非线性方程的拟合，例如 y=x^2+0.5x+1 , 就是将x^2看作X的一个特征值
#准备数据
import numpy as np
import matplotlib.pyplot as plt
x = np.random.uniform(-3,3,size=100)
X = x.reshape(-1,1)
y = 0.5*x**2 + x + 2 +np.random.normal(0,1,size=100)
X2 = np.hstack([X,X**2]) #将X和X^2共同作为特征值构成心得矩阵
from sklearn.linear_model import LinearRegression
lin_reg = LinearRegression()
lin_reg.fit(X2,y)
plt.scatter(x,y)
plt.plot(np.sort(x),lin_reg.predict(X2)[np.argsort(x)],color='r')
scikit-learn中的多项式回归
from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=3)
poly.fit(X)
X2 = poly.transform(X)
# X2.shape = (100,4)
将X拓展为了具有X，X^2,X^3的矩阵
当数据具有多个特征时，即X不止有一列时
多项式回归应用于PipeLine
使用PipeLine管道一次性进行多项式回归、数据归一化、线性回归预测
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
poly_reg = Pipeline([
("poly",PolynomialFeatures(degree=2)),
("std_scaler",StandardScaler()),
("lin_reg",LinearRegression())
])
poly_reg.fit(X,y)
y_predict = poly_reg.predict(X)
plt.scatter(x,y)
plt.plot(np.sort(x),y_predict[np.argsort(x)],color='r')
plt.show()
学习曲线
由学习曲线看欠拟合和过拟合，横轴代表训练用数据数量，纵轴为均方根误差
def plot_learning_curve(algo,X_train,X_test,y_train,y_test):
train_score = []
test_score = []
for i in range(1,len(X_train)+1):
algo.fit(X_train[:i],y_train[:i])
y_train_predict = algo.predict(X_train[:i])
train_score.append(sqrt(mean_squared_error(y_train_predict[:i],y_train[:i])))
y_test_predict = algo.predict(X_test)
test_score.append(sqrt(mean_squared_error(y_test_predict,y_test)))
plt.plot([i for i in range(1,len(X_train)+1)],train_score,label="train")
plt.plot([i for i in range(1,len(X_train)+1)],test_score,label="test")
plt.legend()
plt.axis([0,len(X_train)+1,0,4])
plt.show()
plot_learning_curve(LinearRegression(),X_train,X_test,y_train,y_test)
欠拟合，误差较大
poly2_reg = PolynomialRegression(degree=2)
plot_learning_curve(poly2_reg,X_train,X_test,y_train,y_test)
最佳
poly20_reg = PolynomialRegression(degree=20)
plot_learning_curve(poly20_reg,X_train,X_test,y_train,y_test)
过拟合，泛化能力较差，较好拟合训练集，难以拟合测试集
