Matlab: Fit of histogram data with many Gaussians and AIC evaluation - matlab

Question

Consider this example of code to obtain the best fit from data varying the number of fitting Gaussians according the Akaike criterion
MU1 = [1];
SIGMA1 = [2];
MU2 = [-3];
SIGMA2 = [1 ];
X = [mvnrnd(MU1,SIGMA1,1000);mvnrnd(MU2,SIGMA2,1000)];
AIC = zeros(1,4);
obj = cell(1,4);
options = statset('Display','final');
for k = 1:4
obj{k} = gmdistribution.fit(X,k,'Options',options);
AIC(k)= obj{k}.AIC;
end
[minAIC,numComponents] = min(AIC)
I want to do the same thing but with data that are given in a form of a histogram (consider for example the data http://pastebin.com/embed_js.php?i=1mNRuEHZ).
What is the most direct way to implement the same procedure in matlab in this case?

Answer 1

If I'm getting you right, then your problem is to convert between data that is already compiled as a histogram (so numbers of observations paired with the actual value of an observation) and the original individual observations. Of course, when compiling the histogram, you have lost two things:
Order. You don't know what the order of observations was in the original data, which is probably not important, provided your observations are independent. Also, the way I get gmdistribution.fit() it doesn't take into account order anyway.
Resolution. When you create a histogram, you need to bin your data, which makes you lose precision, so to speak, because it is impossible to recover the precise values of your observations from the bins.
Once you are aware of that you can create a 'vector of observations' from your histogram data. Say, X1 is your histogram data (Nx2 vector). If you do
invX = cell2mat(arrayfun(#(x,y) repmat(y,1,x), abs(int16(1000*X1(:, 2)))', X1(:, 1)', ...
'UniformOutput', false))';
you get a vector that contains individual observations, just like X in your example.
Note that you have to convert the bin counts to integers first. At this step, because the given data's precision is quite high, I had to round to make the computation possible for my machine. However, the final result seems fairly reasonable.
Also note that I used absolute values, there are some cases in your histogram data were your data is actually negative, which, for a histogram obviously doesn't make sense.
Last but not least you have to change the number of iterations for the fit procedure to 1000. The final code to produce the below figure reads
MU1 = [1];
SIGMA1 = [2];
MU2 = [-3];
SIGMA2 = [1 ];
X = [mvnrnd(MU1,SIGMA1,1000);mvnrnd(MU2,SIGMA2,1000)];
X = X1(:, 2);
invX = cell2mat(arrayfun(#(x,y) repmat(y,1,x), abs(int16(1000*X1(:, 2)))', X1(:, 1)', ...
'UniformOutput', false))'; %'
X = invX;
AIC = zeros(1,4);
obj = cell(1,4);
options = statset('Display','final', 'MaxIter', 1000);
for k = 1:4
obj{k} = gmdistribution.fit(X,k,'Options',options);
AIC(k)= obj{k}.AIC;
end
[minAIC,numComponents] = min(AIC);
hold on;
plot(linspace(-1, 2, length(X1(:, 2))), abs(X1(:, 2)), 'LineWidth', 2)
plot(x, pd/max(pd)*double(max(abs(X1(:, 2)))), 'LineWidth', 5);
h = legend('Original data', 'PDF');
set(h,'FontSize',32);
Output looks like this: