什么是拉索回歸
LASSO: Least Absolute Shrinkage and Selection Operator Regression
嶺回歸的目標：
使 $J(\theta) = MSE(Y, \hat{y}; \theta) \alpha \frac{1}{2} \sum_{i=1}^n \theta^2_i $ 盡可能小
LASSO 回歸的目標：
使 $J(\theta) = MSE(Y, \hat{y}; \theta) \alpha \sum_{i=1}^n | \theta_i | $ 盡可能小
LASSO趨向於使得一部分theta值變為0。 所以可作為特征選擇用（也是名字中 Selection 的意義）。
比較 Ridge & LASSO
下面幾組式子 模式比較相像
表達不同的衡量標准，但是背后的數學思想非常相似，表達出來的數學含義近乎一致。
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Ridge & LASSO ：衡量正則化
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MSE & MAE：衡量回歸結果的好壞
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歐拉距離 & 曼哈頓距離：衡量兩點之間距離的大小
L0 正則
$J(\theta)= МЅЕ(у, \hat{y}у; \theta)+ mіn{非0 \theta} $
實際用L1取代，因為L0正則的優化是一個NP難的問題
彈性網 Elastoc Net
結合 L1 和 L2 正則項，添加比例 r。
代碼實現
import numpy as np
import matplotlib.pyplot as plt
np.random.seed(42)
x = np.random.uniform(-3.0, 3.0, size=100)
X = x.reshape(-1, 1)
y = 0.5 * x + 3 + np.random.normal(0, 1, size=100)
plt.scatter(x, y)
plt.show()
from sklearn.model_selection import train_test_split
np.random.seed(666)
X_train, X_test, y_train, y_test = train_test_split(X, y)
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import LinearRegression
def PolynomialRegression(degree):
return Pipeline([
("poly", PolynomialFeatures(degree=degree)),
("std_scaler", StandardScaler()),
("lin_reg", LinearRegression())
])
from sklearn.metrics import mean_squared_error
poly_reg = PolynomialRegression(degree=20)
poly_reg.fit(X_train, y_train)
y_predict = poly_reg.predict(X_test)
mean_squared_error(y_test, y_predict)
# 167.94010867293571
def plot_model(model):
X_plot = np.linspace(-3, 3, 100).reshape(100, 1)
y_plot = model.predict(X_plot)
plt.scatter(x, y)
plt.plot(X_plot[:,0], y_plot, color='r')
plt.axis([-3, 3, 0, 6])
plt.show()
plot_model(poly_reg)
from sklearn.linear_model import Lasso
def LassoRegression(degree, alpha):
return Pipeline([
("poly", PolynomialFeatures(degree=degree)),
("std_scaler", StandardScaler()),
("lasso_reg", Lasso(alpha=alpha))
])
lasso1_reg = LassoRegression(20, 0.01)
lasso1_reg.fit(X_train, y_train)
y1_predict = lasso1_reg.predict(X_test)
mean_squared_error(y_test, y1_predict)
# 1.1496080843259966
plot_model(lasso1_reg)
lasso2_reg = LassoRegression(20, 0.1)
lasso2_reg.fit(X_train, y_train)
y2_predict = lasso2_reg.predict(X_test)
mean_squared_error(y_test, y2_predict)
# 1.1213911351818648
plot_model(lasso2_reg)
lasso3_reg = LassoRegression(20, 1)
lasso3_reg.fit(X_train, y_train)
y3_predict = lasso3_reg.predict(X_test)
mean_squared_error(y_test, y3_predict)
# 1.8408939659515595
plot_model(lasso3_reg)
