评估与选择合适的回归模型

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选择合适的回归模型

学习自： https://blog.csdn.net/zrh_CSDN/article/details/81190001

R-Squard Intuition

线性回归问题中，R-Squared 是用来衡量回归方程与真实样本输出之间的相似程度。其表达式如下所示：

$R^2 = 1-\frac{\sum(y-\hat y)^2}{\sum(y-\overline y)^2}$

上式中，分子部分表示真实值与预测值的平方差之和，类似于 均方差 MSE；分母部分表示真是值与均值的平方差之和，类似于方差Var。

根据 R-Squared 的取值，来判断模型的好坏：如果结果是 0，说明模型拟合效果很差；如果结果是 1，说明模型无错误。

一般来说，R-Squared 越大，表示模型拟合效果越好。R-Squared 反映的是大概有多准，因为，随着特征数量的增加，R-Square必然增加，无法真正定量说明准确程度，只能大概定量。

单独看 R-Squared，并不能推断出增加的特征是否有意义。通常来说，增加一个特征，R-Squared 可能变大也可能保持不变，两者不一定呈正相关。

Adjusted R-Squared

对于 $R-Squared$ 来说，如果加入一个新的 Feature，在回归模型下，系统会自动让 $SS{res}$ 趋向于最小。那么如果这个Feature会让整个模型预测准确性下降的话，算法就会让这个Feature的系数趋于0甚至等于0，那么$R-Squared$ 值就不会变化(甚至还可能有轻微的提升);如果这个Feature 会让整个模型的预测准确性上升，那么 $SS{res}$ 会变小，而$R-Squared$ 值就会变大，于是 这个值变成了一个只增不减的，是有一个局限性的。

所以 $R^2$ 并不能很好的让我们判断一个新的变量对整个模型的影响。

如果使用校正决定系数（Adjusted R-Square）：

$R^2_adjust = 1-(1-R^2)\frac{(n-1)}{n-p-1}$

其中，n是样本数量(sample size)，p是特征数量(number of regressors)。

我们看到如果我们一直增加特征数量的话， $\frac{n-1}{n-p-1}$ 就会增大，$(1-R^2)$ 会减小，所以这相当于是对增加样本做了一个补偿。

如果我们现在添加的特征是对整个模型基本没用的，那么 $\frac{n-1}{n-p-1}$ 对 $R^2-adjust$ 的影响就会高于其本身对$R^2$ 的影响，因此 $R^2-adjust$ 会下降。反之，如果这个变量是对整个模型特别有用的，那么其对$R^2$ 的影响就会大于 对$\frac{n-1}{n-p-1}$ 的影响，那么整个 $R^2-adjust$ 就会上升

$R^2-adjust$ 抵消样本数量对 R-Square 的影响，做到了真正的 $0\sim 1$ 越大越好

选择合适的回归模型

接下来我们通过一个实践，利用刚刚学习的$R^2$或者 $R^2-adjust$ 来选择最准确的模型。

Preparation of the Regression Code

我们使用的是一个大约有10000行数据的csv文件，没有缺失数据。数据来自https://archive.ics.uci.edu/ml/index.php 这里面有很多适合拿来练习data set。 接下来我会提供现在已经学到的五个模型的代码模板.并分别看看这些模板的准确性

multiple linear regression

Importing the libraries

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import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

Importing the dataset

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dataset = pd.read_csv('ENTER_THE_NAME_OF_YOUR_DATASET_HERE.csv')
X = dataset.iloc[:, :-1].values
y = dataset.iloc[:, -1].values

Splitting the dataset into the Training set and Test set

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from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

Training the Multiple Linear Regression model on the Training set

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

Predicting the Test set results

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y_pred = regressor.predict(X_test)
np.set_printoptions(precision=2)
print(np.concatenate((y_pred.reshape(len(y_pred),1), y_test.reshape(len(y_test),1)),1))

Evaluating the Model Performance

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from sklearn.metrics import r2_score
r2_score(y_test, y_pred)

Polynomial Regression

Importing the libraries

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import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

Importing the dataset

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dataset = pd.read_csv('ENTER_THE_NAME_OF_YOUR_DATASET_HERE.csv')
X = dataset.iloc[:, :-1].values
y = dataset.iloc[:, -1].values

Splitting the dataset into the Training set and Test set

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from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

Training the Polynomial Regression model on the Training set

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from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
poly_reg = PolynomialFeatures(degree = 4)
X_poly = poly_reg.fit_transform(X_train)
regressor = LinearRegression()
regressor.fit(X_poly, y_train)

Predicting the Test set results

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y_pred = regressor.predict(poly_reg.transform(X_test))
np.set_printoptions(precision=2)
print(np.concatenate((y_pred.reshape(len(y_pred),1), y_test.reshape(len(y_test),1)),1))

Evaluating the Model Performance

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from sklearn.metrics import r2_score
r2_score(y_test, y_pred)

Support Vector Regression (SVR)

Importing the libraries

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import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

Importing the dataset

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dataset = pd.read_csv('ENTER_THE_NAME_OF_YOUR_DATASET_HERE.csv')
X = dataset.iloc[:, :-1].values
y = dataset.iloc[:, -1].values
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y = y.reshape(len(y),1)

Splitting the dataset into the Training set and Test set

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from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

Feature Scaling

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from sklearn.preprocessing import StandardScaler
sc_X = StandardScaler()
sc_y = StandardScaler()
X_train = sc_X.fit_transform(X_train)
y_train = sc_y.fit_transform(y_train)

Training the SVR model on the Training set

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from sklearn.svm import SVR
regressor = SVR(kernel = 'rbf')
regressor.fit(X_train, y_train)

Predicting the Test set results

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y_pred = sc_y.inverse_transform(regressor.predict(sc_X.transform(X_test)))
np.set_printoptions(precision=2)
print(np.concatenate((y_pred.reshape(len(y_pred),1), y_test.reshape(len(y_test),1)),1))

Evaluating the Model Performance

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from sklearn.metrics import r2_score
r2_score(y_test, y_pred)

Decision Tree Regression

Importing the libraries

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import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

Importing the dataset

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dataset = pd.read_csv('ENTER_THE_NAME_OF_YOUR_DATASET_HERE.csv')
X = dataset.iloc[:, :-1].values
y = dataset.iloc[:, -1].values

Splitting the dataset into the Training set and Test set

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from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

Training the Decision Tree Regression model on the Training set

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

Predicting the Test set results

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y_pred = regressor.predict(X_test)
np.set_printoptions(precision=2)
print(np.concatenate((y_pred.reshape(len(y_pred),1), y_test.reshape(len(y_test),1)),1))

Evaluating the Model Performance

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from sklearn.metrics import r2_score
r2_score(y_test, y_pred)

Random Forest Regression

Importing the libraries

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import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

Importing the dataset

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dataset = pd.read_csv('ENTER_THE_NAME_OF_YOUR_DATASET_HERE.csv')
X = dataset.iloc[:, :-1].values
y = dataset.iloc[:, -1].values

Splitting the dataset into the Training set and Test set

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from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

Training the Random Forest Regression model on the whole dataset

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from sklearn.ensemble import RandomForestRegressor
regressor = RandomForestRegressor(n_estimators = 10, random_state = 0)
regressor.fit(X_train, y_train)

Predicting the Test set results

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y_pred = regressor.predict(X_test)
np.set_printoptions(precision=2)
print(np.concatenate((y_pred.reshape(len(y_pred),1), y_test.reshape(len(y_test),1)),1))

Evaluating the Model Performance

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from sklearn.metrics import r2_score
r2_score(y_test, y_pred)

结果

我们接下来就对上面的模型进行跑分：

  • Multiple Linear Regression : 0.9325315554761303
  • Polynomial Regression : 0.945819297903489
  • Support Vector Regression(SVR) : 0.9480784049986258

  • Decision Tree Regression : 0.9226091050550043

  • Random Forest : 0.9615980699813017

综上我们得到，Random Forest 随机森林模型对这个数据集的预测准确性在五个模型中分数最高，是最适合的回归模型。

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