"""朴素贝叶斯算法的实现"""
"""2019/4/12"""
import numpy as np
import pandas as pd

class NaiveBayes():
    def __init__(self,lambda_):
        self.lambda_=lambda_    #贝叶斯系数 取0时，即为极大似然估计
        self.y_types_count=None #y的（类型：数量）
        self.y_types_proba=None #y的（类型：概率）
        self.x_types_proba=dict() #（xi 的编号,xi的取值，y的类型）：概率

    def fit(self,X_train,y_train):
        self.y_types=np.unique(y_train)  #y的所有取值类型
        X=pd.DataFrame(X_train)          #转化成pandas DataFrame数据格式，下同
        y=pd.DataFrame(y_train)
        # y的（类型：数量）统计
        self.y_types_count=y[0].value_counts()
        # y的（类型：概率）计算
        self.y_types_proba=(self.y_types_count+self.lambda_)/(y.shape[0]+len(self.y_types)*self.lambda_)

        # （xi 的编号,xi的取值，y的类型）：概率的计算
        for idx in X.columns:       # 遍历xi
            for j in self.y_types:  # 选取每一个y的类型
                p_x_y=X[(y==j).values][idx].value_counts() #选择所有y==j为真的数据点的第idx个特征的值，并对这些值进行（类型：数量）统计
                for i in p_x_y.index: #计算（xi 的编号,xi的取值，y的类型）：概率
                    self.x_types_proba[(idx,i,j)]=(p_x_y[i]+self.lambda_)/(self.y_types_count[j]+p_x_y.shape[0]*self.lambda_)

    def predict(self,X_new):
        res=[]
        for y in self.y_types: #遍历y的可能取值
            p_y=self.y_types_proba[y]  #计算y的先验概率P(Y=ck)
            p_xy=1
            for idx,x in enumerate(X_new):
                p_xy*=self.x_types_proba[(idx,x,y)] #计算P(X=(x1,x2...xd)/Y=ck)
            res.append(p_y*p_xy)
        for i in range(len(self.y_types)):
            print("[{}]对应概率：{:.2%}".format(self.y_types[i],res[i]))
        #返回最大后验概率对应的y值
        return self.y_types[np.argmax(res)]

def main():
    X_train=np.array([
                      [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=np.array([-1,-1,1,1,-1,-1,-1,1,1,1,1,1,1,1,-1])
    clf=NaiveBayes(lambda_=0)
    clf.fit(X_train,y_train)
    X_new=np.array([2,"S"])
    y_predict=clf.predict(X_new)
    print("{}被分类为:{}".format(X_new,y_predict))

if __name__=="__main__":
    main()

朴素贝叶斯法通过训练数据集学习联合概率分布P(X,Y)，具体做法是学习先验概率分布P(Y)与条件概率分布P(X|Y)（二者相乘就是联合概率分布），所以它属于生成模型。

image.png
image.png

贝叶斯分布例子

# load the iris dataset
from sklearn.datasets import load_iris
iris = load_iris()

# store the feature matrix (X) and response vector (y)
X = iris.data
y = iris.target

# splitting X and y into training and testing sets
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.4, random_state=1)

# training the model on training set
from sklearn.naive_bayes import GaussianNB
gnb = GaussianNB()
gnb.fit(X_train, y_train)

# making predictions on the testing set
y_pred = gnb.predict(X_test)

# comparing actual response values (y_test) with predicted response values (y_pred)
from sklearn import metrics
print("Gaussian Naive Bayes model accuracy(in %):", metrics.accuracy_score(y_test, y_pred)*100)

image.png image.png

后验概率最大化

image.png image.png image.png