多项式回归 解题 多项式回归系数解释

1 概念

• 多项式回归，回归函数是回归变量多项式的回归。多项式回归模型是线性回归模型的一种，此时回归函数关于回归系数是线性的。
• 多项式回归的最大优点就是可以通过增加x的高次项对实测点进行逼近，直至满意为止。事实上，多项式回归可以处理相当一类非线性问题，它在回归分析中占有重要的地位，因为任一函数都可以分段用多项式来逼近。因此，在通常的实际问题中，不论依变量与其他自变量的关系如何，我们总可以用多项式回归来进行分析。

2 特点

• 多项式回归中，加入了特征的更高次方（例如平方项或立方项），也相当于增加了模型的自由度，用来捕获数据中非线性的变化。添加高阶项的时候，也增加了模型的复杂度。随着模型复杂度的升高，模型的容量以及拟合数据的能力增加，可以进一步降低训练误差，但导致过拟合的风险也随之增加。

3 自己实现的多项式回归

• 思想是添加高次方的特征数来实现线性回归中的非线性预测回归
  1.具体模拟

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, 100)
plt.scatter(x, y)
plt.show()
# 使用直线来模拟
from sklearn.linear_model import LinearRegression

lin_reg = LinearRegression()
lin_reg.fit(X, y)
y_predict = lin_reg.predict(X)
plt.scatter(x, y)
plt.plot(x, y_predict, color='r')
plt.show()



# 加入平方项的特征来实现模拟
X2 = np.hstack([X, X**2])
X2.shape
lin_reg2 = LinearRegression()
lin_reg2.fit(X2, y)
y_predict2 = lin_reg2.predict(X2)
plt.scatter(x, y)
plt.plot(np.sort(x), y_predict2[np.argsort(x)], color='r')
plt.show()
lin_reg2.coef_
lin_reg2.intercept_
lin_reg2.intercept_

直线模拟

多项式模拟

4 scikit-learn中实现多项式回归

1.实现多项式回归模拟

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, 100)
from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=2)
poly.fit(X)
X2 = poly.transform(X)

from sklearn.linear_model import LinearRegression

lin_reg2 = LinearRegression()
lin_reg2.fit(X2, y)
y_predict2 = lin_reg2.predict(X2)
plt.scatter(x, y)
plt.plot(np.sort(x), y_predict2[np.argsort(x)], color='r')
plt.show()
lin_reg2.coef_
lin_reg2.intercept_

poly = PolynomialFeatures(degree=2)添加多项式的最高次方是degree的值

2.PolynomialFeatures的讨论

3. 利用Pipeline实现多项式回归

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, 100)

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()
plt.scatter(x, y)
plt.plot(np.sort(x), y_predict[np.argsort(x)], color='r')
plt.show()

4. 学习曲线：随着样本数的增加，训练，测试集准确率变化曲线

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(666)
x = np.random.uniform(-3.0, 3.0, size=100)
X = x.reshape(-1, 1)
y = 0.5 * x**2 + x + 2 + np.random.normal(0, 1, size=100)
plt.scatter(x, y)
plt.show()

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=10)
X_train.shape
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

train_score = []
test_score = []
for i in range(1, 76):
    lin_reg = LinearRegression()
    lin_reg.fit(X_train[:i], y_train[:i])
    
    y_train_predict = lin_reg.predict(X_train[:i])
    train_score.append(mean_squared_error(y_train[:i], y_train_predict))
    
    y_test_predict = lin_reg.predict(X_test)
    test_score.append(mean_squared_error(y_test, y_test_predict))
plt.plot([i for i in range(1, 76)], np.sqrt(train_score), label="train")
plt.plot([i for i in range(1, 76)], np.sqrt(test_score), label="test")
plt.legend()
plt.show()




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(mean_squared_error(y_train[:i], y_train_predict))
    
        y_test_predict = algo.predict(X_test)
        test_score.append(mean_squared_error(y_test, y_test_predict))
        
    plt.plot([i for i in range(1, len(X_train)+1)], 
                               np.sqrt(train_score), label="train")
    plt.plot([i for i in range(1, len(X_train)+1)], 
                               np.sqrt(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
from sklearn.preprocessing import PolynomialFeatures
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline

def PolynomialRegression(degree):
    return Pipeline([
        ("poly", PolynomialFeatures(degree=degree)),
        ("std_scaler", StandardScaler()),
        ("lin_reg", LinearRegression())
    ])

poly2_reg = PolynomialRegression(degree=2)   # 正常
plot_learning_curve(poly2_reg, X_train, X_test, y_train, y_test)
poly20_reg = PolynomialRegression(degree=20)  # 过拟合
plot_learning_curve(poly20_reg, X_train, X_test, y_train, y_test)

5 训练集，测试集，验证集分离

• 用训练集进行训练，验证集进行测试衡量准确率，测试集去测量测试数据是否符合模型（模型搭建的好坏）也用准确率衡量。
• 具体的示意图
• 衡量方法
  
  
  1 .使用交叉验证方法来查看模型的搭建准确率，以及搜索最优超参数

import numpy as np
from sklearn import datasets
digits = datasets.load_digits()
X = digits.data
y = digits.target

测试train_test_split

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.4, random_state=666)

使用交叉验证

from sklearn.model_selection import cross_val_score

knn_clf = KNeighborsClassifier()
cross_val_score(knn_clf, X_train, y_train)





best_k, best_p, best_score = 0, 0, 0
for k in range(2, 11):
    for p in range(1, 6):
        knn_clf = KNeighborsClassifier(weights="distance", n_neighbors=k, p=p)
        scores = cross_val_score(knn_clf, X_train, y_train)
        score = np.mean(scores)
        if score > best_score:
            best_k, best_p, best_score = k, p, score
            
print("Best K =", best_k)
print("Best P =", best_p)
print("Best Score =", best_score)


best_knn_clf = KNeighborsClassifier(weights="distance", n_neighbors=2, p=2)
best_knn_clf.fit(X_train, y_train)
best_knn_clf.score(X_test, y_test)

网格搜索

from sklearn.model_selection import GridSearchCV

param_grid = [
    {
        'weights': ['distance'],
        'n_neighbors': [i for i in range(2, 11)], 
        'p': [i for i in range(1, 6)]
    }
]

grid_search = GridSearchCV(knn_clf, param_grid, verbose=1)
grid_search.fit(X_train, y_train)
best_knn_clf = grid_search.best_estimator_
best_knn_clf.score(X_test, y_test)

用网格搜索到的最优超参数和用交叉验证搜索的超参数得到的结果是一样的。网格搜索其实质内部就是按照交叉验证方法实现的。

cv参数 cv值代表分成的份数 模型份数

cross_val_score(knn_clf, X_train, y_train, cv=5)
grid_search = GridSearchCV(knn_clf, param_grid, verbose=1, cv=5)

6 解决高方差，模型泛化等问题

1.使用岭回归

from sklearn.linear_model import Ridge

def RidgeRegression(degree, alpha):
    return Pipeline([
        ("poly", PolynomialFeatures(degree=degree)),
        ("std_scaler", StandardScaler()),
        ("ridge_reg", Ridge(alpha=alpha))
    ])
ridge1_reg = RidgeRegression(20, 0.0001)
ridge1_reg.fit(X_train, y_train)

y1_predict = ridge1_reg.predict(X_test)
mean_squared_error(y_test, y1_predict)

alpha 代表后面系数所占的重要程度

2.使用LASSO回归

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)

对比 ：岭回归更加精确。但对于特征维数比较多时候LASSO回归运行时间会更短。LASSO回归因为在训练过程中会使某些维度的权重为0.

7.小结

• 使用多项式回归可以很方便的处理一些线性或非线性模型。pipeline是主要处理方法。但随着多项式特征的升高，会出现过拟合的现象。导致模型不能很好的泛化。模型的均值和方差是一对矛盾的指标，方差较大，模型不能很好泛化，均值过大会导致训练集的误差过大。解决方差过大方法是模型正则化，因此在机器学习中，大多数问题都是此问题。可以用在其他算法，使用岭回归或LASSO回归解决模型方差过大，模型泛化。除此之外，将训练，验证，测试集相分离，进行交叉验证是衡量模型的更好的方法。