线性回归

线性回归算法介绍

 

1. 解决回归问题
2. 思想简单，实现容易
3. 许多强大的非线性模型的基础
4. 结果具有很好的解释性
5. 蕴含机器学习中很多重要的思想

 

简单线性回归代价函数推导

 

 

实现简单线性回归

import numpy as np
import matplotlib.pyplot as plt
x = np.array([1,2,3,4,5])
y = np.array([1,3,2,3,5])
​
x_mean = np.mean(x)
y_mean = np.mean(y)
​
a = np.sum(np.subtract(x,x_mean)*np.subtract(y,y_mean))/np.sum(np.power((x-x_mean),2))
b = y_mean-a*x_mean
y_predict = a*x + b
print("a =",a,"\n","b = ",b)
print(y_predict)
plt.scatter(x,y)
plt.plot(x,y_predict)

　　

 

自己实现简单线性回归

class SimpleLinearRegression:
    def __init__(self):
        """初始化简单线性模型"""
        self.a_ = None
        self.b_ = None
    def fit(self,trainX,trainY):
        """根据数据训练简单线性模型"""
        assert trainX.ndim==1,"SimpleLinearRegression must be input 1D"
        x_mean = np.mean(trainX)
        y_mean = np.mean(trainY)
        self.a_ = np.dot(np.subtract(trainX,x_mean),np.subtract(trainY,y_mean))/np.sum(np.power((trainX-x_mean),2))
        self.b_ = y_mean-self.a_*x_mean
        return self
    def _predict(self,test_x):
        """给定单个待测数，返回预测值"""
        test_y = self.a_*test_x + self.b_
        return test_y
    def predict(self,testX):
        """给定待测数据集，返回结果向量"""
        return np.array([self._predict(i) for i in testX])

　　

simpleLinearRegression = SimpleLinearRegression()
simpleLinearRegression.fit(x,y)
print(simpleLinearRegression.a_)
print(simpleLinearRegression.b_)
simpleLinearRegression.predict(x)

　　

 

衡量线性回归的指标

 

均方误差MSE

 

均方根误差RMSE

 

平均绝对误差MAE

 

 

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


boston = datasets.load_boston()
x = boston.data[:,5] # 只使用房间数量这特征
y = bsoton.target
x = x[y<50]
y = y[y<50]
plt.scatter(x,y)
plt.xlim(2,10)
plt.ylim(0,60)

　

 

 

trainX,testX,trainY,testY = train_test_split(x,y,test_size=0.2)
simpleLinearRegression = SimpleLinearRegression()
simpleLinearRegression.fit(trainX,trainY)
train_y = simpleLinearRegression.predict(trainX)
test_y = simpleLinearRegression.predict(testX)
​
plt.subplot(1,2,1)
plt.scatter(trainX,trainY)
plt.plot(trainX,train_y)
​
plt.subplot(1,2,2)
plt.scatter(testX,testY)
plt.plot(testX,test_y)

　　

 

MSE

 

mse = np.sum(np.power(np.subtract(test_y,testY),2))/test_y.shape[0]
mse
>>>40.95154236464736

　　

RMSE

rmse = np.sqrt(mse)
rmse
>>>6.39933921312563

 

MAE

mae = np.sum(np.abs(test_y-testY))/test_y.shape[0]
mae
 >>>4.399066068730565

　　

R Squared

 

r = 1-mse/np.var(testY)
r
>>>0.2541624531120106


from sklearn.metrics import r2_score
​
r2_score(testY,test_y)
 
>>>0.2541624531120107

　　

 

多元线性回归

 

优点:不需要对数据做归一化处理
缺点：时间复杂度高

 

 

自己实现正规方程法LinearRegressor

from sklearn.metrics import r2_score
​
class LinearRegressor:
    def __init__(self):
        self.coef_ = None # θ向量
        self.intercept_ = None # 截距
        self._theta = None
    def fit(self,trainX,trainY):
        x = np.hstack((np.ones((trainX.shape[0],1)),trainX))
        self._theta = np.linalg.inv(np.dot(x.T,x)).dot(x.T).dot(trainY)
        self.intercept_ = self._theta[0] # 截距
        self.coef_ = self._theta[1:] # 权重参数weights
        return self
    def predict(self,testX):
        x = np.hstack((np.ones((testX.shape[0],1)),testX))
#         y = np.dot(x,self._theta.reshape(-1,1))
        y = np.dot(x,self._theta)
        return y
    def score(self,testX,testY):
        return r2_score(testY,self.predict(testX))

　　

 

trainX,t testX,trainY,testY = train_test_split(x,y,test_size=0.2,random_state=666)
lin_reg = LinearRegressor()
lin_reg.fit(trainX,trainY)
print(lin_reg.predict(testX).shape)
print(lin_reg.coef_)
print(lin_reg.intercept_)
print(lin_reg.score(testX,testY))
 
 
 
>>>(98, 1)
>>>[-1.18919477e-01  3.63991462e-02 -3.56494193e-02  5.66737830e-02
    -1.16195486e+01  3.42022185e+00 -2.31470282e-02 -1.19509560e+00
     2.59339091e-01 -1.40112724e-02 -8.36521175e-01  7.92283639e-03
    -3.81966137e-01]
>>>34.16143549624022
>>>0.8129802602658537

　　

 

使用sklearn中的LinearRegressor

boston = datasets.load_boston()
x = boston.data
y = bsoton.target
x = x[y<50]
y = y[y<50]
x.shape
 
>>>(490, 13)

 
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
​
​
trainX,testX,trainY,testY = train_test_split(x,y,test_size=0.2,random_state=666)
lin_reg = LinearRegression()
lin_reg.fit(trainX,trainY)
print(lin_reg.predict(testX).shape)
print(lin_reg.coef_)
print(lin_reg.intercept_)
print(lin_reg.score(testX,testY))
 
 
 
>>>(98,)
>>>[-1.18919477e-01  3.63991462e-02 -3.56494193e-02  5.66737830e-02
    -1.16195486e+01  3.42022185e+00 -2.31470282e-02 -1.19509560e+00
     2.59339091e-01 -1.40112724e-02 -8.36521175e-01  7.92283639e-03
    -3.81966137e-01]
 >>>34.16143549624665
 >>>0.8129802602658495

　　

　　

KNN Regressor

from sklearn.neighbors import KNeighborsRegressor
from sklearn.metrics import r2_score


kneighborsRegressor = KNeighborsRegressor()
kneighborsRegressor.fit(trainX,trainY)
y_predict = kneighborsRegressor.predict(testX)
r2_score(testY,y_predict)

>>>0.5865412198300899

　　

更多关于线性回归模型的讨论

 

1. 对数据具有强解释性，正负相关影响因素大小，从而使我们有目标性的采集数据
2. 对数据有假设：线性
3. 拿到一组数据先用线性回归试试看，总归是没有坏处的。