[regression] fixed m code

regression/code/errorGradient.m

@@ -1,13 +1,12 @@
-% x, y, slopes, and intercepts from exercise 8.3
+% x, y, slopes, intercepts, ii, ss, and error_surface from exercise 8.3
 
-error_surface = zeros(length(slopes), length(intercepts));
-gradient_m = zeros(size(error_surface));
-gradient_b = zeros(size(error_surface));
-
-for i = 1:length(slopes)
-    for j = 1:length(intercepts)
-	error_surface(i,j) = meanSquaredError(x, y, [slopes(i), intercepts(j)]);
-        grad = meanSquaredGradient(x, y, [slopes(i), intercepts(j)]);
+qslopes = 1.0:0.5:4.0;
+qintercepts = -150:50:150;
+gradient_m = zeros(length(qintercepts), length(qslopes));
+gradient_b = zeros(length(qintercepts), length(qslopes));
+for i = 1:length(qintercepts)
+    for j = 1:length(qslopes)
+        grad = meanSquaredGradient(x, y, [qslopes(j), qintercepts(i)]);
         gradient_m(i,j) = grad(1);
         gradient_b(i,j) = grad(2);
     end
@@ -15,13 +14,12 @@ end
 
 figure()
 hold on
-[N, M] = meshgrid(intercepts, slopes);
-%surface(M, N, error_surface, 'FaceAlpha', 0.5);
-contour(M, N, error_surface, 50);
-quiver(M, N, gradient_m, gradient_b)
+contour(ss, ii, error_surface, 70);
+[qss, qii] = meshgrid(qslopes, qintercepts);
+quiver(qss, qii, gradient_m, gradient_b, 0.01)
 xlabel('Slope m')
 ylabel('Intercept b')
 zlabel('Mean squared error')
-set(gcf, 'paperunits', 'centimeters', 'papersize', [15, 10.5], ... 
-    'paperposition', [0., 0., 15, 10.5])
-saveas(gcf, 'error_gradient', 'pdf')
+%set(gcf, 'paperunits', 'centimeters', 'papersize', [15, 10.5], ... 
+%    'paperposition', [0., 0., 15, 10.5])
+%saveas(gcf, 'error_gradient', 'pdf')

regression/code/errorSurface.m

@@ -1,27 +1,27 @@
 % generate data:
-m = 0.75;
+m = 3.0;
 b = -40.0;
 n = 20;
 x = 120.0*rand(n, 1);
 y = m*x + b + 15.0*randn(n, 1);
 
 % compute mean squared error for a range of slopes and intercepts:
-slopes = -5:0.25:5;
-intercepts = -30:1:30;
-error_surface = zeros(length(slopes), length(intercepts));
-for i = 1:length(slopes)
-    for j = 1:length(intercepts)
-	      error_surf(i,j) = meanSquaredError(x, y, [slopes(i), intercepts(j)]);
+slopes = 0.0:0.1:5.0;
+intercepts = -200:10:200;
+error_surface = zeros(length(intercepts), length(slopes));
+for i = 1:length(intercepts)
+    for j = 1:length(slopes)
+	      error_surface(i,j) = meanSquaredError(x, y, [slopes(j), intercepts(i)]);
     end
 end
 
 % plot the error surface:
 figure()
-[N,M] = meshgrid(intercepts, slopes);
-surface(M, N, error_surface);
+[ss, ii] = meshgrid(slopes, intercepts);
+surface(ss, ii, error_surface);
 xlabel('slope', 'rotation', 7.5)
 ylabel('intercept', 'rotation', -22.5)
-zlabel('error')
+zlabel('mean squared error')
 set(gca,'xtick', (-5:2.5:5))
 grid on
 view(3)

regression/code/gradientDescent.m

@@ -1,13 +1,13 @@
 % x, y from exercise 8.3
 
 % some arbitrary values for the slope and the intercept to start with:
-position = [-2. 10.];  
+position = [-2.0, 10.0];  
 
 % gradient descent:
 gradient = [];
 errors = [];
 count = 1;
-eps = 0.01;
+eps = 0.0001;
 while isempty(gradient) || norm(gradient) > 0.1
     gradient = meanSquaredGradient(x, y, position);
     errors(count) = meanSquaredError(x, y, position);

regression/code/meanSquaredGradient.m

@@ -8,9 +8,9 @@ function gradient = meanSquaredGradient(x, y, parameter)
 %
 % Returns: the gradient as a vector with two elements
 
-  h = 1e-6;   % stepsize for derivatives
+  h = 1e-5;   % stepsize for derivatives
   mse = meanSquaredError(x, y, parameter);
   partial_m = (meanSquaredError(x, y, [parameter(1)+h, parameter(2)]) - mse)/h;
-  partial_n = (meanSquaredError(x, y, [parameter(1), parameter(2)+h]) - mse)/h;
-  gradient = [partial_m, partial_n];
+  partial_b = (meanSquaredError(x, y, [parameter(1), parameter(2)+h]) - mse)/h;
+  gradient = [partial_m, partial_b];
 end

regression/lecture/regression-chapter.tex

@@ -16,6 +16,14 @@
 
 \include{regression}
 
+\subsection{Start with one-dimensional problem!}
+\begin{itemize}
+\item Just the root mean square as a function of the slope
+\item 1-d gradient
+\item 1-d gradient descend
+\item Homework is to do the 2d problem!
+\end{itemize}
+
 \subsection{Linear fits}
 \begin{itemize}
 \item Polyfit is easy: unique solution!