[regression] finished n-dim minimization

regression/code/gradientDescentPower.m

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+function [p, ps, mses] = gradientDescentPower(x, y, p0, epsilon, threshold)
+% Gradient descent for fitting a power-law.
+%
+% Arguments: x, vector of the x-data values.
+%            y, vector of the corresponding y-data values.
+%            p0, vector with initial values for c and alpha. 
+%            epsilon: factor multiplying the gradient.
+%            threshold: minimum value for gradient
+%
+% Returns:   p, vector with the final parameter values.
+%            ps: 2D-vector with all the parameter tuples traversed.
+%            mses: vector with the corresponding mean squared errors
+
+  p = p0;
+  gradient = ones(1, length(p0)) * 1000.0;
+  ps = [];
+  mses = [];
+  while norm(gradient) > threshold
+      ps = [ps, p(:)];
+      mses = [mses, meanSquaredErrorPower(x, y, p)];
+      gradient = meanSquaredGradientPower(x, y, p);
+      p = p - epsilon * gradient;
+  end
+end
+
+function mse = meanSquaredErrorPower(x, y, p)
+    mse = mean((y - p(1)*x.^p(2)).^2);
+end
+
+function gradmse = meanSquaredGradientPower(x, y, p)
+  gradmse = zeros(size(p, 1), size(p, 2));
+  h = 1e-5;               % stepsize for derivatives
+  mse = meanSquaredErrorPower(x, y, p);
+  for i = 1:length(p)     % for each coordinate ...
+    pi = p;
+    pi(i) = pi(i) + h;    %   displace i-th parameter
+    msepi = meanSquaredErrorPower(x, y, pi);
+    gradmse(i) = (msepi - mse)/h;
+  end
+end
+

regression/code/plotgradientdescentpower.m

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+meansquarederrorline;               % generate data
+
+p0 = [2.0, 1.0];
+eps = 0.00001;
+thresh = 50.0;
+[pest, ps, mses] = gradientDescentPower(x, y, p0, eps, thresh);
+pest
+
+subplot(2, 2, 1);                   % top left panel
+hold on;
+plot(ps(1,:), ps(2,:), '.');
+plot(ps(1,end), ps(2,end), 'og');
+plot(c, 3.0, 'or'); % dot indicating true parameter values
+hold off;
+xlabel('Iteration');
+ylabel('C');
+subplot(2, 2, 3);                   % bottom left panel
+plot(mses, '-o');
+xlabel('Iteration steps');
+ylabel('MSE');
+subplot(1, 2, 2);                   % right panel
+hold on;
+% generate x-values for plottig the fit:
+xx = min(x):0.01:max(x);
+yy = pest(1) * xx.^pest(2);
+plot(xx, yy);
+plot(x, y, 'o');                    % plot original data
+xlabel('Size [m]');
+ylabel('Weight [kg]');
+legend('fit', 'data', 'location', 'northwest');
+pause
+