/
githubmirror
/
scikit-learn
Обзор
Документация
Войти
/
githubmirror
/
scikit-learn
Код
Запросы
0
Пакеты
0
Релизы
0
Аналитика
Безопасность
main
sklearn/discriminant_analysis.py
1 198 строк
43 KB
Daniel Herrera-Esposito
FIX Inconsistencies between solvers in `LinearDiscriminantAnalysis` and `QuadraticDiscriminantAnalysis` (#33626)
30 июл 2026, 01:58
Не верифицирован
30 июл 2026, 01:58
76e8196
Код
Авторство
О чём код?
"""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 sklearn.base import ( BaseEstimator, ClassifierMixin, ClassNamePrefixFeaturesOutMixin, TransformerMixin, _fit_context, ) from sklearn.covariance import empirical_covariance, ledoit_wolf, shrunk_covariance from sklearn.linear_model._base import LinearClassifierMixin from sklearn.preprocessing import StandardScaler from sklearn.utils._array_api import ( _expit, array_device, check_same_namespace, get_namespace, size, ) from sklearn.utils._param_validation import HasMethods, Interval, StrOptions from sklearn.utils.extmath import softmax from sklearn.utils.multiclass import check_classification_targets, unique_labels from sklearn.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=array_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` 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 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` 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 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, _ = 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, dtype=X.dtype, device=array_device(X)) # 2) Within variance scaling X = xp.sqrt(fac) * (Xc / std) # SVD of centered (within)scaled data _, 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, n_features = 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(n_samples) 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, n_features) 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) check_same_namespace(X, self, attribute="coef_", method="transform") 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) check_same_namespace(X, self, attribute="coef_", method="predict_proba") xp, _ = get_namespace(X) decision = self.decision_function(X) if size(self.classes_) == 2: proba = _expit(decision, xp=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. """ check_same_namespace(X, self, attribute="coef_", method="predict_log_proba") xp, _ = get_namespace(X) prediction = self.predict_proba(X) smallest_normal = xp.finfo(prediction.dtype).smallest_normal 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 overrides for the docstring. 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 ---------- solver : {'svd', '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. - 'eigen': Eigenvalue decomposition. Can be combined with shrinkage or custom covariance estimator. 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. Enabling shrinkage is expected to improve the model when some classes have a relatively small number of training data points compared to the number of features by mitigating overfitting during the covariance estimation step. This should be left to `None` if `covariance_estimator` is used. Note that shrinkage works only with 'eigen' solver. 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 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 the 'eigen' solver. 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 estimates use the maximum likelihood (biased) covariance estimator. 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`, 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 = { "solver": [StrOptions({"svd", "eigen"})], "shrinkage": [StrOptions({"auto"}), Interval(Real, 0, 1, closed="both"), None], "priors": ["array-like", None], "reg_param": [Interval(Real, 0, 1, closed="both")], "store_covariance": ["boolean"], "tol": [Interval(Real, 0, None, closed="left")], "covariance_estimator": [HasMethods("fit"), None], } def __init__( self, *, solver="svd", shrinkage=None, priors=None, reg_param=0.0, store_covariance=False, tol=1.0e-4, covariance_estimator=None, ): self.solver = solver self.shrinkage = shrinkage self.priors = priors self.reg_param = reg_param self.store_covariance = store_covariance self.tol = tol self.covariance_estimator = covariance_estimator def _solve_eigen(self, X): """Eigenvalue solver. The eigenvalue solver uses the eigen decomposition of the data to compute the rotation and scaling matrices used for scoring new samples. This solver supports use of any covariance estimator. Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. """ n_samples, n_features = X.shape cov = _cov(X, self.shrinkage, self.covariance_estimator) scaling, rotation = linalg.eigh(cov) # scalings are eigenvalues rotation = rotation[:, np.argsort(scaling)[::-1]] # sort eigenvectors scaling = scaling[np.argsort(scaling)[::-1]] # sort eigenvalues return scaling, rotation, cov def _solve_svd(self, X): """SVD solver for Quadratic Discriminant Analysis. Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. """ n_samples, n_features = X.shape mean = X.mean(0) Xc = X - mean # Xc = U * S * V.T _, S, Vt = np.linalg.svd(Xc, full_matrices=False) scaling = (S**2) / n_samples # scalings are squared singular values scaling = ((1 - self.reg_param) * scaling) + self.reg_param rotation = Vt.T cov = None if self.store_covariance: # cov = V * (S^2 / n) * V.T cov = scaling * Vt.T @ Vt return scaling, rotation, cov @_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_ = np.unique(y) 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 " f"{n_classes} class." ) if self.priors is None: _, cnts = np.unique(y, return_counts=True) self.priors_ = cnts / float(n_samples) else: self.priors_ = np.array(self.priors) if self.solver == "svd": if self.shrinkage is not None: # Support for `shrinkage` could be implemented as in # https://github.com/scikit-learn/scikit-learn/issues/32590 raise NotImplementedError("shrinkage not supported with 'svd' solver.") if self.covariance_estimator is not None: raise ValueError( "covariance_estimator is not supported with solver='svd'. " "Try solver='eigen' instead." ) specific_solver = self._solve_svd elif self.solver == "eigen": specific_solver = self._solve_eigen means = [] cov = [] scalings = [] rotations = [] for class_idx, class_label in enumerate(self.classes_): X_class = X[y == class_label, :] if len(X_class) == 1: raise ValueError( "y has only 1 sample in class %s, covariance is ill defined." % str(self.classes_[class_idx]) ) mean_class = X_class.mean(0) means.append(mean_class) scaling_class, rotation_class, cov_class = specific_solver(X_class) rank = np.sum(scaling_class > self.tol) if rank < n_features: n_samples_class = X_class.shape[0] if self.solver == "svd" and n_samples_class <= n_features: raise linalg.LinAlgError( f"The covariance matrix of class {class_label} is not full " f"rank. When using `solver='svd'` the number of samples in " f"each class should be more than the number of features, but " f"class {class_label} has {n_samples_class} samples and " f"{n_features} features. Try using `solver='eigen'` and " f"setting the parameter `shrinkage` for regularization." ) else: msg_param = "shrinkage" if self.solver == "eigen" else "reg_param" raise linalg.LinAlgError( f"The covariance matrix of class {class_label} is not full " f"rank. Increase the value of `{msg_param}` to reduce the " f"collinearity.", ) cov.append(cov_class) scalings.append(scaling_class) rotations.append(rotation_class) if self.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. """ # Only overrides for the docstring. return super().decision_function(X)