Finding the convolution of two histograms
The probability distribution of the sum of two random variables, x and y, is given by the convolution of the individual distributions. I'm having some trouble doing this numerically. In the following example, x and y are uniformly distributed, with their respective distributions approximated as histograms. My reasoning says that the histograms should be convoluted to give the distribution of, x+y.
from numpy.random import uniform
from numpy import ceil,convolve,histogram,sqrt
from pylab import hist,plot,show
n = 10**2
x,y = uniform(-0.5,0.5,n),uniform(-0.5,0.5,n)
bins = ceil(sqrt(n))
pdf_x = histogram(x,bins=bins,normed=True)
pdf_y = histogram(y,bins=bins,normed=True)
s = convolve(pdf_x[0],pdf_y[0])
plot(s)
show()
which giv开发者_如何转开发es the following,
In other words, a triangular distribution, as expected. However, I have no idea how to find the x-values. I would appreciate it if someone could correct me here.
In order to still move on (towards more murky details), I further adapted your code like this:
from numpy.random import uniform
from numpy import convolve, cumsum, histogram, linspace
s, e, n= -0.5, 0.5, 1e3
x, y, bins= uniform(s, e, n), uniform(s, e, n), linspace(s, e, n** .75)
pdf_x= histogram(x, normed= True, bins= bins)[0]
pdf_y= histogram(y, normed= True, bins= bins)[0]
c= convolve(pdf_x, pdf_y); c= c/ c.sum()
bins= linspace(2* s, 2* e, len(c))
# a simulation
xpy= uniform(s, e, 10* n)+ uniform(s, e, 10* n)
c2= histogram(xpy, normed= True, bins= bins)[0]; c2= c2/ c2.sum()
from pylab import grid, plot, show, subplot
subplot(211), plot(bins, c)
plot(linspace(xpy.min(), xpy.max(), len(c2)), c2, 'r'), grid(True)
subplot(212), plot(bins, cumsum(c)), grid(True), show()
Thus, giving plots something like this:
Where the upper part represents thePDF
(blue line), which indeed looks quite triangular and the simulation (red dots), which reflects the triangular shape. Lower part represents the CDF
, which also looks to follow nicely the expected S
-curve.
精彩评论