Python: finding the intersection point of two Gaussian curves

Question

I have two Gaussian graphs:

x = np.linspace(-5,9,10000) plot1=plt.plot(x,mlab.normpdf(x,2.5,1)) plot2=plt.plot(x,mlab.normpdf(x,5,1)) 

and I want to find the point at which two curves intersect. Is there any way to do this? In particular, I want to find the value of the x coordinate where they occur.

Answer 1

You want to find x so that both Gaussian functions have the same height (intersect)

You can do this by equating two Gaussian functions and deciding for x. You end up with a quadratic equation with coefficients related to Gaussian means and variances. Here is the final result:

 import numpy as np def solve(m1,m2,std1,std2): a = 1/(2*std1**2) - 1/(2*std2**2) b = m2/(std2**2) - m1/(std1**2) c = m1**2 /(2*std1**2) - m2**2 / (2*std2**2) - np.log(std2/std1) return np.roots([a,b,c]) m1 = 2.5 std1 = 1.0 m2 = 5.0 std2 = 1.0 result = solve(m1,m2,std1,std2) 

Output:

 array([ 3.75]) 

You can build the intersections found:

 x = np.linspace(-5,9,10000) plot1=plt.plot(x,mlab.normpdf(x,m1,std1)) plot2=plt.plot(x,mlab.normpdf(x,m2,std2)) plot3=plt.plot(result,mlab.normpdf(result,m1,std1),'o') 

The plot will be:

If your Gaussians have multiple intersections, the code will also find all of them (say, m1 = 2.5, std1 = 3.0, m2 = 5.0, std2 = 1.0):