/usr/lib/python2.7/dist-packages/sklearn/lda.py is in python-sklearn 0.14.1-3.
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The :mod:`sklearn.lda` module implements Linear Discriminant Analysis (LDA).
"""
from __future__ import print_function
# Authors: Matthieu Perrot
# Mathieu Blondel
import warnings
import numpy as np
from scipy import linalg
from .base import BaseEstimator, ClassifierMixin, TransformerMixin
from .utils.extmath import logsumexp
from .utils.fixes import unique
from .utils import check_arrays, array2d, column_or_1d
__all__ = ['LDA']
class LDA(BaseEstimator, ClassifierMixin, TransformerMixin):
"""
Linear Discriminant Analysis (LDA)
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.
Parameters
----------
n_components: int
Number of components (< n_classes - 1) for dimensionality reduction
priors : array, optional, shape = [n_classes]
Priors on classes
Attributes
----------
`coef_` : array-like, shape = [rank, n_classes - 1]
Coefficients of the features in the linear decision
function. rank is min(rank_features, n_classes) where
rank_features is the dimensionality of the spaces spanned
by the features (i.e. n_features excluding redundant features).
`covariance_` : array-like, shape = [n_features, n_features]
Covariance matrix (shared by all classes).
`means_` : array-like, shape = [n_classes, n_features]
Class means.
`priors_` : array-like, shape = [n_classes]
Class priors (sum to 1).
`scalings_` : array-like, shape = [rank, n_classes - 1]
Scaling of the features in the space spanned by the class
centroids.
`xbar_` : float, shape = [n_features]
Overall mean.
Examples
--------
>>> import numpy as np
>>> from sklearn.lda import LDA
>>> 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 = LDA()
>>> clf.fit(X, y)
LDA(n_components=None, priors=None)
>>> print(clf.predict([[-0.8, -1]]))
[1]
See also
--------
sklearn.qda.QDA: Quadratic discriminant analysis
"""
def __init__(self, n_components=None, priors=None):
self.n_components = n_components
self.priors = np.asarray(priors) if priors is not None else None
if self.priors is not None:
if (self.priors < 0).any():
raise ValueError('priors must be non-negative')
if self.priors.sum() != 1:
print('warning: the priors do not sum to 1. Renormalizing')
self.priors = self.priors / self.priors.sum()
def fit(self, X, y, store_covariance=False, tol=1.0e-4):
"""
Fit the LDA 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_covariance : boolean
If True the covariance matrix (shared by all classes) is computed
and stored in `self.covariance_` attribute.
"""
X, y = check_arrays(X, y, sparse_format='dense')
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
# Group means n_classes*n_features matrix
means = []
Xc = []
cov = None
if store_covariance:
cov = np.zeros((n_features, n_features))
for ind in range(n_classes):
Xg = X[y == ind, :]
meang = Xg.mean(0)
means.append(meang)
# centered group data
Xgc = Xg - meang
Xc.append(Xgc)
if store_covariance:
cov += np.dot(Xgc.T, Xgc)
if store_covariance:
cov /= (n_samples - n_classes)
self.covariance_ = cov
self.means_ = np.asarray(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 > tol)
if rank < n_features:
warnings.warn("Variables are collinear")
# Scaling of within covariance is: V' 1/S
scalings = (V[:rank] / std).T / S[:rank]
## ----------------------------
## 3) Between variance scaling
# Overall mean
xbar = np.dot(self.priors_, self.means_)
# Scale weighted centers
X = np.dot(((np.sqrt((n_samples * self.priors_) * fac)) *
(means - 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)
rank = np.sum(S > tol * S[0])
# compose the scalings
self.scalings_ = np.dot(scalings, V.T[:, :rank])
self.xbar_ = xbar
# weight vectors / centroids
self.coef_ = np.dot(self.means_ - self.xbar_, self.scalings_)
self.intercept_ = (-0.5 * np.sum(self.coef_ ** 2, axis=1) +
np.log(self.priors_))
return self
@property
def scaling(self): # pragma: no cover
warnings.warn("LDA.scaling is deprecated and will be removed in 0.15."
" Use LDA.scalings_ instead.", DeprecationWarning,
stacklevel=2)
return self.scalings_
def _decision_function(self, X):
X = array2d(X)
# center and scale data
X = np.dot(X - self.xbar_, self.scalings_)
return np.dot(X, self.coef_.T) + self.intercept_
def decision_function(self, X):
"""
This function returns the decision function values related to each
class on an array of test vectors X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
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)
if len(self.classes_) == 2:
return dec_func[:, 1] - dec_func[:, 0]
return dec_func
def transform(self, X):
"""
Project the data so as to maximize class separation (large separation
between projected class means and small variance within each class).
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Returns
-------
X_new : array, shape = [n_samples, n_components]
"""
X = array2d(X)
# center and scale data
X = np.dot(X - self.xbar_, self.scalings_)
n_comp = X.shape[1] if self.n_components is None else self.n_components
return np.dot(X, self.coef_[:n_comp].T)
def predict(self, X):
"""
This function does 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):
"""
This function returns posterior probabilities of classification
according to each class on an array of test vectors X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Returns
-------
C : array, shape = [n_samples, n_classes]
"""
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):
"""
This function returns posterior log-probabilities of classification
according to each class on an array of test vectors X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Returns
-------
C : array, shape = [n_samples, n_classes]
"""
values = self._decision_function(X)
loglikelihood = (values - values.max(axis=1)[:, np.newaxis])
normalization = logsumexp(loglikelihood, axis=1)
return loglikelihood - normalization[:, np.newaxis]
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