Some people can’t bear uncertainty. I can safely be excluded from that list. Uncertainty is rarely a bad thing. Let me explain with one example.
Imagine yourself in a room filled with people maintining conversations all around you. Someone then starts up a conversation with you. You have no problem focusing on your own conversation while simultaneously multiple others are in progress. How do you suppress the background noise in order to concentrate on your dialogue? This is known as the cocktail party effect. There are many ways to solve this problem, but the verdict is still out on how the brain actually does it. Interesting right?
Not too long ago, a friend asked me for some help with solving a problem. The question was: how can I detect multiple peaks in a noisy spectrum?
So, I thought that if the underlying source of each peak could be represented as a Gaussian distribution, the entire spectrum could be just the sum of multiple Gaussians. Effectively, you’re given one signal from which you must estimate multiple sources. Just like above! This formulation is the well known Gaussian Mixture Model (GMM). Let’s break it down.
Every peak is a Gaussian distribution with mean , variance and weight , denoted by the probability density function and is spelled out as
Then, for peaks, the GMM is given by
where all wieghts are greater than 0 and sum to 1. Great! Unfortunately, we are not done. Since this is an inverse problem, the best we can do is to obtain estimates of the parameters: , and . Expectation-Maximization (EM) is one of a few algorithms used to get estimates of the parameters. Thankfully, the gmdistribution.fit function found in Matlab’s statistics toolbox solves this problem. All it asked for is the data and the number of peaks. Here we go.
Let’s create some data. We will generate normally distributed random variables and with known means and standard deviations shown below
then, combine them into one random variable . We then plot a normalized histogram.
N = 2; K = 1000; M1 = 10000; M2 = 2*M1; m = [-3; 6]; s = [1.5; 2.5]; x1 = s(1)*randn(M1,1) + m(1); x2 = s(2)*randn(M2,1) + m(2); x = [x1; x2]; xaxis = [min(x):range(x)/(K-1):max(x)]'; xpdf = hist(x,K)/(M1+M2); figure(1) set(gcf, 'color', 'w'); plot(xaxis, xpdf, 'k'); xlabel('x'); ylabel('f(x)'); axis tight;
How does Matlab now recover the estimates and from ?
% pre-allocation mu = zeros(N,1); sigma = zeros(N,1); weight = zeros(N,1); gaussPdfi = zeros(K,N); gaussPdf = zeros(K,1); % fit options = statset('Display','final'); obj = gmdistribution.fit(x,N,'Options',options); gaussPdf = pdf(obj,xaxis); A = sum(gaussPdf); gaussPdf = gaussPdf/A; % separating N Gaussians for n = 1:N, mu(n) = obj.mu(n); sigma(n) = sqrt(obj.Sigma(1,1,n)); weight(n) = obj.PComponents(n); gaussPdfi(:,n) = weight(n)*normpdf(xaxis,mu(n),sigma(n))/A; end
Let’s see what the distributions look like.
figure(1) set(gcf, 'color', 'w'); set(gca, 'FontSize', 6); plot(xaxis, gaussPdf, 'r', 'linewidth', 1.25); xlabel('x', 'Fontsize', 6); ylabel('f(x), F(x)', 'Fontsize', 6); axis tight; hold off; figure(2) set(gcf, 'color', 'w'); set(gca, 'FontSize', 6); area(xaxis, gaussPdfi(:,1)); h = findobj(gca, 'Type', 'patch'); set(h, 'FaceColor', 'g', 'EdgeColor', 'g', 'facealpha', 0.75); hold on; area(xaxis, gaussPdfi(:,2)); h = findobj(gca, 'Type', 'patch'); set(h,'facealpha', 0.75); plot(xaxis, gaussPdf, 'r', 'linewidth', 1.25); xlabel('x', 'Fontsize', 6); ylabel('F(x), f(x_1), f(x_2)', 'Fontsize', 6); axis tight; hold off;
Looks good! What are the estimates of the mean and standard deviation ?
>> mu mu = -2.9997 5.9863 >> sigma sigma = 1.5144 2.4889
I’ll take that!