%% Convolution of probability distributions
% Nick Hale and Alex Townsend, January 2014
%%
% (Chebfun example stats/ProbabilityConvolution.m)
% [Tags: #Probability distribution, #convolution, #CONV]
LW = 'linewidth'; FS = 'fontsize'; MS = 'markersize';
%%
% It is well known that the probability distribution of the sum of two or more
% independent random variables is the convolution of their individual
% distributions, defined by
%
% $$ h(x) = \int_{-\infty}^\infty f(t) g(x-t) dt. $$
%
% Many standard distributions have simple convolutions, and here we
% investigate some of them before computing the convolution of some more
% exotic distributions.
%% Normal distribution
% The normal distribution has the PDF
%
% $$ f(x; \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}
% e^{-(x-\mu)^2/2\sigma^2}. $$
%
% Let's make an anonymous function that can compute it:
normalDist = @(x, mu, s) exp(-.5*(x-mu).^2/s^2) / (s*sqrt(2*pi));
%%
% Now let's make two normal distributions with differing parameters, `N1` as
% $N(\mu_1, \sigma_1)$ and `N2` as $N(\mu_2, \sigma_2)$. To make things easier,
% we'll truncate to a finite domain.
dom = [-1.2 1.2];
x = chebfun('x', dom);
s1 = .1; m1 = .1;
N1 = normalDist(x, m1, s1);
s2 = .11; m2 = -.3;
N2 = normalDist(x, m2, s2);
%%
% We can compute the distribution of the sum of two variables drawn from these
% distributions by convolving `N1` and `N2`:
N3 = conv(N1, N2);
plot(N1, 'b', N2 ,'r', N3, 'k', LW, 1.6), set(gca, FS, 14)
xlim([-1.5 1.5])
%%
% One can show (see, for example, [1]) that `N3` should be equal to
% $N(\mu_1+\mu_2, \sigma_1^2+\sigma_2^2)$. Let us verify this:
N4 = normalDist(x, m1 + m2, sqrt(s1^2 + s2^2));
norm(N4 - N3{dom(1), dom(2)})
%% Gamma distribution
% We next consider the gamma distribution, which has the PDF
%
% $$ f(x, k, \theta) = \frac{x^{k-1}e^{-x/\theta}}
% {\theta^k\Gamma(k)}, \quad x\ge0. $$
%
% As before, let's compute the distribution for the sum of two variables drawn
% from such distributions by using `conv`:
gammaDist = @(x, k, t) x.^(k-1).*exp(-x./t) / (t^k*gamma(k));
dom = [0 5];
x = chebfun('x', dom);
k1 = 2; t = .3;
G1 = gammaDist(x, k1, t);
k2 = 1;
G2 = gammaDist(x, k2, t);
G3 = conv(G1, G2);
plot(G1, 'b', G2 ,'r', G3, 'k', LW, 1.6)
xlim([0 5])
%%
% In this instance, we should have
%
% $$ G(k_1, \theta) + G(k_2, \theta) = G(k_1 + k_2, \theta): $$
G4 = gammaDist(x, k1 + k2, t);
norm(G4 - G3{dom(1), dom(2)})
%% Exponential distribution
% As our final example of common distributions, let's look at the exponential
% distribution, which has the PDF
%
% $$ f(x, \lambda) = \lambda e^{-\lambda x}, \quad x\ge0. $$
dom = [0 40];
x = chebfun('x', dom);
expDist = @(x, lam) lam*exp(-lam*x);
%%
% This time we make just a single distribution, $E(0.9)$, which we convolve with
% itself to give the distribution for the sum of two variables from $E(0.9)$.
lam = .25;
E = expDist(x, lam);
E2 = conv(E, E);
plot(E, 'b', E2, 'k', LW, 1.6)
xlim([0 40])
%%
% In this instance, the theory tells us we should have a gamma distribution,
% which indeed we do:
E3 = gammaDist(x, 2, 1/lam);
norm(E3 - E2{dom(1), dom(2)})
%% Exotic distributions
% Of course, the relationships verified above are available in closed form
% only for certain special cases. For more exotic distributions the
% convolution _must_ be computed numerically.
%%
% Let's do this for two exotic (and discontinuous) probability distributions,
% arbitrarily defined by summing a Heaviside function with a number of
% Gaussians and then normalising.
rng('default')
x = chebfun('x', [-2 2]);
F = heaviside(x);
G = heaviside(.5-x);
for k = 1:10
F = F + normalDist(x, randn, 2*rand);
G = G + normalDist(x, randn, 2*rand);
end
F = F/sum(F);
G = G./sum(G);
%%
% Here are the distributions themselves:
plot(F,'b', G, 'r', LW, 1.6)
xlim([-4 4]);
%%
% And this is the distribution of the random variable $z=x+y$, where $x$ is
% drawn from $F$ and $y$ is drawn from $G$:
tic
h = conv(F, G);
toc
hold on, plot(h, 'k', LW, 1.6), hold off
%% References
%
% 1. http://en.wikipedia.org/wiki/List_of_convolutions_of_probability_distributions
%
% 2. Nick Hale and Alex Townsend, "Convolution of compactly supported
% functions", in preparation.