局部加权之逻辑回归(1)

• 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:
  egin{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:
  egin{equation}label{eq_2}
  sig( heta(x)) = frac{1}{1 + e^{- heta(x)}}
  end{equation}
  其中, $x$为输入参数, $ heta(x)$为sigmoid函数之相位. 展开该相位, 可获取sigmoid函数之特征线$ heta(x) = c$, 特征线$ heta(x) = 0$也就是我们最常见的分界线. 很显然, 相位$ heta(x)$之具体形式将影响分界线最终的表达能力, 即分界线形状的复杂程度.
  为避免混淆, 现将相位$ heta(x)$拆开:①. $x$ ~ 输入参数; ②. $ heta$ ~ 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:
  egin{equation}label{eq_4}
  h_{ heta}(x) = sig( heta(x))
  end{equation}
  以$y = 1$表示正例, $y = 0$表示负例. 令$h_{ heta}(x)$表示输入参数$x$取正例之概率, $1 - h_{ heta}(x)$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:
  egin{equation}label{eq_5}
  P(right mid x, heta) = h_{ heta}(x)^y(1 - h_{ heta}(x))^{(1 - y)}
  end{equation}
  考虑训练集中存在$n$个样本, 假设相位参数$ heta$已知, 则此n个样本同时预测正确之概率为:
  egin{equation}label{eq_6}
  L( heta) = prod_{i = 1}^{n}P(right mid x^{(i)}, heta)
  end{equation}
  对该似然函数取对数似然:
  egin{equation}label{eq_7}
  lnL( heta) = sum_{i = 1}^{n}lnP(right mid x^{(i)}, heta)
  end{equation}
  定义损失函数:
  egin{equation}label{eq_8}
  J( heta) = -frac{1}{n}lnL( heta)
  end{equation}
  根据最大似然估计, 有如下无约束最优化问题:
  egin{equation}label{eq_9}
  min J( heta)
  end{equation}
  其中, 相位参数$ heta$为优化变量. 对该优化问题进行求解, 即可获取相位参数$ heta$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:
  egin{equation}label{eq_10}
  left{egin{matrix}
  abla h_{ heta} = h_{ heta}(1 - h_{ heta}) abla heta \
  abla lnh_{ heta} = (1 - h_{ heta}) abla heta \
  abla ln(1 - h_{ heta}) = -h_{ heta} abla heta
  end{matrix} ight.
  end{equation}
  目标函数$J( heta)$之gradient:
  egin{equation}label{eq_11}
  abla J = -frac{1}{n} sum_{i = 1}^{n} (y^{(i)} - h_{ heta}(x^{(i)})) abla heta \
  end{equation}
  目标函数$J( heta)$之Hessian矩阵:
  egin{equation}label{eq_12}
  H = abla^2 J = -frac{1}{n} sum_{i = 1}^{n}[(y^{(i)} - h_{ heta}(x^{(i)})) abla^2 heta - h_{ heta}(x^{(i)})(1 - h_{ heta}(x^{(i)})) abla heta ( abla heta)^T]
  end{equation}
  
  Part Ⅳ
  引入局部加权项$exp(-left | x_i - x ight |_2^2 / (2 au^2))$, 引入原则:
  ①. 不改变损失函数$J( heta)$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$ heta(x) = 0$作为分界线, 并令:
  egin{equation}
  left{egin{matrix}
  h_{ heta}(x) geq 0.5 & y = 1 & 正例 \
  h_{ heta}(x) < 0.5 & y = 0 & 负例
  end{matrix} ight.
  end{equation}
• 代码实现:
  

  # 局部加权之逻辑回归
  
  import numpy
  from matplotlib import pyplot as plt
  
  
  def spiral_point(val, center=(0.5, 0.5)):
      rn = 0.4 * (105 - val) / 104
      an = numpy.pi * (val - 1) / 25
      
      x0 = center[0] + rn * numpy.sin(an)
      y0 = center[1] + rn * numpy.cos(an)
      z0 = 0
      x1 = center[0] - rn * numpy.sin(an)
      y1 = center[1] - rn * numpy.cos(an)
      z1 = 1
      
      return (x0, y0, z0), (x1, y1, z1)
  
  
  def spiral_data(valList):
      dataList = list(spiral_point(val) for val in valList)
      data0 = numpy.array(list(item[0] for item in dataList))
      data1 = numpy.array(list(item[1] for item in dataList))
      return data0, data1
  
  
  # 生成训练数据集
  trainingValList = numpy.arange(1, 101, 1)
  trainingData0, trainingData1 = spiral_data(trainingValList)
  trainingSet = numpy.vstack((trainingData0, trainingData1))
  
  # 生成测试数据集
  testValList = numpy.arange(1.5, 101.5, 1)
  testData0, testData1 = spiral_data(testValList)
  testSet = numpy.vstack((testData0, testData1))
  
  
  # 假设模型
  def h_theta(x, theta):
      val = 1. / (1. + numpy.exp(-numpy.sum(x * theta)))
      return val
  
  
  # 假设模型预测正确之概率
  def p_right(x, y_, theta):
      val = h_theta(x, theta) ** y_ * (1. - h_theta(x, theta)) ** (1 - y_)
      return val
  
      
  # 加权损失函数
  def JW(X, Y_, W, theta):
      JVal = 0
      for colIdx in range(X.shape[1]):
          x = X[:, colIdx:colIdx+1]
          y_ = Y_[colIdx, 0]
          w = W[colIdx, 0]
          pVal = p_right(x, y_, theta) if p_right(x, y_, theta) >= 1.e-100 else 1.e-100
          JVal += w * numpy.log(pVal)
      JVal = -JVal / X.shape[1]
      return JVal
      
  
  # 加权损失函数之梯度
  def JW_grad(X, Y_, W, theta):
      grad = numpy.zeros(shape=theta.shape)
      for colIdx in range(X.shape[1]):
          x = X[:, colIdx:colIdx+1]
          y_ = Y_[colIdx, 0]
          w = W[colIdx, 0]
          grad += w * (y_ - h_theta(x, theta)) * x
      grad = -grad / X.shape[1]
      return grad
  
      
  # 加权损失函数之Hessian
  def JW_Hess(X, Y_, W, theta):
      Hess = numpy.zeros(shape=(theta.shape[0], theta.shape[0]))
      for colIdx in range(X.shape[1]):
          x = X[:, colIdx:colIdx+1]
          y_ = Y_[colIdx, 0]
          w = W[colIdx, 0]
          Hess += w * h_theta(x, theta) * (1 - h_theta(x, theta)) * numpy.matmul(x, x.T)
      Hess = Hess / X.shape[1]
      return Hess
  
      
  class LWLR(object):
      
      def __init__(self, trainingSet, tauBound=(0.001, 0.1), epsilon=1.e-3):
          self.__trainingSet = trainingSet                             # 训练集
          self.__tauBound = tauBound                                   # tau之搜索边界
          self.__epsilon = epsilon                                     # tau之搜索精度
          
      
      def search_tau(self):
          '''
          交叉验证返回最优tau
          采用黄金分割法计算最优tau
          '''
          X, Y_ = self.__get_XY_(self.__trainingSet)
          lowerBound, upperBound = self.__tauBound
          lowerTau = self.__calc_lowerTau(lowerBound, upperBound)
          upperTau = self.__calc_upperTau(lowerBound, upperBound)
          lowerErr = self.__calc_generalErr(X, Y_, lowerTau)
          upperErr = self.__calc_generalErr(X, Y_, upperTau)
          
          while (upperTau - lowerTau) > self.__epsilon:
              if lowerErr > upperErr:
                  lowerBound = lowerTau
                  lowerTau = upperTau
                  lowerErr = upperErr
                  upperTau = self.__calc_upperTau(lowerBound, upperBound)
                  upperErr = self.__calc_generalErr(X, Y_, upperTau)
              else:
                  upperBound = upperTau
                  upperTau = lowerTau
                  upperErr = lowerErr
                  lowerTau = self.__calc_lowerTau(lowerBound, upperBound)
                  lowerErr = self.__calc_generalErr(X, Y_, lowerTau)
              # print("{}, {}~{}, {}~{}, {}".format(lowerBound, lowerTau, lowerErr, upperTau, upperErr, upperBound))
          self.bestTau = (upperTau + lowerTau) / 2
          # print(self.bestTau)
          return self.bestTau
          
      
      def __calc_generalErr(self, X, Y_, tau):
          cnt = 0
          generalErr = 0
          
          for idx in range(X.shape[1]):
              tmpx = X[:, idx:idx+1]
              tmpy_ = Y_[idx, 0]
              tmpX = numpy.hstack((X[:, 0:idx], X[:, idx+1:]))
              tmpY_ = numpy.vstack((Y_[0:idx, :], Y_[idx+1:, :]))
              tmpW = self.__get_W(tmpx, tmpX, tau)
              theta, tab = self.__optimize_theta(tmpX, tmpY_, tmpW)
              # print(idx, tab, tmpx.reshape(-1), tmpy_, h_theta(tmpx, theta), theta.reshape(-1))
              if not tab:
                  continue
              cnt += 1
              generalErr -= numpy.log(p_right(tmpx, tmpy_, theta))
          generalErr /= cnt
          # print(tau, generalErr)
          return generalErr
          
          
      def __calc_lowerTau(self, lowerBound, upperBound):
          delta = upperBound - lowerBound
          lowerTau = upperBound - delta * 0.618
          return lowerTau
          
          
      def __calc_upperTau(self, lowerBound, upperBound):
          delta = upperBound - lowerBound
          upperTau = lowerBound + delta * 0.618
          return upperTau
          
          
      def __optimize_theta(self, X, Y_, W, maxIter=1000, epsilon=1.e-9):
          '''
          maxIter: 最大迭代次数
          epsilon: 收敛判据 ~ 梯度趋于0则收敛
          '''
          theta = self.__init_theta((X.shape[0], 1))
          JVal = JW(X, Y_, W, theta)
          grad = JW_grad(X, Y_, W, theta)
          Hess = JW_Hess(X, Y_, W, theta)
          for i in range(maxIter):
              if self.__is_converged1(grad, epsilon):
                  return theta, True
              
              dCurr = -numpy.matmul(numpy.linalg.inv(Hess + 1.e-9 * numpy.identity(Hess.shape[0])), grad)
              alpha = self.__calc_alpha_by_ArmijoRule(theta, JVal, grad, dCurr, X, Y_, W)
              
              thetaNew = theta + alpha * dCurr
              JValNew = JW(X, Y_, W, thetaNew)
              if self.__is_converged2(thetaNew - theta, JValNew - JVal, epsilon):
                  return thetaNew, True
              
              theta = thetaNew
              JVal = JValNew
              grad = JW_grad(X, Y_, W, theta)
              Hess = JW_Hess(X, Y_, W, theta)
              # print(i, JVal, grad.reshape(-1))
          else:
              if self.__is_converged1(grad, epsilon):
                  return theta, True
          
          return theta, False
          
      
      def __calc_alpha_by_ArmijoRule(self, thetaCurr, JCurr, gCurr, dCurr, X, Y_, W, c=1.e-4, v=0.5):
          i = 0
          alpha = v ** i
          thetaNext = thetaCurr + alpha * dCurr
          JNext = JW(X, Y_, W, thetaNext)
          while True:
              if JNext <= JCurr + c * alpha * numpy.matmul(dCurr.T, gCurr)[0, 0]: break
              i += 1
              alpha = v ** i
              thetaNext = thetaCurr + alpha * dCurr
              JNext = JW(X, Y_, W, thetaNext)
          
          return alpha
      
      
      def __is_converged1(self, grad, epsilon):
          if numpy.linalg.norm(grad) <= epsilon:
              return True
          return False
          
          
      def __is_converged2(self, thetaDelta, JValDelta, epsilon):
          if numpy.linalg.norm(thetaDelta) <= epsilon or numpy.linalg.norm(JValDelta) <= epsilon:
              return True
          return False
      
      
      def __init_theta(self, shape):
          '''
          theta之初始化
          '''
          thetaInit = numpy.zeros(shape=shape)
          return thetaInit
          
          
      def __get_XY_(self, dataSet):
          self.X = dataSet[:, 0:2].T                                                 # 注意数值溢出
          self.X = numpy.vstack((self.X, numpy.ones(shape=(1, self.X.shape[1]))))
          self.Y_ = dataSet[:, 2:3]
          return self.X, self.Y_
          
          
      def __get_W(self, x, X, tau):
          term1 = numpy.sum((X - x) ** 2, axis=0)
          term2 = -term1 / (2 * tau ** 2)
          term3 = numpy.exp(term2)
          W = term3.reshape(-1, 1)
          return W
          
          
      def get_cls(self, x, tau=None, midVal=0.5):
          if tau is None:
              if not hasattr(self, "bestTau"):
                  self.search_tau()
              tau = self.bestTau
          if not hasattr(self, "X"):
              self.__get_XY_(self.__trainingSet)
              
          tmpx = numpy.vstack((numpy.array(x).reshape(-1, 1), numpy.ones(shape=(1, 1))))   # 注意数值溢出
          tmpW = self.__get_W(tmpx, self.X, tau)
          theta, tab = self.__optimize_theta(self.X, self.Y_, tmpW)
          
          realVal = h_theta(tmpx, theta)
          if realVal > midVal:                   # 正例
              return 1
          elif realVal == midVal:                # 中间
              return 0.5
          else:                                  # 负例
              return 0
          
          
      def get_accuracy(self, testSet, tau=None, midVal=0.5):
          '''
          正确率计算
          '''
          if tau is None:
              if not hasattr(self, "bestTau"):
                  self.search_tau()
              tau = self.bestTau
          
          rightCnt = 0
          for row in testSet:
              cls = self.get_cls(row[:2], tau, midVal)
              if cls == row[2]:
                  rightCnt += 1
                  
          accuracy = rightCnt / testSet.shape[0]
          return accuracy
          
  
  class SpiralPlot(object):
      
      def spiral_data_plot(self, trainingData0, trainingData1, testData0, testData1):
          fig = plt.figure(figsize=(5, 5))
          ax1 = plt.subplot()
          ax1.scatter(trainingData1[:, 0], trainingData1[:, 1], c="red", marker="o", s=10, label="training data - Positive")
          ax1.scatter(trainingData0[:, 0], trainingData0[:, 1], c="blue", marker="o", s=10, label="training data - Negative")
          
          ax1.scatter(testData1[:, 0], testData1[:, 1], c="red", marker="x", s=10, label="test data - Positive")
          ax1.scatter(testData0[:, 0], testData0[:, 1], c="blue", marker="x", s=10, label="test data - Negative")
          ax1.set(xlim=(0, 1), ylim=(0, 1), xlabel="$x_1$", ylabel="$x_2$")
          plt.legend(fontsize="x-small")
          fig.tight_layout()
          fig.savefig("spiral.png", dpi=100)
          plt.close()
          
          
      def spiral_pred_plot(self, lwlrObj, tau, trainingData0=trainingData0, trainingData1=trainingData1):
          x = numpy.linspace(0, 1, 500)
          y = numpy.linspace(0, 1, 500)
          x, y = numpy.meshgrid(x, y)
          z = numpy.zeros(shape=x.shape)
          for rowIdx in range(x.shape[0]):
              for colIdx in range(x.shape[1]):
                  z[rowIdx, colIdx] = lwlrObj.get_cls((x[rowIdx, colIdx], y[rowIdx, colIdx]), tau=tau)
              # print(rowIdx)
              # print(z)
          cls2color = {0: "blue", 0.5: "white", 1: "red"}
          
          fig = plt.figure(figsize=(5, 5))
          ax1 = plt.subplot()
          ax1.contourf(x, y, z, levels=[-0.25, 0.25, 0.75, 1.25], colors=["blue", "white", "red"], alpha=0.3)
          ax1.scatter(trainingData1[:, 0], trainingData1[:, 1], c="red", marker="o", s=10, label="training data - Positive")
          ax1.scatter(trainingData0[:, 0], trainingData0[:, 1], c="blue", marker="o", s=10, label="training data - Negative")
          ax1.scatter(testData1[:, 0], testData1[:, 1], c="red", marker="x", s=10, label="test data - Positive")
          ax1.scatter(testData0[:, 0], testData0[:, 1], c="blue", marker="x", s=10, label="test data - Negative")
          ax1.set(xlim=(0, 1), ylim=(0, 1), xlabel="$x_1$", ylabel="$x_2$")
          plt.legend(loc="upper left", fontsize="x-small")
          fig.tight_layout()
          fig.savefig("pred.png", dpi=100)
          plt.close()
  
          
          
  if __name__ == "__main__":
      lwlrObj = LWLR(trainingSet)
      # 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
      cls = lwlrObj.get_cls((0.5, 0.5), tau=0.015043232844614693)
      print(cls)
      accuracy = lwlrObj.get_accuracy(testSet, tau=0.015043232844614693)
      print("accuracy: {}".format(accuracy))
      spiralObj = SpiralPlot()
      spiralObj.spiral_data_plot(trainingData0, trainingData1, testData0, testData1)
      # spiralObj.spiral_pred_plot(lwlrObj, tau=0.015043232844614693)       # 运算时间较长

  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$ au$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