Suppose you are given a bunch of points in the form of a vector x (between "a" and "b") and an accompanying density function p(x)>0. For concreteness, consider
x = [0:0.1:1.0]'; px = 0.3 - (x-0.5).^2;
x = [0:0.1:1.0]'; px = 0.3 - (x-0.5).^2;
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| The top subfigure depicts the density function p(x), and the cumulative density function C(x). The bottom subfigure depicts the quadrature points "z" (circles), and the lines mark the end of the intervals, using N=11. |
This density function is shown in the picture above.
We want to create a N*1 vector "z" of quadrature points that are distributed according to the density p(x). We also want a vector "dz" corresponding to the width of strip belonging to a particular quadrature point. As an additional constraint, we assume that we want z(1) = a, and z(N) = b to match the end-points.
The program attached at the end is able to do this relatively efficiently. In fact, I used the following command to generate the picture above:
[z dz] = GridDensity(x,px,11,1);
GNU Octave Code:
%
% PROGRAM:
% Takes in a PDF or density function, and spits out a bunch of points in
% accordance with the PDF
%
% INPUT:
% x = vector of points. It need *not* be equispaced
% px = vector of same size as x: probability distribution or
% density function. It need not be normalized but has to be positive.
% N = Number of points >= 3. The end points of "x" are included
% necessarily
% Pt = Optional argument. If present, then some plotting.
%
% OUTPUT:
% z = Points distributed according to the density
% hz = width of the "intervals" - useful to apportion domain to points
% if you are doing quadrature with the results, for example.
%
% (*) Sachin Shanbhag, March 5, 2012
% GNU Public License
%
function [z h] = GridDensity(x,px,N,Pt)
npts = 100; % can potentially change
xi = linspace(min(x),max(x),npts)'; % reinterpolate on equi-spaced axis
pint = interp1(x,px,xi,'spline'); % smoothen using splines
ci = cumtrapz(xi,pint);
pint = pint/ci(npts);
ci = ci/ci(npts); % normalize ci
alfa = 1/(N-1); % alfa/2 + (N-1)*alfa + alfa/2
zij = zeros(N,1); % quadrature interval end marker
z = zeros(N,1); % quadrature point
z(1) = min(x);
z(N) = max(x);
%
% ci(Z_j,j+1) = (j - 0.5) * alfa
%
beta = [0.5:1:N-1.5]'*alfa;
zij = [z(1); interp1(ci,xi,beta,'spline'); z(N)];
h = diff(zij);
clear beta;
%
% Quadrature points are not the centroids, but rather the center of masses
% of the quadrature intervals
%
beta = [1:1:N-2]'*alfa;
z(2:N-1) = interp1(ci,xi,beta,'spline');
%
% Some plotting if required
%
if(nargin>3)
subplot(2,1,1)
plot(xi,ci,'b-','LineWidth',2,xi,pint,'r-','LineWidth',2);
axis([min(xi),max(xi)]);
legend('CDF','PDF')
title('Visualization')
xlabel('x');
ylabel('CDF(x)/PDF(x)')
subplot(2,1,2)
plot(z,0.5*ones(size(z)),'ro');
axis([min(xi),max(xi)]);
xlabel('z');
hold on;
for i = 2:N
X = [zij(i); zij(i)];
Y = [0; 1];
plot(X,Y,'b')
endfor
hold off;
endif
endfunction