朴素贝叶斯及python实现
朴素贝叶斯
是基于贝叶斯定理与特征条件独立假设的分类方法。对于给定数据集，首先基于特征条件独立假设学习输入输出的联合概率分布；然后基于此模型，对于给定的输入x，利用贝叶斯定理求出后验概率最大的输出y。
式（2）为极大似然估计。
# 先验概率
def pAbusice(self, labels):
N = len(labels)
label = dict(Counter(labels))
pa = {
}
for key, val in label.items():
pa[key] = float(val) / float(N)
return pa
# 极大似然估计
def MIE(self, data):
mie = {
label: [] for label in data}
for label, values in data.items():
n = len(data[label])
for value in zip(*values):
val = Counter(value)
m = {
}
for k, v in val.items():
p = v / n
m[k] = p
mie[label].append(m)
return mie
贝叶斯估计
用极大似然估计可能会出现所要估计得概率值为0的情况。这是会影响到后验概率的计算结果，使分类产生偏差。解决这一问题的方法使采用贝叶斯估计法。
条件概率的贝叶斯方法：
P λ ( X ( j ) = a j l ∣ Y = c k ) = ∑ i = 1 N I ( x i ( j ) = a j l , y i = c k ) + λ ∑ i = 1 N I ( y i = c k ) + S j λ P_\lambda(X^{(j)}=a_{jl}|Y=c_k)=\frac{\sum_{i=1}^{N}I(x_{i}^{(j)}=a_{jl},y_i=c_k)+\lambda}{\sum_{i=1}^{N}I(y_i=c_k)+S_j \lambda} Pλ(X(j)=ajl∣Y=ck)=∑i=1NI(yi=ck)+Sjλ∑i=1NI(xi(j)=ajl,yi=ck)+λ
式中 λ ≥ 0 λ≥0 λ≥0。 λ = 0 λ=0 λ=0时，为极大似然估计； λ = 1 λ=1 λ=1时，称为拉普拉斯平滑。 S j S_j Sj是当 y i = c k y_i=c_k yi=ck时，特征的种类。
先验概率的贝叶斯估计：
P λ ( Y = c k ) = ∑ i = 1 N I ( y i = c i ) + λ N + K λ P_{\lambda}\left(Y=c_{k}\right)=\frac{\sum_{i=1}^{N} I\left(y_{i}=c_{i}\right)+\lambda}{N+K \lambda} Pλ(Y=ck)=N+Kλ∑i=1NI(yi=ci)+λ
N N N为总的标签数， K K K为总的标签种类。
利用上面两式代替算法一中的（1）、（2）得到贝叶斯方法的贝叶斯估计。
# 先验概率
def pAbusice(self, labels):
N = len(labels)
label = dict(Counter(labels))
K = len(label)
pa = {
}
for key, val in label.items():
a = val+self.lamda
b = N+self.lamda*K
pa[key] = a/b
return pa
# 极大似然估计
def MIE(self, data):
mie = {
label: [] for label in data}
for label, values in data.items():
n = len(data[label])
for value in zip(*values):
val = Counter(value)
S = len(val)
m = {
}
for k, v in val.items():
p = (v+self.lamda) / (n+S*self.lamda)
m[k] = p
mie[label].append(m)
return mie
G a u s s i a n N B GaussianNB GaussianNB高斯朴素贝叶斯
特征的可能性被假设为高斯
概率密度函数：
P ( x i ∣ y k ) = 1 2 π σ y k 2 e x p ( − ( x i − μ y k ) 2 2 σ y k 2 ) P(x_i | y_k)=\frac{1}{\sqrt{2\pi\sigma^2_{yk}}}exp(-\frac{(x_i-\mu_{yk})^2}{2\sigma^2_{yk}}) P(xi∣yk)=2πσyk21exp(−2σyk2(xi−μyk)2)
数学期望(mean)： μ \mu μ
方差： σ 2 = ∑ ( X − μ ) 2 N \sigma^2=\frac{\sum(X-\mu)^2}{N} σ2=N∑(X−μ)2
# 数学期望
@staticmethod
def mean(X):
return sum(X) / float(len(X))
# 标准差
def stdev(self, X):
avg = self.mean(X)
return math.sqrt(sum([pow(x - avg, 2) for x in X]) / float(len(X)))
# 概率密度函数
def gaussian_probability(self, x, mean, stdev):
exponent = math.exp(-(math.pow(x - mean, 2) / (2 * math.pow(stdev, 2))))
return (1 / (math.sqrt(2 * math.pi) * stdev)) * exponent
可运行代码
朴素贝叶斯
"""
朴素贝叶斯
"""
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from collections import Counter
import math
class NaiveBayes(object):
def __init__(self):
self.mie = None
self.pabusice = None
# 先验概率
def pAbusice(self, labels):
N = len(labels)
label = dict(Counter(labels))
pa = {
}
for key, val in label.items():
pa[key] = float(val) / float(N)
return pa
# 极大似然估计
def MIE(self, data):
mie = {
label: [] for label in data}
for label, values in data.items():
n = len(data[label])
for value in zip(*values):
val = Counter(value)
m = {
}
for k, v in val.items():
p = v / n
m[k] = p
mie[label].append(m)
return mie
# 训练入口
def fit(self, X_train, y_train):
n = len(X_train)
m = len(X_train[0])
# 数据处理
labels = list(set(y_train))
data = {
label: [] for label in labels}
for f, label in zip(X_train, y_train):
data[label].append(f)
# 先验概率
self.pabusice = self.pAbusice(y_train)
# 极大似然估计
self.mie = self.MIE(data)
# 预测值计算
def calculate_probabilities(self, X):
res = {
label: [] for label, val in self.pabusice.items()}
for label, value in self.pabusice.items():
ans = 1
mi = self.mie[label]
for x in range(len(X)):
ml = mi[x]
a = ml[X[x]]
ans *= a
ans *= value
res[label] = ans
return res
def predict(self, X_test):
label = sorted(
self.calculate_probabilities(X_test).items(),
key=lambda x: x[1]
)[-1][0]
return label
def score(self, X_test, y_test):
right = 0
for X, y in zip(X_test, y_test):
label = self.predict(X)
if label == y:
right += 1
return right / float(len(X_test))
if __name__ == '__main__':
X_train = [(1, 'S'), (1, 'M'), (1, 'M'), (1, 'S'), (1, 'S'),
(2, 'S'), (2, 'M'), (2, 'M'), (2, 'L'), (2, 'L'),
(3, 'L'), (3, 'M'), (3, 'M'), (3, 'L'), (3, 'L')]
y_train = [-1, -1, 1, 1, -1,
-1, -1, 1, 1, 1,
1, 1, 1, 1, -1]
model = NaiveBayes()
model.fit(X_train, y_train)
print(model.predict([2, 'S']))
贝叶斯估计
"""
朴素贝叶斯
"""
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from collections import Counter
import math
class NaiveBayes(object):
def __init__(self,lamda):
self.mie = None
self.pabusice = None
self.lamda = lamda
# 先验概率
def pAbusice(self, labels):
N = len(labels)
label = dict(Counter(labels))
K = len(label)
pa = {
}
for key, val in label.items():
a = val+self.lamda
b = N+self.lamda*K
pa[key] = a/b
return pa
# 极大似然估计
def MIE(self, data):
mie = {
label: [] for label in data}
for label, values in data.items():
n = len(data[label])
for value in zip(*values):
val = Counter(value)
S = len(val)
m = {
}
for k, v in val.items():
p = (v+self.lamda) / (n+S*self.lamda)
m[k] = p
mie[label].append(m)
return mie
# 训练入口
def fit(self, X_train, y_train):
n = len(X_train)
m = len(X_train[0])
# 数据处理
labels = list(set(y_train))
data = {
label: [] for label in labels}
for f, label in zip(X_train, y_train):
data[label].append(f)
# 先验概率
self.pabusice = self.pAbusice(y_train)
# 极大似然估计
self.mie = self.MIE(data)
# 预测值计算
def calculate_probabilities(self, X):
res = {
label: [] for label, val in self.pabusice.items()}
for label, value in self.pabusice.items():
ans = 1
mi = self.mie[label]
for x in range(len(X)):
ml = mi[x]
a = ml[X[x]]
ans *= a
ans *= value
res[label] = ans
return res
def predict(self, X_test):
label = sorted(
self.calculate_probabilities(X_test).items(),
key=lambda x: x[1]
)[-1][0]
return label
def score(self, X_test, y_test):
right = 0
for X, y in zip(X_test, y_test):
label = self.predict(X)
if label == y:
right += 1
return right / float(len(X_test))
if __name__ == '__main__':
X_train = [(1, 'S'), (1, 'M'), (1, 'M'), (1, 'S'), (1, 'S'),
(2, 'S'), (2, 'M'), (2, 'M'), (2, 'L'), (2, 'L'),
(3, 'L'), (3, 'M'), (3, 'M'), (3, 'L'), (3, 'L')]
y_train = [-1, -1, 1, 1, -1,
-1, -1, 1, 1, 1,
1, 1, 1, 1, -1]
lamda = int(input("Please enter you want setting lambda:"))
model = NaiveBayes(lamda)
model.fit(X_train, y_train)
print(model.predict([2, 'S']))
高斯朴素贝叶斯
"""
高斯朴素贝叶斯
"""
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from collections import Counter
import math
def create_data():
iris = load_iris()
df = pd.DataFrame(iris.data, columns=iris.feature_names)
df['label'] = iris.target
df.columns = ['sepal length', 'sepal width', 'petal length', 'petal width', 'label']
data = np.array(df.iloc[:100, :])
return data[:, :-1], data[:, -1]
class NaiveBayes(object):
def __init__(self):
self.model = None
# 数学期望
@staticmethod
def mean(X):
return sum(X) / float(len(X))
# 标准差
def stdev(self, X):
avg = self.mean(X)
return math.sqrt(sum([pow(x - avg, 2) for x in X]) / float(len(X)))
# 概率密度函数
def gaussian_probability(self, x, mean, stdev):
exponent = math.exp(-(math.pow(x - mean, 2) / (2 * math.pow(stdev, 2))))
return (1 / (math.sqrt(2 * math.pi) * stdev)) * exponent
# 处理X_train
def summarize(self, trian_data):
summaries = [(self.mean(i), self.stdev(i)) for i in zip(*trian_data)]
return summaries
# 分别求出数学期望和标准差
def fit(self, X, y):
labels = list(set(y))
data = {
label: [] for label in labels}
for f, label in zip(X, y):
data[label].append(f)
self.model = {
label: self.summarize(value)
for label, value in data.items()
}
return 'gaussianNB train done!'
# 计算概率
def calculate_probabilities(self, input_data):
probabilities = {
}
for label, value in self.model.items():
probabilities[label] = 1
for i in range(len(value)):
mean, stdev = value[i]
probabilities[label] *= self.gaussian_probability(
input_data[i], mean, stdev
)
return probabilities
# 类别，预测
def predict(self, X_test):
label = sorted(
self.calculate_probabilities(X_test).items(),
key=lambda x: x[-1]
)[-1][0]
return label
def score(self, X_test, y_test):
right = 0
for X, y in zip(X_test, y_test):
label = self.predict(X)
if label == y:
right += 1
return right / float(len(X_test))
if __name__ == '__main__':
X, y = create_data()
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)
model = NaiveBayes()
model.fit(X_train, y_train)
print(model.predict([4.4, 3.2, 1.3, 0.2]))
print(model.score(X_test, y_test))