利用python进行一元线性、多项式回归（statsmodels包）
使用python中的statsmodels包进行一元线性回归与二次多项式回归，并使用模型进行预测
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前言
导入包
import pandas as pd
import numpy as np
import statsmodels.api as sm
import matplotlib as mpl
import matplotlib.pyplot as plt
#中文显示问题
plt.rcParams['font.sans-serif']=['SimHei']
plt.rcParams['axes.unicode_minus'] = False
# notebook嵌入图片
%matplotlib inline
# 提高分辨率
%config InlineBackend.figure_format='retina'
# 导入数据
df = pd.read_csv("C:/Users/lenovo/Desktop/附件2.csv", encoding="gbk")
df.head()
一元线性
# 时间与乙醇转化率的一元线性回归
# 添加常数项
X = sm.add_constant(df["时间"])
etoh_reg = sm.OLS(df["乙醇转化率"], X).fit()
print(etoh_reg.summary())
OLS Regression Results
==============================================================================
Dep. Variable: 乙醇转化率 R-squared: 0.934
Model: OLS Adj. R-squared: 0.921
Method: Least Squares F-statistic: 70.59
Date: Sun, 03 Jul 2022 Prob (F-statistic): 0.000391
Time: 08:09:10 Log-Likelihood: -11.145
No. Observations: 7 AIC: 26.29
Df Residuals: 5 BIC: 26.18
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 42.6549 1.098 38.855 0.000 39.833 45.477
时间 -0.0526 0.006 -8.402 0.000 -0.069 -0.037
==============================================================================
Omnibus: nan Durbin-Watson: 1.548
Prob(Omnibus): nan Jarque-Bera (JB): 0.783
Skew: 0.564 Prob(JB): 0.676
Kurtosis: 1.812 Cond. No. 362.
==============================================================================
# 时间与C4烯烃选择性的一元线性回归
# 添加常数项
X = sm.add_constant(df["时间"])
c4_reg = sm.OLS(df["C4烯烃选择性"], X).fit()
print(c4_reg.summary())
OLS Regression Results
==============================================================================
Dep. Variable: C4烯烃选择性 R-squared: 0.046
Model: OLS Adj. R-squared: -0.144
Method: Least Squares F-statistic: 0.2434
Date: Sun, 03 Jul 2022 Prob (F-statistic): 0.643
Time: 08:09:10 Log-Likelihood: -10.336
No. Observations: 7 AIC: 24.67
Df Residuals: 5 BIC: 24.56
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 38.5808 0.978 39.451 0.000 36.067 41.095
时间 0.0028 0.006 0.493 0.643 -0.012 0.017
==============================================================================
Omnibus: nan Durbin-Watson: 2.156
Prob(Omnibus): nan Jarque-Bera (JB): 0.773
Skew: -0.814 Prob(JB): 0.679
Kurtosis: 2.938 Cond. No. 362.
==============================================================================
二次多项式
# 时间与乙醇转化率多项式回归
X = sm.add_constant(df["时间"])
X["时间二次项"] = df["时间"]**2
etoh_ploy = sm.OLS(df["乙醇转化率"],X).fit()
print(etoh_ploy.summary())
OLS Regression Results
==============================================================================
Dep. Variable: 乙醇转化率 R-squared: 0.988
Model: OLS Adj. R-squared: 0.982
Method: Least Squares F-statistic: 163.2
Date: Sun, 03 Jul 2022 Prob (F-statistic): 0.000146
Time: 08:09:10 Log-Likelihood: -5.2009
No. Observations: 7 AIC: 16.40
Df Residuals: 4 BIC: 16.24
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 45.2058 0.800 56.506 0.000 42.985 47.427
时间 -0.1044 0.013 -8.279 0.001 -0.139 -0.069
时间二次项 0.0002 4.15e-05 4.226 0.013 6.02e-05 0.000
==============================================================================
Omnibus: nan Durbin-Watson: 3.379
Prob(Omnibus): nan Jarque-Bera (JB): 0.323
Skew: -0.475 Prob(JB): 0.851
Kurtosis: 2.550 Cond. No. 1.26e+05
==============================================================================
# 时间与C4烯烃选择性多项式回归
# 添加常数项
X = sm.add_constant(df["时间"])
X["时间二次项"] = df["时间"]**2
c4_ploy = sm.OLS(df["C4烯烃选择性"], X).fit()
print(c4_ploy.summary())
OLS Regression Results
==============================================================================
Dep. Variable: C4烯烃选择性 R-squared: 0.241
Model: OLS Adj. R-squared: -0.139
Method: Least Squares F-statistic: 0.6335
Date: Sun, 03 Jul 2022 Prob (F-statistic): 0.577
Time: 08:09:18 Log-Likelihood: -9.5392
No. Observations: 7 AIC: 25.08
Df Residuals: 4 BIC: 24.92
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 39.7151 1.487 26.712 0.000 35.587 43.843
时间 -0.0203 0.023 -0.865 0.436 -0.085 0.045
时间二次项 7.798e-05 7.71e-05 1.011 0.369 -0.000 0.000
==============================================================================
Omnibus: nan Durbin-Watson: 2.491
Prob(Omnibus): nan Jarque-Bera (JB): 0.686
Skew: -0.610 Prob(JB): 0.710
Kurtosis: 2.071 Cond. No. 1.26e+05
==============================================================================
预测与绘图
# 乙醇转化率预测图
temp = np.arange(20,300,1)
c4_x_pre = np.vstack((np.ones_like(temp), temp, temp**2)).T
plt.figure(dpi = 600,figsize = (6,4))
plt.plot(df['时间'],df['乙醇转化率'],'o')
plt.plot(temp,etoh_ploy.predict(c4_x_pre))
plt.legend(['原数据','预测线'])
# C4烯烃选择性预测图
temp = np.arange(20,300,1)
c4_x_pre = np.vstack((np.ones_like(temp), temp, temp**2)).T
plt.figure(dpi = 600,figsize = (6,4))
plt.plot(df['时间'],df['C4烯烃选择性'],'o')
plt.plot(temp,c4_ploy.predict(c4_x_pre))
plt.legend(['原数据','预测线'])
plt.savefig("2.png")
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