python实现简单线性回归

小咪咪 2024-04-17 15:28 72阅读 0赞

1，举例说明和建立模型：

（1）建立模型：

求得最佳的拟合函数yi和mean(y)的差的平方的最小值，即是![sum = min(\\sum (y\_\{i\}-\\bar\{y\})^\{2\})][sum _ min_sum _y_i_-_bar_y_2].

条件：根据数据来拟合函数，拟合的斜率和截距计算如下；

![20190827232906762.png][]                               ![20190827232928382.png][]

从而得出拟合函数是：

![\\hat\{y\} = b\_\{1\}\*\\hat\{x\}+b\_\{0\}][hat_y_ _ b_1_hat_x_b_0] （![\\hat\{y\}][hat_y] = ![y\_\{i\}][y_i]）

（2）举例说明：

汽车卖家做电视广告数量与卖出的汽车数量：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMDA4MzI1_size_16_color_FFFFFF_t_70][]

计算b1和b0:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMDA4MzI1_size_16_color_FFFFFF_t_70 1][]

![20190827235136353.PNG][]

从而得到拟合函数为：![\\hat\{y\} =5\*\\hat\{x\}+10][hat_y_ _5_hat_x_10]

我们从而可以通过x的值和拟合函数来估计y的值了,比如我们输入广告数量6,我们可以得到汽车数量为5\*6+10=40。

3，现在我们用spyder来实现看看，新建一个linenar\_regulation.py：

import numpy as np
    
    def fitSLR(x, y):
        n = len(x)
        dinominator = 0
        numerator = 0
        for i in range(0, n):
            numerator += (x[i] - np.mean(x))*(y[i] - np.mean(y))
            dinominator += (x[i] - np.mean(x))**2
        b1 = numerator/float(dinominator)
        b0 = np.mean(y)/float(np.mean(x))
        return b0, b1
    
    def predict(x, b0, b1):
        return b0 + x*b1
    
    x = [1, 3, 2, 1, 3]
    y = [14, 24, 18, 17, 27]    
    
    
    b0, b1 = fitSLR(x, y)
    
    print ("intercept:", b0, " slope:", b1)
    
    x_test = 6
    
    y_test = predict(6, b0, b1)
    
    print ("y_test:", y_test)

代码运行结果如下：

![20190828000142176.PNG][]

[sum _ min_sum _y_i_-_bar_y_2]: https://private.codecogs.com/gif.latex?sum%20%3D%20min%28%5Csum%20%28y_%7Bi%7D-%5Cbar%7By%7D%29%5E%7B2%7D%29
[20190827232906762.png]: https://img-blog.csdnimg.cn/20190827232906762.png
[20190827232928382.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/17/742c645bfd2e4cacb9cc905f0bb5d3e0.png
[hat_y_ _ b_1_hat_x_b_0]: https://private.codecogs.com/gif.latex?%5Chat%7By%7D%20%3D%20b_%7B1%7D*%5Chat%7Bx%7D&plus;b_%7B0%7D
[hat_y]: https://private.codecogs.com/gif.latex?%5Chat%7By%7D
[y_i]: https://private.codecogs.com/gif.latex?y_%7Bi%7D
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMDA4MzI1_size_16_color_FFFFFF_t_70]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/17/15f1435345114608a332446b7e9c1cd8.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMDA4MzI1_size_16_color_FFFFFF_t_70 1]: https://img-blog.csdnimg.cn/2019082723504476.PNG?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMDA4MzI1,size_16,color_FFFFFF,t_70
[20190827235136353.PNG]: https://img-blog.csdnimg.cn/20190827235136353.PNG
[hat_y_ _5_hat_x_10]: https://private.codecogs.com/gif.latex?%5Chat%7By%7D%20%3D5*%5Chat%7Bx%7D&plus;10
[20190828000142176.PNG]: https://img-blog.csdnimg.cn/20190828000142176.PNG