“机器学习”系列之Logistic Regression (逻辑回归)

本文介绍逻辑回归，这是一种分类算法。它通过Sigmoid函数将线性回归结果映射到[0,1]，以概率形式分类。损失函数为对数似然函数，用随机梯度下降或牛顿法优化。其优势在于输出概率、可解释性强等，应用于CTR预估等场景。还展示了自定义函数及调用sklearn的实现代码与结果。

“机器学习”系列之Logistic Regression (逻辑回归)

0 概念

1 Sigmoid激活函数

import numpy as npimport matplotlib.pyplot as pltx = np.arange(-5.0 , 5.0 , 0.02)y = 1 / (1 + np.exp(-x))plt.xlabel('x')plt.ylabel('y = Sigmoid(x)')plt.title('Sigmoid')plt.plot(x , y)plt.show()

2 损失函数

3 优化

3.1 随机梯度下降

3.2 牛顿法

4 优势

5 应用

6 自定义函数代码实现

from math import expimport numpy as npimport pandas as pdimport matplotlib.pyplot as plt%matplotlib inlinefrom sklearn.datasets import load_irisfrom sklearn.model_selection import train_test_split

6.1 自定义数据

# datadef create_data():    iris = load_iris()    df = pd.DataFrame(iris.data, columns=iris.feature_names)    df['label'] = iris.target    df.columns = ['sepal length', 'sepal width', 'petal length', 'petal width', 'label']    data = np.array(df.iloc[:100, [0,1,-1]])    # print(data)    return data[:,:2], data[:,-1]

X, y = create_data()X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)

import math

6.2 自定义逻辑回归函数

class LogisticReressionClassifier:    def __init__(self, max_iter=200, learning_rate=0.02):        self.max_iter = max_iter        self.learning_rate = learning_rate        # sigmoid激活函数    def sigmoid(self, x):        return 1 / (1 + exp(-x))    def data_matrix(self, X):        data_mat = []        for d in X:            data_mat.append([1.0, *d])        return data_mat    def fit(self, X, y):        # label = np.mat(y)        data_mat = self.data_matrix(X) # m*n        self.weights = np.zeros((len(data_mat[0]),1), dtype=np.float32)        for iter_ in range(self.max_iter):            for i in range(len(X)):                result = self.sigmoid(np.dot(data_mat[i], self.weights))                error = y[i] - result                 self.weights += self.learning_rate * error * np.transpose([data_mat[i]])        print('LogisticRegression Model(learning_rate={},max_iter={})'.format(self.learning_rate, self.max_iter))    # def f(self, x):    #     return -(self.weights[0] + self.weights[1] * x) / self.weights[2]    def score(self, X_test, y_test):        right = 0        X_test = self.data_matrix(X_test)        for x, y in zip(X_test, y_test):            result = np.dot(x, self.weights)            if (result > 0 and y == 1) or (result < 0 and y == 0):                right += 1        return right / len(X_test)

6.3 训练

lr_clf = LogisticReressionClassifier()lr_clf.fit(X_train, y_train)

LogisticRegression Model(learning_rate=0.02,max_iter=200)

6.4 结果展示及可视化

lr_clf.score(X_test, y_test)

0.9666666666666667

x_ponits = np.arange(4, 8)y_ = -(lr_clf.weights[1]*x_ponits + lr_clf.weights[0])/lr_clf.weights[2]plt.plot(x_ponits, y_)#lr_clf.show_graph()plt.scatter(X[:50,0],X[:50,1], label='0')plt.scatter(X[50:,0],X[50:,1], label='1')plt.legend()

7 调用sklearn实现

sklearn.linear_model.LogisticRegression参数

solver参数决定了我们对逻辑回归损失函数的优化方法，有四种算法可以选择，分别是：

from sklearn.linear_model import LogisticRegression

clf = LogisticRegression(max_iter=200)

clf.fit(X_train, y_train)

LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,                   intercept_scaling=1, l1_ratio=None, max_iter=200,                   multi_class='auto', n_jobs=None, penalty='l2',                   random_state=None, solver='lbfgs', tol=0.0001, verbose=0,                   warm_start=False)

clf.score(X_test, y_test)

1.0

print(clf.coef_, clf.intercept_)

[[ 2.69741404 -2.61019199]] [-6.44843344]

x_ponits = np.arange(4, 8)y_ = -(clf.coef_[0][0]*x_ponits + clf.intercept_)/clf.coef_[0][1]plt.plot(x_ponits, y_)plt.plot(X[:50, 0], X[:50, 1], 'bo', color='blue', label='0')plt.plot(X[50:, 0], X[50:, 1], 'bo', color='orange', label='1')plt.xlabel('sepal length')plt.ylabel('sepal width')plt.legend()

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