建模随手记2 --- 最小二乘法实现线性回归

牧云@^-^@ 提交于 2020-01-22 03:08:07

1. 回归分析

1.1. 一元线性回归
一元线性回归可以用来分析一个自变量和因变量之间的关系,通过分散的样本点来得到自变量和因变量之间的线性关系,通过最小二乘法来获得线性回归的系数,计算之后要对获得的回归方程进行检验。

P19 例2.1.1:

import numpy as np
from matplotlib import pyplot as plt
from sklearn.linear_model import LinearRegression


def linear_regression(x, y):
    plt.figure()
    plt.scatter(x, y, alpha=0.5)
    plt.title('weight(y) and temperature(x)')
    plt.xlabel('temperature')
    plt.ylabel('weight')
    lrModel = LinearRegression()
    # 求解模型
    lrModel.fit(x, y)
    # 对x进行预测
    y0 = lrModel.predict(x)
    plt.plot(x, y0)
    plt.show()
    alpha = lrModel.coef_
    # 获得斜率
    beta = lrModel.intercept_
	# 获得截距


def test():
    x = np.array([3.5, 1.8, 2.4, 3.0, 3.5, 3.9, 4.4, 4.8, 5.0])
    y = np.array([8.5, 2.57, 3.0, 5.3, 8.9, 11.69, 13.1, 13.6, 13.3])
    linear_regression(x.reshape([len(x), 1]), y.reshape([len(y), 1]))


test()

使用了sklearn库中的linear_model,其中需注意在使用fit方法时,传参数需要二维数组。

结果:
在这里插入图片描述R2=0.9448786943846488 R^2 = 0.9448786943846488
y=3.99x5.43y=3.99x-5.43

1.2. 检验
在获得模型之后要对模型进行检验,我们的模型拟合程度如何(R2)?我们要检验的使整个模型是否能显著预测因变量的变化(F检验)?每个自变量是否能显著预测因变量的变化(t检验)?

1.2.1 R2 - - - 模型拟合程度
通过计算模型对自变量存在的情况下的因变量的和平均值的差异的解释情况,来判断模型的拟合程度,即计算预测值和平均值差值平方之和(SSA)和实际值和平均值差值的平方之和(SST)的比值。SSA表示对实际值和平均值差异的解释情况,SST表示实际值和平均值之间的差异大小。通过比值可以看出模型对差异的解释情况从而判断模型的拟合情况,也就是R2

1.2.2 F检验 - - - 能否显著预测y值
计算SSA和SSE的比值,所得结果小于5%即线性关系明显。

1.2.3 t检验 - - - 回归系数是否显著
计算Pr要小于0.05即可证明显著。

import numpy as np
import pandas as pd
from statsmodels.formula.api import ols

x = np.array([3.5, 1.8, 2.4, 3.0, 3.5, 3.9, 4.4, 4.8, 5.0])
y = np.array([8.5, 2.57, 3.0, 5.3, 8.9, 11.69, 13.1, 13.6, 13.3])
data = pd.DataFrame({'x': x, 'y': y})
model = ols('y~x', data).fit()
print(model.summary())

使用statsmodels可以更简便的完成回归以及检验,结果如下:

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.945
Model:                            OLS   Adj. R-squared:                  0.937
Method:                 Least Squares   F-statistic:                     120.0
Date:                Tue, 21 Jan 2020   Prob (F-statistic):           1.17e-05
Time:                        23:19:13   Log-Likelihood:                -12.533
No. Observations:                   9   AIC:                             29.07
Df Residuals:                       7   BIC:                             29.46
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -5.4375      1.358     -4.003      0.005      -8.649      -2.226
x              3.9906      0.364     10.954      0.000       3.129       4.852
==============================================================================
Omnibus:                        1.239   Durbin-Watson:                   1.329
Prob(Omnibus):                  0.538   Jarque-Bera (JB):                0.658
Skew:                           0.026   Prob(JB):                        0.720
Kurtosis:                       1.677   Cond. No.                         14.7
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Process finished with exit code 0

可以得到,R2=0.945,说明模型拟合效果显著,F值为120,p值远小于0.05因此通过F检验,自变量和因变量之间存在显著的线性关系,两个参数均通过t检验,所得Pr与那小于0.05.


1.3. 一元非线性回归
在实际问题中许多自变量和因变量之间的关系并不成线性关系,无法做线性回归,但是我们可以通过数学处理实现线性回归,下面几组数据进行处理。

》》》P22 例2.1.2
画图观察:

	x = [1, 2, 3, 4, 4, 6, 6, 8, 8, 9]
	y = [1.85, 1.37, 1.02, 0.75, 0.56, 0.41, 0.31, 0.23, 0.17, 0.10]
	plt.figue()
	plt.scatter(x, y, alpha=0.5)
	plt.show()

在这里插入图片描述
由图推测可能为一下关系:
y=ax+by=ax+b
y=a+bxy=a+\frac{b}{x}
y=axb y=ax^b
y=aebx y=ae^{bx}

i. y=ax+by=ax+b
在这里插入图片描述

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.874
Model:                            OLS   Adj. R-squared:                  0.858
Method:                 Least Squares   F-statistic:                     55.35
Date:                Wed, 22 Jan 2020   Prob (F-statistic):           7.34e-05
Time:                        02:03:50   Log-Likelihood:                 2.1956
No. Observations:                  10   AIC:                           -0.3912
Df Residuals:                       8   BIC:                            0.2139
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      1.6846      0.152     11.093      0.000       1.334       2.035
x             -0.1976      0.027     -7.440      0.000      -0.259      -0.136
==============================================================================
Omnibus:                        0.057   Durbin-Watson:                   0.884
Prob(Omnibus):                  0.972   Jarque-Bera (JB):                0.209
Skew:                           0.132   Prob(JB):                        0.901
Kurtosis:                       2.343   Cond. No.                         13.0
==============================================================================

ii. y=a+bxy=a+\frac{b}{x}
在这里插入图片描述

								OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.890
Model:                            OLS   Adj. R-squared:                  0.877
Method:                 Least Squares   F-statistic:                     65.02
Date:                Wed, 22 Jan 2020   Prob (F-statistic):           4.13e-05
Time:                        02:03:50   Log-Likelihood:                 2.9057
No. Observations:                  10   AIC:                            -1.811
Df Residuals:                       8   BIC:                            -1.206
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      0.0729      0.099      0.740      0.480      -0.154       0.300
x              1.9952      0.247      8.063      0.000       1.425       2.566
==============================================================================
Omnibus:                        1.626   Durbin-Watson:                   1.010
Prob(Omnibus):                  0.443   Jarque-Bera (JB):                1.044
Skew:                           0.541   Prob(JB):                        0.593
Kurtosis:                       1.844   Cond. No.                         4.25
==============================================================================

iii. y=axby=ax^b
在这里插入图片描述
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.875
Model: OLS Adj. R-squared: 0.859
Method: Least Squares F-statistic: 55.94
Date: Wed, 22 Jan 2020 Prob (F-statistic): 7.07e-05
Time: 02:03:50 Log-Likelihood: -2.6834
No. Observations: 10 AIC: 9.367
Df Residuals: 8 BIC: 9.972
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 1.0790 0.269 4.008 0.004 0.458 1.700
x -1.2627 0.169 -7.479 0.000 -1.652 -0.873
==============================================================================
Omnibus: 1.445 Durbin-Watson: 0.825
Prob(Omnibus): 0.486 Jarque-Bera (JB): 1.007
Skew: -0.680 Prob(JB): 0.605
Kurtosis: 2.249 Cond. No. 5.15
==============================================================================

iv. y=aebxy=ae^{bx}
在这里插入图片描述

								OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.973
Model:                            OLS   Adj. R-squared:                  0.969
Method:                 Least Squares   F-statistic:                     286.9
Date:                Wed, 22 Jan 2020   Prob (F-statistic):           1.50e-07
Time:                        02:03:50   Log-Likelihood:                 4.9603
No. Observations:                  10   AIC:                            -5.921
Df Residuals:                       8   BIC:                            -5.315
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      0.9874      0.115      8.573      0.000       0.722       1.253
x             -0.3411      0.020    -16.938      0.000      -0.388      -0.295
==============================================================================
Omnibus:                        0.110   Durbin-Watson:                   2.673
Prob(Omnibus):                  0.947   Jarque-Bera (JB):                0.293
Skew:                           0.169   Prob(JB):                        0.864
Kurtosis:                       2.232   Cond. No.                         13.0
==============================================================================

代码:

import numpy as np
import pandas as pd
from statsmodels.formula.api import ols
from statsmodels.formula.api import ols

def multiple_regression(x, y):
    # prepare data
    linear_data = pd.DataFrame({'x': x, 'y': y})
    inverse_data = pd.DataFrame({'x': 1/x, 'y': y})
    power_data = pd.DataFrame({'x': np.log(x), 'y': np.log(y)})
    exponential_data = pd.DataFrame({'x': x, 'y': np.log(y)})

    # y = ax + b
    Lmodel = ols('y~x', linear_data).fit()
    print(Lmodel.summary())
    plt.figure()
    plt.title('y = ax + b')
    Lx = np.arange(0, 10, 0.1)
    Ly = Lmodel.params[1]*Lx + Lmodel.params[0]
    plt.plot(Lx, Ly)
    plt.scatter(x, y)
    plt.show()

    # y = a + b/x
    Imodel = ols('y~x', inverse_data).fit()
    print(Imodel.summary())
    Ix = np.arange(0, 10, 0.1)
    Iy = Imodel.params[1]/Ix + Imodel.params[0]
    plt.figure()
    plt.title('y = a + b/x')
    plt.plot(Ix, Iy)
    plt.scatter(x, y)
    plt.show()

    # y = a*x^b
    Pmodel = ols('y~x', power_data).fit()
    print(Pmodel.summary())
    Px = np.arange(0, 10, 0.1)
    Py = np.exp(Pmodel.params[0])*np.power(Px, Pmodel.params[1])
    plt.figure()
    plt.title('y = a*x^b')
    plt.plot(Px, Py)
    plt.scatter(x, y)
    plt.show()

    # y = a*e^(b*x)
    Emodel = ols('y~x', exponential_data).fit()
    print(Emodel.summary())
    Ex = np.arange(0, 10, 0.1)
    Ey = np.exp(Emodel.params[0]+Emodel.params[1]*Ex)
    plt.figure()
    plt.title('y = a*e^(b*x)')
    plt.plot(Ex, Ey)
    plt.scatter(x, y)
    plt.show()


def test():
    x = np.array([1, 2, 3, 4, 4, 6, 6, 8, 8, 9])
    y = np.array([1.85, 1.37, 1.02, 0.75, 0.56, 0.41, 0.31, 0.23, 0.17, 0.10])
    multiple_regression(x, y)


test()

总结:
线性规划编写过程:

  1. 如果为非线性问题,先转化为线性问题
  2. 导入需要的工具
import numpy as np
import statsmodels as sm
  1. 导入数据
x = np.array([[a1, a2,..., an],...,[ b1, b2,...,bn]])
y = np.array([y1, y2,..., yn])
  1. 添加常数,完成计算
x = sm.add_constant(x)
res = sm.OLS(y, x)
print(res.summary())
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