一、程序及函数
1.引导脚本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];
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
% 下面的X_poly已经被标准化并且添加了第一列全1
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);
% 下面的X_poly_test已经被标准化并且添加了第一列全1
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);
% 下面的X_poly_val已经被标准化并且添加了第一列全1
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;
2.核心函数 linearRegCostFunction.m
该函数计算了带有正则化项的损失函数J值以及J对各个参数的偏导值。
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
n = length(theta); % number of thetas
% You need to return the following variables correctly
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.
% 初始化累加和
sum = 0;
sum_theta = 0;
sum_grad = zeros(n,1);
for i = 1 : m
sum = sum + (X(i,:) * theta - y(i)).^2;
for j = 1 : n
sum_grad(j) = sum_grad(j) + ((X(i,:) * theta - y(i)) .* X(i,j));
end
end
for j = 2 : n
sum_theta = sum_theta + theta(j).^2;
end
% 计算J值
J = 1 / (2 * m) * sum + lambda / (2 * m) * sum_theta;
% 计算梯度值
grad(1) = 1 / m * sum_grad(1);
grad(2:end) = 1 / m * sum_grad(2:end) + lambda / m * theta(2:end);
% ========================================================================
grad = grad(:);
end
3.trainLinearReg.m
该函数的功能是利用Matlab自带的优化函数训练参数，最后返回最优的theta值。
function [theta] = trainLinearReg(X, y, lambda)
%TRAINLINEARREG Trains linear regression given a dataset (X, y) and a
%regularization parameter lambda
% [theta] = TRAINLINEARREG (X, y, lambda) trains linear regression using
% the dataset (X, y) and regularization parameter lambda. Returns the
% trained parameters theta.
%
% Initialize Theta
initial_theta = zeros(size(X, 2), 1);
% Create "short hand" for the cost function to be minimized
costFunction = @(t) linearRegCostFunction(X, y, t, lambda);
% Now, costFunction is a function that takes in only one argument
options = optimset('MaxIter', 200, 'GradObj', 'on');
% Minimize using fmincg
theta = fmincg(costFunction, initial_theta, options);
end
4.learningCurve.m
给出绘制训练集&交叉验证集的误差随训练集大小而改变的曲线所需要的数值。
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);
% You need to return these values correctly
error_train = zeros(m, 1);
error_val = zeros(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
% 先训练得出theta向量的值
[theta] = trainLinearReg(X(1:i, :), y(1:i), lambda);
% 然后再计算当前训练集和整个验证集的误差
error_train(i) = linearRegCostFunction(X(1:i, :), y(1:i), theta, 0);
error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
% -------------------------------------------------------------
% =======================================================================
end
5.polyFeatures.m
为了得到更好的拟合曲线，我们需要用多项式回归（而不是简单的一次线性回归）。所以我们需要首先把特征X扩展为多维的特征矩阵。
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);
% ====================== 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 = length(X);
for i = 1 : m
for j = 1 : p
X_poly(i,j) = X(i,1).^j;
end
end
% =========================================================================
end
6.validationCurve.m
为了绘制出训练集&交叉验证集的误差随着lambda而改变的曲线，我们首先要计算不同lambda下的两个集的误差。
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
m = size(X, 1);
for i = 1 : length(lambda_vec)
lambda = lambda_vec(i);
% 先训练得出theta向量的值
% 这里的X都已经加上了第一列全1
[theta] = trainLinearReg(X, y, lambda);
error_train(i) = linearRegCostFunction(X, y, theta, 0);
error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
% =========================================================================
end
其他函数都是Andrew Ng已经帮我们写好了的，相对不那么重要，就不贴上来了。
二、运行结果
到当训练样本数量增加时，训练误差和交叉验证误差都很高。这反映了模型中的一个高偏差问题(High Bias)——线性回归模型太简单以至于无法很好地适应训练集，即产生了欠拟合(Underfitting)的问题。
多项式回归的学习曲线：
观察曲线我们可以发现，训练误差一直很低但是交叉验证误差比训练误差要大很多，即训练误差和交叉验证误差之间存在较大差距。这表明当前的多项式回归模型存在高方差(High Variance)问题，也即模型存在过拟合(Overfitting)的问题。
一个好的lambda值要在减小J值和防止模型出现过拟合之间达到一个较好的平衡状态（这取决于实际需求）。在本问题中我们可以发现当lambda = 3时，验证集上的误差最小，说明lambda = 3是一个比较理想的值。
