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‎CH01/CH01_SEC02.m‎

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clear all, close all, clc
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A=imread('../DATA/dog.jpg');
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X=double(rgb2gray(A)); % Convert RBG->gray, 256 bit->double.
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nx = size(X,1); ny = size(X,2);
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imagesc(X), axis off, colormap gray
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[U,S,V] = svd(X);
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for r=[5 20 100]; % Truncation value
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Xapprox = U(:,1:r)*S(1:r,1:r)*V(:,1:r)'; % Approx. image
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figure, imagesc(Xapprox), axis off
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title(['r=',num2str(r,'%d'),']);
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end
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%% f_ch01_ex02_2
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subplot(1,2,1), semilogy(diag(S),'k')
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subplot(1,2,2), plot(cumsum(diag(S))/sum(diag(S)),'k')

‎CH01/CH01_SEC02_production.m‎

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clear all, close all, clc
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A=imread('../DATA/dog.jpg');
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X=double(rgb2gray(A)); % Convert RBG to gray, 256 bit to double.
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nx = size(X,1); ny = size(X,2);
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[U,S,V] = svd(X);
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figure, subplot(2,2,1)
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imagesc(X), axis off, colormap gray
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title('Original')
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plotind = 2;
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for r=[5 20 100]; % Truncation value
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Xapprox = U(:,1:r)*S(1:r,1:r)*V(:,1:r)'; % Approx. image
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subplot(2,2,plotind), plotind = plotind + 1;
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imagesc(Xapprox), axis off
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title(['r=',num2str(r,'%d'),', ',num2str(100*r*(nx+ny)/(nx*ny),'%2.2f'),'% storage']);
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end
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set(gcf,'Position',[100 100 550 400])
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%% f_ch01_ex02_2
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figure, subplot(1,2,1)
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semilogy(diag(S),'k','LineWidth',1.2), grid on
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xlabel('r')
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ylabel('Singular value, \sigma_r')
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xlim([-50 1550])
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subplot(1,2,2)
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plot(cumsum(diag(S))/sum(diag(S)),'k','LineWidth',1.2), grid on
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xlabel('r')
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ylabel('Cumulative Energy')
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xlim([-50 1550]); ylim([0 1.1])
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set(gcf,'Position',[100 100 550 240])

‎CH01/CH01_SEC03_Rotation.m‎

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clear all, close all, clc
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theta = [pi/15; -pi/9; -pi/20];
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Sigma = diag([3; 1; 0.5]); % scale x, then y, then z
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Rx = [1 0 0; % rotate about x-axis
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0 cos(theta(1)) -sin(theta(1));
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0 sin(theta(1)) cos(theta(1))];
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Ry = [cos(theta(2)) 0 sin(theta(2)); % rotate about y-axis
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0 1 0;
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-sin(theta(2)) 0 cos(theta(2))];
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Rz = [cos(theta(3)) -sin(theta(3)) 0; % rotate about z-axis
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sin(theta(3)) cos(theta(3)) 0;
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0 0 1];
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X = Rz*Ry*Rx*Sigma; % rotate and scale
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%% Plot sphere
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subplot(1,2,1), hold on
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[x,y,z] = sphere(25);
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h1=surf(x,y,z);
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set(h1,'FaceAlpha',.7)
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colormap jet, lighting phong, axis equal
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axis([-2 2 -2 2 -2 2]), view([45 26])
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%%
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xR = 0*x; yR = 0*y; zR = 0*z;
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for i=1:size(x,1)
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for j=1:size(x,2)
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vec = [x(i,j); y(i,j); z(i,j)];
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vecR = X*vec;
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xR(i,j) = vecR(1);
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yR(i,j) = vecR(2);
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zR(i,j) = vecR(3);
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end
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end
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subplot(1,2,2), hold on
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h2=surf(xR,yR,zR,z); % using the z-position of sphere for color
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set(h2,'FaceAlpha',.7)
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colormap jet, lighting phong, axis equal
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axis([-2 2 -2 2 -2 2]), view([45 26])
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set(gcf,'Position',[100 100 800 350])

‎CH01/CH01_SEC03_Rotation_production.m‎

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clear all, close all, clc
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xlim([-2 2])
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ylim([-2 2])
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zlim([-2 2])
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% view([-72 18])
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view([45 26])
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% axis off
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set(gca,'FontSize',13);
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set(gcf,'Position',[100 100 300 300])
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set(gcf,'PaperPositionMode','auto')
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print('-depsc2', '-loose', 'CH01_EX09_1_axis.eps')
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%% Define rotation matrix
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t1 = pi/15;
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t2 = -pi/9;
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t3 = -pi/20;
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Rx = [1 0 0;
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0 cos(t1) -sin(t1);
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0 sin(t1) cos(t1)];
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Ry = [cos(t2) 0 sin(t2);
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0 1 0;
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-sin(t2) 0 cos(t2)];
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Rz = [cos(t3) -sin(t3) 0;
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sin(t3) cos(t3) 0;
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0 0 1];
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Sigma = diag([3; 1; 0.5]);
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R = Rz*Ry*Rx*Sigma;
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%% Plot sphere and great circles
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[x,y,z] = sphere(250);
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% subplot(1,2,1)
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plot3([-2 2],[0 0],[0 0],'k','LineWidth',6);
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hold on
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plot3([0 0],[-2 2],[0 0],'k','LineWidth',6);
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plot3([0 0],[0 0],[-2 2],'k','LineWidth',6);
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h1=surf(x,y,z);
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hold on
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set(h1,'EdgeColor','none','FaceColor',[.5 .5 .5],'FaceAlpha',.75)
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% set(h1,'FaceColor','none')
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lighting phong
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shading interp
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axis equal
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theta = (0:.001:1)*2*pi;
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x1 = cos(theta);
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y1 = sin(theta);
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z1 = 0*theta;
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x2 = 0*theta;
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y2 = cos(theta);
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z2 = sin(theta);
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x3 = cos(theta);
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y3 = 0*theta;
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z3 = sin(theta);
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plot3(x1,y1,z1,'k','LineWidth',6);
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plot3(x2,y2,z2,'k','LineWidth',6);
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plot3(x3,y3,z3,'k','LineWidth',6);
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xlim([-2 2])
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ylim([-2 2])
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zlim([-2 2])
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% view([-72 18])
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view([45 26])
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% axis off
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set(gca,'FontSize',13);
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set(gcf,'Position',[100 100 100 100])
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set(gcf,'PaperPositionMode','auto')
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set(gcf,'renderer','opengl')
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axis off
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% save2pdf('CH01_EX09_1b.pdf',gcf,600)
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% print('-depsc2', '-loose', 'CH01_EX09_1b.eps')
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print('-dpng','-r1800' ,'-loose', 'CH01_EX09_1a.png')
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%%
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xR = 0*x;
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yR = 0*y;
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zR = 0*z;
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for i=1:size(x,1)
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for j=1:size(x,2)
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vec = [x(i,j); y(i,j); z(i,j)];
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vecR = R*vec;
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xR(i,j) = vecR(1);
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yR(i,j) = vecR(2);
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zR(i,j) = vecR(3);
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end
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end
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vec1 = [x1; y1; z1];
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vec2 = [x2; y2; z2];
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vec3 = [x3; y3; z3];
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vec1R = R*vec1;
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vec2R = R*vec2;
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vec3R = R*vec3;
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eX = [2; 0; 0];
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eY = [0; 2; 0];
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eZ = [0; 0; 2];
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eXR = R*eX;
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eYR = R*eY;
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eZR = R*eZ;
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x1R = vec1R(1,:);
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y1R = vec1R(2,:);
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z1R = vec1R(3,:);
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x2R = vec2R(1,:);
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y2R = vec2R(2,:);
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z2R = vec2R(3,:);
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x3R = vec3R(1,:);
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y3R = vec3R(2,:);
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z3R = vec3R(3,:);
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figure
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% plot3([0 -2],[0 0],[0 0],'Color',[.3 .3 .3],'LineWidth',6);
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% hold on
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% plot3([0 0],[0 -2],[0 0],'Color',[.3 .3 .3],'LineWidth',6);
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% plot3([0 0],[0 0],[0 2],'Color',[.3 .3 .3],'LineWidth',6);
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h1=surf(xR,yR,zR,z);
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hold on
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set(h1,'EdgeColor','none','FaceColor',[.5 .5 .5],'FaceAlpha',.75)
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% set(h1,'FaceColor','none')
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axis equal
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shading interp
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lighting phong
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plot3(x1R,y1R,z1R,'k','LineWidth',6);
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plot3(x2R,y2R,z2R,'k','LineWidth',6);
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plot3(x3R,y3R,z3R,'k','LineWidth',6);
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plot3([-eXR(1) eXR(1)],[-eXR(2) eXR(2)],[-eXR(3) eXR(3)],'k','LineWidth',6);
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plot3([-eYR(1) eYR(1)],[-eYR(2) eYR(2)],[-eYR(3) eYR(3)],'k','LineWidth',6);
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plot3([-eZR(1) eZR(1)],[-eZR(2) eZR(2)],[-eZR(3) eZR(3)],'k','LineWidth',6);
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xlim([-2 2])
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ylim([-2 2])
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zlim([-2 2])
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grid off
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% view([-72 18])
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view([45 26])
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% axis off
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set(gca,'FontSize',13);
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set(gcf,'Position',[100 100 100 100])
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set(gcf,'PaperPositionMode','auto')
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set(gcf,'renderer','opengl')
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axis off
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% save2pdf('CH01_EX09_1b.pdf',gcf,600)
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% print('-depsc2', '-loose', 'CH01_EX09_1b.eps')
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print('-dpng','-r1800' ,'-loose', 'CH01_EX09_1b.png')

‎CH01/CH01_SEC04_1_Linear.m‎

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clear all, close all, clc
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x = 3; % True slope
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a = [-2:.25:2]';
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b = a*x + 1*randn(size(a)); % Add noise
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plot(a,x*a,'k','LineWidth',1.5) % True relationship
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hold on
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plot(a,b,'rx','LineWidth',1.5) % Noisy measurements
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[U,S,V] = svd(a,'econ');
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xtilde = V*inv(S)*U'*b; % Least-square fit
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plot(a,xtilde*a,'b--','LineWidth',1.5) % Plot fit
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l1=legend('True line','Noisy data','Regression line');
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set(l1,'String',{' ','',''})
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set(l1,'Location','NorthWest')
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grid on
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set(gcf,'Position',[100 100 300 300])
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set(gcf,'PaperPositionMode','auto')
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set(gca,'FontSize',13)
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print('-depsc2', '-loose', 'figures/CH01_EX_PseudoInv')
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%% Three methods of computing regression
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xtilde1 = V*inv(S)*U'*b
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xtilde2 = pinv(a)*b
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xtilde3 = regress(b,a)
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‎CH01/CH01_SEC04_2_Cement.m‎

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clear all, close all, clc
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load hald; % Load Portlant Cement dataset
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A = ingredients;
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b = heat;
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[U,S,V] = svd(A,'econ');
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x = V*inv(S)*U'*b; % Solve Ax=b using the SVD
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plot(b,'k','LineWidth',2); hold on % Plot data
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plot(A*x,'r-o','LineWidth',1.,'MarkerSize',2); % Plot regression
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l1 = legend('Heat data','Regression')
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%% Alternative 1 (regress)
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x = regress(b,A);
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%% Alternative 2 (pinv)
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x = pinv(A)*b;

‎CH01/CH01_SEC04_3_Housing.m‎

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clear all, close all, clc
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load housing.data
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b = housing(:,14); % housing values in $1000s
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A = housing(:,1:13); % other factors,
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A = [A ones(size(A,1),1)]; % Pad with ones for nonzero offset
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x = regress(b,A);
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subplot(1,2,1)
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plot(b,'k-o');
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hold on, grid on
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plot(A*x,'r-o');
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set(gca,'FontSize',13)
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xlim([0 size(A,1)])
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subplot(1,2,2)
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[b sortind] = sort(housing(:,14)); % sorted values
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plot(b,'k-o')
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hold on, grid on
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plot(A(sortind,:)*x,'r-o')
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l1=legend('Housing value','Regression')
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set(l1,'Location','NorthWest')
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set(gca,'FontSize',13)
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xlim([0 size(A,1)])
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set(gcf,'Position',[100 100 600 250])
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%%
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A2 = A-ones(size(A,1),1)*mean(A,1);
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for i=1:size(A,2)-1
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A2std = std(A2(:,i));
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A2(:,i) = A2(:,i)/A2std;
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end
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A2(:,end) = ones(size(A,1),1);
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x = regress(b,A2)
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figure
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bar(x(1:13))
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xlabel('Attribute'), ylabel('Correlation')
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set(gca,'FontSize',13)
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set(gcf,'Position',[100 100 600 250])

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