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- %% Machine Learning Online Class
- % Exercise 5 | Regularized Linear Regression and Bias-Variance
- %
- % Instructions
- % ------------
- %
- % This file contains code that helps you get started on the
- % exercise. You will need to complete the following functions:
- %
- % linearRegCostFunction.m
- % learningCurve.m
- % validationCurve.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
- %% =========== Part 1: Loading and Visualizing Data =============
- % We start the exercise by first loading and visualizing the dataset.
- % The following code will load the dataset into your environment and plot
- % the data.
- %
- % Load Training Data
- fprintf('Loading and Visualizing Data ...\n')
- % Load from ex5data1:
- % You will have X, y, Xval, yval, Xtest, ytest in your environment
- load ('ex5data1.mat');
- % m = Number of examples
- m = size(X, 1);
- % Plot training data
- plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
- xlabel('Change in water level (x)');
- ylabel('Water flowing out of the dam (y)');
- fprintf('Program paused. Press enter to continue.\n');
- pause;
- %% =========== Part 2: Regularized Linear Regression Cost =============
- % You should now implement the cost function for regularized linear
- % regression.
- %
- theta = [1 ; 1];
- J = linearRegCostFunction([ones(m, 1) X], y, theta, 1);
- fprintf(['Cost at theta = [1 ; 1]: %f '...
- '\n(this value should be about 303.993192)\n'], J);
- fprintf('Program paused. Press enter to continue.\n');
- pause;
- %% =========== Part 3: Regularized Linear Regression Gradient =============
- % You should now implement the gradient for regularized linear
- % regression.
- %
- theta = [1 ; 1];
- [J, grad] = linearRegCostFunction([ones(m, 1) X], y, theta, 1);
- fprintf(['Gradient at theta = [1 ; 1]: [%f; %f] '...
- '\n(this value should be about [-15.303016; 598.250744])\n'], ...
- grad(1), grad(2));
- fprintf('Program paused. Press enter to continue.\n');
- pause;
- %% =========== Part 4: Train Linear Regression =============
- % Once you have implemented the cost and gradient correctly, the
- % trainLinearReg function will use your cost function to train
- % regularized linear regression.
- %
- % Write Up Note: The data is non-linear, so this will not give a great
- % fit.
- %
- % Train linear regression with lambda = 0
- lambda = 0;
- [theta] = trainLinearReg([ones(m, 1) X], y, lambda);
- % Plot fit over the data
- plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
- xlabel('Change in water level (x)');
- ylabel('Water flowing out of the dam (y)');
- hold on;
- plot(X, [ones(m, 1) X]*theta, '--', 'LineWidth', 2)
- hold off;
- fprintf('Program paused. Press enter to continue.\n');
- pause;
- %% =========== Part 5: Learning Curve for Linear Regression =============
- % Next, you should implement the learningCurve function.
- %
- % Write Up Note: Since the model is underfitting the data, we expect to
- % see a graph with "high bias" -- Figure 3 in ex5.pdf
- %
- lambda = 0;
- [error_train, error_val] = ...
- learningCurve([ones(m, 1) X], y, ...
- [ones(size(Xval, 1), 1) Xval], yval, ...
- lambda);
- plot(1:m, error_train, 1:m, error_val);
- title('Learning curve for linear regression')
- legend('Train', 'Cross Validation')
- xlabel('Number of training examples')
- ylabel('Error')
- axis([0 13 0 150])
- fprintf('# Training Examples\tTrain Error\tCross Validation Error\n');
- for i = 1:m
- fprintf(' \t%d\t\t%f\t%f\n', i, error_train(i), error_val(i));
- end
- fprintf('Program paused. Press enter to continue.\n');
- pause;
- %% =========== Part 6: Feature Mapping for Polynomial Regression =============
- % One solution to this is to use polynomial regression. You should now
- % complete polyFeatures to map each example into its powers
- %
- p = 8;
- % Map X onto Polynomial Features and Normalize
- X_poly = polyFeatures(X, p);
- [X_poly, mu, sigma] = featureNormalize(X_poly); % Normalize
- X_poly = [ones(m, 1), X_poly]; % Add Ones
- % Map X_poly_test and normalize (using mu and sigma)
- X_poly_test = polyFeatures(Xtest, p);
- X_poly_test = bsxfun(@minus, X_poly_test, mu);
- X_poly_test = bsxfun(@rdivide, X_poly_test, sigma);
- X_poly_test = [ones(size(X_poly_test, 1), 1), X_poly_test]; % Add Ones
- % Map X_poly_val and normalize (using mu and sigma)
- X_poly_val = polyFeatures(Xval, p);
- X_poly_val = bsxfun(@minus, X_poly_val, mu);
- X_poly_val = bsxfun(@rdivide, X_poly_val, sigma);
- X_poly_val = [ones(size(X_poly_val, 1), 1), X_poly_val]; % Add Ones
- fprintf('Normalized Training Example 1:\n');
- fprintf(' %f \n', X_poly(1, :));
- fprintf('\nProgram paused. Press enter to continue.\n');
- pause;
- %% =========== Part 7: Learning Curve for Polynomial Regression =============
- % Now, you will get to experiment with polynomial regression with multiple
- % values of lambda. The code below runs polynomial regression with
- % lambda = 0. You should try running the code with different values of
- % lambda to see how the fit and learning curve change.
- %
- lambda = 0;
- [theta] = trainLinearReg(X_poly, y, lambda);
- % Plot training data and fit
- figure(1);
- plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
- plotFit(min(X), max(X), mu, sigma, theta, p);
- xlabel('Change in water level (x)');
- ylabel('Water flowing out of the dam (y)');
- title (sprintf('Polynomial Regression Fit (lambda = %f)', lambda));
- figure(2);
- [error_train, error_val] = ...
- learningCurve(X_poly, y, X_poly_val, yval, lambda);
- plot(1:m, error_train, 1:m, error_val);
- title(sprintf('Polynomial Regression Learning Curve (lambda = %f)', lambda));
- xlabel('Number of training examples')
- ylabel('Error')
- axis([0 13 0 100])
- legend('Train', 'Cross Validation')
- fprintf('Polynomial Regression (lambda = %f)\n\n', lambda);
- fprintf('# Training Examples\tTrain Error\tCross Validation Error\n');
- for i = 1:m
- fprintf(' \t%d\t\t%f\t%f\n', i, error_train(i), error_val(i));
- end
- fprintf('Program paused. Press enter to continue.\n');
- pause;
- %% =========== Part 8: Validation for Selecting Lambda =============
- % You will now implement validationCurve to test various values of
- % lambda on a validation set. You will then use this to select the
- % "best" lambda value.
- %
- [lambda_vec, error_train, error_val] = ...
- validationCurve(X_poly, y, X_poly_val, yval);
- close all;
- plot(lambda_vec, error_train, lambda_vec, error_val);
- legend('Train', 'Cross Validation');
- xlabel('lambda');
- ylabel('Error');
- fprintf('lambda\t\tTrain Error\tValidation Error\n');
- for i = 1:length(lambda_vec)
- fprintf(' %f\t%f\t%f\n', ...
- lambda_vec(i), error_train(i), error_val(i));
- end
- fprintf('Program paused. Press enter to continue.\n');
- pause;
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