->
  [> tag] Obj.t ->
  float

Computes the log-likelihood of a Gaussian data set with self.covariance_ as an estimator of its covariance matrix.

Parameters

X_test : array-like of shape (n_samples, n_features) Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).
y : Ignored Not used, present for API consistence purpose.

Returns

res : float The likelihood of the data set with self.covariance_ as an estimator of its covariance matrix.

set_params¶

method set_params

val set_params :
  ?params:(string * Py.Object.t) list ->
  [> tag] Obj.t ->
  t

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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 : object Estimator instance.

covariance_¶

attribute covariance_

val covariance_ : t -> [>`ArrayLike] Np.Obj.t
val covariance_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

location_¶

attribute location_

val location_ : t -> [>`ArrayLike] Np.Obj.t
val location_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

precision_¶

attribute precision_

val precision_ : t -> [>`ArrayLike] Np.Obj.t
val precision_opt : t -> ([>`ArrayLike] Np.Obj.t) option

This attribute is documented in create above. The first version raises Not_found if the attribute is None. The _opt version returns an option.

to_string¶

method to_string

val to_string: t -> string

Print the object to a human-readable representation.

show¶

method show

val show: t -> string

Print the object to a human-readable representation.

pp¶

method pp

val pp: Format.formatter -> t -> unit

Pretty-print the object to a formatter.

empirical_covariance¶

function empirical_covariance

val empirical_covariance :
  ?assume_centered:bool ->
  x:[>`ArrayLike] Np.Obj.t ->
  unit ->
  [>`ArrayLike] Np.Obj.t

Computes the Maximum likelihood covariance estimator

Parameters

X : ndarray of shape (n_samples, n_features) Data from which to compute the covariance estimate
assume_centered : bool, default=False If True, data will not be centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data will be centered before computation.

Returns

covariance : ndarray of shape (n_features, n_features) Empirical covariance (Maximum Likelihood Estimator).

Examples

>>> from sklearn.covariance import empirical_covariance
>>> X = [[1,1,1],[1,1,1],[1,1,1],
...      [0,0,0],[0,0,0],[0,0,0]]
>>> empirical_covariance(X)
array([[0.25, 0.25, 0.25],
       [0.25, 0.25, 0.25],
       [0.25, 0.25, 0.25]])

fast_mcd¶

function fast_mcd

val fast_mcd :
  ?support_fraction:float ->
  ?cov_computation_method:Py.Object.t ->
  ?random_state:int ->
  x:[>`ArrayLike] Np.Obj.t ->
  unit ->
  ([>`ArrayLike] Np.Obj.t * [>`ArrayLike] Np.Obj.t * Py.Object.t)

Estimates the Minimum Covariance Determinant matrix.

Read more in the :ref:User Guide <robust_covariance>.

Parameters

X : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples.
support_fraction : float, default=None The proportion of points to be included in the support of the raw MCD estimate. Default is None, which implies that the minimum value of support_fraction will be used within the algorithm: (n_sample + n_features + 1) / 2. This parameter must be in the range (0, 1).
cov_computation_method : callable, default=:func:sklearn.covariance.empirical_covariance The function which will be used to compute the covariance. Must return an array of shape (n_features, n_features).
random_state : int or RandomState instance, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls.
See :term: Glossary <random_state>.

Returns

location : ndarray of shape (n_features,) Robust location of the data.
covariance : ndarray of shape (n_features, n_features) Robust covariance of the features.
support : ndarray of shape (n_samples,), dtype=bool A mask of the observations that have been used to compute the robust location and covariance estimates of the data set.

Notes

The FastMCD algorithm has been introduced by Rousseuw and Van Driessen in 'A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS'. The principle is to compute robust estimates and random subsets before pooling them into a larger subsets, and finally into the full data set. Depending on the size of the initial sample, we have one, two or three such computation levels.

Note that only raw estimates are returned. If one is interested in the correction and reweighting steps described in [RouseeuwVan]_, see the MinCovDet object.

References

.. [RouseeuwVan] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS

.. [Butler1993] R. W. Butler, P. L. Davies and M. Jhun, Asymptotics For The Minimum Covariance Determinant Estimator, The Annals of Statistics, 1993, Vol. 21, No. 3, 1385-1400

graphical_lasso¶

function graphical_lasso

val graphical_lasso :
  ?cov_init:[>`ArrayLike] Np.Obj.t ->
  ?mode:[`Cd | `Lars] ->
  ?tol:float ->
  ?enet_tol:float ->
  ?max_iter:int ->
  ?verbose:int ->
  ?return_costs:bool ->
  ?eps:float ->
  ?return_n_iter:bool ->
  emp_cov:[>`ArrayLike] Np.Obj.t ->
  alpha:float ->
  unit ->
  ([>`ArrayLike] Np.Obj.t * [>`ArrayLike] Np.Obj.t * Py.Object.t * int)

l1-penalized covariance estimator

Read more in the :ref:User Guide <sparse_inverse_covariance>.

.. versionchanged:: v0.20 graph_lasso has been renamed to graphical_lasso

Parameters

emp_cov : ndarray of shape (n_features, n_features) Empirical covariance from which to compute the covariance estimate.
alpha : float The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf].
cov_init : array of shape (n_features, n_features), default=None The initial guess for the covariance.
mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where p > n. Elsewhere prefer cd which is more numerically stable.
tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf].
enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. Range is (0, inf].
max_iter : int, default=100 The maximum number of iterations.
verbose : bool, default=False If verbose is True, the objective function and dual gap are printed at each iteration.
return_costs : bool, default=Flase If return_costs is True, the objective function and dual gap at each iteration are returned.
eps : float, default=eps The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Default is np.finfo(np.float64).eps.
return_n_iter : bool, default=False Whether or not to return the number of iterations.

Returns

covariance : ndarray of shape (n_features, n_features) The estimated covariance matrix.
precision : ndarray of shape (n_features, n_features) The estimated (sparse) precision matrix.
costs : list of (objective, dual_gap) pairs The list of values of the objective function and the dual gap at each iteration. Returned only if return_costs is True.
n_iter : int Number of iterations. Returned only if return_n_iter is set to True.

Notes

The algorithm employed to solve this problem is the GLasso algorithm, from the Friedman 2008 Biostatistics paper. It is the same algorithm as in the R glasso package.

One possible difference with the glasso R package is that the diagonal coefficients are not penalized.

ledoit_wolf¶

function ledoit_wolf

val ledoit_wolf :
  ?assume_centered:bool ->
  ?block_size:int ->
  x:[>`ArrayLike] Np.Obj.t ->
  unit ->
  ([>`ArrayLike] Np.Obj.t * float)

Estimates the shrunk Ledoit-Wolf covariance matrix.

Read more in the :ref:User Guide <shrunk_covariance>.

Parameters

X : array-like of shape (n_samples, n_features) Data from which to compute the covariance estimate
assume_centered : bool, default=False If True, data will not be centered before computation. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data will be centered before computation.
block_size : int, default=1000 Size of the blocks into which the covariance matrix will be split. This is purely a memory optimization and does not affect results.

Returns

shrunk_cov : ndarray of shape (n_features, n_features) Shrunk covariance.
shrinkage : float Coefficient in the convex combination used for the computation of the shrunk estimate.

Notes

The regularized (shrunk) covariance is:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features

ledoit_wolf_shrinkage¶

function ledoit_wolf_shrinkage

val ledoit_wolf_shrinkage :
  ?assume_centered:bool ->
  ?block_size:int ->
  x:[>`ArrayLike] Np.Obj.t ->
  unit ->
  float

Estimates the shrunk Ledoit-Wolf covariance matrix.

Read more in the :ref:User Guide <shrunk_covariance>.

Parameters

X : array-like of shape (n_samples, n_features) Data from which to compute the Ledoit-Wolf shrunk covariance shrinkage.
assume_centered : bool, default=False If True, data will not be centered before computation. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data will be centered before computation.
block_size : int, default=1000 Size of the blocks into which the covariance matrix will be split.

Returns

shrinkage : float Coefficient in the convex combination used for the computation of the shrunk estimate.

Notes

The regularized (shrunk) covariance is:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features

log_likelihood¶

function log_likelihood

val log_likelihood :
  emp_cov:[>`ArrayLike] Np.Obj.t ->
  precision:[>`ArrayLike] Np.Obj.t ->
  unit ->
  float

Computes the sample mean of the log_likelihood under a covariance model

computes the empirical expected log-likelihood (accounting for the normalization terms and scaling), allowing for universal comparison (beyond this software package)

Parameters

emp_cov : ndarray of shape (n_features, n_features) Maximum Likelihood Estimator of covariance.
precision : ndarray of shape (n_features, n_features) The precision matrix of the covariance model to be tested.

Returns

log_likelihood_ : float Sample mean of the log-likelihood.

oas¶

function oas

val oas :
  ?assume_centered:bool ->
  x:[>`ArrayLike] Np.Obj.t ->
  unit ->
  ([>`ArrayLike] Np.Obj.t * float)

Estimate covariance with the Oracle Approximating Shrinkage algorithm.

Parameters

X : array-like of shape (n_samples, n_features) Data from which to compute the covariance estimate.
assume_centered : bool, default=False If True, data will not be centered before computation. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data will be centered before computation.

Returns

shrunk_cov : array-like of shape (n_features, n_features) Shrunk covariance.
shrinkage : float Coefficient in the convex combination used for the computation of the shrunk estimate.

Notes

The regularised (shrunk) covariance is:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features

The formula we used to implement the OAS is slightly modified compared to the one given in the article. See :class:OAS for more details.

shrunk_covariance¶

function shrunk_covariance

val shrunk_covariance :
  ?shrinkage:float ->
  emp_cov:[>`ArrayLike] Np.Obj.t ->
  unit ->
  [>`ArrayLike] Np.Obj.t

Calculates a covariance matrix shrunk on the diagonal

Read more in the :ref:User Guide <shrunk_covariance>.

Parameters

emp_cov : array-like of shape (n_features, n_features) Covariance matrix to be shrunk
shrinkage : float, default=0.1 Coefficient in the convex combination used for the computation of the shrunk estimate. Range is [0, 1].

Returns

shrunk_cov : ndarray of shape (n_features, n_features) Shrunk covariance.

Notes

The regularized (shrunk) covariance is given by:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features

Covariance

EllipticEnvelope¶

create¶

Parameters

Attributes

Examples

See Also

Notes

References

correct_covariance¶

Parameters

Returns

References

decision_function¶

Parameters

Returns

error_norm¶

Parameters

Returns

fit¶

Parameters

fit_predict¶

Parameters

Returns

get_params¶

Parameters

Returns

get_precision¶

Returns

mahalanobis¶

Parameters

Returns

predict¶

Parameters

Returns

reweight_covariance¶

Parameters

Returns

References

score¶

Parameters

Returns

score_samples¶

Parameters

Returns

set_params¶

Parameters

Returns

location_¶

covariance_¶

precision_¶

support_¶

offset_¶

raw_location_¶

raw_covariance_¶

raw_support_¶

dist_¶

to_string¶

show¶

pp¶

EmpiricalCovariance¶

create¶

Parameters

Attributes

Examples

error_norm¶

Parameters

Returns

fit¶

Parameters

Returns

get_params¶

Parameters

Returns

get_precision¶

Returns

mahalanobis¶

Parameters

Returns

score¶