How to calculate the confidence intervals for prediction in Regression? and also how to plot it in python

Question

Fig 7.1, An Introduction To Statistical Learning

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I am currently studying a book named Introduction to Statistical Learning with applications in R, and also converting the solutions to python language.
I am not able to get how to get the confidence intervals and plot them as shown in the above image(dashed lines). I have plotted the line. Here's my code for that - (I am using polynomial regression with predictiors - 'age' and response - 'wage',degree is 4)

poly = PolynomialFeatures(4)
X = poly.fit_transform(data['age'].to_frame())
y = data['wage']
# X.shape

model = sm.OLS(y,X).fit()
print(model.summary())

# So, what we want here is not only the final line, but also the standart error related to the line
# TO find that we need to calcualte the predictions for some values of age
test_ages = np.linspace(data['age'].min(),data['age'].max(),100)

X_test = poly.transform(test_ages.reshape(-1,1))
pred = model.predict(X_test)

plt.figure(figsize = (12,8))
plt.scatter(data['age'],data['wage'],facecolors='none', edgecolors='darkgray')
plt.plot(test_ages,pred)

Here data is WAGE data which is available in R. This is the resulting graph i get -

Answer 1

I have used bootstraping to calculate the confidence intervals, for this i have used a self customed module -

import numpy as np
import pandas as pd
from tqdm import tqdm

class Bootstrap_ci:


    def boot(self,X_data,y_data,R,test_data,model):
        predictions = []
        for i in tqdm(range(R)):
            predictions.append(self.alpha(X_data,y_data,self.get_indices(X_data,200),test_data,model))
           
        return np.percentile(predictions,2.5,axis = 0),np.percentile(predictions,97.5,axis = 0)

    def alpha(self,X_data,y_data,index,test_data,model):
        X = X_data.loc[index]
        y = y_data.loc[index]
        
        lr = model
        lr.fit(pd.DataFrame(X),y)
        
        return lr.predict(pd.DataFrame(test_data))


    def get_indices(self,data,num_samples):
        return  np.random.choice(data.index, num_samples, replace=True)

The above module can be used as -

poly = PolynomialFeatures(4)
X = poly.fit_transform(data['age'].to_frame())
y = data['wage']

X_test = np.linspace(min(data['age']),max(data['age']),100)
X_test_poly = poly.transform(X_test.reshape(-1,1))

from bootstrap import Bootstrap_ci

bootstrap = Bootstrap_ci()

li,ui = bootstrap.boot(pd.DataFrame(X),y,1000,X_test_poly,LinearRegression())

This will give us the lower confidence interval, and upper confidence interval. To plot the graph -

plt.scatter(data['age'],data['wage'],facecolors='none', edgecolors='darkgray')
plt.plot(X_test,pred,label = 'Fitted Line')
plt.plot(X_test,ui,linestyle = 'dashed',color = 'r',label = 'Confidence Intervals')
plt.plot(X_test,li,linestyle = 'dashed',color = 'r')

The resultant graph is

Answer 2

Following code results in the 95% confidence interval

from scipy import stats

confidence = 0.95
squared_errors = (<<predicted values>> - <<true y_test values>>) ** 2
np.sqrt(stats.t.interval(confidence, len(squared_errors) - 1,
                         loc=squared_errors.mean(),
                         scale=stats.sem(squared_errors)))