Question

Firstly my maths is limited, so this question may have a simple answer. So, I am using the following equation to make guassian distributions:

height * np.exp( - ((x-mean)/width)**2 )

When I make gussians with the above equation where is the width of the peak applied? Is it at full width half maximum? I made the following gaussian with the following values:

height = 5
mean = 100
width = 10

When I then calculate the FWHM it is 16.6510941453 so the peak width cannot be applied at the FWHM. Where is it applied?

I am trying to constrain the FWHM so the FWHM is 10x smaller than that of the mean. So in the above example I would of liked the gaussian to have a FWHM of 10 at the mean of 100 at a peak height of 5.

Answer 1

In your equation, the width parameter is actually sigma, which is the standard deviation of a Gaussian, not FWHM. Below are functions to convert between the two of these properties

from numpy import sqrt, log

def sigma2Gamma(sigma):
    '''Function to convert standard deviation (sigma) to FWHM (Gamma)'''
    return sigma * sqrt(2 * log(2)) * 2 / sqrt(2)

Gamma = sigma2Gamma(10)
print Gamma
# prints 16.651092223153956, which is what you saw in your graph

def Gamma2sigma(Gamma):
    '''Function to convert FWHM (Gamma) to standard deviation (sigma)'''
    return Gamma * sqrt(2) / ( sqrt(2 * log(2)) * 2 )

sigma = Gamma2sigma(10)
print sigma
# prints 6.0056120439322491, which is the standard deviation that will
# give a FWHM of 10

I would recommend changing your equation to

height * np.exp( - ((x-mean)/Gamma2sigma(width))**2 )

if you want to input the FWHM and not the standard deviation