{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 条件随机场" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 马尔可夫过程" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 定义" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "假设一个随机过程中，$t_n$ 时刻的状态$x_n$的条件发布，只与其前一状态$x_{n-1}$ 相关，即：\n", "\n", "$$\n", " P(x_n|x_1,x_2,...,x_{n-1}) = P(x_n|x_{n-1})\n", "$$\n", "\n", "则将其称为 马尔可夫过程。\n", "\n", "![](img/马尔可夫过程.png)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 隐马尔科夫算法" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 定义" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "隐马尔科夫算法是对含有未知参数（隐状态）的马尔可夫链进行建模的生成模型，如下图所示：\n", "\n", "![](img/隐马尔科夫算法.png)\n", "\n", "在隐马尔科夫模型中，包含隐状态 和 观察状态，隐状态 $x_i$ 对于观察者而言是不可见的，而观察状态 $y_i$ 对于观察者而言是可见的。隐状态间存在转移概率，隐状态 $x_i$到对应的观察状态 $y_i$ 间存在输出概率。\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 假设" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. 假设隐状态$x_i$ 的状态满足马尔可夫过程，i时刻的状态$x_i$ 的条件分布，仅与其前一个状态$x_{i-1}$相关，即：\n", "\n", "$$\n", " P(x_i|x_1,x_2,...,x_{i-1}) = P(x_i|x_{i-1})\n", "$$\n", "\n", "2. 假设观测序列中各个状态仅取决于它所对应的隐状态，即：\n", "\n", "$$\n", " P(y_i|x_1,x_2,...,x_{i-1},y_1,y_2,...,y_{i-1},y_{i+1},...) = P(y_i|x_{i})\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 存在问题" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在序列标注问题中，隐状态（标注）不仅和单个观测状态相关，还和观察序列的长度、上下文等信息相关。例如词性标注问题中，一个词被标注为动词还是名词，不仅与它本身以及它前一个词的标注有关，还依赖于上下文中的其他词。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 条件随机场 （以线性链条件随机场为例）" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 定义" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "给定 $X=(x_1,x_2,...,x_n)$ ，$Y=(y_1,y_2,...,y_n)$ 均为线性链表示的随机变量序列，若在给随机变量序列 X 的条件下，随机变量序列 Y 的条件概率分布 $P(Y|X)$ 构成条件随机场，即满足马尔可夫性：\n", "\n", "$$\n", " P(y_i|x_1,x_2,...,x_{i-1},y_1,y_2,...,y_{i-1},y_{i+1})\n", " = P(y_i|x,y_{i-1},y_{i+1})\n", "$$\n", "\n", "则称为 P(Y|X) 为线性链条件随机场。\n", "\n", "通过去除了隐马尔科夫算法中的观测状态相互独立假设，使算法在计算当前隐状态$x_i$时，会考虑整个观测序列，从而获得更高的表达能力，并进行全局归一化解决标注偏置问题。\n", "\n", "\n", "![条件随机场图片](img/条件随机场.png)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 参数化形式" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "p\\left(y | x\\right)=\\frac{1}{Z\\left(x\\right)} \\prod_{i=1}^{n} \\exp \\left(\\sum_{i, k} \\lambda_{k} t_{k}\\left(y_{i-1}, y_{i}, x, i\\right)+\\sum_{i, l} \\mu_{l} s_{l}\\left(y_{i}, x, i\\right)\\right)\n", "$$\n", "\n", "其中：\n", "\n", "> $Z(x)$ 为归一化因子，是在全局范围进行归一化，枚举了整个隐状态序列$x_{1…n}$的全部可能，从而解决了局部归一化带来的标注偏置问题。\n", "\n", "$$\n", "Z(x)=\\sum_{y} \\exp \\left(\\sum_{i, k} \\lambda_{x} t_{k}\\left(y_{i-1}, y_{i}, x, i\\right)+\\sum_{i, l} \\mu_{l} s_{l}\\left(y_{i}, x, i\\right)\\right)\n", "$$\n", "\n", "> $t_k$ 为定义在边上的特征函数，转移特征，依赖于前一个和当前位置\n", "\n", "> $s_1$ 为定义在节点上的特征函数，状态特征，依赖于当前位置。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 简化形式" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "因为条件随机场中同一特征在各个位置都有定义，所以可以对同一个特征在各个位置求和，将局部特征函数转化为一个全局特征函数，这样就可以将条件随机场写成权值向量和特征向量的内积形式，即条件随机场的简化形式。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### step 1\n", "\n", "将转移特征和状态特征及其权值用统一的符号表示，设有k1个转移特征，$k_2$个状态特征，$K=k_1+k_2$,记\n", "\n", " \"图片名称\"\n", "\n", "##### step 2\n", "\n", "对转移与状态特征在各个位置i求和，记作\n", "\n", " \"图片名称\"\n", "\n", "##### step 3\n", "\n", "将 $\\lambda_{x}$ 和 $\\mu_{l}$ 用统一的权重表示，记作\n", "\n", " \"图片名称\"\n", "\n", "##### step 4\n", "\n", "转化后的条件随机场可表示为：\n", "\n", " \"图片名称\"\n", "\n", "##### step 5\n", "\n", "若 $w$ 表示权重向量：\n", "\n", "$$\n", " w = (w_1,w_2,...,w_K)^T\n", "$$\n", "\n", "以 $F(y,x)$ 表示特征向量，即\n", "\n", " \"图片名称\"\n", "\n", "则，条件随机场写成内积形式为：\n", "\n", " \"图片名称\"\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 矩阵形式" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> 推导 begin\n", "\n", "\n", "\n", "\n", "> 推导 end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 基本问题" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "条件随机场包含概率计算问题、学习问题和预测问题三个问题。\n", "\n", "> 1. 概率计算问题：已知模型的所有参数，计算观测序列 $Y$ 出现的概率，常用方法：前向和后向算法；\n", "\n", "> 2. 学习问题：已知观测序列 $Y$，求解使得该观测序列概率最大的模型参数，包括隐状态序列、隐状态间的转移概率分布和从隐状态到观测状态的概率分布，常用方法：Baum-Wehch 算法；\n", "\n", "> 3. 预测问题：一直模型所有参数和观测序列 $Y$ ，计算最可能的隐状态序列 $X$,常用算法：维特比算法。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 概率计算问题" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> 给定条件随机场$P(Y|X)$，输入序列 $x$ 和 输出序列 $y$;\n", "\n", "> 计算条件概率\n", "\n", "$$\n", " P(Y_i=y_i|x), P(Y_{i-1} = y_{i-1},Y_i = y_i|x)\n", "$$\n", "\n", "> 计算相应的数学期望问题；" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 前向-后向算法" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###### step 1 前向计算\n", "\n", "对观测序列 $x$ 的每个位置 $i=1,2,...,n+1$ ，定义一个 $m$ 阶矩阵（$m$ 为标记$Y_i$取值的个数）\n", "\n", " \"图片名称\"\n", "\n", "对每个指标 $i=0,1,...,n+1$，定义前向向量 $\\alpha_{i}(x)$，则递推公式:\n", "\n", " \"图片名称\"\n", "\n", "其中，\n", "\n", " \"图片名称\"\n", " \n", " \n", "\n", "###### step 2 后向计算\n", "\n", "对每个指标 $i=0,1,...,n+1$，定义前向向量 $\\beta_{i}(x)$，则递推公式:\n", "\n", "\"图片名称\"\n", " \n", "\"图片名称\"\n", "\n", "\n", "###### step 3\n", "\n", " \"图片名称\"\n", "\n", "###### step 4 概率计算\n", "\n", "所以，标注序列在位置 $i$ 是标注 $y_i$ 的条件概率为：\n", "\n", "\"图片名称\"\n", " \n", "\"图片名称\"\n", "\n", "其中，\n", "\n", "\"图片名称\"\n", "\n", "###### step 5 期望值计算\n", "\n", "通过利用前向-后向向量，计算特征函数关于联合概率分布 $P(X,Y)$ 和 条件概率分布 $P(Y|X)$ 的数学期望，即特征函数 $f_k$ 关于条件概率分布 $P(Y|X)$ 的数学期望：\n", "\n", "\"图片名称\"\n", "\n", "其中：\n", "\n", "\"图片名称\"\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 学习问题" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "这里主要介绍一下 BFGS 算法的思路。\n", "\n", "\n", " 输入：特征函数 $f_1,f_2,...,f_n$：经验分布 $\\widetilde{P}(X,Y)$；\n", " \n", " 输出：最优参数值 $\\widehat{w}$，最优模型$P_{\\widehat{w}}(y|x)$。\n", " \n", " 1. 选定初始点 w^{(0)}， 取 $B_0$ 为正定对称矩阵，k = 0;\n", " 2. 计算 $g_k = g(w^(k))$，若 $g_k = 0$ ，则停止计算，否则转 (3) ；\n", " 3. 利用 $B_k p_k = -g_k$ 计算 $p_k$；\n", " 4. 一维搜索：求 $\\lambda_k$使得\n", " \n", " \"图片名称\"\n", " \n", " 5. 设 $w^{(k+1)} = w^{(k)} + \\lambda_k * p_k$\n", " 6. 计算 $g_{k+1}$ = g(w^{(k+1)}),\n", " \n", " 若 $g_k = 0$， 则停止计算；否则，利用下面公式计算 $B_{k+1}$:\n", " \n", " \"图片名称\"\n", " \n", " 7. 令 $k=k+1$，转步骤（3）；\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 预测问题" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "对于预测问题，常用的方法是维特比算法，其思路如下：\n", "\n", "输入：模型特征向量 $F(y,x)$ 和权重向量 $w$，输入序列（观测序列） $x={x_1,x_2,...,x_n}$；\n", "\n", "输出：条件概率最大的输出序列（标记序列）$y^{*}= (y_1^*,y_2^*,...,y_n^*)$，也就是最优路径；\n", "\n", "1. 初始化\n", "\n", "\"图片名称\"\n", " \n", "2. 递推，对$i=2,3,...,n$\n", "\n", "\"图片名称\"\n", "\n", "3. 终止\n", "\n", "\"图片名称\"\n", "\n", "4. 返回路径\n", "\n", "\"图片名称\"\n", "\n", "求得最优路径 $y^{*}= (y_1^*,y_2^*,...,y_n^*)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 例子说明" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "利用维特比算法计算给定输入序列$x$ 对应的最优输出序列$y^*$：\n", "\n", "\"图片名称\"\n", "\n", "1. 初始化\n", "\n", "\"图片名称\"\n", " \n", "2. 递推，对$i=2,3,...,n$\n", "\n", "\"图片名称\"\n", "\n", "\"图片名称\"\n", "\n", "3. 终止\n", "\n", "\"图片名称\"\n", "\n", "4. 返回路径\n", "\n", "\"图片名称\"\n", "\n", "求得最优路径 $y^{*}= (y_1^*,y_2^*,...,y_n^*) = (1,2,1)$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "self.V[0]= [1 1] self.VW[0]= [1. 0.5] self.D[0]= [1. 0.5]\n", "self.P: [[0. 0.]\n", " [0. 0.]\n", " [0. 0.]]\n", "(x0,y=0)-->(x1,y=0):1.00 + 0.60 + 0.80= 2.4000000000000004\n", "(x0,y=1)-->(x1,y=0):0.50 + 1.00 + 0.80= 2.3\n", "self.D[x1,y=0]=2.40\n", "\n", "(x0,y=0)-->(x1,y=1):1.00 + 1.00 + 0.50= 2.5\n", "(x0,y=1)-->(x1,y=1):0.50 + 0.00 + 0.50= 1.0\n", "self.D[x1,y=1]=2.50\n", "\n", "(x1,y=0)-->(x2,y=0):2.40 + 0.00 + 0.80= 3.2\n", "(x1,y=1)-->(x2,y=0):2.50 + 1.00 + 0.80= 4.3\n", "self.D[x2,y=0]=4.30\n", "\n", "(x1,y=0)-->(x2,y=1):2.40 + 1.00 + 0.50= 3.9000000000000004\n", "(x1,y=1)-->(x2,y=1):2.50 + 0.20 + 0.50= 3.2\n", "self.D[x2,y=1]=3.90\n", "\n", "self.Delta:\n", " [[1. 0.5]\n", " [2.4 2.5]\n", " [4.3 3.9]]\n", "self.Psi:\n", " [[0. 0.]\n", " [0. 0.]\n", " [1. 0.]]\n", "最优状态路径为： [1. 2. 1.]\n", "最优状态路径为: [1. 2. 1.]\n" ] } ], "source": [ "import numpy as np\n", " \n", "class CRF(object):\n", " '''实现条件随机场预测问题的维特比算法\n", " '''\n", " def __init__(self, V, VW, E, EW):\n", " '''\n", " :param V:是定义在节点上的特征函数，称为状态特征\n", " :param VW:是V对应的权值\n", " :param E:是定义在边上的特征函数，称为转移特征\n", " :param EW:是E对应的权值\n", " '''\n", " self.V = V #点分布表\n", " self.VW = VW #点权值表\n", " self.E = E #边分布表\n", " self.EW = EW #边权值表\n", " self.D = [] #Delta表，最大非规范化概率的局部状态路径概率\n", " self.P = [] #Psi表，当前状态和最优前导状态的索引表s\n", " self.BP = [] #BestPath，最优路径\n", " return \n", " \n", " def Viterbi(self):\n", " '''\n", " 条件随机场预测问题的维特比算法，此算法一定要结合CRF参数化形式对应的状态路径图来理解，更容易理解.\n", " '''\n", " self.D = np.full(shape=(np.shape(self.V)), fill_value=.0)\n", " self.P = np.full(shape=(np.shape(self.V)), fill_value=.0)\n", " for i in range(np.shape(self.V)[0]):\n", " #初始化\n", " if 0 == i:\n", " self.D[i] = np.multiply(self.V[i], self.VW[i])\n", " self.P[i] = np.array([0, 0])\n", " print('self.V[%d]='%i, self.V[i], 'self.VW[%d]='%i, self.VW[i], 'self.D[%d]='%i, self.D[i])\n", " print('self.P:', self.P)\n", " pass\n", " #递推求解布局最优状态路径\n", " else:\n", " for y in range(np.shape(self.V)[1]): #delta[i][y=1,2...]\n", " for l in range(np.shape(self.V)[1]): #V[i-1][l=1,2...]\n", " delta = 0.0\n", " delta += self.D[i-1, l] #前导状态的最优状态路径的概率\n", " delta += self.E[i-1][l,y]*self.EW[i-1][l,y] #前导状态到当前状体的转移概率\n", " delta += self.V[i,y]*self.VW[i,y] #当前状态的概率\n", " print('(x%d,y=%d)-->(x%d,y=%d):%.2f + %.2f + %.2f='%(i-1, l, i, y, \\\n", " self.D[i-1, l], \\\n", " self.E[i-1][l,y]*self.EW[i-1][l,y], \\\n", " self.V[i,y]*self.VW[i,y]), delta)\n", " if 0 == l or delta > self.D[i, y]:\n", " self.D[i, y] = delta\n", " self.P[i, y] = l\n", " print('self.D[x%d,y=%d]=%.2f\\n'%(i, y, self.D[i,y]))\n", " print('self.Delta:\\n', self.D)\n", " print('self.Psi:\\n', self.P)\n", " \n", " #返回，得到所有的最优前导状态\n", " N = np.shape(self.V)[0]\n", " self.BP = np.full(shape=(N,), fill_value=0.0)\n", " t_range = -1 * np.array(sorted(-1*np.arange(N)))\n", " for t in t_range:\n", " if N-1 == t:#得到最优状态\n", " self.BP[t] = np.argmax(self.D[-1])\n", " else: #得到最优前导状态\n", " self.BP[t] = self.P[t+1, int(self.BP[t+1])]\n", " \n", " #最优状态路径表现在存储的是状态的下标，我们执行存储值+1转换成示例中的状态值\n", " #也可以不用转换，只要你能理解，self.BP中存储的0是状态1就可以~~~~\n", " self.BP += 1\n", " \n", " print('最优状态路径为：', self.BP)\n", " return self.BP\n", " \n", "def CRF_manual(): \n", " S = np.array([[1,1], #X1:S(Y1=1), S(Y1=2)\n", " [1,1], #X2:S(Y2=1), S(Y2=2)\n", " [1,1]]) #X3:S(Y3=1), S(Y3=1)\n", " SW = np.array([[1.0, 0.5], #X1:SW(Y1=1), SW(Y1=2)\n", " [0.8, 0.5], #X2:SW(Y2=1), SW(Y2=2)\n", " [0.8, 0.5]])#X3:SW(Y3=1), SW(Y3=1)\n", " E = np.array([[[1, 1], #Edge:Y1=1--->(Y2=1, Y2=2)\n", " [1, 0]], #Edge:Y1=2--->(Y2=1, Y2=2)\n", " [[0, 1], #Edge:Y2=1--->(Y3=1, Y3=2) \n", " [1, 1]]])#Edge:Y2=2--->(Y3=1, Y3=2)\n", " EW= np.array([[[0.6, 1], #EdgeW:Y1=1--->(Y2=1, Y2=2)\n", " [1, 0.0]], #EdgeW:Y1=2--->(Y2=1, Y2=2)\n", " [[0.0, 1], #EdgeW:Y2=1--->(Y3=1, Y3=2)\n", " [1, 0.2]]])#EdgeW:Y2=2--->(Y3=1, Y3=2)\n", " \n", " crf = CRF(S, SW, E, EW)\n", " ret = crf.Viterbi()\n", " print('最优状态路径为:', ret)\n", " return\n", " \n", "if __name__=='__main__':\n", " CRF_manual()\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.3" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }