局部加权之逻辑回归(1) - Python实现

2023-08-17 15:54:04

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                           # 训练集文件名
 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_converged(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.
参考文档:
螺旋数据集来源: https://www.cnblogs.com/tornadomeet/archive/2012/06/05/2537360.html

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