## Plotting confidence intervals of linear regression in Python

After a friendly tweet from @tomstafford who mentioned that this script was useful I’ve re-posted it here in preparation for the removal of my Newcastle University pages.

This script calculates and plots confidence intervals around a linear regression based on new observations. After I couldn’t find anything similar on the internet I developed my own implementation based on Statistics in Geography by David Ebdon (ISBN: 978-0631136880).

Plot of linear regression with confidence intervals

# linfit.py - example of confidence limit calculation for linear regression fitting.
# References:
# - Statistics in Geography by David Ebdon (ISBN: 978-0631136880)
# - Reliability Engineering Resource Website:
# - http://www.weibull.com/DOEWeb/confidence_intervals_in_simple_linear_regression.htm
# - University of Glascow, Department of Statistics:
# - http://www.stats.gla.ac.uk/steps/glossary/confidence_intervals.html#conflim
import numpy as np
import matplotlib.pyplot as plt
# example data
x = np.array([4.0,2.5,3.2,5.8,7.4,4.4,8.3,8.5])
y = np.array([2.1,4.0,1.5,6.3,5.0,5.8,8.1,7.1])
# fit a curve to the data using a least squares 1st order polynomial fit
z = np.polyfit(x,y,1)
p = np.poly1d(z)
fit = p(x)
# get the coordinates for the fit curve
c_y = [np.min(fit),np.max(fit)]
c_x = [np.min(x),np.max(x)]
# predict y values of origional data using the fit
p_y = z[0] * x + z[1]
# calculate the y-error (residuals)
y_err = y -p_y
# create series of new test x-values to predict for
p_x = np.arange(np.min(x),np.max(x)+1,1)
# now calculate confidence intervals for new test x-series
mean_x = np.mean(x) # mean of x
n = len(x) # number of samples in origional fit
t = 2.31 # appropriate t value (where n=9, two tailed 95%)
s_err = np.sum(np.power(y_err,2)) # sum of the squares of the residuals
confs = t * np.sqrt((s_err/(n-2))*(1.0/n + (np.power((p_x-mean_x),2)/
((np.sum(np.power(x,2)))-n*(np.power(mean_x,2))))))
# now predict y based on test x-values
p_y = z[0]*p_x+z[0]
# get lower and upper confidence limits based on predicted y and confidence intervals
lower = p_y - abs(confs)
upper = p_y + abs(confs)
# set-up the plot
plt.axes().set_aspect('equal')
plt.xlabel('X values')
plt.ylabel('Y values')
plt.title('Linear regression and confidence limits')
# plot sample data
plt.plot(x,y,'bo',label='Sample observations')
# plot line of best fit
plt.plot(c_x,c_y,'r-',label='Regression line')
# plot confidence limits
plt.plot(p_x,lower,'b--',label='Lower confidence limit (95%)')
plt.plot(p_x,upper,'b--',label='Upper confidence limit (95%)')
# set coordinate limits
plt.xlim(0,11)
plt.ylim(0,11)
# configure legend
plt.legend(loc=0)
leg = plt.gca().get_legend()
ltext = leg.get_texts()
plt.setp(ltext, fontsize=10)
# show the plot
plt.show()