/usr/lib/python2.7/dist-packages/sklearn/qda.py is in python-sklearn 0.14.1-3.
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Quadratic Discriminant Analysis
"""
# Author: Matthieu Perrot <matthieu.perrot@gmail.com>
#
# License: BSD 3 clause
import warnings
import numpy as np
from .base import BaseEstimator, ClassifierMixin
from .externals.six.moves import xrange
from .utils.fixes import unique
from .utils import check_arrays, array2d, column_or_1d
__all__ = ['QDA']
class QDA(BaseEstimator, ClassifierMixin):
"""
Quadratic Discriminant Analysis (QDA)
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.
Parameters
----------
priors : array, optional, shape = [n_classes]
Priors on classes
reg_param : float, optional
Regularizes the covariance estimate as
``(1-reg_param)*Sigma + reg_param*np.eye(n_features)``
Attributes
----------
`covariances_` : list of array-like, shape = [n_features, n_features]
Covariance matrices of each class.
`means_` : array-like, shape = [n_classes, n_features]
Class means.
`priors_` : array-like, shape = [n_classes]
Class priors (sum to 1).
`rotations_` : list of arrays
For each class an array of shape [n_samples, n_samples], the
rotation of the Gaussian distribution, i.e. its principal axis.
`scalings_` : array-like, shape = [n_classes, n_features]
Contains the scaling of the Gaussian
distributions along the principal axes for each
class, i.e. the variance in the rotated coordinate system.
Examples
--------
>>> from sklearn.qda import QDA
>>> 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 = QDA()
>>> clf.fit(X, y)
QDA(priors=None, reg_param=0.0)
>>> print(clf.predict([[-0.8, -1]]))
[1]
See also
--------
sklearn.lda.LDA: Linear discriminant analysis
"""
def __init__(self, priors=None, reg_param=0.):
self.priors = np.asarray(priors) if priors is not None else None
self.reg_param = reg_param
def fit(self, X, y, store_covariances=False, tol=1.0e-4):
"""
Fit the QDA model according to the given training data and parameters.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Training vector, where n_samples in the number of samples and
n_features is the number of features.
y : array, shape = [n_samples]
Target values (integers)
store_covariances : boolean
If True the covariance matrices are computed and stored in the
`self.covariances_` attribute.
"""
X, y = check_arrays(X, y)
y = column_or_1d(y, warn=True)
self.classes_, y = unique(y, return_inverse=True)
n_samples, n_features = X.shape
n_classes = len(self.classes_)
if n_classes < 2:
raise ValueError('y has less than 2 classes')
if self.priors is None:
self.priors_ = np.bincount(y) / float(n_samples)
else:
self.priors_ = self.priors
cov = None
if store_covariances:
cov = []
means = []
scalings = []
rotations = []
for ind in xrange(n_classes):
Xg = X[y == ind, :]
meang = Xg.mean(0)
means.append(meang)
Xgc = Xg - meang
# Xgc = U * S * V.T
U, S, Vt = np.linalg.svd(Xgc, full_matrices=False)
rank = np.sum(S > 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 store_covariances:
# cov = V * (S^2 / (n-1)) * V.T
cov.append(np.dot(S2 * Vt.T, Vt))
scalings.append(S2)
rotations.append(Vt.T)
if store_covariances:
self.covariances_ = cov
self.means_ = np.asarray(means)
self.scalings_ = np.asarray(scalings)
self.rotations_ = rotations
return self
@property
def scalings(self): # pragma: no cover
warnings.warn("QDA.scalings is deprecated and will be removed in 0.15."
" Use QDA.scalings_ instead.", DeprecationWarning,
stacklevel=2)
return self.scalings_
@property
def rotations(self): # pragma: no cover
warnings.warn("QDA.rotations is deprecated and will be removed in "
"0.15. Use QDA.rotations_ instead.", DeprecationWarning,
stacklevel=2)
return self.rotations_
def _decision_function(self, X):
X = array2d(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, 1))
norm2 = np.array(norm2).T # shape = [len(X), n_classes]
return (-0.5 * (norm2 + np.sum(np.log(self.scalings_), 1))
+ np.log(self.priors_))
def decision_function(self, X):
"""Apply decision function to an array of samples.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Array of samples (test vectors).
Returns
-------
C : array, shape = [n_samples, n_classes] or [n_samples,]
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, shape = [n_samples, n_features]
Returns
-------
C : array, 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, shape = [n_samples, n_features]
Array of samples/test vectors.
Returns
-------
C : array, 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 posterior probabilities of classification.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Array of samples/test vectors.
Returns
-------
C : array, 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_)
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