一、 观察波士顿房价数据并加载数据集

1、加载数据集

from sklearn.datasets import load_boston
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np

boston=load_boston()
df=pd.DataFrame(boston.data,columns=boston.feature_names)
df['target']=boston.target

数据集共506条，包含有13个与房价相关的特征，分别是：

name	释义
CRIM	城镇人均犯罪率
ZN	住宅用地所占比例
INDUS	城镇中非住宅用地所占比例
CHAS	虚拟变量,用于回归分析
NOX	环保指数
RM	每栋住宅的房间数
AGE	1940 年以前建成的自住单位的比例
DIS	距离 5 个波士顿的就业中心的加权距离
RAD	距离高速公路的便利指数
TAX	每一万美元的不动产税率
PTRATIO	城镇中的教师学生比例
B	城镇中的黑人比例
LSTAT	地区中有多少房东属于低收入人群

2、查看数据项

#查看数据项
df.head()

二、 特征选择

1、 画出各数据项和房价的散点图
2、 根据散点图粗略选择CRIM, RM, LSTAT三个特征值

features=df[['RM','CRIM', 'LSTAT']]
target=df['target']
 

三、 模型选择

利用多元线性回归模型，其中自变量为数据集中的 feature_names 的维度（13维度），因变量为数据集中的 target 维度（房价）

#数据集划分
split_num=int(len(features)*0.8)
X_train=features[:split_num]
Y_train=target[:split_num]
X_test=features[split_num:]
Y_test=target[split_num:]

设置标签字段，切分数据集：训练集80%，测试集20%

四、 模型训练和测试

1、 训练模型

split_num=int(len(features)*0.8)
X_train=features[:split_num]
Y_train=target[:split_num]
X_test=features[split_num:]
Y_test=target[split_num:]

2、打印线性方程参数

print(model.coef_,model.intercept_)

3、模型预测

preds=model.predict(X_test)

4、 计算mae、mse

def mae_value(y_true,y_pred):
    n=len(y_true)
    mae=sum(np.abs(y_true-y_pred))/n
return mae

def mse_value(y_true,y_pred):
    n=len(y_true)
    mse=sum(np.square(y_true-y_pred))/n
return mse

mae=mae_value(Y_test.values,preds)
mse=mse_value(Y_test.values,preds)
print("MAE",mae)
print("MSE",mse)
 

5、 画出学习曲线

from sklearn.model_selection import learning_curve
from sklearn.model_selection import ShuffleSplit
import matplotlib.pyplot as plt
import numpy as np
def plot_learning_curve(plt,estimator,title,X,y,ylim=None,cv=None,n_jobs=1,train_sizes=np.linspace(.1,1.0,5)):
   plt.title(title)
   if ylim is not None:
     plt.ylim(ylim)
   plt.xlabel("Training examples")
   plt.ylabel("Score")
   train_sizes,train_scores,test_scores=learning_curve(estimator,X,y,cv=cv,n_jobs=n_jobs,train_sizes=train_sizes)
   train_scores_mean=np.mean(train_scores,axis=1)
   train_scores_std=np.std(train_scores,axis=1)
   test_scores_mean=np.mean(test_scores,axis=1)
   test_scores_std=np.std(test_scores,axis=1)
   plt.grid()
   plt.fill_between(train_sizes,train_scores_mean-train_scores_std,train_scores_mean+train_scores_std,alpha=0.1,color="r")
   plt.fill_between(train_sizes,test_scores_mean-test_scores_std,test_scores_mean+test_scores_std,alpha=0.1,color="g")
   plt.plot(train_sizes,train_scores_mean,'o--',color="r",label="Training scores")
   plt.plot(train_sizes,test_scores_mean,'o-',color="g",label="Cross-validation score")
   plt.legend(loc="best")
   return plt


cv=ShuffleSplit(n_splits=10,test_size=0.2,random_state=0)
plt.figure(figsize=(10,6))
plot_learning_curve(plt,model,"Learn Curve for LinearRegression",features,target,ylim=None,cv=cv)
plt.show()

五、 模型性能评估和优化

1、 模型优化，考虑用二项式和三项式优化

import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
from sklearn.model_selection import ShuffleSplit
from sklearn.model_selection import learning_curve

2、 划分数据集函数

def split_data():
    boston = load_boston()
    x = boston.data
    y = boston.target
    print(boston.feature_names)
    x_train, x_test, y_train, y_test = train_test_split(x, y, test_size = 0.2,random_state=2)
    return (x, y, x_train, x_test, y_train, y_test)

3、定义MAE、MSE函数

def mae_value(y_true,y_pred):
    n=len(y_true)
    mae=sum(np.abs(y_true-y_pred))/n
    return mae

def mse_value(y_true,y_pred):
    n=len(y_true)
    mse=sum(np.square(y_true-y_pred))/n
return mse

4、定义多项式模型函数

def polynomial_regression(degree=1):
    polynomial_features = PolynomialFeatures(degree=degree, include_bias=False)
    #模型开启数据归一化
    linear_regression_model = LinearRegression(normalize=True)
    model = Pipeline([("polynomial_features", polynomial_features),
                         ("linear_regression", linear_regression_model)])
    return model

5、 训练模型

def train_model(x_train, x_test, y_train, y_test, degrees):   
    res = []
    for degree in degrees:
        model = polynomial_regression(degree)
        model.fit(x_train, y_train)
        train_score = model.score(x_train, y_train)
        test_score = model.score(x_test, y_test)
        res.append({"model": model, "degree": degree, "train_score": train_score, "test_score": test_score})
		preds=model.predict(x_test)
        mae=mae_value(y_test,preds)
        mse=mse_value(y_test,preds)
        print(" degree: " ,degree, "  MAE:",mae,"    MSE",mse)
    for r in res:
        print("degree: {}; train score: {}; test_score: {}".format(r["degree"], r["train_score"], r["test_score"]))
    return res

6、 定义画出学习曲线的函数

def plot_learning_curve(plt,estimator,title,X,y,ylim=None,cv=None,n_jobs=1,train_sizes=np.linspace(.1,1.0,5)):
   plt.title(title)
   if ylim is not None:
     plt.ylim(ylim)
   plt.xlabel("Training examples")
   plt.ylabel("Score")
   train_sizes,train_scores,test_scores=learning_curve(estimator,X,y,cv=cv,n_jobs=n_jobs,train_sizes=train_sizes)
   train_scores_mean=np.mean(train_scores,axis=1)
   train_scores_std=np.std(train_scores,axis=1)
   test_scores_mean=np.mean(test_scores,axis=1)
   test_scores_std=np.std(test_scores,axis=1)
   plt.grid()
   plt.fill_between(train_sizes,train_scores_mean-train_scores_std,train_scores_mean+train_scores_std,alpha=0.1,color="r")
   plt.fill_between(train_sizes,test_scores_mean-test_scores_std,test_scores_mean+test_scores_std,alpha=0.1,color="g")
   plt.plot(train_sizes,train_scores_mean,'o--',color="r",label="Training scores")
   plt.plot(train_sizes,test_scores_mean,'o-',color="g",label="Cross-validation score")
   plt.legend(loc="best")
   return plt

7、定义1、2、3次多项式

degrees = [1,2,3]

8、划分数据集

x, y, x_train, x_test, y_train, y_test = split_data()

9、训练模型，并打印train score

res = train_model(x_train, x_test, y_train, y_test, degrees)

10、画出学习曲线

cv = ShuffleSplit(n_splits=10, test_size=0.2, random_state=0)
plt.figure(figsize=(10, 6))

for index, data in enumerate(res):
    plot_learning_curve(plt, data["model"], "degree %d"%data["degree"], x, y, cv=cv)
plt.show()
 

六、 结论与分析

通过对波士顿房价数据的分析预测练习，运用多元回归模型（一共十三个维度），前期训练量不足导致拟合程度不理想。经过模型的参数优化，采用了全部特征值，结果显示一次多项式训练准确度72%，测试准确度76%。二次多项式训练准确度92%，测试准确度89%，mae=2.36, mse=8.67。综上所述，采用二次多项式回归方法优化效果较好。