%%%% Example for introductory (tutorial) manual pages.%% Fits a 1-dimensional sine wave.%%%%%% Prepare figures.%%figure(1)pos = get(1, 'Position');set(1, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample1', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(2)pos = get(2, 'Position');set(2, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample3', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(3)pos = get(3, 'Position');set(3, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample4', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(4)pos = get(4, 'Position');set(4, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample5', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(5)pos = get(5, 'Position');set(5, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample6', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(6)pos = get(6, 'Position');set(6, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample7', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(7)pos = get(7, 'Position');set(7, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample8', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])figure(8)pos = get(8, 'Position');set(8, ...  'Position', [pos(1) pos(2) 400 400], ...  'NumberTitle', 'off', ...  'Name', 'introExample9', ...  'PaperType', 'a4letter', ...  'InvertHardCopy', 'on', ...  'PaperPosition', [0.5 0.5 4 4])%%%% Get training data and plot it.%%p = 50;sigma = 0.2;NEW = 0;if NEW  x = 1 - rand(1,p);  y = sin(10 * x)' + sigma * randn(p,1);  save introExample x yelse  load introExampleendfigure(1)hold offplot(x, y, 'c+')axis([0 1 -1.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [-1 0 1])title('Figure 1')drawnow%%%% Get test data and plot target.%%pt = 500;xt = linspace(0,1,pt);yt = sin(10 * xt)';hold onplot(xt, yt, 'b--')%%%% Plot single Gaussian radial basis function.%%cj = 0.5;rj = 0.1;ht = exp(-(xt - cj).^2 / rj^2);figure(2)hold offplot(xt, ht, 'g-')axis([0 1 -0.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [0 1])title('Figure 3')drawnow%%%% Design matrices.%%c = x;m = p;r = 0.1;H = rbfDesign(x, c, r, 'c');Ht = rbfDesign(xt, c, r, 'c');%%%% Train with normal equation.%%w = inv(H' * H) * H' * y;%%%% Plot the output.%%ft = Ht * w;figure(3)hold offplot(x, y, 'c+')axis([0 1 -1.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [-1 0 1])hold onplot(xt, yt, 'b--')plot(xt, ft, 'g-')title('Figure 4')drawnow%%%% Forward selection: predict and plot.%%subset = forwardSelect(H, y, '-t BIC -v');Hs = H(:,subset);w = inv(Hs' * Hs) * Hs' * y;ft = Ht(:,subset) * w;figure(4)plot(x, y, 'c+')axis([0 1 -1.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [-1 0 1])hold onplot(xt, yt, 'b--')plot(xt, ft, 'g-')title('Figure 5')drawnow%%%% Mean square error.%%(ft - yt)' * (ft - yt) / pt%%%% Global ridge with guessed regularisation paremeter.%%lambda = 1e-4;w = inv(H' * H + lambda * eye(m)) * H' * y;ft = Ht * w;figure(5)hold offplot(x, y, 'c+')axis([0 1 -1.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [-1 0 1])hold onplot(xt, yt, 'b--')plot(xt, ft, 'g-')title('Figure 6')drawnow%%%% Mean square error.%%(ft - yt)' * (ft - yt) / pt%%%% Actively choose the right regularisation parameter.%%lambda = globalRidge(H, y, 0.1, 'BIC -v')%%%% Global ridge with optimal lambda.%%w = inv(H' * H + lambda * eye(m)) * H' * y;ft = Ht * w;figure(6)hold offplot(x, y, 'c+')axis([0 1 -1.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [-1 0 1])hold onplot(xt, yt, 'b--')plot(xt, ft, 'g-')title('Figure 7')drawnow%%%% Mean square error.%%(ft - yt)' * (ft - yt) / pt%%%% Explicit calculation and plot of BIC for range of lambda.%%lambdas = logspace(-4, 2, 100);bics = zeros(1,100);b = 0;for i = 1:100  b = overWrite(b, i);  bics(i) = predictError(H, y, lambdas(i), 'BIC');endoverWrite(b);bic = predictError(H, y, lambda, 'BIC');figure(7)hold offloglog(lambdas, bics, 'm-')hold onloglog(lambda, bic, 'r*')title('Figure 8')drawnow%%%% Try local ridge regression.%%lambdas = localRidge(H, y, lambda, '-v');finite = find(lambdas ~= Inf)%%%% Work out fit and plot it.%%Hl = H(:,finite);lambdas = lambdas(finite);w = inv(Hl' * Hl + diag(lambdas)) * Hl' * y;ft = Ht(:,finite) * w;figure(8)hold offplot(x, y, 'c+')axis([0 1 -1.5 1.5])set(gca, 'XTick', [0 0.5 1])set(gca, 'YTick', [-1 0 1])hold onplot(xt, yt, 'b--')plot(xt, ft, 'g-')title('Figure 9')drawnow%%%% Mean square error.%%(ft - yt)' * (ft - yt) / pt