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naive_bayes.py
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naive_bayes.py
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# -*- coding: utf-8 -*-
"""
The :mod:`sklearn.naive_bayes` module implements Naive Bayes algorithms. These
are supervised learning methods based on applying Bayes' theorem with strong
(naive) feature independence assumptions.
"""
# Author: Vincent Michel <vincent.michel@inria.fr>
# Minor fixes by Fabian Pedregosa
# Amit Aides <amitibo@tx.technion.ac.il>
# Yehuda Finkelstein <yehudaf@tx.technion.ac.il>
# Lars Buitinck <L.J.Buitinck@uva.nl>
# (parts based on earlier work by Mathieu Blondel)
#
# License: BSD Style.
from abc import ABCMeta, abstractmethod
import numpy as np
from scipy.sparse import issparse
import warnings
from .base import BaseEstimator, ClassifierMixin
from .preprocessing import binarize, LabelBinarizer
from .utils import array2d, atleast2d_or_csr
from .utils.extmath import safe_sparse_dot, logsumexp
from .utils import check_arrays
__all__ = ['BernoulliNB', 'GaussianNB', 'MultinomialNB']
class BaseNB(BaseEstimator, ClassifierMixin):
"""Abstract base class for naive Bayes estimators"""
__metaclass__ = ABCMeta
@abstractmethod
def _joint_log_likelihood(self, X):
"""Compute the unnormalized posterior log probability of X
I.e. ``log P(c) + log P(x|c)`` for all rows x of X, as an array-like of
shape [n_classes, n_samples].
Input is passed to _joint_log_likelihood as-is by predict,
predict_proba and predict_log_proba.
"""
def predict(self, X):
"""
Perform classification on an array of test vectors X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Returns
-------
C : array, shape = [n_samples]
Predicted target values for X
"""
jll = self._joint_log_likelihood(X)
return self.classes_[np.argmax(jll, axis=1)]
def predict_log_proba(self, X):
"""
Return log-probability estimates for the test vector X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Returns
-------
C : array-like, shape = [n_samples, n_classes]
Returns the log-probability of the sample for each class
in the model, where classes are ordered arithmetically.
"""
jll = self._joint_log_likelihood(X)
# normalize by P(x) = P(f_1, ..., f_n)
log_prob_x = logsumexp(jll, axis=1)
return jll - np.atleast_2d(log_prob_x).T
def predict_proba(self, X):
"""
Return probability estimates for the test vector X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Returns
-------
C : array-like, shape = [n_samples, n_classes]
Returns the probability of the sample for each class in
the model, where classes are ordered arithmetically.
"""
return np.exp(self.predict_log_proba(X))
class GaussianNB(BaseNB):
"""
Gaussian Naive Bayes (GaussianNB)
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 vector relative to X
Attributes
----------
`class_prior_` : array, shape = [n_classes]
probability of each class.
`theta_` : array, shape = [n_classes, n_features]
mean of each feature per class
`sigma_` : array, shape = [n_classes, n_features]
variance of each feature per class
Examples
--------
>>> 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])
>>> from sklearn.naive_bayes import GaussianNB
>>> clf = GaussianNB()
>>> clf.fit(X, Y)
GaussianNB()
>>> print(clf.predict([[-0.8, -1]]))
[1]
"""
def fit(self, X, y):
"""Fit Gaussian Naive Bayes according to X, y
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples
and n_features is the number of features.
y : array-like, shape = [n_samples]
Target values.
Returns
-------
self : object
Returns self.
"""
X, y = check_arrays(X, y, sparse_format='dense')
n_samples, n_features = X.shape
if n_samples != y.shape[0]:
raise ValueError("X and y have incompatible shapes")
self.classes_ = unique_y = np.unique(y)
n_classes = unique_y.shape[0]
self.theta_ = np.zeros((n_classes, n_features))
self.sigma_ = np.zeros((n_classes, n_features))
self.class_prior_ = np.zeros(n_classes)
epsilon = 1e-9
for i, y_i in enumerate(unique_y):
self.theta_[i, :] = np.mean(X[y == y_i, :], axis=0)
self.sigma_[i, :] = np.var(X[y == y_i, :], axis=0) + epsilon
self.class_prior_[i] = np.float(np.sum(y == y_i)) / n_samples
return self
def _joint_log_likelihood(self, X):
X = array2d(X)
joint_log_likelihood = []
for i in xrange(np.size(self.classes_)):
jointi = np.log(self.class_prior_[i])
n_ij = - 0.5 * np.sum(np.log(np.pi * self.sigma_[i, :]))
n_ij -= 0.5 * np.sum(((X - self.theta_[i, :]) ** 2) /
(self.sigma_[i, :]), 1)
joint_log_likelihood.append(jointi + n_ij)
joint_log_likelihood = np.array(joint_log_likelihood).T
return joint_log_likelihood
class BaseDiscreteNB(BaseNB):
"""Abstract base class for naive Bayes on discrete/categorical data
Any estimator based on this class should provide:
__init__
_joint_log_likelihood(X) as per BaseNB
"""
def fit(self, X, y, sample_weight=None, class_prior=None):
"""Fit Naive Bayes classifier according to X, y
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples and
n_features is the number of features.
y : array-like, shape = [n_samples]
Target values.
sample_weight : array-like, shape = [n_samples], optional
Weights applied to individual samples (1. for unweighted).
Returns
-------
self : object
Returns self.
"""
X = atleast2d_or_csr(X)
labelbin = LabelBinarizer()
Y = labelbin.fit_transform(y)
self.classes_ = labelbin.classes_
n_classes = len(self.classes_)
if Y.shape[1] == 1:
Y = np.concatenate((1 - Y, Y), axis=1)
if X.shape[0] != Y.shape[0]:
msg = "X and y have incompatible shapes."
if issparse(X):
msg += "\nNote: Sparse matrices cannot be indexed w/ boolean \
masks (use `indices=True` in CV)."
raise ValueError(msg)
if sample_weight is not None:
Y *= array2d(sample_weight).T
if class_prior is not None:
warnings.warn('class_prior has been made an ``__init__`` parameter'
' and will be removed from fit in version 0.15.',
DeprecationWarning)
else:
class_prior = self.class_prior
if class_prior:
if len(class_prior) != n_classes:
raise ValueError("Number of priors must match number of"
" classes.")
self.class_log_prior_ = np.log(class_prior)
elif self.fit_prior:
# empirical prior, with sample_weight taken into account
y_freq = Y.sum(axis=0)
self.class_log_prior_ = np.log(y_freq) - np.log(y_freq.sum())
else:
self.class_log_prior_ = np.zeros(n_classes) - np.log(n_classes)
# N_c_i is the count of feature i in all samples of class c.
# N_c is the denominator.
N_c, N_c_i = self._count(X, Y)
self.feature_log_prob_ = np.log(N_c_i) - np.log(N_c.reshape(-1, 1))
return self
# XXX The following is a stopgap measure; we need to set the dimensions
# of class_log_prior_ and feature_log_prob_ correctly.
def _get_coef(self):
return (self.feature_log_prob_[1]
if len(self.classes_) == 2 else self.feature_log_prob_)
def _get_intercept(self):
return (self.class_log_prior_[1]
if len(self.classes_) == 2 else self.class_log_prior_)
coef_ = property(_get_coef)
intercept_ = property(_get_intercept)
class MultinomialNB(BaseDiscreteNB):
"""
Naive Bayes classifier for multinomial models
The multinomial Naive Bayes classifier is suitable for classification with
discrete features (e.g., word counts for text classification). The
multinomial distribution normally requires integer feature counts. However,
in practice, fractional counts such as tf-idf may also work.
Parameters
----------
alpha : float, optional (default=1.0)
Additive (Laplace/Lidstone) smoothing parameter
(0 for no smoothing).
fit_prior : boolean
Whether to learn class prior probabilities or not.
If false, a uniform prior will be used.
class_prior : array-like, size=[n_classes,]
Prior probabilities of the classes. If specified the priors are not
adjusted according to the data.
Attributes
----------
`intercept_`, `class_log_prior_` : array, shape = [n_classes]
Smoothed empirical log probability for each class.
`feature_log_prob_`, `coef_` : array, shape = [n_classes, n_features]
Empirical log probability of features
given a class, P(x_i|y).
(`intercept_` and `coef_` are properties
referring to `class_log_prior_` and
`feature_log_prob_`, respectively.)
Examples
--------
>>> import numpy as np
>>> X = np.random.randint(5, size=(6, 100))
>>> Y = np.array([1, 2, 3, 4, 5, 6])
>>> from sklearn.naive_bayes import MultinomialNB
>>> clf = MultinomialNB()
>>> clf.fit(X, Y)
MultinomialNB(alpha=1.0, class_prior=None, fit_prior=True)
>>> print(clf.predict(X[2]))
[3]
Notes
-----
For the rationale behind the names `coef_` and `intercept_`, i.e.
naive Bayes as a linear classifier, see J. Rennie et al. (2003),
Tackling the poor assumptions of naive Bayes text classifiers, ICML.
"""
def __init__(self, alpha=1.0, fit_prior=True, class_prior=None):
self.alpha = alpha
self.fit_prior = fit_prior
self.class_prior = class_prior
def _count(self, X, Y):
"""Count and smooth feature occurrences."""
if np.any((X.data if issparse(X) else X) < 0):
raise ValueError("Input X must be non-negative.")
N_c_i = safe_sparse_dot(Y.T, X) + self.alpha
N_c = np.sum(N_c_i, axis=1)
return N_c, N_c_i
def _joint_log_likelihood(self, X):
"""Calculate the posterior log probability of the samples X"""
X = atleast2d_or_csr(X)
return (safe_sparse_dot(X, self.feature_log_prob_.T)
+ self.class_log_prior_)
class BernoulliNB(BaseDiscreteNB):
"""Naive Bayes classifier for multivariate Bernoulli models.
Like MultinomialNB, this classifier is suitable for discrete data. The
difference is that while MultinomialNB works with occurrence counts,
BernoulliNB is designed for binary/boolean features.
Parameters
----------
alpha : float, optional (default=1.0)
Additive (Laplace/Lidstone) smoothing parameter
(0 for no smoothing).
binarize : float or None, optional
Threshold for binarizing (mapping to booleans) of sample features.
If None, input is presumed to already consist of binary vectors.
fit_prior : boolean
Whether to learn class prior probabilities or not.
If false, a uniform prior will be used.
class_prior : array-like, size=[n_classes,]
Prior probabilities of the classes. If specified the priors are not
adjusted according to the data.
Attributes
----------
`class_log_prior_` : array, shape = [n_classes]
Log probability of each class (smoothed).
`feature_log_prob_` : array, shape = [n_classes, n_features]
Empirical log probability of features given a class, P(x_i|y).
Examples
--------
>>> import numpy as np
>>> X = np.random.randint(2, size=(6, 100))
>>> Y = np.array([1, 2, 3, 4, 4, 5])
>>> from sklearn.naive_bayes import BernoulliNB
>>> clf = BernoulliNB()
>>> clf.fit(X, Y)
BernoulliNB(alpha=1.0, binarize=0.0, class_prior=None, fit_prior=True)
>>> print(clf.predict(X[2]))
[3]
References
----------
C.D. Manning, P. Raghavan and H. Schütze (2008). Introduction to
Information Retrieval. Cambridge University Press, pp. 234–265.
A. McCallum and K. Nigam (1998). A comparison of event models for naive
Bayes text classification. Proc. AAAI/ICML-98 Workshop on Learning for
Text Categorization, pp. 41–48.
V. Metsis, I. Androutsopoulos and G. Paliouras (2006). Spam filtering with
naive Bayes -- Which naive Bayes? 3rd Conf. on Email and Anti-Spam (CEAS).
"""
def __init__(self, alpha=1.0, binarize=.0, fit_prior=True,
class_prior=None):
self.alpha = alpha
self.binarize = binarize
self.fit_prior = fit_prior
self.class_prior = class_prior
def _count(self, X, Y):
"""Count and smooth feature occurrences."""
if self.binarize is not None:
X = binarize(X, threshold=self.binarize)
N_c_i = safe_sparse_dot(Y.T, X) + self.alpha
N_c = Y.sum(axis=0) + self.alpha * Y.shape[1]
return N_c, N_c_i
def _joint_log_likelihood(self, X):
"""Calculate the posterior log probability of the samples X"""
X = atleast2d_or_csr(X)
if self.binarize is not None:
X = binarize(X, threshold=self.binarize)
n_classes, n_features = self.feature_log_prob_.shape
n_samples, n_features_X = X.shape
if n_features_X != n_features:
raise ValueError("Expected input with %d features, got %d instead"
% (n_features, n_features_X))
neg_prob = np.log(1 - np.exp(self.feature_log_prob_))
# Compute neg_prob · (1 - X).T as ∑neg_prob - X · neg_prob
X_neg_prob = (neg_prob.sum(axis=1)
- safe_sparse_dot(X, neg_prob.T))
jll = safe_sparse_dot(X, self.feature_log_prob_.T) + X_neg_prob
return jll + self.class_log_prior_