如何在Python中对具有2个变量且具有高斯分布的函数进行加权？

Question

I've been working with this for the last days and I couldn't see yet where is the problem.

I'm trying to weight a function with 2 variables f(q,r) within a Gaussian distribution g(r) with a specific mean value ( R0 ) and deviation ( sigma ). This is needed because the theoretical function f(q) has a certain dispersity in its r variable when analyzed experimentally. Therefore, we use a probability density function to weigh our function in the r variable.

I include the code, which works, but doesn't give the expected result (the weighted curve should be smoother as the polydispersity grows (higher sigma ) as it is shown below. As you can see, I integrated the convolution of the 2 functions f(r,q)*g(r) from r = 0 to r = +inf .

The result is plotted to compare the weigh result with the simple function:

from scipy.integrate import quad, quadrature
import numpy as np
import math as m
import matplotlib.pyplot as plt 

#function weighted with a probability density function (gaussian)
def integrand(r,q):
    #gaussian function normalized
    def gauss_nor(r):
        #gaussian function
        def gauss(r):
            return m.exp(-((r-R0)**2)/(2*sigma**2))
        return (m.exp(-((r-R0)**2)/(2*sigma**2)))/(quad(gauss,0,np.inf)[0])
    #function f(r,q)
    def f(r,q):
        return 3*(np.sin(q*r)-q*r*np.cos(q*r))/((r*q)**3)
    return gauss_nor(r)*f(r,q)

#quadratic integration of the integrand (from 0 to +inf)
#integrand is function*density_function (gauss) 
def function(q):
    return quad(integrand, 0, np.inf, args=(q))[0]

#parameters used in the function
R0=20
sigma=5

#range to plot q
q=np.arange(0.001,2.0,0.005)

#vector where the result of the integral will be saved
function_vec = np.vectorize(function)

#vector for the squared power of the integral
I=[]
I=(function_vec(q))**2

#function without density function
I0=[]
I0=(3*(np.sin(q*R0)-q*R0*np.cos(q*R0))/((R0*q)**3))**2

#plot of weighted and non-weighted functions
p1,=plt.plot(q,I,'b')
p3,=plt.plot(q,I0,'r')
plt.legend([p1,p3],('Weighted','No weighted'))
plt.yscale('log')
plt.xscale('log')
plt.show()

Thank you very much. I've been with this problems for some days already and I haven't found the mistake.

Maybe somebody know how to weigh a function with a PDF in an easier way.

Answer 1

I simplified your code, the output is the same as yours. I think it's already very smooth, there are some very sharp peak in the log-log graph, just because the curve has zero points. So it's not smooth in a log-log graph, but it's smooth in a normal XY graph.

import numpy as np

def gauss(r):
    return np.exp(-((r-R0)**2)/(2*sigma**2))

def f(r,q):
    return 3*(np.sin(q*r)-q*r*np.cos(q*r))/((r*q)**3)

R0=20
sigma=5

qm, rm = np.ogrid[0.001:2.0:0.005, 0.001:40:1000j]
gr = gauss(rm)
gr /= np.sum(gr)
fm = f(rm, qm)
fm *= gr

plot(qm.ravel(), fm.sum(axis=1)**2)
plt.yscale('log')
plt.xscale('log')