p1

assignments/hwk04/HW4_Q1.m

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+%generate two-class data
+rng(1); % For reproducibility
+N = 20;
+r = sqrt(1*rand(N,1)); % Radius
+t = 2*pi*rand(N,1);  % Angle
+data1 = [r.*cos(t), r.*sin(t)]; % Points
+
+r2 = sqrt(8*rand(N,1)+1); % Radius
+t2 = 2*pi*rand(N,1);      % Angle
+data2 = [r2.*cos(t2), r2.*sin(t2)]; % points
+
+%combine the two class data
+data3 = [data1;data2];
+theclass = ones(2*N,1);
+theclass(1:N) = -1;
+figure;
+
+for degree = [1 2 4]
+    %TO DO: replace fitcsvm with kernel_perceptron
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    %SVM classification demo;
+    % alphas = fitcsvm(data3,theclass,'KernelFunction','polynomial','PolynomialOrder',degree);%tune the sigma parameter here, 1, 3, 0.1
+    [alphas, b] = kernel_perceptron(data3, theclass, degree);
+
+    %[alphas,b] = kernel_perceptron(data3,theclass,degree); 
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    
+    d = 0.02;
+    [x1Grid,x2Grid] = meshgrid(min(data3(:,1)):d:max(data3(:,1)),...
+        min(data3(:,2)):d:max(data3(:,2)));
+    xGrid = [x1Grid(:),x2Grid(:)];
+    
+    %TO DO: replace predict with kp_predict
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    %SVM prediction demo; replace with kernel perceptron
+
+    % [~,scores1] = predict(alphas,data3); %replace it with mypredict(...)
+    % [~,scores2] = predict(alphas,xGrid); %replace it with mypredict(...)
+
+    % scores1 = kp_predict(alphas, b, degree, data3, theclass, data3)
+
+    % scores2 = scores2(:,2);
+    % training_error = CalculateErrorRate(theclass,sign(scores1(:,2)));
+    
+    scores1 = kp_predict(alphas,b,degree,data3,theclass,data3); %return the prediction scores
+    scores2 = kp_predict(alphas,b,degree,data3,theclass,xGrid); %return the prediction scores
+    training_error = CalculateErrorRate(theclass,sign(scores1));
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    
+    subplot(5,1,degree);
+    h(1:2) = gscatter(data3(:,1),data3(:,2),theclass,'rb','.');
+    hold on
+    contour(x1Grid,x2Grid,reshape(scores2,size(x1Grid)),[0 0],'k');
+    plottitle = sprintf("Degree = %d; Training Error = %f",degree,training_error);
+    title(plottitle);
+end
+
+function error_rate = CalculateErrorRate(y_pred,y_label)
+    N = length(y_label);
+    error_rate = sum(y_pred ~= y_label)/N;
+end

assignments/hwk04/HW4_sol.typ

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+== Problem 1
+
+Results:
+
+#image("prob1.png")
+
+== Problem 2
+
+Original:
+
+$
+  frac(1, 2) w^T S w - nu rho + sum_t C^t xi^t
+$
+
+Find the Lagrangian first:
+
+$
+  frac(1, 2) w^T S w - nu rho + sum_t C^t xi^t
+$

assignments/hwk04/kernel_perceptron.m

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+
+function [alphas, b] = kernel_perceptron(X,Y,d); 
+
+N = size(X,1);
+err = 1; %error rate
+max_iter = 10000; %maximum number of iterations
+alphas = zeros(N,1);
+b = 0;
+t = 1;
+
+while t<=max_iter && err>0
+  for ii = 1 : N         %cycle through training set
+  %TO DO: update alpha and b
+    xt = X(ii, :);
+    yt = Y(ii, :);
+
+    sum = 0;
+    for i = 1 : N
+      ai = alphas(i);
+      yi = Y(i);
+      xi = X(i, :);
+      sum = sum + ai * yi * dot(xi, xt) ^ d;
+    end
+    sum = sum + b;
+
+    if sum * yt <= 0
+      alphas(ii) = alphas(ii) + 1;
+      b = b + yt;
+    end
+  
+  %*******************************
+  end
+  t = t+1;
+  %TO DO: calculate the error rate
+
+  %*******************************
+end

assignments/hwk04/kp_predict.m

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+function scores = kp_predict(alphas, b, d, X, Y, tst_data)
+% alphas, b: sample weights and bias term for classification
+% d: degree of polynomial kernel <x,y>^d
+% X,Y: training data and label
+% tst_data: test data for prediction 
+% scores: the perdiction scores. 
+
+N = size(X,1);
+%TO DO: calculate the predictions
+
+[tst_size, ~] = size(tst_data);
+scores = zeros(tst_size, 1);
+for t = 1 : tst_size
+  xt = tst_data(t, :);
+
+  sum = 0;
+  for i = 1 : N
+    ai = alphas(i);
+    yi = Y(i);
+    xi = X(i, :);
+    sum = sum + ai * yi * dot(xi, xt) ^ d;
+  end
+  sum = sum + b;
+
+  scores(t) = sign(sum);
+
+end
+
+
+%********************************