1 最熟悉的陌生人：多重共线性

逆矩阵存在的充分必要条件 

行列式不为0的充分必要条件

 

 

 

 

 

 

 

 

 

 

 

 

 

 

矩阵满秩的充分必要条件

 

 

 

 

 

 

 

 

 

 

 

 

2 岭回归

2.1 岭回归解决多重共线性问题

 

 

 

 

 

 

 

 

2.2 linear_model.Ridge 

 

 

import numpy as np
import pandas as pd
from sklearn.linear_model import Ridge, LinearRegression, Lasso
from sklearn.model_selection import train_test_split as TTS
from sklearn.datasets import fetch_california_housing as fch
import matplotlib.pyplot as plt
housevalue = fch()
X = pd.DataFrame(housevalue.data) y = housevalue.target
X.columns = ["住户收入中位数","房屋使用年代中位数","平均房间数目"
           ,"平均卧室数目","街区人口","平均入住率","街区的纬度","街区的经度"] X.head()
Xtrain,Xtest,Ytrain,Ytest = TTS(X,y,test_size=0.3,random_state=420) #数据集索引恢复
for i in [Xtrain,Xtest]:
    i.index = range(i.shape[0])
#使用岭回归来进行建模
reg = Ridge(alpha=1).fit(Xtrain,Ytrain)
reg.score(Xtest,Ytest) #交叉验证下，与线性回归相比，岭回归的结果如何变化？
alpharange = np.arange(1,1001,100)
ridge, lr = [], []
for alpha in alpharange:
    reg = Ridge(alpha=alpha)
    linear = LinearRegression()
    regs = cross_val_score(reg,X,y,cv=5,scoring = "r2").mean()
    linears = cross_val_score(linear,X,y,cv=5,scoring = "r2").mean()
    ridge.append(regs)
    lr.append(linears)
plt.plot(alpharange,ridge,color="red",label="Ridge")
plt.plot(alpharange,lr,color="orange",label="LR")
plt.title("Mean")
plt.legend()
plt.show()
#细化一下学习曲线
alpharange = np.arange(1,201,10)

#模型方差如何变化？
alpharange = np.arange(1,1001,100)
ridge, lr = [], []
for alpha in alpharange:
    reg = Ridge(alpha=alpha)
    linear = LinearRegression()
    varR = cross_val_score(reg,X,y,cv=5,scoring="r2").var()
    varLR = cross_val_score(linear,X,y,cv=5,scoring="r2").var()
    ridge.append(varR)
    lr.append(varLR)
plt.plot(alpharange,ridge,color="red",label="Ridge")
plt.plot(alpharange,lr,color="orange",label="LR")
plt.title("Variance")
plt.legend()
plt.show()

 

 

from sklearn.datasets import load_boston
from sklearn.model_selection import cross_val_score
X = load_boston().data
y = load_boston().target
Xtrain,Xtest,Ytrain,Ytest = TTS(X,y,test_size=0.3,random_state=420) #先查看方差的变化
alpharange = np.arange(1,1001,100)
ridge, lr = [], []
for alpha in alpharange:
    reg = Ridge(alpha=alpha)
    linear = LinearRegression()
    varR = cross_val_score(reg,X,y,cv=5,scoring="r2").var()
    varLR = cross_val_score(linear,X,y,cv=5,scoring="r2").var()
    ridge.append(varR)
    lr.append(varLR)
plt.plot(alpharange,ridge,color="red",label="Ridge")
plt.plot(alpharange,lr,color="orange",label="LR")
plt.title("Variance")
plt.legend()
plt.show()
#查看R2的变化
alpharange = np.arange(1,1001,100)
ridge, lr = [], []
for alpha in alpharange:
    reg = Ridge(alpha=alpha)
    linear = LinearRegression()
    regs = cross_val_score(reg,X,y,cv=5,scoring = "r2").mean()
    linears = cross_val_score(linear,X,y,cv=5,scoring = "r2").mean()
    ridge.append(regs)
    lr.append(linears)
plt.plot(alpharange,ridge,color="red",label="Ridge")
plt.plot(alpharange,lr,color="orange",label="LR")
plt.title("Mean")
plt.legend()
plt.show()
#细化学习曲线
alpharange = np.arange(100,300,10)
ridge, lr = [], []
for alpha in alpharange:
    reg = Ridge(alpha=alpha)
    #linear = LinearRegression()
    regs = cross_val_score(reg,X,y,cv=5,scoring = "r2").mean()
    #linears = cross_val_score(linear,X,y,cv=5,scoring = "r2").mean()
    ridge.append(regs)
    lr.append(linears)
plt.plot(alpharange,ridge,color="red",label="Ridge")
#plt.plot(alpharange,lr,color="orange",label="LR")
plt.title("Mean")
plt.legend()
plt.show()

2.3 选取最佳的正则化参数取值 

 

 

 

import numpy as np
import matplotlib.pyplot as plt
from sklearn import linear_model
#创造10*10的希尔伯特矩阵
X = 1. / (np.arange(1, 11) + np.arange(0, 10)[:, np.newaxis])
y = np.ones(10) #计算横坐标
n_alphas = 200
alphas = np.logspace(-10, -2, n_alphas) #建模，获取每一个正则化取值下的系数组合
coefs = []
for a in alphas:
    ridge = linear_model.Ridge(alpha=a, fit_intercept=False)
    ridge.fit(X, y)
    coefs.append(ridge.coef_) #绘图展示结果
ax = plt.gca()
ax.plot(alphas, coefs)
ax.set_xscale('log')
ax.set_xlim(ax.get_xlim()[::-1])  #将横坐标逆转
plt.xlabel('正则化参数alpha')
plt.ylabel('系数w')
plt.title('岭回归下的岭迹图')
plt.axis('tight')
plt.show()

 

 

 

 

 

 

这个类的使用也非常容易，依然使用我们之前建立的加利佛尼亚房屋价值数据集： 

import numpy as np
import pandas as pd
from sklearn.linear_model import RidgeCV, LinearRegression
from sklearn.model_selection import train_test_split as TTS
from sklearn.datasets import fetch_california_housing as fch
import matplotlib.pyplot as plt
housevalue = fch()
X = pd.DataFrame(housevalue.data) y = housevalue.target
X.columns = ["住户收入中位数","房屋使用年代中位数","平均房间数目"
           ,"平均卧室数目","街区人口","平均入住率","街区的纬度","街区的经度"]
Ridge_ = RidgeCV(alphas=np.arange(1,1001,100)
                #,scoring="neg_mean_squared_error"
                 ,store_cv_values=True
                #,cv=5
               ).fit(X, y)
#无关交叉验证的岭回归结果
Ridge_.score(X,y) #调用所有交叉验证的结果
Ridge_.cv_values_.shape
#进行平均后可以查看每个正则化系数取值下的交叉验证结果
Ridge_.cv_values_.mean(axis=0) #查看被选择出来的最佳正则化系数
Ridge_.alpha_