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python 實(shí)現(xiàn)邏輯回歸

更新時(shí)間：2020年12月30日 10:04:16 作者：呱唧_T_呱唧

這篇文章主要介紹了python 實(shí)現(xiàn)邏輯回歸的方法，幫助大家更好的理解和使用python，感興趣的朋友可以了解下

邏輯回歸

適用類型：解決二分類問題

邏輯回歸的出現(xiàn)：線性回歸可以預(yù)測(cè)連續(xù)值，但是不能解決分類問題，我們需要根據(jù)預(yù)測(cè)的結(jié)果判定其屬于正類還是負(fù)類。所以邏輯回歸就是將線性回歸的結(jié)果，通過Sigmoid函數(shù)映射到（0，1）之間

線性回歸的決策函數(shù)：數(shù)據(jù)與θ的乘法，數(shù)據(jù)的矩陣格式(樣本數(shù)×列數(shù))，θ的矩陣格式(列數(shù)×1)

將其通過Sigmoid函數(shù)，獲得邏輯回歸的決策函數(shù)

使用Sigmoid函數(shù)的原因：

可以對(duì)(-∞, +∞)的結(jié)果，映射到(0, 1)之間作為概率

可以將1/2作為決策邊界

數(shù)學(xué)特性好，求導(dǎo)容易

邏輯回歸的損失函數(shù)

線性回歸的損失函數(shù)維平方損失函數(shù)，如果將其用于邏輯回歸的損失函數(shù)，則其數(shù)學(xué)特性不好，有很多局部極小值，難以用梯度下降法求解最優(yōu)

這里使用對(duì)數(shù)損失函數(shù)

解釋：如果一個(gè)樣本為正樣本，那么我們希望將其預(yù)測(cè)為正樣本的概率p越大越好，也就是決策函數(shù)的值越大越好，則logp越大越好，邏輯回歸的決策函數(shù)值就是樣本為正的概率；如果一個(gè)樣本為負(fù)樣本，那么我們希望將其預(yù)測(cè)為負(fù)樣本的概率越大越好，也就是（1-p）越大越好，即log(1-p)越大越好

為什么使用對(duì)數(shù)函數(shù)：樣本集中有很多樣本，要求其概率連乘，概率為0-1之間的數(shù)，連乘越來越小，利用log變換將其變?yōu)檫B加，不會(huì)溢出，不會(huì)超出計(jì)算精度

損失函數(shù):： y(1->m)表示Sigmoid值(樣本數(shù)×1)，hθx(1->m)表示決策函數(shù)值(樣本數(shù)×1)，所以中括號(hào)的值(1×1)

二分類邏輯回歸直線編碼實(shí)現(xiàn)

import numpy as np
from matplotlib import pyplot as plt

from scipy.optimize import minimize
from sklearn.preprocessing import PolynomialFeatures


class MyLogisticRegression:
  def __init__(self):
    plt.rcParams["font.sans-serif"] = ["SimHei"]
    # 包含數(shù)據(jù)和標(biāo)簽的數(shù)據(jù)集
    self.data = np.loadtxt("./data2.txt", delimiter=",")
    self.data_mat = self.data[:, 0:2]
    self.label_mat = self.data[:, 2]
    self.thetas = np.zeros((self.data_mat.shape[1]))

    # 生成多項(xiàng)式特征，最高6次項(xiàng)
    self.poly = PolynomialFeatures(6)
    self.p_data_mat = self.poly.fit_transform(self.data_mat)

  def cost_func_reg(self, theta, reg):
    """
    損失函數(shù)具體實(shí)現(xiàn)
    :param theta: 邏輯回歸系數(shù)
    :param data_mat: 帶有截距項(xiàng)的數(shù)據(jù)集
    :param label_mat: 標(biāo)簽數(shù)據(jù)集
    :param reg:
    :return:
    """
    m = self.label_mat.size
    label_mat = self.label_mat.reshape(-1, 1)
    h = self.sigmoid(self.p_data_mat.dot(theta))

    J = -1 * (1/m)*(np.log(h).T.dot(label_mat) + np.log(1-h).T.dot(1-label_mat))\
      + (reg / (2*m)) * np.sum(np.square(theta[1:]))
    if np.isnan(J[0]):
      return np.inf
    return J[0]

  def gradient_reg(self, theta, reg):
    m = self.label_mat.size
    h = self.sigmoid(self.p_data_mat.dot(theta.reshape(-1, 1)))
    label_mat = self.label_mat.reshape(-1, 1)

    grad = (1 / m)*self.p_data_mat.T.dot(h-label_mat) + (reg/m)*np.r_[[[0]], theta[1:].reshape(-1, 1)]
    return grad

  def gradient_descent_reg(self, alpha=0.01, reg=0, iterations=200):
    """
    邏輯回歸梯度下降收斂函數(shù)
    :param alpha: 學(xué)習(xí)率
    :param reg:
    :param iterations: 最大迭代次數(shù)
    :return: 邏輯回歸系數(shù)組
    """
    m, n = self.p_data_mat.shape
    theta = np.zeros((n, 1))
    theta_set = []

    for i in range(iterations):
      grad = self.gradient_reg(theta, reg)
      theta = theta - alpha*grad.reshape(-1, 1)
      theta_set.append(theta)
    return theta, theta_set

  def plot_data_reg(self, x_label=None, y_label=None, neg_text="negative", pos_text="positive", thetas=None):
    neg = self.label_mat == 0
    pos = self.label_mat == 1
    fig1 = plt.figure(figsize=(12, 8))
    ax1 = fig1.add_subplot(111)
    ax1.scatter(self.p_data_mat[neg][:, 1], self.p_data_mat[neg][:, 2], marker="o", s=100, label=neg_text)
    ax1.scatter(self.p_data_mat[pos][:, 1], self.p_data_mat[pos][:, 2], marker="+", s=100, label=pos_text)
    ax1.set_xlabel(x_label, fontsize=14)

    # 描繪邏輯回歸直線（曲線）
    if isinstance(thetas, type(np.array([]))):
      x1_min, x1_max = self.p_data_mat[:, 1].min(), self.p_data_mat[:, 1].max()
      x2_min, x2_max = self.p_data_mat[:, 2].min(), self.p_data_mat[:, 2].max()
      xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))
      h = self.sigmoid(self.poly.fit_transform(np.c_[xx1.ravel(), xx2.ravel()]).dot(thetas))
      h = h.reshape(xx1.shape)
      ax1.contour(xx1, xx2, h, [0.5], linewidths=3)
    ax1.legend(fontsize=14)
    plt.show()

  @staticmethod
  def sigmoid(z):
    return 1.0 / (1 + np.exp(-z))


if __name__ == '__main__':
  my_logistic_regression = MyLogisticRegression()
  # my_logistic_regression.plot_data(x_label="線性不可分?jǐn)?shù)據(jù)集")

  thetas, theta_set = my_logistic_regression.gradient_descent_reg(alpha=0.5, reg=0, iterations=500)
  my_logistic_regression.plot_data_reg(thetas=thetas, x_label="$\\lambda$ = {}".format(0))

  thetas = np.zeros((my_logistic_regression.p_data_mat.shape[1], 1))
  # 未知錯(cuò)誤，有大佬解決可留言
  result = minimize(my_logistic_regression.cost_func_reg, thetas,
           args=(0, ),
           method=None,
           jac=my_logistic_regression.gradient_reg)
  my_logistic_regression.plot_data_reg(thetas=result.x, x_label="$\\lambda$ = {}".format(0))

二分類問題邏輯回歸曲線編碼實(shí)現(xiàn)

import numpy as np
from matplotlib import pyplot as plt

from scipy.optimize import minimize
from sklearn.preprocessing import PolynomialFeatures


class MyLogisticRegression:
  def __init__(self):
    plt.rcParams["font.sans-serif"] = ["SimHei"]
    # 包含數(shù)據(jù)和標(biāo)簽的數(shù)據(jù)集
    self.data = np.loadtxt("./data2.txt", delimiter=",")
    self.data_mat = self.data[:, 0:2]
    self.label_mat = self.data[:, 2]
    self.thetas = np.zeros((self.data_mat.shape[1]))

    # 生成多項(xiàng)式特征，最高6次項(xiàng)
    self.poly = PolynomialFeatures(6)
    self.p_data_mat = self.poly.fit_transform(self.data_mat)

  def cost_func_reg(self, theta, reg):
    """
    損失函數(shù)具體實(shí)現(xiàn)
    :param theta: 邏輯回歸系數(shù)
    :param data_mat: 帶有截距項(xiàng)的數(shù)據(jù)集
    :param label_mat: 標(biāo)簽數(shù)據(jù)集
    :param reg:
    :return:
    """
    m = self.label_mat.size
    label_mat = self.label_mat.reshape(-1, 1)
    h = self.sigmoid(self.p_data_mat.dot(theta))

    J = -1 * (1/m)*(np.log(h).T.dot(label_mat) + np.log(1-h).T.dot(1-label_mat))\
      + (reg / (2*m)) * np.sum(np.square(theta[1:]))
    if np.isnan(J[0]):
      return np.inf
    return J[0]

  def gradient_reg(self, theta, reg):
    m = self.label_mat.size
    h = self.sigmoid(self.p_data_mat.dot(theta.reshape(-1, 1)))
    label_mat = self.label_mat.reshape(-1, 1)

    grad = (1 / m)*self.p_data_mat.T.dot(h-label_mat) + (reg/m)*np.r_[[[0]], theta[1:].reshape(-1, 1)]
    return grad

  def gradient_descent_reg(self, alpha=0.01, reg=0, iterations=200):
    """
    邏輯回歸梯度下降收斂函數(shù)
    :param alpha: 學(xué)習(xí)率
    :param reg:
    :param iterations: 最大迭代次數(shù)
    :return: 邏輯回歸系數(shù)組
    """
    m, n = self.p_data_mat.shape
    theta = np.zeros((n, 1))
    theta_set = []

    for i in range(iterations):
      grad = self.gradient_reg(theta, reg)
      theta = theta - alpha*grad.reshape(-1, 1)
      theta_set.append(theta)
    return theta, theta_set

  def plot_data_reg(self, x_label=None, y_label=None, neg_text="negative", pos_text="positive", thetas=None):
    neg = self.label_mat == 0
    pos = self.label_mat == 1
    fig1 = plt.figure(figsize=(12, 8))
    ax1 = fig1.add_subplot(111)
    ax1.scatter(self.p_data_mat[neg][:, 1], self.p_data_mat[neg][:, 2], marker="o", s=100, label=neg_text)
    ax1.scatter(self.p_data_mat[pos][:, 1], self.p_data_mat[pos][:, 2], marker="+", s=100, label=pos_text)
    ax1.set_xlabel(x_label, fontsize=14)

    # 描繪邏輯回歸直線（曲線）
    if isinstance(thetas, type(np.array([]))):
      x1_min, x1_max = self.p_data_mat[:, 1].min(), self.p_data_mat[:, 1].max()
      x2_min, x2_max = self.p_data_mat[:, 2].min(), self.p_data_mat[:, 2].max()
      xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))
      h = self.sigmoid(self.poly.fit_transform(np.c_[xx1.ravel(), xx2.ravel()]).dot(thetas))
      h = h.reshape(xx1.shape)
      ax1.contour(xx1, xx2, h, [0.5], linewidths=3)
    ax1.legend(fontsize=14)
    plt.show()

  @staticmethod
  def sigmoid(z):
    return 1.0 / (1 + np.exp(-z))


if __name__ == '__main__':
  my_logistic_regression = MyLogisticRegression()
  # my_logistic_regression.plot_data(x_label="線性不可分?jǐn)?shù)據(jù)集")

  thetas, theta_set = my_logistic_regression.gradient_descent_reg(alpha=0.5, reg=0, iterations=500)
  my_logistic_regression.plot_data_reg(thetas=thetas, x_label="$\\lambda$ = {}".format(0))

  thetas = np.zeros((my_logistic_regression.p_data_mat.shape[1], 1))
  # 未知錯(cuò)誤，有大佬解決可留言
  result = minimize(my_logistic_regression.cost_func_reg, thetas,
           args=(0, ),
           method=None,
           jac=my_logistic_regression.gradient_reg)
  my_logistic_regression.plot_data_reg(thetas=result.x, x_label="$\\lambda$ = {}".format(0))

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