Added matlab code to mle chapter

statistics/code/mlemeanstd.m

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+% draw random numbers:
+n = 500;
+mu = 3.0;
+sigma =2.0;
+x = randn(n,1)*sigma+mu;
+fprintf('              mean of the data is %.2f\n', mean(x))
+fprintf('standard deviation of the data is %.2f\n', std(x))
+
+% mean as parameter:
+pmus = 2.0:0.01:4.0;
+% matrix with the probabilities for each x and pmus:
+lms = zeros(length(x), length(pmus));
+for i=1:length(pmus)
+    pmu = pmus(i);
+    p = exp(-0.5*((x-pmu)/sigma).^2.0)/sqrt(2.0*pi)/sigma;
+    lms(:,i) = p;
+end
+lm = prod(lms, 1);          % likelihood
+loglm = sum(log(lms), 1);   % log likelihood
+
+% plot likelihood of mean:
+subplot(2, 2, 1);
+plot(pmus, lm );
+xlabel('mean')
+ylabel('likelihood')
+subplot(2, 2, 2);
+plot(pmus, loglm );
+xlabel('mean')
+ylabel('log likelihood')
+
+% standard deviation as parameter:
+psigs = 1.0:0.01:3.0;
+% matrix with the probabilities for each x and psigs:
+lms = zeros(length(x), length(psigs));
+for i=1:length(psigs)
+    psig = psigs(i);
+    p = exp(-0.5*((x-mu)/psig).^2.0)/sqrt(2.0*pi)/psig;
+    lms(:,i) = p;
+end
+lm = prod(lms, 1);          % likelihood
+loglm = sum(log(lms), 1);   % log likelihood
+
+% plot likelihood of standard deviation:
+subplot(2, 2, 3);
+plot(psigs, lm );
+xlabel('standard deviation')
+ylabel('likelihood')
+subplot(2, 2, 4);
+plot(psigs, loglm);
+xlabel('standard deviation')
+ylabel('log likelihood')

statistics/code/mlepdffit.m

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+% plot gamma pdfs:
+xx = 0.0:0.1:10.0;
+shapes = [ 1.0, 2.0, 3.0, 5.0];
+cc = jet(length(shapes) );
+for i=1:length(shapes)
+    yy = gampdf(xx, shapes(i), 1.0);
+    plot(xx, yy, '-', 'linewidth', 3, 'color', cc(i,:), ...
+        'DisplayName', sprintf('s=%.0f', shapes(i)) );
+    hold on;
+end
+
+% generate gamma distributed random numbers:
+n = 50;
+x = gamrnd(3.0, 1.0, n, 1);
+
+% histogram:
+[h,b] = hist(x, 15);
+h = h/sum(h)/(b(2)-b(1));
+bar(b, h, 1.0, 'DisplayName', 'data');
+
+% maximum likelihood estimate:
+p = mle(x, 'distribution', 'gamma');
+yy = gampdf(xx, p(1), p(2));
+plot(xx, yy, '-k', 'linewidth', 5, 'DisplayName', 'mle' );
+
+hold off;
+legend('show');

statistics/code/mlepropfit.m

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+m = 2.0;      % slope
+sigma = 1.0;  % standard deviation
+n = 100;      % number of data pairs
+
+% data pairs:
+x = 5.0*rand(n, 1);
+y = m*x + sigma*randn(n, 1);
+
+% fit:
+slope = mleslope(x, y);
+fprintf('slopes:\n');
+fprintf('original = %.2f\n', m);
+fprintf('     fit = %.2f\n', slope);
+
+% lines:
+xx = 0.0:0.1:5.0;     % x-axis values
+yorg = m*xx;
+yfit = slope*xx;
+
+% plot:
+plot(xx, yorg, '-r', 'linewidth', 5);
+hold on;
+plot(xx, yfit, '-g', 'linewidth', 2);
+plot(x, y, 'ob');
+hold off;
+legend('data', 'original', 'fit', 'Location', 'NorthWest');
+legend('boxoff')
+xlabel('x');
+ylabel('y');

statistics/code/mleslope.m

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+function slope = mleslope(x, y )
+% Compute the maximum likelihood estimate of the slope
+% of a line through the origin 
+% given the data pairs in the vectors x and y.
+    slope = sum(x.*y)/sum(x.*x);
+end