CRF的参数学习同HMM一样,采用的极大似然估计的方式,其对数似然函数为:
$$ L(w)=L_{\tilde{P}}(P_w)\\ =log\prod_{x,y}P_w(y\mid x)^{\tilde{P}(x,y)}\\ =\sum_{x,y}\tilde{P}(x,y)logP_w(y\mid x)\\ =\sum_{j=1}^NlogP_w(y_j\mid x_j)(假设样本量为N) $$将CRF的势函数:
$$ P_w(y\mid x)=\frac{exp(\sum_{k=1}^Kw_kf_k(y,x))}{Z_w(x)} $$带入上面的对数似然函数可得:
$$ L(w)=\sum_{j=1}^N\sum_{k=1}^Kw_kf_k(y_j,x_j)-\sum_{j=1}^NlogZ_w(x_j) $$最终,我们的问题等价于:
$$ w_k^*=arg\min_{w_k}\sum_{j=1}^N(logZ_w(x_j)-\sum_{k=1}^Kw_kf_k(y_j,x_j)) $$最简单的就是梯度下降,可推导出其梯度函数为:
$$ g(w^t)=\sum_{j=1}^N(P_{w^t}(y_j\mid x_j)-1)F(y_j,x_j)) $$这里,$F(y_j,x_j)=[f_1(y_j,x_j),f_2(y_j,x_j),...,f_K(y_j,x_j)]^T$,所以梯度更新公式也就出来了:
$$ w^{t+1}=w^t-\eta g(w^t) $$这里,$\eta$为学习率,用梯度更新公式来理解一下训练过程,每次迭代会将特征函数输出为1的权重增加,而输出为0的保持不变,则看起来也蛮合理的~~~
import os
os.chdir('../')
from ml_models.pgm import CRFFeatureFunction
import numpy as np
"""
线性链CRF的实现,封装到ml_models.pgm
"""
class CRF(object):
def __init__(self, epochs=10, lr=1e-3, tol=1e-5, output_status_num=None, input_status_num=None, unigram_rulers=None,
bigram_rulers=None):
"""
:param epochs: 迭代次数
:param lr: 学习率
:param tol:梯度更新的阈值
:param output_status_num:标签状态数
:param input_status_num:输入状态数
:param unigram_rulers: 状态特征规则
:param bigram_rulers: 状态转移规则
"""
self.epochs = epochs
self.lr = lr
self.tol = tol
# 为输入序列和标签状态序列添加一个头尾id
self.output_status_num = output_status_num + 2
self.input_status_num = input_status_num + 2
self.input_status_head_tail = [input_status_num, input_status_num + 1]
self.output_status_head_tail = [output_status_num, output_status_num + 1]
# 特征函数
self.FF = CRFFeatureFunction(unigram_rulers, bigram_rulers)
# 模型参数
self.w = None
def fit(self, x, y):
"""
:param x: [[...],[...],...,[...]]
:param y: [[...],[...],...,[...]]
:return
"""
# 为 x,y加头尾
x = [[self.input_status_head_tail[0]] + xi + [self.input_status_head_tail[1]] for xi in x]
y = [[self.output_status_head_tail[0]] + yi + [self.output_status_head_tail[1]] for yi in y]
self.FF.fit(x, y)
self.w = np.ones(len(self.FF.feature_funcs)) * 1e-5
for _ in range(0, self.epochs):
# 偷个懒,用随机梯度下降
for i in range(0, len(x)):
xi = x[i]
yi = y[i]
"""
1.求F(yi \mid xi)以及P_w(yi \mid xi)
"""
F_y_x = []
Z_x = np.ones(shape=(self.output_status_num, 1)).T
for j in range(1, len(xi)):
F_y_x.append(self.FF.map(yi[j - 1], yi[j], xi, j))
# 构建M矩阵
M = np.zeros(shape=(self.output_status_num, self.output_status_num))
for k in range(0, self.output_status_num):
for t in range(0, self.output_status_num):
M[k, t] = np.exp(np.dot(self.w, self.FF.map(k, t, xi, j)))
# 前向算法求 Z(x)
Z_x = Z_x.dot(M)
F_y_x = np.sum(F_y_x, axis=0)
Z_x = np.sum(Z_x)
# 求P_w(yi \mid xi)
P_w = np.exp(np.dot(self.w, F_y_x)) / Z_x
"""
2.求梯度,并更新
"""
dw = (P_w - 1) * F_y_x
self.w = self.w - self.lr * dw
if (np.sqrt(np.dot(dw, dw) / len(dw))) < self.tol:
break
# 随便测试一下
x = [
[1, 2, 3, 0, 1, 3, 4],
[1, 2, 3],
[0, 2, 4, 2],
[4, 3, 2, 1],
[3, 1, 1, 1, 1],
[2, 1, 3, 2, 1, 3, 4]
]
y = x
crf = CRF(output_status_num=5, input_status_num=5)
crf.fit(x, y)
crf.w
array([1.00000000e-05, 1.00009053e-01, 7.00089277e-02, 7.00091106e-02, 2.00098984e-02, 4.00097839e-02, 6.00090103e-02, 1.00000000e-05, 2.00092452e-02, 2.00092452e-02, 1.00099996e-02, 1.00099996e-02, 3.00099986e-02, 2.00099991e-02, 2.00099991e-02, 2.00092452e-02, 2.00092452e-02, 2.00092452e-02, 1.00099996e-02, 1.00099996e-02, 3.00099986e-02, 2.00099991e-02, 2.00099991e-02, 1.00000000e-05, 2.00092452e-02, 2.00092452e-02, 1.00099996e-02, 1.00099996e-02, 3.00099986e-02, 2.00099991e-02, 2.00099991e-02, 2.00092452e-02, 2.00092452e-02, 2.00092452e-02, 1.00099996e-02, 1.00099996e-02, 3.00099986e-02, 2.00099991e-02, 2.00099991e-02, 2.00092452e-02, 2.00092452e-02, 2.00092452e-02, 1.00099996e-02, 1.00099996e-02, 3.00099986e-02, 2.00099991e-02, 2.00099991e-02, 1.00092456e-02, 1.00092456e-02, 1.00092456e-02, 1.00092456e-02, 1.00092456e-02, 1.00000000e-05, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00000000e-05, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00098988e-02, 1.00000000e-05, 1.00098860e-02, 2.00098855e-02, 3.00098850e-02, 2.00098668e-02, 1.00098860e-02, 1.00098860e-02, 2.00098855e-02, 3.00098850e-02, 2.00098668e-02, 1.00000000e-05, 1.00098860e-02, 2.00098855e-02, 3.00098850e-02, 2.00098668e-02, 1.00098860e-02, 1.00098860e-02, 2.00098855e-02, 3.00098850e-02, 2.00098668e-02, 1.00098860e-02, 1.00098860e-02, 2.00098855e-02, 3.00098850e-02, 2.00098668e-02, 1.00000000e-05, 1.00099808e-02, 3.00099424e-02, 1.00099808e-02, 1.00099808e-02, 3.00099424e-02, 1.00000000e-05, 1.00099808e-02, 3.00099424e-02, 1.00099808e-02, 1.00099808e-02, 3.00099424e-02, 1.00099808e-02, 1.00099808e-02, 3.00099424e-02, 1.00000000e-05, 1.00099995e-02, 1.00000000e-05, 1.00099995e-02, 1.00099995e-02])
len(crf.w)
122
可知一共有122个特征函数,接下来还剩最后一个问题,那就是标签预测的问题,见下一小节~~~