## 1. 载入数据

```import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import cross_val_score
from collections import Counter
sns.set_style('darkgrid')
dataset = pd.DataFrame(boston_dataset.data, columns = boston_dataset.feature_names)

```CRIM    ZN  INDUS  CHAS    NOX  ...  RAD    TAX  PTRATIO       B  LSTAT
0  0.00632  18.0   2.31   0.0  0.538  ...  1.0  296.0     15.3  396.90   4.98
1  0.02731   0.0   7.07   0.0  0.469  ...  2.0  242.0     17.8  396.90   9.14
2  0.02729   0.0   7.07   0.0  0.469  ...  2.0  242.0     17.8  392.83   4.03
3  0.03237   0.0   2.18   0.0  0.458  ...  3.0  222.0     18.7  394.63   2.94
4  0.06905   0.0   2.18   0.0  0.458  ...  3.0  222.0     18.7  396.90   5.33
[5 rows x 13 columns]```

### 列解释Columns:

1. CRIM: 按城镇划分的人均犯罪率

1. ZN: 大于25000平方英尺的住宅用地比例

1. INDUS: 每个城镇非零售业务英亩比例

1. CHAS : 查尔斯河流 哑变量 (靠近河流为1; 否则为0)

1. NOX: 一氧化氮浓度 (百万分之)

1. RM: 每个住宅的房间数

1. AGE: 1940年之前建造的自由单位

1. DIS: 与5个波士顿就业中心的加权距离

1. PTRATIO: 按城镇划分的师生比例

1. B: 1000(Bk — 0.63)², 按城镇划分的非裔人口结构比例

1. LSTAT: 低收入人口百分比

1. MEDV: 自有住房的中位数价值(单位：1000美元)

`dataset['MEDV'] = boston_dataset.target`

## 2. 数据分析

### 2.1 预处理

`dataset.isnull().sum()`

```X = dataset.iloc[:, 0:13].values
y = dataset.iloc[:, 13].values.reshape(-1,1)```

```from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state = 25)```

```print("Shape of X_train: ",X_train.shape)
print("Shape of X_test: ", X_test.shape)
print("Shape of y_train: ",y_train.shape)
print("Shape of y_test",y_test.shape)

Shape of X_train:  (354, 13)
Shape of X_test:  (152, 13)
Shape of y_train:  (354, 1)
Shape of y_test (152, 1)```

### 2.2 可视化

```#相关系数矩阵，即给出了任意两个变量之间的相关系数
corr = dataset.corr()
import matplotlib.pyplot as plt
import seaborn as sns
fig, ax = plt.subplots(figsize=(10, 10))
sns.heatmap(corr, cmap='RdBu', annot=True, fmt=".2f")
plt.xticks(range(len(corr.columns)), corr.columns)
plt.yticks(range(len(corr.columns)), corr.columns)
plt.show()```

```sns.pairplot(dataset)
plt.show()```

## 3. 训练模型

### 3.1 线性拟合

```from sklearn.linear_model import LinearRegression
regressor_linear = LinearRegression()
regressor_linear.fit(X_train, y_train)```

```from sklearn.metrics import r2_score
# 交叉验证：将数据集分为10折，做一次交叉验证，实际上它是计算了十次，将每一折都当做一次测试集，其余九折当做训练集，这样循环十次。通过传入的模型，训练十次，最后将十次结果求平均值。
cv_linear = cross_val_score(estimator = regressor_linear, X = X_train, y = y_train, cv = 10)
# R2 score，即决定系数，反映因变量的全部变异能通过回归关系被自变量解释的比例
y_pred_linear_train = regressor_linear.predict(X_train)
r2_score_linear_train = r2_score(y_train, y_pred_linear_train)
y_pred_linear_test = regressor_linear.predict(X_test)
r2_score_linear_test = r2_score(y_test, y_pred_linear_test)
# RMSE一般指均方根误差。均方根误差亦称标准误差。
rmse_linear = (np.sqrt(mean_squared_error(y_test, y_pred_linear_test)))
print("CV: ", cv_linear.mean())
print('R2_score (train): ', r2_score_linear_train)
print('R2_score (test): ', r2_score_linear_test)
print("RMSE: ", rmse_linear)```

```CV:  0.6984854476156042
R2_score (train):  0.7435787589010061
R2_score (test):  0.7133593313710366
RMSE:  4.6472797457242```

### 3.2 多项式回归(二次)

```from sklearn.preprocessing import PolynomialFeatures
poly_reg = PolynomialFeatures(degree = 2)
X_poly = poly_reg.fit_transform(X_train)
poly_reg.fit(X_poly, y_train)
regressor_poly2 = LinearRegression()
regressor_poly2.fit(X_poly, y_train)```

```from sklearn.metrics import r2_score
cv_poly2 = cross_val_score(estimator = regressor_poly2, X = X_train, y = y_train, cv = 10)
y_pred_poly2_train = regressor_poly2.predict(poly_reg.fit_transform(X_train))
r2_score_poly2_train = r2_score(y_train, y_pred_poly2_train)
y_pred_poly2_test = regressor_poly2.predict(poly_reg.fit_transform(X_test))
r2_score_poly2_test = r2_score(y_test, y_pred_poly2_test)
rmse_poly2 = (np.sqrt(mean_squared_error(y_test, y_pred_poly2_test)))
print('CV: ', cv_poly2.mean())
print('R2_score (train): ', r2_score_poly2_train)
print('R2_score (test): ', r2_score_poly2_test)
print("RMSE: ", rmse_poly2)```

### 3.3 脊回归(Ridge Regression),又叫岭回归

lasso 回归和岭回归（ridge regression）其实就是在标准线性回归的基础上分别加入 L1 和 L2 正则化（regularization）

```from sklearn.linear_model import Ridge
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
steps = [
('scalar', StandardScaler()),
('poly', PolynomialFeatures(degree=2)),
('model', Ridge(alpha=3.8, fit_intercept=True))
]
ridge_pipe = Pipeline(steps)
ridge_pipe.fit(X_train, y_train)```

```from sklearn.metrics import r2_score
cv_ridge = cross_val_score(estimator = ridge_pipe, X = X_train, y = y_train.ravel(), cv = 10)
y_pred_ridge_train = ridge_pipe.predict(X_train)
r2_score_ridge_train = r2_score(y_train, y_pred_ridge_train)
y_pred_ridge_test = ridge_pipe.predict(X_test)
r2_score_ridge_test = r2_score(y_test, y_pred_ridge_test)
rmse_ridge = (np.sqrt(mean_squared_error(y_test, y_pred_ridge_test)))
print('CV: ', cv_ridge.mean())
print('R2_score (train): ', r2_score_ridge_train)
print('R2_score (test): ', r2_score_ridge_test)
print("RMSE: ", rmse_ridge)```

### 3.4 Lasso 回归

```from sklearn.linear_model import Lasso
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
steps = [
('scalar', StandardScaler()),
('poly', PolynomialFeatures(degree=2)),
('model', Lasso(alpha=0.012, fit_intercept=True, max_iter=3000))
]
lasso_pipe = Pipeline(steps)
lasso_pipe.fit(X_train, y_train)```

```from sklearn.metrics import r2_score
# Predicting Cross Validation Score
cv_lasso = cross_val_score(estimator = lasso_pipe, X = X_train, y = y_train, cv = 10)
# Predicting R2 Score the Test set results
y_pred_lasso_train = lasso_pipe.predict(X_train)
r2_score_lasso_train = r2_score(y_train, y_pred_lasso_train)
# Predicting R2 Score the Test set results
y_pred_lasso_test = lasso_pipe.predict(X_test)
r2_score_lasso_test = r2_score(y_test, y_pred_lasso_test)
# Predicting RMSE the Test set results
rmse_lasso = (np.sqrt(mean_squared_error(y_test, y_pred_lasso_test)))
print('CV: ', cv_lasso.mean())
print('R2_score (train): ', r2_score_lasso_train)
print('R2_score (test): ', r2_score_lasso_test)
print("RMSE: ", rmse_lasso)```

### 3.5 支持向量回归 Support Vector Regression

```# Feature Scaling
from sklearn.preprocessing import StandardScaler
sc_X = StandardScaler()
sc_y = StandardScaler()
X_scaled = sc_X.fit_transform(X_train)
y_scaled = sc_y.fit_transform(y_train.reshape(-1,1))
# Fitting the SVR Model to the dataset
from sklearn.svm import SVR
regressor_svr = SVR(kernel = 'rbf', gamma = 'scale')
regressor_svr.fit(X_scaled, y_scaled.ravel())```

```from sklearn.metrics import r2_score
# Predicting Cross Validation Score
cv_svr = cross_val_score(estimator = regressor_svr, X = X_scaled, y = y_scaled.ravel(), cv = 10)
# Predicting R2 Score the Train set results
y_pred_svr_train = sc_y.inverse_transform(regressor_svr.predict(sc_X.transform(X_train)))
r2_score_svr_train = r2_score(y_train, y_pred_svr_train)
# Predicting R2 Score the Test set results
y_pred_svr_test = sc_y.inverse_transform(regressor_svr.predict(sc_X.transform(X_test)))
r2_score_svr_test = r2_score(y_test, y_pred_svr_test)
# Predicting RMSE the Test set results
rmse_svr = (np.sqrt(mean_squared_error(y_test, y_pred_svr_test)))
print('CV: ', cv_svr.mean())
print('R2_score (train): ', r2_score_svr_train)
print('R2_score (test): ', r2_score_svr_test)
print("RMSE: ", rmse_svr)```

### 3.6 决策树回归 Decision Tree Regression

```from sklearn.tree import DecisionTreeRegressor
regressor_dt = DecisionTreeRegressor(random_state = 0)
regressor_dt.fit(X_train, y_train)```

```from sklearn.metrics import r2_score
# Predicting Cross Validation Score
cv_dt = cross_val_score(estimator = regressor_dt, X = X_train, y = y_train, cv = 10)
# Predicting R2 Score the Train set results
y_pred_dt_train = regressor_dt.predict(X_train)
r2_score_dt_train = r2_score(y_train, y_pred_dt_train)
# Predicting R2 Score the Test set results
y_pred_dt_test = regressor_dt.predict(X_test)
r2_score_dt_test = r2_score(y_test, y_pred_dt_test)
# Predicting RMSE the Test set results
rmse_dt = (np.sqrt(mean_squared_error(y_test, y_pred_dt_test)))
print('CV: ', cv_dt.mean())
print('R2_score (train): ', r2_score_dt_train)
print('R2_score (test): ', r2_score_dt_test)
print("RMSE: ", rmse_dt)```

### 3.7 随机森林回归 Random Forest Regression

```from sklearn.ensemble import RandomForestRegressor
regressor_rf = RandomForestRegressor(n_estimators = 500, random_state = 0)
regressor_rf.fit(X_train, y_train.ravel())```

```from sklearn.metrics import r2_score
# Predicting Cross Validation Score
cv_rf = cross_val_score(estimator = regressor_rf, X = X_scaled, y = y_train.ravel(), cv = 10)
# Predicting R2 Score the Train set results
y_pred_rf_train = regressor_rf.predict(X_train)
r2_score_rf_train = r2_score(y_train, y_pred_rf_train)
# Predicting R2 Score the Test set results
y_pred_rf_test = regressor_rf.predict(X_test)
r2_score_rf_test = r2_score(y_test, y_pred_rf_test)
# Predicting RMSE the Test set results
rmse_rf = (np.sqrt(mean_squared_error(y_test, y_pred_rf_test)))
print('CV: ', cv_rf.mean())
print('R2_score (train): ', r2_score_rf_train)
print('R2_score (test): ', r2_score_rf_test)
print("RMSE: ", rmse_rf)```

## 4. 评估结果汇总

```models = [('Linear Regression', rmse_linear, r2_score_linear_train, r2_score_linear_test, cv_linear.mean()),
('Polynomial Regression (2nd)', rmse_poly2, r2_score_poly2_train, r2_score_poly2_test, cv_poly2.mean()),
('Ridge Regression', rmse_ridge, r2_score_ridge_train, r2_score_ridge_test, cv_ridge.mean()),
('Lasso Regression', rmse_lasso, r2_score_lasso_train, r2_score_lasso_test, cv_lasso.mean()),
('Support Vector Regression', rmse_svr, r2_score_svr_train, r2_score_svr_test, cv_svr.mean()),
('Decision Tree Regression', rmse_dt, r2_score_dt_train, r2_score_dt_test, cv_dt.mean()),
('Random Forest Regression', rmse_rf, r2_score_rf_train, r2_score_rf_test, cv_rf.mean())
]```

```predict = pd.DataFrame(data = models, columns=['Model', 'RMSE', 'R2_Score(training)', 'R2_Score(test)', 'Cross-Validation'])
predict```

No Model RMSE R2_Score(training) R2_Score(test) Cross-Validation
0 Linear Regression 4.647280 0.743579 0.713359 0.698485
1 Polynomial Regression (2nd) 4.194313 0.930656 0.766513 0.698485
2 Ridge Regression 2.853062 0.922818 0.891965 0.763563
3 Lasso Regression 2.811451 0.923402 0.895094 0.750544
4 Support Vector Regression 3.838898 0.874272 0.804407 0.782601
5 Decision Tree Regression 5.723785 1.000000 0.565183 0.613295
6 Random Forest Regression 3.211470 0.976717 0.863118 0.818677

## 5. 可视化评估结果

```f, axe = plt.subplots(1,1, figsize=(18,6))
predict.sort_values(by=['Cross-Validation'], ascending=False, inplace=True)
sns.barplot(x='Cross-Validation', y='Model', data = predict, ax = axe)
#axes[0].set(xlabel='Region', ylabel='Charges')
axe.set_xlabel('Cross-Validaton Score', size=16)
axe.set_ylabel('Model')
axe.set_xlim(0,1.0)
plt.show()```

```f, axes = plt.subplots(2,1, figsize=(14,10))
predict.sort_values(by=['R2_Score(training)'], ascending=False, inplace=True)
sns.barplot(x='R2_Score(training)', y='Model', data = predict, palette='Blues_d', ax = axes[0])
#axes[0].set(xlabel='Region', ylabel='Charges')
axes[0].set_xlabel('R2 Score (Training)', size=16)
axes[0].set_ylabel('Model')
axes[0].set_xlim(0,1.0)
predict.sort_values(by=['R2_Score(test)'], ascending=False, inplace=True)
sns.barplot(x='R2_Score(test)', y='Model', data = predict, palette='Reds_d', ax = axes[1])
#axes[0].set(xlabel='Region', ylabel='Charges')
axes[1].set_xlabel('R2 Score (Test)', size=16)
axes[1].set_ylabel('Model')
axes[1].set_xlim(0,1.0)
plt.show()```

```predict.sort_values(by=['RMSE'], ascending=False, inplace=True)
f, axe = plt.subplots(1,1, figsize=(18,6))
sns.barplot(x='Model', y='RMSE', data=predict, ax = axe)
axe.set_xlabel('Model', size=16)
axe.set_ylabel('RMSE', size=16)
plt.show()```