- 算法特征:
利用sigmoid函数的概率含义, 借助回归之手段达到分类之目的. - 算法推导:
Part Ⅰ
sigmoid函数之定义:
\begin{equation}\label{eq_1}
sig(x) = \frac{1}{1 + e^{-x}}
\end{equation}
相关函数图像:
由此可见, sigmoid函数将整个实数域$(-\infty, +\infty)$映射至$(0, 1)$区间内, 反映了一种良好概率意义下的映射关系. 对该函数进行如下扩展:
\begin{equation}\label{eq_2}
sig(\theta(x)) = \frac{1}{1 + e^{-\theta(x)}}
\end{equation}
其中, $x$为输入参数, $\theta(x)$为sigmoid函数之相位. 展开该相位, 可获取sigmoid函数之特征线$\theta(x) = c$, 特征线$\theta(x) = 0$也就是我们最常见的分界线. 很显然, 相位$\theta(x)$之具体形式将影响分界线最终的表达能力, 即分界线形状的复杂程度.
为避免混淆, 现将相位$\theta(x)$拆开:①. $x$ ~ 输入参数; ②. $\theta$ ~ 相位参数. 在具有严格数理要求的情形下, 相位参数按实际要求选取; 否则, 经过训练集上的交叉验证选择合适的形式即可.
Part Ⅱ
现建立如下假设模型:
\begin{equation}\label{eq_4}
h_{\theta}(x) = sig(\theta(x))
\end{equation}
以$y = 1$表示正例, $y = 0$表示负例. 令$h_{\theta}(x)$表示输入参数$x$取正例之概率, $1 - h_{\theta}(x)$表示输入参数$x$取负例之概率, 则假设模型对输入参数$x$预测正确之概率(正例情况下预测正例之概率、负例情况下预测负例之概率)为:
\begin{equation}\label{eq_5}
P(right \mid x, \theta) = h_{\theta}(x)^y(1 - h_{\theta}(x))^{(1 - y)}
\end{equation}
考虑训练集中存在$n$个样本, 假设相位参数$\theta$已知, 则此n个样本同时预测正确之概率为:
\begin{equation}\label{eq_6}
L(\theta) = \prod_{i = 1}^{n}P(right \mid x^{(i)}, \theta)
\end{equation}
对该似然函数取对数似然:
\begin{equation}\label{eq_7}
lnL(\theta) = \sum_{i = 1}^{n}lnP(right \mid x^{(i)}, \theta)
\end{equation}
定义损失函数:
\begin{equation}\label{eq_8}
J(\theta) = -\frac{1}{n}lnL(\theta)
\end{equation}
根据最大似然估计, 有如下无约束最优化问题:
\begin{equation}\label{eq_9}
min\ J(\theta)
\end{equation}
其中, 相位参数$\theta$为优化变量. 对该优化问题进行求解, 即可获取相位参数$\theta$之最优解, 也即当前训练集下的最优分类模型.
Part Ⅲ
为方便优化, 给出如下协助信息:
\begin{equation}\label{eq_10}
\left\{\begin{matrix}
\nabla h_{\theta} = h_{\theta}(1 - h_{\theta})\nabla \theta \\
\nabla lnh_{\theta} = (1 - h_{\theta})\nabla \theta \\
\nabla ln(1 - h_{\theta}) = -h_{\theta}\nabla \theta
\end{matrix}\right.
\end{equation}
目标函数$J(\theta)$之gradient:
\begin{equation}\label{eq_11}
\nabla J = -\frac{1}{n} \sum_{i = 1}^{n} (y^{(i)} - h_{\theta}(x^{(i)}))\nabla \theta \\
\end{equation}
目标函数$J(\theta)$之Hessian矩阵:
\begin{equation}\label{eq_12}
H = \nabla^2 J = -\frac{1}{n} \sum_{i = 1}^{n}[(y^{(i)} - h_{\theta}(x^{(i)}))\nabla^2 \theta - h_{\theta}(x^{(i)})(1 - h_{\theta}(x^{(i)}))\nabla \theta (\nabla \theta)^T]
\end{equation}
Part Ⅳ
引入局部加权项$exp(-\left \| x_i - x\right \|_2^2 / (2\tau^2))$, 引入原则:
①. 不改变损失函数$J(\theta)$中各样本的损失贡献计算方式;
②. 对样本的损失贡献独立进行加权.
③. 泛化误差之计算仅依赖于样本的损失贡献.
Part Ⅴ
现以一个二分类为例进行算法实施.
以常见的$\theta(x) = 0$作为分界线, 并令:
\begin{equation}
\left\{\begin{matrix}
h_{\theta}(x) \geq 0.5 & y = 1 & 正例 \\
h_{\theta}(x) < 0.5 & y = 0 & 负例
\end{matrix}\right.
\end{equation}- 代码实现:
1 # 局部加权之逻辑回归
2
3 import numpy
4 from matplotlib import pyplot as plt
5
6
7 def spiral_point(val, center=(0.5, 0.5)):
8 rn = 0.4 * (105 - val) / 104
9 an = numpy.pi * (val - 1) / 25
10
11 x0 = center[0] + rn * numpy.sin(an)
12 y0 = center[1] + rn * numpy.cos(an)
13 z0 = 0
14 x1 = center[0] - rn * numpy.sin(an)
15 y1 = center[1] - rn * numpy.cos(an)
16 z1 = 1
17
18 return (x0, y0, z0), (x1, y1, z1)
19
20
21 def spiral_data(valList):
22 dataList = list(spiral_point(val) for val in valList)
23 data0 = numpy.array(list(item[0] for item in dataList))
24 data1 = numpy.array(list(item[1] for item in dataList))
25 return data0, data1
26
27
28 # 生成训练数据集
29 trainingValList = numpy.arange(1, 101, 1)
30 trainingData0, trainingData1 = spiral_data(trainingValList)
31 trainingSet = numpy.vstack((trainingData0, trainingData1))
32
33 # 生成测试数据集
34 testValList = numpy.arange(1.5, 101.5, 1)
35 testData0, testData1 = spiral_data(testValList)
36 testSet = numpy.vstack((testData0, testData1))
37
38
39 # 假设模型
40 def h_theta(x, theta):
41 val = 1. / (1. + numpy.exp(-numpy.sum(x * theta)))
42 return val
43
44
45 # 假设模型预测正确之概率
46 def p_right(x, y_, theta):
47 val = h_theta(x, theta) ** y_ * (1. - h_theta(x, theta)) ** (1 - y_)
48 return val
49
50
51 # 加权损失函数
52 def JW(X, Y_, W, theta):
53 JVal = 0
54 for colIdx in range(X.shape[1]):
55 x = X[:, colIdx:colIdx+1]
56 y_ = Y_[colIdx, 0]
57 w = W[colIdx, 0]
58 pVal = p_right(x, y_, theta) if p_right(x, y_, theta) >= 1.e-100 else 1.e-100
59 JVal += w * numpy.log(pVal)
60 JVal = -JVal / X.shape[1]
61 return JVal
62
63
64 # 加权损失函数之梯度
65 def JW_grad(X, Y_, W, theta):
66 grad = numpy.zeros(shape=theta.shape)
67 for colIdx in range(X.shape[1]):
68 x = X[:, colIdx:colIdx+1]
69 y_ = Y_[colIdx, 0]
70 w = W[colIdx, 0]
71 grad += w * (y_ - h_theta(x, theta)) * x
72 grad = -grad / X.shape[1]
73 return grad
74
75
76 # 加权损失函数之Hessian
77 def JW_Hess(X, Y_, W, theta):
78 Hess = numpy.zeros(shape=(theta.shape[0], theta.shape[0]))
79 for colIdx in range(X.shape[1]):
80 x = X[:, colIdx:colIdx+1]
81 y_ = Y_[colIdx, 0]
82 w = W[colIdx, 0]
83 Hess += w * h_theta(x, theta) * (1 - h_theta(x, theta)) * numpy.matmul(x, x.T)
84 Hess = Hess / X.shape[1]
85 return Hess
86
87
88 class LWLR(object):
89
90 def __init__(self, trainingSet, tauBound=(0.001, 0.1), epsilon=1.e-3):
91 self.__trainingSet = trainingSet # 训练集
92 self.__tauBound = tauBound # tau之搜索边界
93 self.__epsilon = epsilon # tau之搜索精度
94
95
96 def search_tau(self):
97 '''
98 交叉验证返回最优tau
99 采用黄金分割法计算最优tau
100 '''
101 X, Y_ = self.__get_XY_(self.__trainingSet)
102 lowerBound, upperBound = self.__tauBound
103 lowerTau = self.__calc_lowerTau(lowerBound, upperBound)
104 upperTau = self.__calc_upperTau(lowerBound, upperBound)
105 lowerErr = self.__calc_generalErr(X, Y_, lowerTau)
106 upperErr = self.__calc_generalErr(X, Y_, upperTau)
107
108 while (upperTau - lowerTau) > self.__epsilon:
109 if lowerErr > upperErr:
110 lowerBound = lowerTau
111 lowerTau = upperTau
112 lowerErr = upperErr
113 upperTau = self.__calc_upperTau(lowerBound, upperBound)
114 upperErr = self.__calc_generalErr(X, Y_, upperTau)
115 else:
116 upperBound = upperTau
117 upperTau = lowerTau
118 upperErr = lowerErr
119 lowerTau = self.__calc_lowerTau(lowerBound, upperBound)
120 lowerErr = self.__calc_generalErr(X, Y_, lowerTau)
121 # print("{}, {}~{}, {}~{}, {}".format(lowerBound, lowerTau, lowerErr, upperTau, upperErr, upperBound))
122 self.bestTau = (upperTau + lowerTau) / 2
123 # print(self.bestTau)
124 return self.bestTau
125
126
127 def __calc_generalErr(self, X, Y_, tau):
128 cnt = 0
129 generalErr = 0
130
131 for idx in range(X.shape[1]):
132 tmpx = X[:, idx:idx+1]
133 tmpy_ = Y_[idx, 0]
134 tmpX = numpy.hstack((X[:, 0:idx], X[:, idx+1:]))
135 tmpY_ = numpy.vstack((Y_[0:idx, :], Y_[idx+1:, :]))
136 tmpW = self.__get_W(tmpx, tmpX, tau)
137 theta, tab = self.__optimize_theta(tmpX, tmpY_, tmpW)
138 # print(idx, tab, tmpx.reshape(-1), tmpy_, h_theta(tmpx, theta), theta.reshape(-1))
139 if not tab:
140 continue
141 cnt += 1
142 generalErr -= numpy.log(p_right(tmpx, tmpy_, theta))
143 generalErr /= cnt
144 # print(tau, generalErr)
145 return generalErr
146
147
148 def __calc_lowerTau(self, lowerBound, upperBound):
149 delta = upperBound - lowerBound
150 lowerTau = upperBound - delta * 0.618
151 return lowerTau
152
153
154 def __calc_upperTau(self, lowerBound, upperBound):
155 delta = upperBound - lowerBound
156 upperTau = lowerBound + delta * 0.618
157 return upperTau
158
159
160 def __optimize_theta(self, X, Y_, W, maxIter=1000, epsilon=1.e-9):
161 '''
162 maxIter: 最大迭代次数
163 epsilon: 收敛判据 ~ 梯度趋于0则收敛
164 '''
165 theta = self.__init_theta((X.shape[0], 1))
166 JVal = JW(X, Y_, W, theta)
167 grad = JW_grad(X, Y_, W, theta)
168 Hess = JW_Hess(X, Y_, W, theta)
169 for i in range(maxIter):
170 if self.__is_converged1(grad, epsilon):
171 return theta, True
172
173 dCurr = -numpy.matmul(numpy.linalg.inv(Hess + 1.e-9 * numpy.identity(Hess.shape[0])), grad)
174 alpha = self.__calc_alpha_by_ArmijoRule(theta, JVal, grad, dCurr, X, Y_, W)
175
176 thetaNew = theta + alpha * dCurr
177 JValNew = JW(X, Y_, W, thetaNew)
178 if self.__is_converged2(thetaNew - theta, JValNew - JVal, epsilon):
179 return thetaNew, True
180
181 theta = thetaNew
182 JVal = JValNew
183 grad = JW_grad(X, Y_, W, theta)
184 Hess = JW_Hess(X, Y_, W, theta)
185 # print(i, JVal, grad.reshape(-1))
186 else:
187 if self.__is_converged1(grad, epsilon):
188 return theta, True
189
190 return theta, False
191
192
193 def __calc_alpha_by_ArmijoRule(self, thetaCurr, JCurr, gCurr, dCurr, X, Y_, W, c=1.e-4, v=0.5):
194 i = 0
195 alpha = v ** i
196 thetaNext = thetaCurr + alpha * dCurr
197 JNext = JW(X, Y_, W, thetaNext)
198 while True:
199 if JNext <= JCurr + c * alpha * numpy.matmul(dCurr.T, gCurr)[0, 0]: break
200 i += 1
201 alpha = v ** i
202 thetaNext = thetaCurr + alpha * dCurr
203 JNext = JW(X, Y_, W, thetaNext)
204
205 return alpha
206
207
208 def __is_converged1(self, grad, epsilon):
209 if numpy.linalg.norm(grad) <= epsilon:
210 return True
211 return False
212
213
214 def __is_converged2(self, thetaDelta, JValDelta, epsilon):
215 if numpy.linalg.norm(thetaDelta) <= epsilon or numpy.linalg.norm(JValDelta) <= epsilon:
216 return True
217 return False
218
219
220 def __init_theta(self, shape):
221 '''
222 theta之初始化
223 '''
224 thetaInit = numpy.zeros(shape=shape)
225 return thetaInit
226
227
228 def __get_XY_(self, dataSet):
229 self.X = dataSet[:, 0:2].T # 注意数值溢出
230 self.X = numpy.vstack((self.X, numpy.ones(shape=(1, self.X.shape[1]))))
231 self.Y_ = dataSet[:, 2:3]
232 return self.X, self.Y_
233
234
235 def __get_W(self, x, X, tau):
236 term1 = numpy.sum((X - x) ** 2, axis=0)
237 term2 = -term1 / (2 * tau ** 2)
238 term3 = numpy.exp(term2)
239 W = term3.reshape(-1, 1)
240 return W
241
242
243 def get_cls(self, x, tau=None, midVal=0.5):
244 if tau is None:
245 if not hasattr(self, "bestTau"):
246 self.search_tau()
247 tau = self.bestTau
248 if not hasattr(self, "X"):
249 self.__get_XY_(self.__trainingSet)
250
251 tmpx = numpy.vstack((numpy.array(x).reshape(-1, 1), numpy.ones(shape=(1, 1)))) # 注意数值溢出
252 tmpW = self.__get_W(tmpx, self.X, tau)
253 theta, tab = self.__optimize_theta(self.X, self.Y_, tmpW)
254
255 realVal = h_theta(tmpx, theta)
256 if realVal > midVal: # 正例
257 return 1
258 elif realVal == midVal: # 中间
259 return 0.5
260 else: # 负例
261 return 0
262
263
264 def get_accuracy(self, testSet, tau=None, midVal=0.5):
265 '''
266 正确率计算
267 '''
268 if tau is None:
269 if not hasattr(self, "bestTau"):
270 self.search_tau()
271 tau = self.bestTau
272
273 rightCnt = 0
274 for row in testSet:
275 cls = self.get_cls(row[:2], tau, midVal)
276 if cls == row[2]:
277 rightCnt += 1
278
279 accuracy = rightCnt / testSet.shape[0]
280 return accuracy
281
282
283 class SpiralPlot(object):
284
285 def spiral_data_plot(self, trainingData0, trainingData1, testData0, testData1):
286 fig = plt.figure(figsize=(5, 5))
287 ax1 = plt.subplot()
288 ax1.scatter(trainingData1[:, 0], trainingData1[:, 1], c="red", marker="o", s=10, label="training data - Positive")
289 ax1.scatter(trainingData0[:, 0], trainingData0[:, 1], c="blue", marker="o", s=10, label="training data - Negative")
290
291 ax1.scatter(testData1[:, 0], testData1[:, 1], c="red", marker="x", s=10, label="test data - Positive")
292 ax1.scatter(testData0[:, 0], testData0[:, 1], c="blue", marker="x", s=10, label="test data - Negative")
293 ax1.set(xlim=(0, 1), ylim=(0, 1), xlabel="$x_1$", ylabel="$x_2$")
294 plt.legend(fontsize="x-small")
295 fig.tight_layout()
296 fig.savefig("spiral.png", dpi=100)
297 plt.close()
298
299
300 def spiral_pred_plot(self, lwlrObj, tau, trainingData0=trainingData0, trainingData1=trainingData1):
301 x = numpy.linspace(0, 1, 500)
302 y = numpy.linspace(0, 1, 500)
303 x, y = numpy.meshgrid(x, y)
304 z = numpy.zeros(shape=x.shape)
305 for rowIdx in range(x.shape[0]):
306 for colIdx in range(x.shape[1]):
307 z[rowIdx, colIdx] = lwlrObj.get_cls((x[rowIdx, colIdx], y[rowIdx, colIdx]), tau=tau)
308 # print(rowIdx)
309 # print(z)
310 cls2color = {0: "blue", 0.5: "white", 1: "red"}
311
312 fig = plt.figure(figsize=(5, 5))
313 ax1 = plt.subplot()
314 ax1.contourf(x, y, z, levels=[-0.25, 0.25, 0.75, 1.25], colors=["blue", "white", "red"], alpha=0.3)
315 ax1.scatter(trainingData1[:, 0], trainingData1[:, 1], c="red", marker="o", s=10, label="training data - Positive")
316 ax1.scatter(trainingData0[:, 0], trainingData0[:, 1], c="blue", marker="o", s=10, label="training data - Negative")
317 ax1.scatter(testData1[:, 0], testData1[:, 1], c="red", marker="x", s=10, label="test data - Positive")
318 ax1.scatter(testData0[:, 0], testData0[:, 1], c="blue", marker="x", s=10, label="test data - Negative")
319 ax1.set(xlim=(0, 1), ylim=(0, 1), xlabel="$x_1$", ylabel="$x_2$")
320 plt.legend(loc="upper left", fontsize="x-small")
321 fig.tight_layout()
322 fig.savefig("pred.png", dpi=100)
323 plt.close()
324
325
326
327 if __name__ == "__main__":
328 lwlrObj = LWLR(trainingSet)
329 # tau = lwlrObj.search_tau() # 运算时间较长 ~ 目前搜索精度下最优值大约在tau=0.015043232844614693
330 cls = lwlrObj.get_cls((0.5, 0.5), tau=0.015043232844614693)
331 print(cls)
332 accuracy = lwlrObj.get_accuracy(testSet, tau=0.015043232844614693)
333 print("accuracy: {}".format(accuracy))
334 spiralObj = SpiralPlot()
335 spiralObj.spiral_data_plot(trainingData0, trainingData1, testData0, testData1)
336 # spiralObj.spiral_pred_plot(lwlrObj, tau=0.015043232844614693) # 运算时间较长- View Code 笔者所用训练集与测试集数据分布如下:
很显然, 此螺旋数据集并非线性可分.
- 结果展示:
此局部加权模型在训练集与测试集上的正确率均达到100%.
- 使用建议:
①. 权重参数$\tau$的获取极为关键, 很大程度上将影响模型的表达能力.
②. 泛化误差可能存在多个局部极值, 需要指定合适的范围, 再行采用黄金分割法进行搜索.
本文章为转载内容，我们尊重原作者对文章享有的著作权。如有内容错误或侵权问题，欢迎原作者联系我们进行内容更正或删除文章。
