Question: I need to plot prediction and confidence intervals and need to use python and only the following packages. How do I plot both intervals in the same model. I managed to plot the prediction intervals with the help of ChatGPT. this is the code including the data set. 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()  How should I continue to be able to plot both intervals?

Answer: Explanation:  The summary_frame method provides both the prediction intervals (obs_ci_lower, obs_ci_upper) and the confidence intervals (mean_ci_lower, mean_ci_upper). The fill_between function is used twice to plot both intervals: once for the prediction interval and once for the confidence interval.  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 ax = df.plot(kind='scatter', x='X', y='Y', color='black', label='Data') # Regression line ax.plot(df['X'], results.fittedvalues, color='#58C9F4', label='Regression Line') # Prediction interval ax.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 ax.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') ax.set(xlabel='X', ylabel='Y', title='Linear Regression with Prediction and Confidence Intervals') ax.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.    In the original code, the intercept at x = 0 is correct, but to visually represent it, the x-axis limits need to be expanded. Additionally, the slope appears distorted due to an unequal aspect ratio between the X and Y axes, causing a misrepresentation of the linear relationship. # OLS model = smf.ols(formula='Y ~ X', data=df) results = model.fit() # Extract coefficients intercept = results.params['Intercept'] slope = results.params['X'] # Calculate regression line values x_values = np.linspace(0, 59037, 100) y_values = intercept + slope * x_values # 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 ax = df.plot(kind='scatter', x='X', y='Y', color='black', label='Data', s=1, figsize=(12, 10)) # Regression line ax.plot(x_values, y_values, color='#58C9F4', label='Regression Line') # Adding the point where the regression line intersects the y-axis ax.scatter(0, intercept, color='red', label='Y-intercept (x=0)', s=2) ax.annotate(f'(0, {intercept:.2f})', xy=(0, intercept), xytext=(2000, intercept + 1000), arrowprops=dict(facecolor='black', shrink=0.05)) # Annotating the regression line with the slope ax.text(x_values[len(x_values)//2], y_values[len(y_values)//2], f'Slope: {slope:.2f}', fontsize=12, verticalalignment='bottom') # Prediction interval ax.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 ax.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') # Setting equal aspect ratio ax.set_aspect('equal', 'box') ax.set(xlabel='X', ylabel='Y', title='Linear Regression with Prediction and Confidence Intervals') ax.legend(bbox_to_anchor=(1, 0.5), loc='center left', frameon=False) plt.show()