Chapter4-Naive-Bayes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 第4章 朴素贝叶斯\n",
    "朴素贝叶斯的基本思想是： 基于特征条件独立假设学习输入和输出的联合概率分布，然后，基于此模型，对给定的输入$x$，利用贝叶斯订立求出后验概率最大的输出$y$，根据《统计学习方法》书中的规划，这里将简述朴素贝叶斯的学习和分类、朴素贝叶斯的参数估计方法。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.1 朴素贝叶斯的学习和分类\n",
    "### 4.1.1简述\n",
    "输入:训练数据集\n",
    "$$T=\\{ \\left( x_1,y_1 \\right),\\left( x_2,y_2 \\right),\\left( x_i,y_i \\right),...,\\left( x_N,y_N \\right) \\}$$\n",
    "\n",
    "朴素贝叶斯的目的是： 通过训练数据集$T$学习到联合概率分布$P(X,Y)$，具体来讲，这个联合概率分布的求解根据联合概率密度求解的公式$P(x|y)=\\frac{P(x,y)}{P_Y(y)}$又分为两个部分：\n",
    ">先验概率分布：\n",
    "$$P(Y=c_k),k=1,2,...,K$$\n",
    ">条件概率分布：\n",
    "$$P(X=x|Y=c_k)=P(X^\\left(1\\right)=x^\\left(1\\right),...,X^\\left(n\\right)=x^\\left(n\\right)|Y=c_k),k=1,2,...,K$$\n",
    "\n",
    "其中，条件独立性假设\n",
    ">$$ P\\left( X=x|Y=c_k \\right)=P( X^\\left(1\\right)=x^\\left(1\\right),...,X^\\left(n\\right)=x^\\left(n\\right) )\n",
    "=\\prod_{j=1}^n P \\left(X^\\left(j \\right)=x^\\left( j \\right)| Y=c_k\\right)$$\n",
    "\n",
    "关于**生成模型**和**判别模型**：  \n",
    "根据上面的式子，这里由于要求解的是$P(x|y)=\\frac{P(x,y)}{P_Y(y)}$，实质上是一种生成数据的机制，所以属于**生成模型**，可能有的同学会听说过生成对抗网络（GAN）的概念，没错，里面的生成也是生成模型，在GAN中，生成模型也是用来生成数据的，但是可能它生成的数据更加复杂，不再单单是一个高斯分布了，很有可能是一张图什么的，这个是后话了，这里只要知道朴素贝叶斯是生成模型、它学习的是一种生成数据的机制，就可以了。但是话又说回来了，你都知道如何生成数据了，当然就可以知道如何对数据进行区分了，不是么哈哈～"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.1.2算法描述\n",
    "好了，这里简言之，直接描述朴素贝叶斯的算法过程  \n",
    "输入：  \n",
    ">* 训练数据$T=\\{ \\left( x_1,y_1 \\right),\\left( x_2,y_2 \\right),\\left( x_i,y_i \\right),...,\\left( x_N,y_N \\right) \\}$，其中，$x_i = \\left(x_i^\\left(1\\right),x_i^\\left(2\\right),...,x_i^\\left(n\\right) \\right)^T$;  \n",
    "* $x_i^\\left(j\\right)$是第$i$个样本的第$j$个特征，$x_i^\\left(j\\right) \\in \\left\\{a_j1,a_j2,...,a_{jS_j} \\right\\}$，$a_{jl}$是第$j$个特征可能取的第$l$个值，$j=1,2,...,n$，$l=1,2,...,S_j$，$y_i \\in \\left\\{c_1,c_2,...,c_K \\right\\}$;  \n",
    "* 实例$x$;\n",
    "\n",
    "输出：  \n",
    ">* 实例$x$的分类结果"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "算法过程：  \n",
    ">1. 计算先验概率以及条件概率  \n",
    "$$P\\left( Y = c_k \\right) = \\frac{\\sum_{i=1}^N I(y_i=c_i)}{N},k=1,2,...,K$$  \n",
    "$$P\\left( X^\\left(j\\right)=a_{jl} | Y=c_k \\right) = \\frac{\\sum_{i=1}^N I(x^\\left(j\\right)=a_{jl},y_i=c_k)}{\\sum_{i=1}^N I(y_i=c_i)}$$  \n",
    "$$j=1,2,...,n; l=1,2,...,S_j,k=1,2,...,K$$  \n",
    "2. 对于给定的实例$x$，计算\n",
    "$$P(Y=c_k)\\prod_{j=1}^n P(X^\\left(j\\right) = x^\\left(j\\right)|Y=c_k),k=1,2,...,K $$\n",
    "3. 确定实例$x$的类\n",
    "$$y=\\arg\\underset{c_k}{\\max} P(Y=c_k) P(X^\\left(j\\right) = x^\\left(j\\right)|Y=c_k)$$\n",
    "\n",
    "### 4.1.3编写代码测试朴素贝叶斯算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 给定数据集,其中SML分别替换为1，2，3\n",
    "import numpy as np\n",
    "x = np.asarray([[1,1],[1,2],[1,2],[1,1],[1,1],\n",
    "      [2,1],[2,2],[2,2],[2,3],[2,3],\n",
    "      [3,3],[3,2],[3,2],[3,3],[3,3]])\n",
    "y = np.asarray([-1,-1,1,1,-1,\n",
    "     -1,-1,1,1,1,\n",
    "     1,1,1,1,-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">计算$P(Y=1)$和$P（Y=-1）$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(Y=1)= 0.6 , P(Y=-1)= 0.4\n"
     ]
    }
   ],
   "source": [
    "AllNum = np.size(y)\n",
    "# P(Y=1)\n",
    "PosY = np.sum(y==1)/AllNum\n",
    "#P(Y=-1)\n",
    "NegY = np.sum(y==-1)/AllNum\n",
    "print('P(Y=1)=',PosY,', P(Y=-1)=',NegY)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">计算$P\\left( X^\\left(j\\right)=a_{jl} | Y=c_k \\right) = \\frac{\\sum_{i=1}^N I(x^\\left(j\\right)=a_{jl},y_i=c_k)}{\\sum_{i=1}^N I(y_i=c_i)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(Y = -1 ) = 0.4\n",
      "P(X特征维度= 0 特征值= 1 | Y= -1 )= 0.5\n",
      "P(X特征维度= 0 特征值= 2 | Y= -1 )= 0.3333333333333333\n",
      "P(X特征维度= 0 特征值= 3 | Y= -1 )= 0.16666666666666666\n",
      "P(X特征维度= 1 特征值= 1 | Y= -1 )= 0.5\n",
      "P(X特征维度= 1 特征值= 2 | Y= -1 )= 0.3333333333333333\n",
      "P(X特征维度= 1 特征值= 3 | Y= -1 )= 0.16666666666666666\n",
      "P(Y = 1 ) = 0.6\n",
      "P(X特征维度= 0 特征值= 1 | Y= 1 )= 0.2222222222222222\n",
      "P(X特征维度= 0 特征值= 2 | Y= 1 )= 0.3333333333333333\n",
      "P(X特征维度= 0 特征值= 3 | Y= 1 )= 0.4444444444444444\n",
      "P(X特征维度= 1 特征值= 1 | Y= 1 )= 0.1111111111111111\n",
      "P(X特征维度= 1 特征值= 2 | Y= 1 )= 0.4444444444444444\n",
      "P(X特征维度= 1 特征值= 3 | Y= 1 )= 0.4444444444444444\n"
     ]
    }
   ],
   "source": [
    "# y 取值范围\n",
    "y_ = np.unique(y)\n",
    "# x 的维度\n",
    "feanum = np.size(x,1)\n",
    "min_result_dictionary = {}\n",
    "for k in range(len(y_)):# 对于每个Y类别\n",
    "    P_Y = np.sum(y==y_[k])/np.size(y) # 计算P(Y=k)的值\n",
    "    print('P(Y =',y_[k],') =',P_Y)\n",
    "    k_x = x[y==y_[k],:]# 保留在P(Y=k)条件下所有的x样本\n",
    "    for j in range(feanum): # 对于每个特征\n",
    "        fea_val_range = np.unique(x[:,j])# 该维度下属性值的取值范围\n",
    "        for l in range(len(fea_val_range)): # 对于特征下的第l个取值\n",
    "            P_X_Y = np.sum(k_x[:,j] == fea_val_range[l])/np.sum(y==y_[k])\n",
    "            print('P(X特征维度=',j,'特征值=',fea_val_range[l],'| Y=',y_[k],')=',P_X_Y)\n",
    "            split = ','\n",
    "            key = split.join([str(y_[k]),str(j),str(fea_val_range[l])])\n",
    "            min_result_dictionary[key] = P_X_Y\n",
    "# min_result_dictionary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">当给定$x=(2,S)^T$，也即$x=(2,1)^T$，计算$P(Y=c_k)\\prod_{j=1}^n P(X^\\left(j\\right) = x^\\left(j\\right)|Y=c_k)$："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "testx = np.array([2,1])\n",
    "labelvalue =[]\n",
    "for k in range(len(y_)):\n",
    "    P_Y = np.sum(y==y_[k])/np.size(y) # 计算P(Y=k)的值\n",
    "    j_p = 1# 记录每个类别下的概率值\n",
    "    for j in range(feanum):\n",
    "        split = ','\n",
    "        key = split.join([str(y_[k]),str(j),str(testx[j])])\n",
    "        j_p = j_p * min_result_dictionary[key]\n",
    "    labelvalue.append(P_Y*j_p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">返回值最大时候得到的类别标签$y=\\arg\\underset{c_k}{\\max} P(Y=c_k) P(X^\\left(j\\right) = x^\\left(j\\right)|Y=c_k)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_[labelvalue.index(max(labelvalue))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "综上可以得知，实际上朴素贝叶斯是在给定每个类别标签的情况下，查看这个数据“合理性”有多高，“合理性”越高，说明越有可能是这个类别，而这个“合理性”的度量，就是使用了概率来衡量，最后动过一个$\\arg \\max$操作，得到了对应的类别标签。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.1.4贝叶斯估计\n",
    "\n",
    "用心的读者也许会发现，如果用以上的方法（实际是“极大似然估计”）可能会得到概率值为0的情况，然后会影响后续一系列操作（如求最大值等）。根据《统计学习方法》一书，我们在这里加一个步骤，将原来的式子做小的修改：\n",
    "$$P\\left( X^\\left(j\\right)=a_{jl} | Y=c_k \\right) = \\frac{\\sum_{i=1}^N I(x^\\left(j\\right)=a_{jl},y_i=c_k)}{\\sum_{i=1}^N I(y_i=c_i)} \\rightarrow P\\left( X^\\left(j\\right)=a_{jl} | Y=c_k \\right) = \\frac{\\sum_{i=1}^N I(x^\\left(j\\right)=a_{jl},y_i=c_k)+\\lambda}{\\sum_{i=1}^N I(y_i=c_i)+S_j\\lambda}$$  \n",
    "其中，$\\lambda \\geq 0$，当$\\lambda = 0$的时候等同于原始的极大似然估计，而通常，这里取$\\lambda = 1$，称为**拉普拉斯平滑**，这时，对于$l=1,2,...,S_j$，$k=1,2,...,K$，有\n",
    "$$P_\\lambda(X^\\left(j\\right)=a_{jl}|Y=c_k)>0$$\n",
    "$$\\sum_{l=1}^{S_j}P_\\lambda(X^\\left(j\\right)=a_{jl}|Y=c_k) = 1$$\n",
    "\n",
    "这两个式子说明拉普拉斯平滑之后的式子确实是一个概率分布（即有每个概率大于0,总概率和等于1），此时，得到鲜艳概率贝叶斯估计为：\n",
    "$$P_\\lambda(Y=c_k)=\\frac{\\sum_{i=1}^N I(y_i=c_k)+\\lambda}{N+K\\lambda}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "此时我们修改上面的代码，运行可以得到如下结果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(Y = -1 ) = 0.4117647058823529\n",
      "P(X特征维度= 0 特征值= 1 | Y= -1 )= 0.4444444444444444\n",
      "P(X特征维度= 0 特征值= 2 | Y= -1 )= 0.3333333333333333\n",
      "P(X特征维度= 0 特征值= 3 | Y= -1 )= 0.2222222222222222\n",
      "P(X特征维度= 1 特征值= 1 | Y= -1 )= 0.4444444444444444\n",
      "P(X特征维度= 1 特征值= 2 | Y= -1 )= 0.3333333333333333\n",
      "P(X特征维度= 1 特征值= 3 | Y= -1 )= 0.2222222222222222\n",
      "P(Y = 1 ) = 0.5882352941176471\n",
      "P(X特征维度= 0 特征值= 1 | Y= 1 )= 0.25\n",
      "P(X特征维度= 0 特征值= 2 | Y= 1 )= 0.3333333333333333\n",
      "P(X特征维度= 0 特征值= 3 | Y= 1 )= 0.4166666666666667\n",
      "P(X特征维度= 1 特征值= 1 | Y= 1 )= 0.16666666666666666\n",
      "P(X特征维度= 1 特征值= 2 | Y= 1 )= 0.4166666666666667\n",
      "P(X特征维度= 1 特征值= 3 | Y= 1 )= 0.4166666666666667\n"
     ]
    }
   ],
   "source": [
    "# y 取值范围\n",
    "y_ = np.unique(y)\n",
    "# x 的维度\n",
    "feanum = np.size(x,1)\n",
    "lmda = 1\n",
    "min_result_dictionary = {}\n",
    "for k in range(len(y_)):# 对于每个Y类别\n",
    "    P_Y = (np.sum(y==y_[k])+lmda)/(np.size(y)+lmda*len(y_)) # 计算P(Y=k)的值\n",
    "    print('P(Y =',y_[k],') =',P_Y)\n",
    "    k_x = x[y==y_[k],:]# 保留在P(Y=k)条件下所有的x样本\n",
    "    for j in range(feanum): # 对于每个特征\n",
    "        fea_val_range = np.unique(x[:,j])# 该维度下属性值的取值范围\n",
    "        for l in range(len(fea_val_range)): # 对于特征下的第l个取值\n",
    "            P_X_Y = (np.sum(k_x[:,j] == fea_val_range[l])+lmda)/(np.sum(y==y_[k])+lmda*len(fea_val_range))\n",
    "            print('P(X特征维度=',j,'特征值=',fea_val_range[l],'| Y=',y_[k],')=',P_X_Y)\n",
    "            split = ','\n",
    "            key = split.join([str(y_[k]),str(j),str(fea_val_range[l])])\n",
    "            min_result_dictionary[key] = P_X_Y\n",
    "# min_result_dictionary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.05925925925925926, 0.03333333333333333]\n"
     ]
    }
   ],
   "source": [
    "testx = np.array([2,1])\n",
    "labelvalue =[]\n",
    "for k in range(len(y_)):\n",
    "    P_Y = np.sum(y==y_[k])/np.size(y) # 计算P(Y=k)的值\n",
    "    j_p = 1# 记录每个类别下的概率值\n",
    "    for j in range(feanum):\n",
    "        split = ','\n",
    "        key = split.join([str(y_[k]),str(j),str(testx[j])])\n",
    "        j_p = j_p * min_result_dictionary[key]\n",
    "    labelvalue.append(P_Y*j_p)\n",
    "print(labelvalue)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">返回值最大时候得到的类别标签"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_[labelvalue.index(max(labelvalue))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注：上面由于浮点精度问题，这里的结果与《统计学习方法》书中的结果存在一点差异，但是原理就是这样了！"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## 4.2 定义自己的朴素贝叶斯算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 4.2定义自己的朴素贝叶斯算法\n",
    "import numpy as np\n",
    "class myNaiveBayes():\n",
    "    def train(self,x,y,lmda=None):\n",
    "        \"\"\"\n",
    "        x： 训练集  \n",
    "        y： 类别标签向量\n",
    "        lmda： 拉普拉斯平滑参数值，默认为1\n",
    "        \"\"\"\n",
    "        self.x = x\n",
    "        self.y = y\n",
    "        self.lmda = lmda\n",
    "        model = []\n",
    "        # y 取值范围\n",
    "        y_ = np.unique(y)\n",
    "        # x 的维度\n",
    "        feanum = np.size(x,1)\n",
    "        min_result_dictionary = {}\n",
    "        P_Y = np.empty(len(y_))\n",
    "        for k in range(len(y_)):# 对于每个Y类别\n",
    "            P_Y[k] = np.sum(y==y_[k])/np.size(y) # 计算P(Y=k)的值\n",
    "            k_x = x[y==y_[k],:]# 保留在P(Y=k)条件下所有的x样本\n",
    "            for j in range(feanum): # 对于每个特征\n",
    "                fea_val_range = np.unique(x[:,j])# 该维度下属性值的取值范围\n",
    "                for l in range(len(fea_val_range)): # 对于特征下的第l个取值\n",
    "                    P_X_Y = (np.sum(k_x[:,j] == fea_val_range[l])+lmda)/(np.sum(y==y_[k])+lmda*len(fea_val_range))\n",
    "                    split = ','\n",
    "                    key = split.join([str(y_[k]),str(j),str(fea_val_range[l])])\n",
    "                    min_result_dictionary[key] = P_X_Y\n",
    "        \n",
    "        model.append(feanum) # 特征维度数目  0\n",
    "        model.append(y_) # 类别取值范围 1\n",
    "        model.append(min_result_dictionary) # dictionary保存的概率值 2\n",
    "        model.append(np.size(y)) # 样本个数 3\n",
    "        model.append(P_Y)\n",
    "        return model\n",
    "    \n",
    "    def predict(self,x,model):\n",
    "        \"\"\"\n",
    "        x 应该与train中的x具有相同的维度数\n",
    "        y 为预测类别标签\n",
    "        \"\"\"\n",
    "        y = []\n",
    "        self.model = model\n",
    "        self.x = x\n",
    "        if self.x.ndim==1:\n",
    "            self.x = self.x.reshape(1,len(x))\n",
    "        for i in range(len(self.x)):\n",
    "            testx = np.asarray(self.x[i,:])\n",
    "            labelvalue =[]\n",
    "            for k in range(len(self.model[1])):\n",
    "                P_Y = model[4][k]\n",
    "                j_p = 1# 记录每个类别下的概率值\n",
    "                for j in range(model[0]):\n",
    "                    split = ','\n",
    "                    key = split.join([str(model[1][k]),str(j),str(testx[j])])\n",
    "                    j_p = j_p * model[2][key]\n",
    "                labelvalue.append(P_Y*j_p)\n",
    "            y.append(y_[labelvalue.index(max(labelvalue))])\n",
    "        return y\n",
    "    \n",
    "    def accuracy(self,y,realy):\n",
    "        self.y = y\n",
    "        self.realy = realy\n",
    "        return 1-sum(np.sign(np.abs(y-realy)))/len(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">定义课本的15个实例"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.asarray([[1,1],[1,2],[1,2],[1,1],[1,1],\n",
    "      [2,1],[2,2],[2,2],[2,3],[2,3],\n",
    "      [3,3],[3,2],[3,2],[3,3],[3,3]])\n",
    "y = np.asarray([-1,-1,1,1,-1,\n",
    "     -1,-1,1,1,1,\n",
    "     1,1,1,1,-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">验证课本的15个实例"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "正确率： 0.7333333333333334\n"
     ]
    }
   ],
   "source": [
    "mynb = myNaiveBayes()\n",
    "model = mynb.train(x,y,lmda=1)\n",
    "newy = mynb.predict(x,model)\n",
    "print('正确率：',mynb.accuracy(y,newy))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">课本测试用例"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1]\n"
     ]
    }
   ],
   "source": [
    "temp = np.asarray([2,1])\n",
    "newy = mynb.predict(temp,model)\n",
    "print(newy)"
   ]
  }
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