Add files via upload

costFunction.m

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+function [J, grad] = costFunction(theta, X, y)
+%COSTFUNCTION Compute cost and gradient for logistic regression
+%   J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the
+%   parameter for logistic regression and the gradient of the cost
+%   w.r.t. to the parameters.
+
+% Initialize some useful values
+m = length(y); % number of training examples
+
+% You need to return the following variables correctly 
+J = 0;
+grad = zeros(size(theta));
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the cost of a particular choice of theta.
+%               You should set J to the cost.
+%               Compute the partial derivatives and set grad to the partial
+%               derivatives of the cost w.r.t. each parameter in theta
+%
+% Note: grad should have the same dimensions as theta
+%
+
+J = mean((-y).* log(sigmoid(X*theta))- (1-y).* log(1 - sigmoid(X*theta)));
+
+grad = 1/m * X' * (sigmoid(X*theta) - y);
+
+
+
+
+% =============================================================
+
+end

costFunctionReg.m

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+function [J, grad] = costFunctionReg(theta, X, y, lambda)
+%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
+%   J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using
+%   theta as the parameter for regularized logistic regression and the
+%   gradient of the cost w.r.t. to the parameters. 
+
+% Initialize some useful values
+m = length(y); % number of training examples
+
+% You need to return the following variables correctly 
+J = 0;
+grad = zeros(size(theta));
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the cost of a particular choice of theta.
+%               You should set J to the cost.
+%               Compute the partial derivatives and set grad to the partial
+%               derivatives of the cost w.r.t. each parameter in theta
+
+theta0 = [0; theta(2:end)];
+
+J = mean((-y).* log(sigmoid(X*theta))- (1-y).* log(1 - sigmoid(X*theta)))...
+    + lambda/(2*m)*sum(theta0.*theta0);
+
+
+
+grad = 1/m * X' * (sigmoid(X*theta) - y) + lambda/m*theta0;
+
+
+
+
+
+% =============================================================
+
+end

ex2.m

@@ -0,0 +1,151 @@
+%% Machine Learning Online Class - Exercise 2: Logistic Regression
+%
+%  Instructions
+%  ------------
+% 
+%  This file contains code that helps you get started on the logistic
+%  regression exercise. You will need to complete the following functions 
+%  in this exericse:
+%
+%     sigmoid.m
+%     costFunction.m
+%     predict.m
+%     costFunctionReg.m
+%
+%  For this exercise, you will not need to change any code in this file,
+%  or any other files other than those mentioned above.
+%
+
+%% Initialization
+clear ; close all; clc
+
+%% Load Data
+%  The first two columns contains the exam scores and the third column
+%  contains the label.
+
+data = load('ex2data1.txt');
+X = data(:, [1, 2]); y = data(:, 3);
+
+%% ==================== Part 1: Plotting ====================
+%  We start the exercise by first plotting the data to understand the 
+%  the problem we are working with.
+
+fprintf(['Plotting data with + indicating (y = 1) examples and o ' ...
+         'indicating (y = 0) examples.\n']);
+
+plotData(X, y);
+
+% Put some labels 
+hold on;
+% Labels and Legend
+xlabel('Exam 1 score')
+ylabel('Exam 2 score')
+
+% Specified in plot order
+legend('Admitted', 'Not admitted')
+hold off;
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ============ Part 2: Compute Cost and Gradient ============
+%  In this part of the exercise, you will implement the cost and gradient
+%  for logistic regression. You neeed to complete the code in 
+%  costFunction.m
+
+%  Setup the data matrix appropriately, and add ones for the intercept term
+[m, n] = size(X);
+
+% Add intercept term to x and X_test
+X = [ones(m, 1) X];
+
+% Initialize fitting parameters
+initial_theta = zeros(n + 1, 1);
+
+% Compute and display initial cost and gradient
+[cost, grad] = costFunction(initial_theta, X, y);
+
+fprintf('Cost at initial theta (zeros): %f\n', cost);
+fprintf('Expected cost (approx): 0.693\n');
+fprintf('Gradient at initial theta (zeros): \n');
+fprintf(' %f \n', grad);
+fprintf('Expected gradients (approx):\n -0.1000\n -12.0092\n -11.2628\n');
+
+% Compute and display cost and gradient with non-zero theta
+test_theta = [-24; 0.2; 0.2];
+[cost, grad] = costFunction(test_theta, X, y);
+
+fprintf('\nCost at test theta: %f\n', cost);
+fprintf('Expected cost (approx): 0.218\n');
+fprintf('Gradient at test theta: \n');
+fprintf(' %f \n', grad);
+fprintf('Expected gradients (approx):\n 0.043\n 2.566\n 2.647\n');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ============= Part 3: Optimizing using fminunc  =============
+%  In this exercise, you will use a built-in function (fminunc) to find the
+%  optimal parameters theta.
+
+%  Set options for fminunc
+options = optimset('GradObj', 'on', 'MaxIter', 400);
+
+%  Run fminunc to obtain the optimal theta
+%  This function will return theta and the cost 
+[theta, cost] = ...
+	fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);
+
+% Print theta to screen
+fprintf('Cost at theta found by fminunc: %f\n', cost);
+fprintf('Expected cost (approx): 0.203\n');
+fprintf('theta: \n');
+fprintf(' %f \n', theta);
+fprintf('Expected theta (approx):\n');
+fprintf(' -25.161\n 0.206\n 0.201\n');
+
+% Plot Boundary
+plotDecisionBoundary(theta, X, y);
+
+% Put some labels 
+hold on;
+% Labels and Legend
+xlabel('Exam 1 score')
+ylabel('Exam 2 score')
+
+% Specified in plot order
+legend('Admitted', 'Not admitted')
+hold off;
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+%% ============== Part 4: Predict and Accuracies ==============
+%  After learning the parameters, you'll like to use it to predict the outcomes
+%  on unseen data. In this part, you will use the logistic regression model
+%  to predict the probability that a student with score 45 on exam 1 and 
+%  score 85 on exam 2 will be admitted.
+%
+%  Furthermore, you will compute the training and test set accuracies of 
+%  our model.
+%
+%  Your task is to complete the code in predict.m
+
+%  Predict probability for a student with score 45 on exam 1 
+%  and score 85 on exam 2 
+
+prob = sigmoid([1 45 85] * theta);
+fprintf(['For a student with scores 45 and 85, we predict an admission ' ...
+         'probability of %f\n'], prob);
+fprintf('Expected value: 0.775 +/- 0.002\n\n');
+
+% Compute accuracy on our training set
+p = predict(theta, X);
+
+fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);
+fprintf('Expected accuracy (approx): 89.0\n');
+fprintf('\n');
+
+

ex2_reg.m

@@ -0,0 +1,136 @@
+%% Machine Learning Online Class - Exercise 2: Logistic Regression
+%
+%  Instructions
+%  ------------
+%
+%  This file contains code that helps you get started on the second part
+%  of the exercise which covers regularization with logistic regression.
+%
+%  You will need to complete the following functions in this exericse:
+%
+%     sigmoid.m
+%     costFunction.m
+%     predict.m
+%     costFunctionReg.m
+%
+%  For this exercise, you will not need to change any code in this file,
+%  or any other files other than those mentioned above.
+%
+
+%% Initialization
+clear ; close all; clc
+
+%% Load Data
+%  The first two columns contains the X values and the third column
+%  contains the label (y).
+
+data = load('ex2data2.txt');
+X = data(:, [1, 2]); y = data(:, 3);
+
+plotData(X, y);
+
+% Put some labels
+hold on;
+
+% Labels and Legend
+xlabel('Microchip Test 1')
+ylabel('Microchip Test 2')
+
+% Specified in plot order
+legend('y = 1', 'y = 0')
+hold off;
+
+
+%% =========== Part 1: Regularized Logistic Regression ============
+%  In this part, you are given a dataset with data points that are not
+%  linearly separable. However, you would still like to use logistic
+%  regression to classify the data points.
+%
+%  To do so, you introduce more features to use -- in particular, you add
+%  polynomial features to our data matrix (similar to polynomial
+%  regression).
+%
+
+% Add Polynomial Features
+
+% Note that mapFeature also adds a column of ones for us, so the intercept
+% term is handled
+X = mapFeature(X(:,1), X(:,2));
+
+% Initialize fitting parameters
+initial_theta = zeros(size(X, 2), 1);
+
+% Set regularization parameter lambda to 1
+lambda = 1;
+
+% Compute and display initial cost and gradient for regularized logistic
+% regression
+[cost, grad] = costFunctionReg(initial_theta, X, y, lambda);
+
+fprintf('Cost at initial theta (zeros): %f\n', cost);
+fprintf('Expected cost (approx): 0.693\n');
+fprintf('Gradient at initial theta (zeros) - first five values only:\n');
+fprintf(' %f \n', grad(1:5));
+fprintf('Expected gradients (approx) - first five values only:\n');
+fprintf(' 0.0085\n 0.0188\n 0.0001\n 0.0503\n 0.0115\n');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+% Compute and display cost and gradient
+% with all-ones theta and lambda = 10
+test_theta = ones(size(X,2),1);
+[cost, grad] = costFunctionReg(test_theta, X, y, 10);
+
+fprintf('\nCost at test theta (with lambda = 10): %f\n', cost);
+fprintf('Expected cost (approx): 3.16\n');
+fprintf('Gradient at test theta - first five values only:\n');
+fprintf(' %f \n', grad(1:5));
+fprintf('Expected gradients (approx) - first five values only:\n');
+fprintf(' 0.3460\n 0.1614\n 0.1948\n 0.2269\n 0.0922\n');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+%% ============= Part 2: Regularization and Accuracies =============
+%  Optional Exercise:
+%  In this part, you will get to try different values of lambda and
+%  see how regularization affects the decision coundart
+%
+%  Try the following values of lambda (0, 1, 10, 100).
+%
+%  How does the decision boundary change when you vary lambda? How does
+%  the training set accuracy vary?
+%
+
+% Initialize fitting parameters
+initial_theta = zeros(size(X, 2), 1);
+
+% Set regularization parameter lambda to 1 (you should vary this)
+lambda = 1;
+
+% Set Options
+options = optimset('GradObj', 'on', 'MaxIter', 400);
+
+% Optimize
+[theta, J, exit_flag] = ...
+	fminunc(@(t)(costFunctionReg(t, X, y, lambda)), initial_theta, options);
+
+% Plot Boundary
+plotDecisionBoundary(theta, X, y);
+hold on;
+title(sprintf('lambda = %g', lambda))
+
+% Labels and Legend
+xlabel('Microchip Test 1')
+ylabel('Microchip Test 2')
+
+legend('y = 1', 'y = 0', 'Decision boundary')
+hold off;
+
+% Compute accuracy on our training set
+p = predict(theta, X);
+
+fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);
+fprintf('Expected accuracy (with lambda = 1): 83.1 (approx)\n');
+

ex2data1.txt

@@ -0,0 +1,100 @@
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ex2data2.txt

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mapFeature.m

@@ -0,0 +1,21 @@
+function out = mapFeature(X1, X2)
+% MAPFEATURE Feature mapping function to polynomial features
+%
+%   MAPFEATURE(X1, X2) maps the two input features
+%   to quadratic features used in the regularization exercise.
+%
+%   Returns a new feature array with more features, comprising of 
+%   X1, X2, X1.^2, X2.^2, X1*X2, X1*X2.^2, etc..
+%
+%   Inputs X1, X2 must be the same size
+%
+
+degree = 6;
+out = ones(size(X1(:,1)));
+for i = 1:degree
+    for j = 0:i
+        out(:, end+1) = (X1.^(i-j)).*(X2.^j);
+    end
+end
+
+end

plotData.m

@@ -0,0 +1,33 @@
+function plotData(X, y)
+%PLOTDATA Plots the data points X and y into a new figure 
+%   PLOTDATA(x,y) plots the data points with + for the positive examples
+%   and o for the negative examples. X is assumed to be a Mx2 matrix.
+
+% Create New Figure
+figure; hold on;
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Plot the positive and negative examples on a
+%               2D plot, using the option 'k+' for the positive
+%               examples and 'ko' for the negative examples.
+%
+
+pos = find(y==1);
+neg = find(y==0);
+
+plot(X(pos, 1), X(pos, 2), 'K+', 'LineWidth', 2, 'MarkerSize', 7);
+plot(X(neg, 1), X(neg, 2), 'ro', 'MarkerSize', 7)
+
+
+
+
+
+
+
+% =========================================================================
+
+
+
+hold off;
+
+end

plotDecisionBoundary.m

@@ -0,0 +1,48 @@
+function plotDecisionBoundary(theta, X, y)
+%PLOTDECISIONBOUNDARY Plots the data points X and y into a new figure with
+%the decision boundary defined by theta
+%   PLOTDECISIONBOUNDARY(theta, X,y) plots the data points with + for the 
+%   positive examples and o for the negative examples. X is assumed to be 
+%   a either 
+%   1) Mx3 matrix, where the first column is an all-ones column for the 
+%      intercept.
+%   2) MxN, N>3 matrix, where the first column is all-ones
+
+% Plot Data
+plotData(X(:,2:3), y);
+hold on
+
+if size(X, 2) <= 3
+    % Only need 2 points to define a line, so choose two endpoints
+    plot_x = [min(X(:,2))-2,  max(X(:,2))+2];
+
+    % Calculate the decision boundary line
+    plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));
+
+    % Plot, and adjust axes for better viewing
+    plot(plot_x, plot_y)
+
+    % Legend, specific for the exercise
+    legend('Admitted', 'Not admitted', 'Decision Boundary')
+    axis([30, 100, 30, 100])
+else
+    % Here is the grid range
+    u = linspace(-1, 1.5, 50);
+    v = linspace(-1, 1.5, 50);
+
+    z = zeros(length(u), length(v));
+    % Evaluate z = theta*x over the grid
+    for i = 1:length(u)
+        for j = 1:length(v)
+            z(i,j) = mapFeature(u(i), v(j))*theta;
+        end
+    end
+    z = z'; % important to transpose z before calling contour
+
+    % Plot z = 0
+    % Notice you need to specify the range [0, 0]
+    contour(u, v, z, [0, 0], 'LineWidth', 2)
+end
+hold off
+
+end

predict.m

@@ -0,0 +1,33 @@
+function p = predict(theta, X)
+%PREDICT Predict whether the label is 0 or 1 using learned logistic 
+%regression parameters theta
+%   p = PREDICT(theta, X) computes the predictions for X using a 
+%   threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)
+
+m = size(X, 1); % Number of training examples
+
+% You need to return the following variables correctly
+p = zeros(m, 1);
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Complete the following code to make predictions using
+%               your learned logistic regression parameters. 
+%               You should set p to a vector of 0's and 1's
+%
+y = sigmoid(X*theta);
+
+pos = y>=0.5;
+neg = y<0.5;
+
+p(pos) = 1;
+p(neg) = 0;
+
+
+
+
+
+
+% =========================================================================
+
+
+end

sigmoid.m

@@ -0,0 +1,18 @@
+function g = sigmoid(z)
+%SIGMOID Compute sigmoid function
+%   g = SIGMOID(z) computes the sigmoid of z.
+
+% You need to return the following variables correctly 
+g = zeros(size(z));
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the sigmoid of each value of z (z can be a matrix,
+%               vector or scalar).
+
+
+g = ones(size(z))./(1 + exp(-z));
+
+
+% =============================================================
+
+end