import numpy as np
from matplotlib import pyplot as plt
# 创建线性回归数据集
def create_dataset():
X = 2 * np.random.rand(100, 1)
# 结果加上高斯噪声
y = 4 + 3*X + np.random.randn(100, 1)
return X, y
# 线性回归解析法：使用正态方程求解，直接得到全局最优解
def linear_regression_analysis(X, y):
# 特征向量为参数b添加值为1的特征
X_b = np.c_[np.ones((100, 1)), X]
# 用正态方程解得全局最优解
theta_best = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y)
print("线性回归解析解为：", theta_best)
# 预测
sample = np.array([[0], [2]])
sample_b = np.c_[np.ones((2, 1)), sample]
predict = sample_b.dot(theta_best)
# print('解析解方程预测为：', predict)
# 绘制线性回归模型图像
plt.plot(sample, predict, 'r-')
plt.plot(X, y, 'b.')
plt.axis([0, 2, 0, 15])
plt.show()
return X_b
# 使用sk-learn的线性回归模型，默认使用解析法
def linear_regression_sk(X, y):
from sklearn.linear_model import LinearRegression
# 创建线性回归模型实例
lin_reg = LinearRegression()
lin_reg.fit(X, y)
print('sk-learn线性回归解析解：', 'b：', lin_reg.intercept_, 'w：', lin_reg.coef_)
# 线性回归批量梯度下降法（batch gradient descent）
def linear_regression_batch_gd(X_b, y):
# 学习率不变、迭代次数和样本数
learning_rate = 0.1
max_iterations = 1000
m = 100
# 随机初始值
theta = np.random.randn(2, 1)
# 开始迭代
for n in range(max_iterations):
gradients = 2/m * X_b.T.dot(X_b.dot(theta)-y)
theta = theta - learning_rate*gradients
print('线性回归批量梯度下降法解：', theta)
# 线性回归随机梯度下降法（stochastic gradient descent）
def linear_regression_stochastic_gd(X_b, y):
# epoch次数，样本数
n_epochs = 50
m = 100
theta = np.random.randn(2, 1)
for epoch in range(n_epochs):
for i in range(m):
random_index = np.random.randint(m)
xi = X_b[random_index:random_index+1]
yi = y[random_index:random_index+1]
gradients = 2 * xi.T.dot(xi.dot(theta) - yi)
learning_rate = 1.0/(epoch*m + i + 10)
theta = theta - learning_rate*gradients
print('线性回归随机梯度下降法解：', theta)
# sk-learn 线性回归随机梯度下降
def linear_regression_stochastic_gd_sk(X, y):
from sklearn.linear_model import SGDRegressor
sgd_reg = SGDRegressor(n_iter=50, penalty=None, eta0=0.1)
sgd_reg.fit(X, y.ravel())
print('sk-learn线性回归随机梯度下降法解：', 'b：', sgd_reg.intercept_, 'w：', sgd_reg.coef_)
# 创建多项式回归数据集
def create_dataset_poly():
m = 100
X1 = 6 * np.random.rand(m, 1) - 3
y1 = 0.5 * X1 ** 2 + X1 + 2 + np.random.randn(m, 1)
return X1, y1
# 多项式回归
def polynomial_regression(X, y):
# 添加二次特征
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
poly_features = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly_features.fit_transform(X)
lin_reg_poly = LinearRegression()
lin_reg_poly.fit(X_poly, y)
print('多项式回归解：', 'b：', lin_reg_poly.intercept_, 'w：', lin_reg_poly.coef_)
return lin_reg_poly
# 绘制关于训练集规模的学习曲线
def plot_learning_curves(model, X, y):
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split
X_train, X_val, y_train, y_val = train_test_split(X, y, test_size=0.2)
train_errors, val_errors = [], []
for m in range(1, len(X_train)):
model.fit(X_train[:m], y_train[:m])
y_train_predict = model.predict(X_train[:m])
y_val_predict = model.predict(X_val)
train_errors.append(mean_squared_error(y_train_predict, y_train[:m]))
val_errors.append(mean_squared_error(y_val_predict, y_val))
plt.plot(np.sqrt(train_errors), "r-+", linewidth=2, label="train")
plt.plot(np.sqrt(val_errors), "b-", linewidth=3, label="val")
plt.show()
# 岭回归，l2正则化，封闭方程求解
def ridge_regression_analysis(X, y):
from sklearn.linear_model import Ridge
ridge_reg = Ridge(alpha=1, solver="cholesky")
ridge_reg.fit(X, y)
print('岭回归解：', 'b：', ridge_reg.intercept_, 'w：', ridge_reg.coef_)
# Lasso 回归，l2正则化，封闭方程求解
def lasso_regression_analysis(X, y):
from sklearn.linear_model import Lasso
lasso_reg = Lasso(alpha=0.1)
lasso_reg.fit(X, y)
print('Lasso 回归解：', 'b：', lasso_reg.intercept_, 'w：', lasso_reg.coef_)
# l2,l1正则化，梯度下降求解
def regularization_regression_gd(X, y):
from sklearn.linear_model import SGDRegressor
# l1正则化把 penalty="l2" 改为 penalty="l1"
sgd_reg = SGDRegressor(penalty="l2")
sgd_reg.fit(X, y.ravel())
print('l2梯度下降法解：', 'b：', sgd_reg.intercept_, 'w：', sgd_reg.coef_)
# 弹性网路正则化，即l1、l2混合正则化
def elasticnet_regression_gd(X, y):
from sklearn.linear_model import ElasticNet
# l1_ratio 指的就是混合率, 即l1正则化占的比例
elastic_net = ElasticNet(alpha=0.1, l1_ratio=0.5)
elastic_net.fit(X, y)
print('弹性网络解：', 'b：', elastic_net.intercept_, 'w：', elastic_net.coef_)
# 早期停止法（Early Stopping）
def early_stoping(X, y):
from sklearn.base import clone
from sklearn.linear_model import SGDRegressor
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split
# 当warm_start=True时，调用fit()方法后，训练会从停下来的地方继续，而不是从头重新开始。
sgd_reg = SGDRegressor(max_iter=1, warm_start=True, penalty=None, learning_rate="constant", eta0=0.0005)
X_train, X_val, y_train, y_val = train_test_split(X, y, test_size=0.2)
minimum_val_error = float("inf")
best_epoch = None
best_model = None
for epoch in range(1000):
sgd_reg.fit(X_train, y_train.ravel())
y_val_predict = sgd_reg.predict(X_val)
val_error = mean_squared_error(y_val_predict, y_val)
if val_error < minimum_val_error:
minimum_val_error = val_error
best_epoch = epoch
best_model = clone(sgd_reg)
print('stopping in:', best_epoch)
# 加载鸢尾花数据集
def load_dataset_flower():
from sklearn import datasets
iris = datasets.load_iris()
# X_f = iris['data']
# y_f = iris['target']
# print('加载鸢尾花数据集成功：', iris)
return iris
# logistic 回归
def logistic_classify(iris):
from sklearn.linear_model import LogisticRegression
X = iris["data"][:, 3:] # petal width
y = (iris["target"] == 2).astype(np.int)
log_reg = LogisticRegression()
log_reg.fit(X, y)
# 绘图
X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
y_proba = log_reg.predict_proba(X_new)
plt.plot(X_new, y_proba[:, 1], "g-", label="Iris-Virginica")
plt.plot(X_new, y_proba[:, 0], "b--", label="Not Iris-Virginica")
plt.show()
# softmax 回归多分类
def softmax_classify(iris):
from sklearn.linear_model import LogisticRegression
# 划分数据集
X = iris["data"][:, (2, 3)] # petal length, petal width
y = iris["target"]
# 创建 softmax 回归实例
softmax_reg = LogisticRegression(multi_class="multinomial", solver="lbfgs", C=10)
softmax_reg.fit(X, y)
# 预测
predict = softmax_reg.predict([[5, 2]])
predict_pro = softmax_reg.predict_proba([[5, 2]])
print('softmax回归预测为：', predict, '各类概率为', predict_pro)
if __name__ == '__main__':
# 获得线性回归数据集
X, y = create_dataset()
# 线性回归解析法
# X_b = linear_regression_analysis(X, y)
# sk-learn线性回归解
# linear_regression_sk(X, y)
# 线性回归批量梯度下降法
# linear_regression_batch_gd(X_b, y)
# 线性回归随机梯度下降法
# linear_regression_stochastic_gd(X_b, y)
# sk-learn线性回归随机梯度下降法
# linear_regression_stochastic_gd_sk(X, y)
# 获得多项式回归数据集
# X1, y1 = create_dataset_poly()
# 多项式回归解
# lin_reg_poly = polynomial_regression(X1, y1)
# 获得关于训练集规模的学习曲线
# plot_learning_curves(lin_reg_poly, X1, y1)
# 岭回归，l2正则化
# ridge_regression_analysis(X, y)
# lasso回归，l1正则化
# lasso_regression_analysis(X, y)
# 梯度下降法的正则化
# regularization_regression_gd(X, y)
# 弹性网络
# elasticnet_regression_gd(X, y)
# 早期停止
# early_stoping(X1, y1)
# 加载花的数据集
iris = load_dataset_flower()
# logistic 回归二分类
logistic_classify(iris)
# softmax 多分类
softmax_classify(iris)