线性回归实现及线性回归中衡量算法模型的标准(R-Squared,MSE.RMSE,MAE)

﹏ヽ暗。殇╰゛Y 2022-05-19 08:57 215阅读 0赞

MSE:均方误差，mean\_squared\_error  
RMSE：均方根误差，root\_mean\_squared\_error  
MAE：平均绝对误差，mean\_absolute\_error

## 05 衡量回归算法的标准，MSE vs MAE ##

import numpy as np
    import matplotlib.pyplot as plt
    from sklearn import datasets

### 波士顿房产数据 ###

boston = datasets.load_boston()

boston.keys()

dict_keys(['data', 'target', 'feature_names', 'DESCR'])

特征向量名称：

boston.feature_names

array(['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD',
           'TAX', 'PTRATIO', 'B', 'LSTAT'], 
          dtype='<U7')

x = boston.data[:,5] # 只使用房间数量这个特征,所有行，第五列

x.shape

(506,)

y = boston.target

y.shape

(506,)

plt.scatter(x, y)
    plt.show()

![这里写图片描述][70]

np.max(y)

50.0

x = x[y < 50.0]#采用最大值的点可能不是真实的点
    y = y[y < 50.0]

x.shape

(490,)

y.shape

(490,)

plt.scatter(x, y)
    plt.show()

![这里写图片描述][70 1]

### 使用简单线性回归法 ###

封装：

import numpy as np
    class LinearRegression1:
        def __init__(self):
            self.a_=None
            self.b_=None
    
        def fit(self,x_train,y_train):
            num=0.0
            d=0.0
            x_mean=np.mean(x_train)
            y_mean=np.mean(y_train)
            for x_i,y_i in zip(x_train,y_train):
                num+=(x_i-x_mean)*(y_i-y_mean)
                d+=(x_i-x_mean)**2
            self.a_=num/d
            self.b_=y_mean-self.a_*(x_mean)
            return self
    
        def _predict(self,X_train):
            return np.array([self.predict(x) for x in X_train])
    
        def predict(self,x_single):
            return self.a_*x_single+self.b_

import numpy as np
    
    
    class LinearRegression2:
        def __init__(self):
            self.a_=None
            self.b_=None
    
        def fit(self,x_train,y_train):
            x_mean=np.mean(x_train)
            y_mean=np.mean(y_train)
            num=(x_train-x_mean).dot(y_train-y_mean)
            d=(x_train-x_mean).dot(x_train-x_mean)
            self.a_=num/d
            self.b_=y_mean-self.a_*x_mean
            return self
    
        def _predict(self,X_train):
            return np.array([self.predict(x) for x in X_train])
    
        def predict(self,x_single):
            return self.a_*x_single+self.b_

随机方法：

import numpy as np
    
    
    def train_test_split(X, y, test_ratio=0.2, seed=None):
        """将数据 X 和 y 按照test_ratio分割成X_train, X_test, y_train, y_test"""
        assert X.shape[0] == y.shape[0], \
            "the size of X must be equal to the size of y"
        assert 0.0 <= test_ratio <= 1.0, \
            "test_ration must be valid"
    
        if seed:
            np.random.seed(seed)
    
        shuffled_indexes = np.random.permutation(len(X))
    
        test_size = int(len(X) * test_ratio)
        test_indexes = shuffled_indexes[:test_size]
        train_indexes = shuffled_indexes[test_size:]
    
        X_train = X[train_indexes]
        y_train = y[train_indexes]
    
        X_test = X[test_indexes]
        y_test = y[test_indexes]
    
        return X_train, X_test, y_train, y_test

测试：

from playML.model_selection import train_test_split
    
    x_train, x_test, y_train, y_test = train_test_split(x, y, seed=666)

x_train.shape

(392,)

y_train.shape

(392,)

x_test.shape

(98,)

y_test.shape

(98,)

from playML.SimpleLinearRegression import SimpleLinearRegression

reg = SimpleLinearRegression()
    reg.fit(x_train, y_train)

SimpleLinearRegression()

reg.a_#斜率

7.8608543562689555

reg.b_#截距

-27.459342806705543

plt.scatter(x_train, y_train)
    plt.plot(x_train, reg.predict(x_train), color='r')
    plt.show()

![这里写图片描述][70 2]

plt.scatter(x_train, y_train)
    plt.scatter(x_test, y_test, color="c")
    plt.plot(x_train, reg.predict(x_train), color='r')
    plt.show()

![这里写图片描述][70 3]

y_predict = reg.predict(x_test)

### MSE ###

mse_test = np.sum((y_predict - y_test)**2) / len(y_test)
    mse_test#MSE和实际的y的量纲不一样

24.156602134387438

### RMSE ###

from math import sqrt
    
    rmse_test = sqrt(mse_test)
    rmse_test#平均误差在4.91左右

4.914936635846635

### MAE ###

mae_test = np.sum(np.absolute(y_predict - y_test))/len(y_test)
    mae_test#MAE得到的结果比RMSE得到的结果小，因为RMSE有平方操作，相当于放大了操作

3.5430974409463873

### 封装我们自己的评测函数 ###

代码为：

import numpy as np
    from math import sqrt
    
    
    def accuracy_score(y_true, y_predict):
        """计算y_true和y_predict之间的准确率"""
        assert len(y_true) == len(y_predict), \
            "the size of y_true must be equal to the size of y_predict"
    
        return np.sum(y_true == y_predict) / len(y_true)
    
    #计算MSE
    def mean_squared_error(y_true, y_predict):
        """计算y_true和y_predict之间的MSE"""
        assert len(y_true) == len(y_predict), \
            "the size of y_true must be equal to the size of y_predict"
    
        return np.sum((y_true - y_predict)**2) / len(y_true)
    
    #计算RMSE
    def root_mean_squared_error(y_true, y_predict):
        """计算y_true和y_predict之间的RMSE"""
    
        return sqrt(mean_squared_error(y_true, y_predict))
    
    #计算MAE
    def mean_absolute_error(y_true, y_predict):
        """计算y_true和y_predict之间的MAE"""
        assert len(y_true) == len(y_predict), \
            "the size of y_true must be equal to the size of y_predict"
    
        return np.sum(np.absolute(y_true - y_predict)) / len(y_true)

测试前面写的算法：

from playML.metrics import mean_squared_error
    from playML.metrics import root_mean_squared_error
    from playML.metrics import mean_absolute_error

mean_squared_error(y_test, y_predict)

24.156602134387438

root_mean_squared_error(y_test, y_predict)

4.914936635846635

mean_absolute_error(y_test, y_predict)

3.5430974409463873

### scikit-learn中的MSE和MAE ###

没有RMSE，对MSE求方根即可得到

from sklearn.metrics import mean_squared_error
    from sklearn.metrics import mean_absolute_error

mean_squared_error(y_test, y_predict)

24.156602134387438

mean_absolute_error(y_test, y_predict)

3.5430974409463873

### R Squared ###

R^2 越靠近1算法精确度越好

MSE
    mse_test=np.sum((y_predict-y_test)**2)/len(y_test)
    
    RSME
    rmse_test=sqrt(mse_test)
    
    MAE
    mae_test=np.sum((np.absolute(y_predict-y_test))/len(y_test))

### R Square ###

from playML.metrics import mean_squared_error
    
    1 - mean_squared_error(y_test, y_predict)/np.var(y_test)

### 封装我们自己的 R Score ###

def r2_score(y_true, y_predict):
        """计算y_true和y_predict之间的R Square"""
    
        return 1 - mean_squared_error(y_true, y_predict)/np.var(y_true)

from playML.metrics import r2_score
    
    r2_score(y_test, y_predict)

0.61293168039373225

### scikit-learn中的 r2\_score ###

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

0.61293168039373236

scikit-learn中的LinearRegression中的score返回r2\_score:[http://scikit-learn.org/stable/modules/generated/sklearn.linear\_model.LinearRegression.html][http_scikit-learn.org_stable_modules_generated_sklearn.linear_model.LinearRegression.html]

### 在我们的SimpleRegression中添加score ###

reg.score(x_test, y_test)

0.61293168039373225

[70]: /images/20220519/d58dc678979d42ceb2f36e26e45f1f1f.png
[70 1]: /images/20220519/748abedc6f6c4b048e6e984b1a9f86b3.png
[70 2]: /images/20220519/ba9af37665054e25b6ceffefa2deb192.png
[70 3]: /images/20220519/3078f102085b46668bb354a6d27c2663.png
[http_scikit-learn.org_stable_modules_generated_sklearn.linear_model.LinearRegression.html]: http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html