如何计算和绘制线性回归的预测和置信区间

Question

我需要绘制预测和置信区间，并且需要使用 python 并且仅使用以下包。如何在同一模型中绘制两个区间。我在 ChatGPT 的帮助下成功绘制了预测区间。这是包含数据集的代码。

import statsmodels.api as sm
import statsmodels.formula.api as smf
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

#dataframe
data = {
    'X': [55641, 55681, 55637, 55825, 55772, 55890, 56068, 56299, 56825, 57205, 
          57562, 57850, 57975, 57992, 58240, 58414, 58561, 59066, 58596, 58631, 
          58758, 59037],
    'Y': [21886, 21934, 21699, 21901, 21812, 21714, 21932, 22086, 22265, 22551, 
          22736, 22301, 22518, 22580, 22618, 22890, 23112, 23315, 22865, 22788, 
          22949, 23149]
}

df = pd.DataFrame(data)

#OLS
model = smf.ols(formula='Y ~ X', data=df)
results = model.fit()

print(results.summary())

#calculating prediction intevals
predictions = results.get_prediction(df)
prediction_summary_frame = predictions.summary_frame(alpha=0.05)

#data points
plt.scatter(df['X'], df['Y'], color='black', label='Data')

#regression line
plt.plot(df['X'], results.fittedvalues, color='#58C9F4', label='Regression Line')

#prediction invterval
plt.fill_between(df['X'], prediction_summary_frame['obs_ci_lower'], prediction_summary_frame['obs_ci_upper'], color='grey', alpha=0.2, label='95% Prediction Interval')

plt.xlabel('X')
plt.ylabel('Y')
plt.title('Linear Regression with Prediction Intervals')
plt.legend()
plt.show()

我应该如何继续绘制两个区间？

Answer 1

import statsmodels.api as sm
import statsmodels.formula.api as smf
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Dataframe
data = {
    'X': [55641, 55681, 55637, 55825, 55772, 55890, 56068, 56299, 56825, 57205, 
          57562, 57850, 57975, 57992, 58240, 58414, 58561, 59066, 58596, 58631, 
          58758, 59037],
    'Y': [21886, 21934, 21699, 21901, 21812, 21714, 21932, 22086, 22265, 22551, 
          22736, 22301, 22518, 22580, 22618, 22890, 23112, 23315, 22865, 22788, 
          22949, 23149]
}

df = pd.DataFrame(data)

# OLS
model = smf.ols(formula='Y ~ X', data=df)
results = model.fit()

print(results.summary())

# Calculating prediction intervals
predictions = results.get_prediction(df)
prediction_summary_frame = predictions.summary_frame(alpha=0.05)

# Calculating confidence intervals
confidence_intervals = results.conf_int(alpha=0.05)

# Data points
plt.scatter(df['X'], df['Y'], color='black', label='Data')

# Regression line
plt.plot(df['X'], results.fittedvalues, color='#58C9F4', label='Regression Line')

# Prediction interval
plt.fill_between(df['X'], prediction_summary_frame['obs_ci_lower'], prediction_summary_frame['obs_ci_upper'], color='grey', alpha=0.2, label='95% Prediction Interval')

# Confidence interval
plt.fill_between(df['X'], prediction_summary_frame['mean_ci_lower'], prediction_summary_frame['mean_ci_upper'], color='blue', alpha=0.2, label='95% Confidence Interval')

plt.xlabel('X')
plt.ylabel('Y')
plt.title('Linear Regression with Prediction and Confidence Intervals')
plt.legend()
plt.show()

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      Y   R-squared:                       0.919
Model:                            OLS   Adj. R-squared:                  0.915
Method:                 Least Squares   F-statistic:                     227.5
Date:                Tue, 11 Jun 2024   Prob (F-statistic):           2.17e-12
Time:                        06:30:58   Log-Likelihood:                -140.06
No. Observations:                  22   AIC:                             284.1
Df Residuals:                      20   BIC:                             286.3
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept    559.4600   1450.698      0.386      0.704   -2466.642    3585.562
X              0.3815      0.025     15.084      0.000       0.329       0.434
==============================================================================
Omnibus:                        0.314   Durbin-Watson:                   1.479
Prob(Omnibus):                  0.855   Jarque-Bera (JB):                0.390
Skew:                          -0.242   Prob(JB):                        0.823
Kurtosis:                       2.562   Cond. No.                     2.64e+06
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.64e+06. This might indicate that there are
strong multicollinearity or other numerical problems.