一、ex5.m
%% 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]; % 2*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" -- slide 8 in ML-advice.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;
二、linearRegCostFunction.m
function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear
%regression with multiple variables
% [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the
% cost of using theta as the parameter for linear regression to fit the
% data points in X and y. Returns the cost in J and the gradient in grad
% Initialize some useful values
m = length(y); % number of training examples % m
% You need to return the following variables correctly
J = 0; % 1*1
grad = zeros(size(theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost and gradient of regularized linear
% regression for a particular choice of theta.
%
% You should set J to the cost and grad to the gradient.
%
% linearReg
h = X * theta; % m*(n+1) × (n+1)* -> m*
J_part1 = sum((h-y).^2) /2 /m;
theta2 = theta(2:end, :);
J_part2 = sum(theta2.^2)/2/m*lambda;
J = J_part1 + J_part2;
% gradient
grad_ori = X' * (h-y) / m; %(n+1)*m × m*1 -> (n+1)*1
grad(1) = grad_ori(1);
grad(2:end) = grad_ori(2:end) + lambda/m*theta(2:end);
% =========================================================================
grad = grad(:);
end
三、learningCurve.m
function [error_train, error_val] = ...
learningCurve(X, y, Xval, yval, lambda)
%LEARNINGCURVE Generates the train and cross validation set errors needed
%to plot a learning curve
% [error_train, error_val] = ...
% LEARNINGCURVE(X, y, Xval, yval, lambda) returns the train and
% cross validation set errors for a learning curve. In particular,
% it returns two vectors of the same length - error_train and
% error_val. Then, error_train(i) contains the training error for
% i examples (and similarly for error_val(i)).
%
% In this function, you will compute the train and test errors for
% dataset sizes from 1 up to m. In practice, when working with larger
% datasets, you might want to do this in larger intervals.
%
% Number of training examples
m = size(X, 1); % m
% You need to return these values correctly
error_train = zeros(m, 1); % m*1
error_val = zeros(m, 1); % m*1
% ====================== YOUR CODE HERE ======================
% Instructions: Fill in this function to return training errors in
% error_train and the cross validation errors in error_val.
% i.e., error_train(i) and
% error_val(i) should give you the errors
% obtained after training on i examples.
%
% Note: You should evaluate the training error on the first i training
% examples (i.e., X(1:i, :) and y(1:i)).
%
% For the cross-validation error, you should instead evaluate on
% the _entire_ cross validation set (Xval and yval).
%
% Note: If you are using your cost function (linearRegCostFunction)
% to compute the training and cross validation error, you should
% call the function with the lambda argument set to 0.
% Do note that you will still need to use lambda when running
% the training to obtain the theta parameters.
%
% Hint: You can loop over the examples with the following:
%
% for i = 1:m
% % Compute train/cross validation errors using training examples
% % X(1:i, :) and y(1:i), storing the result in
% % error_train(i) and error_val(i)
% ....
%
% end
%
% ---------------------- Sample Solution ----------------------
for i = 1:m
X_train = X(1:i, :);
y_train = y(1:i);
[theta] = trainLinearReg(X_train, y_train, lambda);
lambda0 = 0;
[J_train grad_train] = linearRegCostFunction(X_train, y_train, theta, lambda0);
[J_val grad_val] = linearRegCostFunction(Xval, yval, theta, lambda0);
error_train(i) = J_train;
error_val(i) = J_val;
end
% -------------------------------------------------------------
% =========================================================================
end
四、validationCurve.m
function [lambda_vec, error_train, error_val] = ...
validationCurve(X, y, Xval, yval)
%VALIDATIONCURVE Generate the train and validation errors needed to
%plot a validation curve that we can use to select lambda
% [lambda_vec, error_train, error_val] = ...
% VALIDATIONCURVE(X, y, Xval, yval) returns the train
% and validation errors (in error_train, error_val)
% for different values of lambda. You are given the training set (X,
% y) and validation set (Xval, yval).
%
% Selected values of lambda (you should not change this)
lambda_vec = [0 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10]';
% You need to return these variables correctly.
error_train = zeros(length(lambda_vec), 1);
error_val = zeros(length(lambda_vec), 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Fill in this function to return training errors in
% error_train and the validation errors in error_val. The
% vector lambda_vec contains the different lambda parameters
% to use for each calculation of the errors, i.e,
% error_train(i), and error_val(i) should give
% you the errors obtained after training with
% lambda = lambda_vec(i)
%
% Note: You can loop over lambda_vec with the following:
%
% for i = 1:length(lambda_vec)
% lambda = lambda_vec(i);
% % Compute train / val errors when training linear
% % regression with regularization parameter lambda
% % You should store the result in error_train(i)
% % and error_val(i)
% ....
%
% end
%
%
for i = 1:length(lambda_vec)
lambda = lambda_vec(i);
[theta] = trainLinearReg(X, y, lambda);
lambda0 = 0;
[J_train grad_train] = linearRegCostFunction(X, y, theta, lambda0);
[J_val grad_val] = linearRegCostFunction(Xval, yval, theta, lambda0);
error_train(i) = J_train;
error_val(i) = J_val;
end
% =========================================================================
end
五、polyFeatures.m
function [X_poly] = polyFeatures(X, p)
%POLYFEATURES Maps X (1D vector) into the p-th power
% [X_poly] = POLYFEATURES(X, p) takes a data matrix X (size m x 1) and
% maps each example into its polynomial features where
% X_poly(i, :) = [X(i) X(i).^2 X(i).^3 ... X(i).^p];
%
% You need to return the following variables correctly.
X_poly = zeros(numel(X), p); % m*p
% ====================== YOUR CODE HERE ======================
% Instructions: Given a vector X, return a matrix X_poly where the p-th
% column of X contains the values of X to the p-th power.
%
%
m = numel(X);
for i = 1:m
for j = 1:p
X_poly(i, j) = X(i).^j;
end
end
% =========================================================================
end
六、submit results
