[regression] finished main text and exercises

regression/code/gradientDescent.m

@@ -1,32 +1,42 @@
-% x, y from exercise 8.3
+function [p, ps, mses] = gradientDescent(x, y, func, p0, epsilon, threshold)
+% Gradient descent for fitting a function to data pairs.
+%
+% Arguments: x, vector of the x-data values.
+%            y, vector of the corresponding y-data values.
+%            func, function handle func(x, p)
+%            p0, vector with initial parameter values
+%            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 vectors traversed.
+%            mses: vector with the corresponding mean squared errors
 
-% some arbitrary values for the slope and the intercept to start with:
-position = [-2.0, 10.0];  
-
-% gradient descent:
-gradient = [];
-errors = [];
-count = 1;
-eps = 0.0001;
-while isempty(gradient) || norm(gradient) > 0.1
-    gradient = meanSquaredGradient(x, y, position);
-    errors(count) = meanSquaredError(x, y, position);
-    position = position - eps .* gradient; 
-    count = count + 1;
+  p = p0;
+  gradient = ones(1, length(p0)) * 1000.0;
+  ps = [];
+  mses = [];
+  while norm(gradient) > threshold
+      ps = [ps, p(:)];
+      mses = [mses, meanSquaredError(x, y, func, p)];
+      gradient = meanSquaredGradient(x, y, func, p);
+      p = p - epsilon * gradient;
+  end
+end
+
+function mse = meanSquaredError(x, y, func, p)
+    mse = mean((y - func(x, p)).^2);
+end
+
+function gradmse = meanSquaredGradient(x, y, func, p)
+  gradmse = zeros(size(p, 1), size(p, 2));
+  h = 1e-5;               % stepsize for derivatives
+  mse = meanSquaredError(x, y, func, p);
+  for i = 1:length(p)     % for each coordinate ...
+    pi = p;
+    pi(i) = pi(i) + h;    %   displace i-th parameter
+    msepi = meanSquaredError(x, y, func, pi);
+    gradmse(i) = (msepi - mse)/h;
+  end
 end
 
-figure()
-subplot(2,1,1)
-hold on
-scatter(x, y, 'displayname', 'data')
-xx = min(x):0.01:max(x);
-yy = position(1).*xx + position(2);
-plot(xx, yy, 'displayname', 'fit')
-xlabel('Input')
-ylabel('Output')
-grid on
-legend show
-subplot(2,1,2)
-plot(errors)
-xlabel('optimization steps')
-ylabel('error')

regression/code/gradientDescentPower.m

@@ -1,41 +0,0 @@
-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

@@ -3,7 +3,7 @@ meansquarederrorline;               % generate data
 p0 = [2.0, 1.0];
 eps = 0.00001;
 thresh = 50.0;
-[pest, ps, mses] = gradientDescentPower(x, y, p0, eps, thresh);
+[pest, ps, mses] = gradientDescent(x, y, @powerLaw, p0, eps, thresh);
 pest
 
 subplot(2, 2, 1);                   % top left panel
@@ -22,7 +22,7 @@ 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);
+yy = powerLaw(xx, pest);
 plot(xx, yy);
 plot(x, y, 'o');                    % plot original data
 xlabel('Size [m]');