""" Linear Discriminant Analysis and Quadratic Discriminant Analysis """ # Authors: Clemens Brunner # Martin Billinger # Matthieu Perrot # Mathieu Blondel # License: BSD 3-Clause import warnings import numpy as np from scipy import linalg from scipy.special import expit from .base import BaseEstimator, TransformerMixin, ClassifierMixin from .linear_model._base import LinearClassifierMixin from .covariance import ledoit_wolf, empirical_covariance, shrunk_covariance from .utils.multiclass import unique_labels from .utils import check_array from .utils.validation import check_is_fitted from .utils.multiclass import check_classification_targets from .utils.extmath import softmax from .preprocessing import StandardScaler from .utils.validation import _deprecate_positional_args __all__ = ['LinearDiscriminantAnalysis', 'QuadraticDiscriminantAnalysis'] def _cov(X, shrinkage=None): """Estimate covariance matrix (using optional shrinkage). 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. Returns ------- s : ndarray of shape (n_features, n_features) Estimated covariance matrix. """ 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) else: raise ValueError('unknown shrinkage parameter') elif isinstance(shrinkage, float) or isinstance(shrinkage, int): if shrinkage < 0 or shrinkage > 1: raise ValueError('shrinkage parameter must be between 0 and 1') s = shrunk_covariance(empirical_covariance(X), shrinkage) else: raise TypeError('shrinkage must be of string or int type') 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. """ classes, y = np.unique(y, return_inverse=True) cnt = np.bincount(y) means = np.zeros(shape=(len(classes), X.shape[1])) np.add.at(means, y, X) means /= cnt[:, None] return means def _class_cov(X, y, priors, shrinkage=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. 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)) return cov class LinearDiscriminantAnalysis(BaseEstimator, LinearClassifierMixin, TransformerMixin): """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 *LinearDiscriminantAnalysis*. Read more in the :ref:`User Guide `. 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. - 'eigen': Eigenvalue decomposition, can be combined with shrinkage. 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. Note that shrinkage works only with 'lsqr' and 'eigen' solvers. 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. store_covariance : bool, default=False If True, explicitely 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 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. See also -------- sklearn.discriminant_analysis.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] """ @_deprecate_positional_args def __init__(self, *, solver='svd', shrinkage=None, priors=None, n_components=None, store_covariance=False, tol=1e-4): 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 def _solve_lsqr(self, X, y, shrinkage): """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 optional shrinkage), 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. 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) 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): """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 optional shrinkage). 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. 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) Sw = self.covariance_ # within scatter St = _cov(X, shrinkage) # 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. """ n_samples, n_features = X.shape n_classes = len(self.classes_) 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_ = np.dot(self.priors_, self.means_) Xc = np.concatenate(Xc, axis=0) # 1) within (univariate) scaling by with classes std-dev std = Xc.std(axis=0) # avoid division by zero in normalization std[std == 0] = 1. fac = 1. / (n_samples - n_classes) # 2) Within variance scaling X = np.sqrt(fac) * (Xc / std) # SVD of centered (within)scaled data U, S, V = linalg.svd(X, full_matrices=False) rank = np.sum(S > self.tol) # Scaling of within covariance is: V' 1/S scalings = (V[:rank] / std).T / S[:rank] # 3) Between variance scaling # Scale weighted centers X = np.dot(((np.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, V = linalg.svd(X, full_matrices=0) self.explained_variance_ratio_ = (S**2 / np.sum( S**2))[:self._max_components] rank = np.sum(S > self.tol * S[0]) self.scalings_ = np.dot(scalings, V.T[:, :rank]) coef = np.dot(self.means_ - self.xbar_, self.scalings_) self.intercept_ = (-0.5 * np.sum(coef ** 2, axis=1) + np.log(self.priors_)) self.coef_ = np.dot(coef, self.scalings_.T) self.intercept_ -= np.dot(self.xbar_, self.coef_.T) def fit(self, X, y): """Fit LinearDiscriminantAnalysis model according to the given training data and parameters. .. versionchanged:: 0.19 *store_covariance* has been moved to main constructor. .. versionchanged:: 0.19 *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. """ X, y = self._validate_data(X, y, ensure_min_samples=2, estimator=self, dtype=[np.float64, np.float32]) self.classes_ = unique_labels(y) n_samples, _ = X.shape n_classes = len(self.classes_) 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 _, y_t = np.unique(y, return_inverse=True) # non-negative ints self.priors_ = np.bincount(y_t) / float(len(y)) else: self.priors_ = np.asarray(self.priors) if (self.priors_ < 0).any(): raise ValueError("priors must be non-negative") if not np.isclose(self.priors_.sum(), 1.0): 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(len(self.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') self._solve_svd(X, y) elif self.solver == 'lsqr': self._solve_lsqr(X, y, shrinkage=self.shrinkage) elif self.solver == 'eigen': self._solve_eigen(X, y, shrinkage=self.shrinkage) else: raise ValueError("unknown solver {} (valid solvers are 'svd', " "'lsqr', and 'eigen').".format(self.solver)) if self.classes_.size == 2: # treat binary case as a special case self.coef_ = np.array(self.coef_[1, :] - self.coef_[0, :], ndmin=2, dtype=X.dtype) self.intercept_ = np.array(self.intercept_[1] - self.intercept_[0], ndmin=1, dtype=X.dtype) 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) Transformed data. """ if self.solver == 'lsqr': raise NotImplementedError("transform not implemented for 'lsqr' " "solver (use 'svd' or 'eigen').") check_is_fitted(self) X = check_array(X) if self.solver == 'svd': X_new = np.dot(X - self.xbar_, self.scalings_) elif self.solver == 'eigen': X_new = np.dot(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) decision = self.decision_function(X) if self.classes_.size == 2: proba = expit(decision) return np.vstack([1-proba, proba]).T 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. """ return np.log(self.predict_proba(X)) 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 override for the doc return super().decision_function(X) class QuadraticDiscriminantAnalysis(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 *QuadraticDiscriminantAnalysis* Read more in the :ref:`User Guide `. Parameters ---------- priors : ndarray 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 explicitely computed and stored in the `self.covariance_` attribute. .. versionadded:: 0.17 tol : float, default=1.0e-4 Absolute threshold for a singular value to be considered significant, used to estimate the rank of `Xk` where `Xk` is the centered matrix of samples in class k. This parameter does not affect the predictions. It only controls a warning that is raised when features are considered to be colinear. .. 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. 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] See also -------- sklearn.discriminant_analysis.LinearDiscriminantAnalysis: Linear Discriminant Analysis """ @_deprecate_positional_args def __init__(self, *, priors=None, reg_param=0., store_covariance=False, tol=1.0e-4): self.priors = np.asarray(priors) if priors is not None else None self.reg_param = reg_param self.store_covariance = store_covariance self.tol = tol 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) """ X, y = self._validate_data(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_ = 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) rank = np.sum(S > self.tol) if rank < n_features: warnings.warn("Variables are collinear") S2 = (S ** 2) / (len(Xg) - 1) S2 = ((1 - self.reg_param) * S2) + self.reg_param 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 = check_array(X) 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. """ dec_func = self._decision_function(X) # handle special case of two classes if len(self.classes_) == 2: return dec_func[:, 1] - dec_func[:, 0] return dec_func 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) Returns ------- C : ndarray of shape (n_samples,) """ d = self._decision_function(X) y_pred = self.classes_.take(d.argmax(1)) return y_pred 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. """ values = self._decision_function(X) # compute the likelihood of the underlying gaussian models # up to a multiplicative constant. likelihood = np.exp(values - values.max(axis=1)[:, np.newaxis]) # compute posterior probabilities return likelihood / likelihood.sum(axis=1)[:, np.newaxis] 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 probas_ = self.predict_proba(X) return np.log(probas_)