机器学习算法练习之（一）：Python实现logistic回归

第一步：生成数据并可视化

import numpy as npimport matplotlib.pyplot as pltnp.random.seed(12)num_observations=5000#生成二维高斯分布数据x1 = np.random.multivariate_normal([0, 0], [[1, .75],[.75, 1]], num_observations)x2 = np.random.multivariate_normal([1, 4], [[1, .75],[.75, 1]], num_observations)simulated_separableish_features = np.vstack((x1, x2)).astype(np.float32)simulated_labels = np.hstack((np.zeros(num_observations),                              np.ones(num_observations)))plt.figure(figsize=(12,8))plt.scatter(simulated_separableish_features[:, 0], simulated_separableish_features[:, 1],            c = simulated_labels, alpha = .4)

plt.figure(figsize=(12,8))plt.scatter(simulated_separableish_features[:, 0], simulated_separableish_features[:, 1],            c = simulated_labels, alpha = .4)

第二步：定义sigmoid函数以及对数似然函数

#定义sigmoid函数def sigmoid(scores):    return 1 / (1 + np.exp(-scores))

#对数似然估计def log_likelihood(features, target, weights):    scores = np.dot(features, weights)    ll = np.sum( target*scores - np.log(1 + np.exp(scores)) )    return ll

第三步：定义对数似然回归

#对数似然回归def logistic_regression(features, target, num_steps, learning_rate, add_intercept = False):    if add_intercept:        intercept = np.ones((features.shape[0], 1))        features = np.hstack((intercept, features))    weights = np.zeros(features.shape[1])    for step in range(num_steps):        scores = np.dot(features, weights)        predictions = sigmoid(scores)        # Update weights with gradient        output_error_signal = target - predictions        gradient = np.dot(features.T, output_error_signal)        weights += learning_rate * gradient        # Print log-likelihood every so often        if step % 10000 == 0:            print (log_likelihood(features, target, weights))    return weights

weights = logistic_regression(simulated_separableish_features, simulated_labels,                     num_steps = 300000, learning_rate = 5e-5, add_intercept=True)

weights：-4346.26477915
[…]
-140.725421362
-140.725421357
-140.725421355

从sklearn包导入LogisticRegression,得到权值

from sklearn.linear_model import LogisticRegressionclf = LogisticRegression(fit_intercept=True, C = 1e15)clf.fit(simulated_separableish_features, simulated_labels)print (clf.intercept_, clf.coef_)print (weights)

[-13.99400797] [[-5.02712572 8.23286799]]
[-14.09225541 -5.05899648 8.28955762]
第四步：与sklearn得到的训练准确率相比较

data_with_intercept = np.hstack((np.ones((simulated_separableish_features.shape[0], 1)),                                 simulated_separableish_features))final_scores = np.dot(data_with_intercept, weights)preds = np.round(sigmoid(final_scores))print ('Accuracy from scratch: {0}'.format((preds == simulated_labels).sum().astype(float) / len(preds)))print ('Accuracy from sk-learn: {0}'.format(clf.score(simulated_separableish_features, simulated_labels)))

Accuracy from scratch: 0.9948
Accuracy from sk-learn: 0.9948

plt.figure(figsize = (12, 8))plt.scatter(simulated_separableish_features[:, 0], simulated_separableish_features[:, 1],            c = preds == simulated_labels - 1, alpha = .8, s = 50)

蓝色代表预测正确的数据，红色代表预测错误的数据