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"""Linear and quadratic discriminant analysis."""
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import warnings
from numbers import Integral, Real
import numpy as np
import scipy.linalg
from scipy import linalg
from .base import (
BaseEstimator,
ClassifierMixin,
ClassNamePrefixFeaturesOutMixin,
TransformerMixin,
_fit_context,
)
from .covariance import empirical_covariance, ledoit_wolf, shrunk_covariance
from .linear_model._base import LinearClassifierMixin
from .preprocessing import StandardScaler
from .utils._array_api import _expit, device, get_namespace, size
from .utils._param_validation import HasMethods, Interval, StrOptions
from .utils.extmath import softmax
from .utils.multiclass import check_classification_targets, unique_labels
from .utils.validation import check_is_fitted, validate_data
__all__ = ["LinearDiscriminantAnalysis", "QuadraticDiscriminantAnalysis"]
def _cov(X, shrinkage=None, covariance_estimator=None):
"""Estimate covariance matrix (using optional covariance_estimator).
Parameters
----------
X : array-like of shape (n_samples, n_features)
Input data.
shrinkage : {'empirical', 'auto'} or float, default=None
Shrinkage parameter, possible values:
- None or 'empirical': no shrinkage (default).
- 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
- float between 0 and 1: fixed shrinkage parameter.
Shrinkage parameter is ignored if `covariance_estimator`
is not None.
covariance_estimator : estimator, default=None
If not None, `covariance_estimator` is used to estimate
the covariance matrices instead of relying on the empirical
covariance estimator (with potential shrinkage).
The object should have a fit method and a ``covariance_`` attribute
like the estimators in :mod:`sklearn.covariance``.
if None the shrinkage parameter drives the estimate.
.. versionadded:: 0.24
Returns
-------
s : ndarray of shape (n_features, n_features)
Estimated covariance matrix.
"""
if covariance_estimator is None:
shrinkage = "empirical" if shrinkage is None else shrinkage
if isinstance(shrinkage, str):
if shrinkage == "auto":
sc = StandardScaler() # standardize features
X = sc.fit_transform(X)
s = ledoit_wolf(X)[0]
# rescale
s = sc.scale_[:, np.newaxis] * s * sc.scale_[np.newaxis, :]
elif shrinkage == "empirical":
s = empirical_covariance(X)
elif isinstance(shrinkage, Real):
s = shrunk_covariance(empirical_covariance(X), shrinkage)
else:
if shrinkage is not None and shrinkage != 0:
raise ValueError(
"covariance_estimator and shrinkage parameters "
"are not None. Only one of the two can be set."
)
covariance_estimator.fit(X)
if not hasattr(covariance_estimator, "covariance_"):
raise ValueError(
"%s does not have a covariance_ attribute"
% covariance_estimator.__class__.__name__
)
s = covariance_estimator.covariance_
return s
def _class_means(X, y):
"""Compute class means.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Input data.
y : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values.
Returns
-------
means : array-like of shape (n_classes, n_features)
Class means.
"""
xp, is_array_api_compliant = get_namespace(X)
classes, y = xp.unique_inverse(y)
means = xp.zeros((classes.shape[0], X.shape[1]), device=device(X), dtype=X.dtype)
if is_array_api_compliant:
for i in range(classes.shape[0]):
means[i, :] = xp.mean(X[y == i], axis=0)
else:
# TODO: Explore the choice of using bincount + add.at as it seems sub optimal
# from a performance-wise
cnt = np.bincount(y)
np.add.at(means, y, X)
means /= cnt[:, None]
return means
def _class_cov(X, y, priors, shrinkage=None, covariance_estimator=None):
"""Compute weighted within-class covariance matrix.
The per-class covariance are weighted by the class priors.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Input data.
y : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values.
priors : array-like of shape (n_classes,)
Class priors.
shrinkage : 'auto' or float, default=None
Shrinkage parameter, possible values:
- None: no shrinkage (default).
- 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
- float between 0 and 1: fixed shrinkage parameter.
Shrinkage parameter is ignored if `covariance_estimator` is not None.
covariance_estimator : estimator, default=None
If not None, `covariance_estimator` is used to estimate
the covariance matrices instead of relying the empirical
covariance estimator (with potential shrinkage).
The object should have a fit method and a ``covariance_`` attribute
like the estimators in sklearn.covariance.
If None, the shrinkage parameter drives the estimate.
.. versionadded:: 0.24
Returns
-------
cov : array-like of shape (n_features, n_features)
Weighted within-class covariance matrix
"""
classes = np.unique(y)
cov = np.zeros(shape=(X.shape[1], X.shape[1]))
for idx, group in enumerate(classes):
Xg = X[y == group, :]
cov += priors[idx] * np.atleast_2d(_cov(Xg, shrinkage, covariance_estimator))
return cov
class DiscriminantAnalysisPredictionMixin:
"""Mixin class for QuadraticDiscriminantAnalysis and NearestCentroid."""
def decision_function(self, X):
"""Apply decision function to an array of samples.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Array of samples (test vectors).
Returns
-------
y_scores : ndarray of shape (n_samples,) or (n_samples, n_classes)
Decision function values related to each class, per sample.
In the two-class case, the shape is `(n_samples,)`, giving the
log likelihood ratio of the positive class.
"""
y_scores = self._decision_function(X)
if len(self.classes_) == 2:
return y_scores[:, 1] - y_scores[:, 0]
return y_scores
def predict(self, X):
"""Perform classification on an array of vectors `X`.
Returns the class label for each sample.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Input vectors, where `n_samples` is the number of samples and
`n_features` is the number of features.
Returns
-------
y_pred : ndarray of shape (n_samples,)
Class label for each sample.
"""
scores = self._decision_function(X)
return self.classes_.take(scores.argmax(axis=1))
def predict_proba(self, X):
"""Estimate class probabilities.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Input data.
Returns
-------
y_proba : ndarray of shape (n_samples, n_classes)
Probability estimate of the sample for each class in the
model, where classes are ordered as they are in `self.classes_`.
"""
return np.exp(self.predict_log_proba(X))
def predict_log_proba(self, X):
"""Estimate log class probabilities.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Input data.
Returns
-------
y_log_proba : ndarray of shape (n_samples, n_classes)
Estimated log probabilities.
"""
scores = self._decision_function(X)
log_likelihood = scores - scores.max(axis=1)[:, np.newaxis]
return log_likelihood - np.log(
np.exp(log_likelihood).sum(axis=1)[:, np.newaxis]
)
class LinearDiscriminantAnalysis(
ClassNamePrefixFeaturesOutMixin,
LinearClassifierMixin,
TransformerMixin,
BaseEstimator,
):
"""Linear Discriminant Analysis.
A classifier with a linear decision boundary, generated by fitting class
conditional densities to the data and using Bayes' rule.
The model fits a Gaussian density to each class, assuming that all classes
share the same covariance matrix.
The fitted model can also be used to reduce the dimensionality of the input
by projecting it to the most discriminative directions, using the
`transform` method.
.. versionadded:: 0.17
For a comparison between
:class:`~sklearn.discriminant_analysis.LinearDiscriminantAnalysis`
and :class:`~sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis`, see
:ref:`sphx_glr_auto_examples_classification_plot_lda_qda.py`.
Read more in the :ref:`User Guide <lda_qda>`.
Parameters
----------
solver : {'svd', 'lsqr', 'eigen'}, default='svd'
Solver to use, possible values:
- 'svd': Singular value decomposition (default).
Does not compute the covariance matrix, therefore this solver is
recommended for data with a large number of features.
- 'lsqr': Least squares solution.
Can be combined with shrinkage or custom covariance estimator.
- 'eigen': Eigenvalue decomposition.
Can be combined with shrinkage or custom covariance estimator.
.. versionchanged:: 1.2
`solver="svd"` now has experimental Array API support. See the
:ref:`Array API User Guide <array_api>` for more details.
shrinkage : 'auto' or float, default=None
Shrinkage parameter, possible values:
- None: no shrinkage (default).
- 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
- float between 0 and 1: fixed shrinkage parameter.
This should be left to None if `covariance_estimator` is used.
Note that shrinkage works only with 'lsqr' and 'eigen' solvers.
For a usage example, see
:ref:`sphx_glr_auto_examples_classification_plot_lda.py`.
priors : array-like of shape (n_classes,), default=None
The class prior probabilities. By default, the class proportions are
inferred from the training data.
n_components : int, default=None
Number of components (<= min(n_classes - 1, n_features)) for
dimensionality reduction. If None, will be set to
min(n_classes - 1, n_features). This parameter only affects the
`transform` method.
For a usage example, see
:ref:`sphx_glr_auto_examples_decomposition_plot_pca_vs_lda.py`.
store_covariance : bool, default=False
If True, explicitly compute the weighted within-class covariance
matrix when solver is 'svd'. The matrix is always computed
and stored for the other solvers.
.. versionadded:: 0.17
tol : float, default=1.0e-4
Absolute threshold for a singular value of X to be considered
significant, used to estimate the rank of X. Dimensions whose
singular values are non-significant are discarded. Only used if
solver is 'svd'.
.. versionadded:: 0.17
covariance_estimator : covariance estimator, default=None
If not None, `covariance_estimator` is used to estimate
the covariance matrices instead of relying on the empirical
covariance estimator (with potential shrinkage).
The object should have a fit method and a ``covariance_`` attribute
like the estimators in :mod:`sklearn.covariance`.
if None the shrinkage parameter drives the estimate.
This should be left to None if `shrinkage` is used.
Note that `covariance_estimator` works only with 'lsqr' and 'eigen'
solvers.
.. versionadded:: 0.24
Attributes
----------
coef_ : ndarray of shape (n_features,) or (n_classes, n_features)
Weight vector(s).
intercept_ : ndarray of shape (n_classes,)
Intercept term.
covariance_ : array-like of shape (n_features, n_features)
Weighted within-class covariance matrix. It corresponds to
`sum_k prior_k * C_k` where `C_k` is the covariance matrix of the
samples in class `k`. The `C_k` are estimated using the (potentially
shrunk) biased estimator of covariance. If solver is 'svd', only
exists when `store_covariance` is True.
explained_variance_ratio_ : ndarray of shape (n_components,)
Percentage of variance explained by each of the selected components.
If ``n_components`` is not set then all components are stored and the
sum of explained variances is equal to 1.0. Only available when eigen
or svd solver is used.
means_ : array-like of shape (n_classes, n_features)
Class-wise means.
priors_ : array-like of shape (n_classes,)
Class priors (sum to 1).
scalings_ : array-like of shape (rank, n_classes - 1)
Scaling of the features in the space spanned by the class centroids.
Only available for 'svd' and 'eigen' solvers.
xbar_ : array-like of shape (n_features,)
Overall mean. Only present if solver is 'svd'.
classes_ : array-like of shape (n_classes,)
Unique class labels.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
See Also
--------
QuadraticDiscriminantAnalysis : Quadratic Discriminant Analysis.
Examples
--------
>>> import numpy as np
>>> from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
>>> y = np.array([1, 1, 1, 2, 2, 2])
>>> clf = LinearDiscriminantAnalysis()
>>> clf.fit(X, y)
LinearDiscriminantAnalysis()
>>> print(clf.predict([[-0.8, -1]]))
[1]
"""
_parameter_constraints: dict = {
"solver": [StrOptions({"svd", "lsqr", "eigen"})],
"shrinkage": [StrOptions({"auto"}), Interval(Real, 0, 1, closed="both"), None],
"n_components": [Interval(Integral, 1, None, closed="left"), None],
"priors": ["array-like", None],
"store_covariance": ["boolean"],
"tol": [Interval(Real, 0, None, closed="left")],
"covariance_estimator": [HasMethods("fit"), None],
}
def __init__(
self,
solver="svd",
shrinkage=None,
priors=None,
n_components=None,
store_covariance=False,
tol=1e-4,
covariance_estimator=None,
):
self.solver = solver
self.shrinkage = shrinkage
self.priors = priors
self.n_components = n_components
self.store_covariance = store_covariance # used only in svd solver
self.tol = tol # used only in svd solver
self.covariance_estimator = covariance_estimator
def _solve_lstsq(self, X, y, shrinkage, covariance_estimator):
"""Least squares solver.
The least squares solver computes a straightforward solution of the
optimal decision rule based directly on the discriminant functions. It
can only be used for classification (with any covariance estimator),
because
estimation of eigenvectors is not performed. Therefore, dimensionality
reduction with the transform is not supported.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data.
y : array-like of shape (n_samples,) or (n_samples, n_classes)
Target values.
shrinkage : 'auto', float or None
Shrinkage parameter, possible values:
- None: no shrinkage.
- 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
- float between 0 and 1: fixed shrinkage parameter.
Shrinkage parameter is ignored if `covariance_estimator` i
not None
covariance_estimator : estimator, default=None
If not None, `covariance_estimator` is used to estimate
the covariance matrices instead of relying the empirical
covariance estimator (with potential shrinkage).
The object should have a fit method and a ``covariance_`` attribute
like the estimators in sklearn.covariance.
if None the shrinkage parameter drives the estimate.
.. versionadded:: 0.24
Notes
-----
This solver is based on [1]_, section 2.6.2, pp. 39-41.
References
----------
.. [1] R. O. Duda, P. E. Hart, D. G. Stork. Pattern Classification
(Second Edition). John Wiley & Sons, Inc., New York, 2001. ISBN
0-471-05669-3.
"""
self.means_ = _class_means(X, y)
self.covariance_ = _class_cov(
X, y, self.priors_, shrinkage, covariance_estimator
)
self.coef_ = linalg.lstsq(self.covariance_, self.means_.T)[0].T
self.intercept_ = -0.5 * np.diag(np.dot(self.means_, self.coef_.T)) + np.log(
self.priors_
)
def _solve_eigen(self, X, y, shrinkage, covariance_estimator):
"""Eigenvalue solver.
The eigenvalue solver computes the optimal solution of the Rayleigh
coefficient (basically the ratio of between class scatter to within
class scatter). This solver supports both classification and
dimensionality reduction (with any covariance estimator).
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data.
y : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values.
shrinkage : 'auto', float or None
Shrinkage parameter, possible values:
- None: no shrinkage.
- 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
- float between 0 and 1: fixed shrinkage constant.
Shrinkage parameter is ignored if `covariance_estimator` i
not None
covariance_estimator : estimator, default=None
If not None, `covariance_estimator` is used to estimate
the covariance matrices instead of relying the empirical
covariance estimator (with potential shrinkage).
The object should have a fit method and a ``covariance_`` attribute
like the estimators in sklearn.covariance.
if None the shrinkage parameter drives the estimate.
.. versionadded:: 0.24
Notes
-----
This solver is based on [1]_, section 3.8.3, pp. 121-124.
References
----------
.. [1] R. O. Duda, P. E. Hart, D. G. Stork. Pattern Classification
(Second Edition). John Wiley & Sons, Inc., New York, 2001. ISBN
0-471-05669-3.
"""
self.means_ = _class_means(X, y)
self.covariance_ = _class_cov(
X, y, self.priors_, shrinkage, covariance_estimator
)
Sw = self.covariance_ # within scatter
St = _cov(X, shrinkage, covariance_estimator) # total scatter
Sb = St - Sw # between scatter
evals, evecs = linalg.eigh(Sb, Sw)
self.explained_variance_ratio_ = np.sort(evals / np.sum(evals))[::-1][
: self._max_components
]
evecs = evecs[:, np.argsort(evals)[::-1]] # sort eigenvectors
self.scalings_ = evecs
self.coef_ = np.dot(self.means_, evecs).dot(evecs.T)
self.intercept_ = -0.5 * np.diag(np.dot(self.means_, self.coef_.T)) + np.log(
self.priors_
)
def _solve_svd(self, X, y):
"""SVD solver.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data.
y : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values.
"""
xp, is_array_api_compliant = get_namespace(X)
if is_array_api_compliant:
svd = xp.linalg.svd
else:
svd = scipy.linalg.svd
n_samples, n_features = X.shape
n_classes = self.classes_.shape[0]
self.means_ = _class_means(X, y)
if self.store_covariance:
self.covariance_ = _class_cov(X, y, self.priors_)
Xc = []
for idx, group in enumerate(self.classes_):
Xg = X[y == group]
Xc.append(Xg - self.means_[idx, :])
self.xbar_ = self.priors_ @ self.means_
Xc = xp.concat(Xc, axis=0)
# 1) within (univariate) scaling by with classes std-dev
std = xp.std(Xc, axis=0)
# avoid division by zero in normalization
std[std == 0] = 1.0
fac = xp.asarray(1.0 / (n_samples - n_classes), dtype=X.dtype)
# 2) Within variance scaling
X = xp.sqrt(fac) * (Xc / std)
# SVD of centered (within)scaled data
U, S, Vt = svd(X, full_matrices=False)
rank = xp.sum(xp.astype(S > self.tol, xp.int32))
# Scaling of within covariance is: V' 1/S
scalings = (Vt[:rank, :] / std).T / S[:rank]
fac = 1.0 if n_classes == 1 else 1.0 / (n_classes - 1)
# 3) Between variance scaling
# Scale weighted centers
X = (
(xp.sqrt((n_samples * self.priors_) * fac)) * (self.means_ - self.xbar_).T
).T @ scalings
# Centers are living in a space with n_classes-1 dim (maximum)
# Use SVD to find projection in the space spanned by the
# (n_classes) centers
_, S, Vt = svd(X, full_matrices=False)
if self._max_components == 0:
self.explained_variance_ratio_ = xp.empty((0,), dtype=S.dtype)
else:
self.explained_variance_ratio_ = (S**2 / xp.sum(S**2))[
: self._max_components
]
rank = xp.sum(xp.astype(S > self.tol * S[0], xp.int32))
self.scalings_ = scalings @ Vt.T[:, :rank]
coef = (self.means_ - self.xbar_) @ self.scalings_
self.intercept_ = -0.5 * xp.sum(coef**2, axis=1) + xp.log(self.priors_)
self.coef_ = coef @ self.scalings_.T
self.intercept_ -= self.xbar_ @ self.coef_.T
@_fit_context(
# LinearDiscriminantAnalysis.covariance_estimator is not validated yet
prefer_skip_nested_validation=False
)
def fit(self, X, y):
"""Fit the Linear Discriminant Analysis model.
.. versionchanged:: 0.19
`store_covariance` and `tol` has been moved to main constructor.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data.
y : array-like of shape (n_samples,)
Target values.
Returns
-------
self : object
Fitted estimator.
"""
xp, _ = get_namespace(X)
X, y = validate_data(
self, X, y, ensure_min_samples=2, dtype=[xp.float64, xp.float32]
)
self.classes_ = unique_labels(y)
n_samples, _ = X.shape
n_classes = self.classes_.shape[0]
if n_samples == n_classes:
raise ValueError(
"The number of samples must be more than the number of classes."
)
if self.priors is None: # estimate priors from sample
_, cnts = xp.unique_counts(y) # non-negative ints
self.priors_ = xp.astype(cnts, X.dtype) / float(y.shape[0])
else:
self.priors_ = xp.asarray(self.priors, dtype=X.dtype)
if xp.any(self.priors_ < 0):
raise ValueError("priors must be non-negative")
if xp.abs(xp.sum(self.priors_) - 1.0) > 1e-5:
warnings.warn("The priors do not sum to 1. Renormalizing", UserWarning)
self.priors_ = self.priors_ / self.priors_.sum()
# Maximum number of components no matter what n_components is
# specified:
max_components = min(n_classes - 1, X.shape[1])
if self.n_components is None:
self._max_components = max_components
else:
if self.n_components > max_components:
raise ValueError(
"n_components cannot be larger than min(n_features, n_classes - 1)."
)
self._max_components = self.n_components
if self.solver == "svd":
if self.shrinkage is not None:
raise NotImplementedError("shrinkage not supported with 'svd' solver.")
if self.covariance_estimator is not None:
raise ValueError(
"covariance estimator "
"is not supported "
"with svd solver. Try another solver"
)
self._solve_svd(X, y)
elif self.solver == "lsqr":
self._solve_lstsq(
X,
y,
shrinkage=self.shrinkage,
covariance_estimator=self.covariance_estimator,
)
elif self.solver == "eigen":
self._solve_eigen(
X,
y,
shrinkage=self.shrinkage,
covariance_estimator=self.covariance_estimator,
)
if size(self.classes_) == 2: # treat binary case as a special case
coef_ = xp.asarray(self.coef_[1, :] - self.coef_[0, :], dtype=X.dtype)
self.coef_ = xp.reshape(coef_, (1, -1))
intercept_ = xp.asarray(
self.intercept_[1] - self.intercept_[0], dtype=X.dtype
)
self.intercept_ = xp.reshape(intercept_, (1,))
self._n_features_out = self._max_components
return self
def transform(self, X):
"""Project data to maximize class separation.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Input data.
Returns
-------
X_new : ndarray of shape (n_samples, n_components) or \
(n_samples, min(rank, n_components))
Transformed data. In the case of the 'svd' solver, the shape
is (n_samples, min(rank, n_components)).
"""
if self.solver == "lsqr":
raise NotImplementedError(
"transform not implemented for 'lsqr' solver (use 'svd' or 'eigen')."
)
check_is_fitted(self)
xp, _ = get_namespace(X)
X = validate_data(self, X, reset=False)
if self.solver == "svd":
X_new = (X - self.xbar_) @ self.scalings_
elif self.solver == "eigen":
X_new = X @ self.scalings_
return X_new[:, : self._max_components]
def predict_proba(self, X):
"""Estimate probability.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Input data.
Returns
-------
C : ndarray of shape (n_samples, n_classes)
Estimated probabilities.
"""
check_is_fitted(self)
xp, is_array_api_compliant = get_namespace(X)
decision = self.decision_function(X)
if size(self.classes_) == 2:
proba = _expit(decision, xp)
return xp.stack([1 - proba, proba], axis=1)
else:
return softmax(decision)
def predict_log_proba(self, X):
"""Estimate log probability.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Input data.
Returns
-------
C : ndarray of shape (n_samples, n_classes)
Estimated log probabilities.
"""
xp, _ = get_namespace(X)
prediction = self.predict_proba(X)
info = xp.finfo(prediction.dtype)
if hasattr(info, "smallest_normal"):
smallest_normal = info.smallest_normal
else:
# smallest_normal was introduced in NumPy 1.22
smallest_normal = info.tiny
prediction[prediction == 0.0] += smallest_normal
return xp.log(prediction)
def decision_function(self, X):
"""Apply decision function to an array of samples.
The decision function is equal (up to a constant factor) to the
log-posterior of the model, i.e. `log p(y = k | x)`. In a binary
classification setting this instead corresponds to the difference
`log p(y = 1 | x) - log p(y = 0 | x)`. See :ref:`lda_qda_math`.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Array of samples (test vectors).
Returns
-------
y_scores : ndarray of shape (n_samples,) or (n_samples, n_classes)
Decision function values related to each class, per sample.
In the two-class case, the shape is `(n_samples,)`, giving the
log likelihood ratio of the positive class.
"""
# Only override for the doc
return super().decision_function(X)
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.array_api_support = True
return tags
class QuadraticDiscriminantAnalysis(
DiscriminantAnalysisPredictionMixin, ClassifierMixin, BaseEstimator
):
"""Quadratic Discriminant Analysis.
A classifier with a quadratic decision boundary, generated
by fitting class conditional densities to the data
and using Bayes' rule.
The model fits a Gaussian density to each class.
.. versionadded:: 0.17
For a comparison between
:class:`~sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis`
and :class:`~sklearn.discriminant_analysis.LinearDiscriminantAnalysis`, see
:ref:`sphx_glr_auto_examples_classification_plot_lda_qda.py`.
Read more in the :ref:`User Guide <lda_qda>`.
Parameters
----------
priors : array-like of shape (n_classes,), default=None
Class priors. By default, the class proportions are inferred from the
training data.
reg_param : float, default=0.0
Regularizes the per-class covariance estimates by transforming S2 as
``S2 = (1 - reg_param) * S2 + reg_param * np.eye(n_features)``,
where S2 corresponds to the `scaling_` attribute of a given class.
store_covariance : bool, default=False
If True, the class covariance matrices are explicitly computed and
stored in the `self.covariance_` attribute.
.. versionadded:: 0.17
tol : float, default=1.0e-4
Absolute threshold for the covariance matrix to be considered rank
deficient after applying some regularization (see `reg_param`) to each
`Sk` where `Sk` represents covariance matrix for k-th class. This
parameter does not affect the predictions. It controls when a warning
is raised if the covariance matrix is not full rank.
.. versionadded:: 0.17
Attributes
----------
covariance_ : list of len n_classes of ndarray \
of shape (n_features, n_features)
For each class, gives the covariance matrix estimated using the
samples of that class. The estimations are unbiased. Only present if
`store_covariance` is True.
means_ : array-like of shape (n_classes, n_features)
Class-wise means.
priors_ : array-like of shape (n_classes,)
Class priors (sum to 1).
rotations_ : list of len n_classes of ndarray of shape (n_features, n_k)
For each class k an array of shape (n_features, n_k), where
``n_k = min(n_features, number of elements in class k)``
It is the rotation of the Gaussian distribution, i.e. its
principal axis. It corresponds to `V`, the matrix of eigenvectors
coming from the SVD of `Xk = U S Vt` where `Xk` is the centered
matrix of samples from class k.
scalings_ : list of len n_classes of ndarray of shape (n_k,)
For each class, contains the scaling of
the Gaussian distributions along its principal axes, i.e. the
variance in the rotated coordinate system. It corresponds to `S^2 /
(n_samples - 1)`, where `S` is the diagonal matrix of singular values
from the SVD of `Xk`, where `Xk` is the centered matrix of samples
from class k.
classes_ : ndarray of shape (n_classes,)
Unique class labels.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
See Also
--------
LinearDiscriminantAnalysis : Linear Discriminant Analysis.
Examples
--------
>>> from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis
>>> import numpy as np
>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
>>> y = np.array([1, 1, 1, 2, 2, 2])
>>> clf = QuadraticDiscriminantAnalysis()
>>> clf.fit(X, y)
QuadraticDiscriminantAnalysis()
>>> print(clf.predict([[-0.8, -1]]))
[1]
"""
_parameter_constraints: dict = {
"priors": ["array-like", None],
"reg_param": [Interval(Real, 0, 1, closed="both")],
"store_covariance": ["boolean"],
"tol": [Interval(Real, 0, None, closed="left")],
}
def __init__(
self, *, priors=None, reg_param=0.0, store_covariance=False, tol=1.0e-4
):
self.priors = priors
self.reg_param = reg_param
self.store_covariance = store_covariance
self.tol = tol
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y):
"""Fit the model according to the given training data and parameters.
.. versionchanged:: 0.19
``store_covariances`` has been moved to main constructor as
``store_covariance``.
.. versionchanged:: 0.19
``tol`` has been moved to main constructor.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples and
`n_features` is the number of features.
y : array-like of shape (n_samples,)
Target values (integers).
Returns
-------
self : object
Fitted estimator.
"""
X, y = validate_data(self, X, y)
check_classification_targets(y)
self.classes_, y = np.unique(y, return_inverse=True)
n_samples, n_features = X.shape
n_classes = len(self.classes_)
if n_classes < 2:
raise ValueError(
"The number of classes has to be greater than one; got %d class"
% (n_classes)
)
if self.priors is None:
self.priors_ = np.bincount(y) / float(n_samples)
else:
self.priors_ = np.array(self.priors)
cov = None
store_covariance = self.store_covariance
if store_covariance:
cov = []
means = []
scalings = []
rotations = []
for ind in range(n_classes):
Xg = X[y == ind, :]
meang = Xg.mean(0)
means.append(meang)
if len(Xg) == 1:
raise ValueError(
"y has only 1 sample in class %s, covariance is ill defined."
% str(self.classes_[ind])
)
Xgc = Xg - meang
# Xgc = U * S * V.T
_, S, Vt = np.linalg.svd(Xgc, full_matrices=False)
S2 = (S**2) / (len(Xg) - 1)
S2 = ((1 - self.reg_param) * S2) + self.reg_param
rank = np.sum(S2 > self.tol)
if rank < n_features:
warnings.warn(
f"The covariance matrix of class {ind} is not full rank. "
"Increasing the value of parameter `reg_param` might help"
" reducing the collinearity.",
linalg.LinAlgWarning,
)
if self.store_covariance or store_covariance:
# cov = V * (S^2 / (n-1)) * V.T
cov.append(np.dot(S2 * Vt.T, Vt))
scalings.append(S2)
rotations.append(Vt.T)
if self.store_covariance or store_covariance:
self.covariance_ = cov
self.means_ = np.asarray(means)
self.scalings_ = scalings
self.rotations_ = rotations
return self
def _decision_function(self, X):
# return log posterior, see eq (4.12) p. 110 of the ESL.
check_is_fitted(self)
X = validate_data(self, X, reset=False)
norm2 = []
for i in range(len(self.classes_)):
R = self.rotations_[i]
S = self.scalings_[i]
Xm = X - self.means_[i]
X2 = np.dot(Xm, R * (S ** (-0.5)))
norm2.append(np.sum(X2**2, axis=1))
norm2 = np.array(norm2).T # shape = [len(X), n_classes]
u = np.asarray([np.sum(np.log(s)) for s in self.scalings_])
return -0.5 * (norm2 + u) + np.log(self.priors_)
def decision_function(self, X):
"""Apply decision function to an array of samples.
The decision function is equal (up to a constant factor) to the
log-posterior of the model, i.e. `log p(y = k | x)`. In a binary
classification setting this instead corresponds to the difference
`log p(y = 1 | x) - log p(y = 0 | x)`. See :ref:`lda_qda_math`.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Array of samples (test vectors).
Returns
-------
C : ndarray of shape (n_samples,) or (n_samples, n_classes)
Decision function values related to each class, per sample.
In the two-class case, the shape is `(n_samples,)`, giving the
log likelihood ratio of the positive class.
"""
return super().decision_function(X)
def predict(self, X):
"""Perform classification on an array of test vectors X.
The predicted class C for each sample in X is returned.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Vector to be scored, where `n_samples` is the number of samples and
`n_features` is the number of features.
Returns
-------
C : ndarray of shape (n_samples,)
Estimated probabilities.
"""
return super().predict(X)
def predict_proba(self, X):
"""Return posterior probabilities of classification.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Array of samples/test vectors.
Returns
-------
C : ndarray of shape (n_samples, n_classes)
Posterior probabilities of classification per class.
"""
# compute the likelihood of the underlying gaussian models
# up to a multiplicative constant.
return super().predict_proba(X)
def predict_log_proba(self, X):
"""Return log of posterior probabilities of classification.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Array of samples/test vectors.
Returns
-------
C : ndarray of shape (n_samples, n_classes)
Posterior log-probabilities of classification per class.
"""
# XXX : can do better to avoid precision overflows
return super().predict_log_proba(X)
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