Coursera-Machine Learning-Andrew Ng-Programming Exercise 5

【Exercise 5 Regularized Linear Regression and Bias v.s. Variance】

【代码】

ex5.m

-> 加载数据(上升曲线)可视化
-> 实现成本 实现梯度

不进行特征匹配:(只用原始的一维x)
-> 训练,绘制结果
-> 画学习曲线:样本个数-误差关系

进行特征匹配(p维x,有高次特征)
-> 特征匹配
-> 特征匹配产生了“多个”特征,故归一化、并输出均值标准差、
   cv集 测试集需要做同样处理,但是归一化要减去训练集的均值、除以训练集的标准差
-> 训练,绘制结果
-> 画学习曲线:样本个数-误差关系

-> 误差-λ曲线

不计分部分:
-> 选取cv集上表现最优的λ,利用这个值训练,计算测试集误差

-> 作图显示训练集、预测、测试集关系

note:原始数据没有x_0,不论匹配不匹配特征,学习前都要加全1列(预测前也要)

%% 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.


% 计时开始(画平滑学习曲线耗时较大)
tic
% 原始取0,平滑时为更好效果lambda取0.01
lambda = 0.01;
%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


% 计时结束
toc


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;


%% =========== Part 9: Ungraded Exercises =============


% Train using the lambda of the best performance
[~,lambda_index]=min(error_val);
lambda=lambda_vec(lambda_index);
[theta] = trainLinearReg(X_poly, y, lambda);


% Compute the test error
[error_test,~] = linearRegCostFunction(X_poly_test, ytest, theta, 0); 
% Notice that lambda is assigned 0, according to the formula
fprintf('(UNGRADED PART)\nTest error:%.4f for lambda=%f\n',error_test,lambda);


% Plotting
plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
hold on
plot(Xtest, ytest, 'yx', 'MarkerSize', 10, 'LineWidth', 1.5);
plotFit(min(X), max(X), mu, sigma, theta, p);


legend('Train','Test','Fitting')


xlabel('Change in water level (x)');
ylabel('Water flowing out of the dam (y)');
title (sprintf('Polynomial Regression Fit (lambda = %f)', lambda));


linearRegCostFunction.m

套公式,注意向量化,注意θ_0特例

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

% You need to return the following variables correctly 
J = 0;
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.
%
%
    J=1/2/m*sum((X*theta-y).^2);
    J=J+lambda/2/m*sum(theta(2:end).^2);
    grad=1/m*(X'*(X*theta-y));
    grad(2:end)=grad(2:end)+lambda/m*theta(2:end);
% =========================================================================

grad = grad(:);

end

trainLinearReg.m

全0初始化,把成本+梯度函数、option送入求解器。

Coursera-Machine Learning-Andrew Ng-Programming Exercise 5_第1张图片

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

learningCurve.m【基础】

for循环,从只用1个样本、只用2个样本到全部m个样本,各情况下分别训练,再计算训练集误差、cv集误差。

计算误差可以列公式,也可以调用已经实现好的linearRegCostFunction函数。

注意:1、列公式或调用函数时,有关位置要改成前i个而非全部m个。尤其注意1/2m。

2、cv集上的误差总是对全体cv集来说,不受i的影响

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
        [theta] = trainLinearReg(X(1:i,:), y(1:i), lambda);
        error_train(i) = 1/2/i*sum((X(1:i,:)*theta-y(1:i)).^2);
        error_val(i) = 1/2/size(Xval,1)*sum((Xval*theta-yval).^2);
             
        [error_train(i),~] = linearRegCostFunction(X(1:i,:), y(1:i), theta, 0);
        [error_val(i),~] = linearRegCostFunction(Xval, yval, theta, 0); 
        % 注意lambda=0,根据定义,误差值没有正则项;
        % 外层传进来的lambda是用来训练参数theta的
    end
% -------------------------------------------------------------

% =========================================================================

end

learningCurve.m【平滑】

原始版本简单地取前i个样本;这里对每个i重复进行若干次计算,每次计算重新随机取i个样本,最后重复计算的误差取平均,作为这个i值的误差。

第一层循环同前为1到m。第二层循环为重复计算。随机取i个样本:

            random_index=randperm(m);

            random_index=random_index(1:i);  

            Xtrain=X(random_index,:);

            ytrain=y(random_index);

代码内部设置了变量smooth,手动修改其为0时,实现功能与基础版本相同;修改为1时,作平滑的学习曲线,无法Coursera通过在线测试

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 ----------------------


%说明:以下为绘制平滑learning curve有关变量,为使其它部分能正常通过在线测
%      试,仅采用手动修改参数方法简单演示。
% - smooth变量设置为0,即不进行平滑,可正常提交通过测试;
% - smooth变量设置为1,即进行平滑。lambda_smooth设置正则化系数。
%   repeat_times为重复计算次数,_temp变量存储重复计算误差数据。
%   (ex5.m中传入的lambda应做相应修改,以得到更好效果)
% - 绘制多项式的平滑learning curve需要接近6分钟
    smooth=0;
    repeat_times=50;
    error_train_temp=zeros(repeat_times,1);
    error_val_temp=zeros(repeat_times,1);
%-------------------------------------------------------------------------
%设置smooth=0,不进行平滑,可通过在线测试
if smooth==0
    for i=1:m
        [theta] = trainLinearReg(X(1:i,:), y(1:i), lambda);
        error_train(i) = 1/2/i*sum((X(1:i,:)*theta-y(1:i)).^2);
        error_val(i) = 1/2/size(Xval,1)*sum((Xval*theta-yval).^2);
        
        [error_train(i),~] = linearRegCostFunction(X(1:i,:), y(1:i), theta, 0);
        [error_val(i),~] = linearRegCostFunction(Xval, yval, theta, 0); 
        % 注意lambda=0,根据定义,误差值没有正则项;
        % 外层传进来的lambda是用来训练参数theta的
    end
%设置smooth=1,进行平滑,
else
    for i=1:m
        for j=1:repeat_times
            random_index=randperm(m);
            random_index=random_index(1:i);          
            Xtrain=X(random_index,:);
            ytrain=y(random_index);
            [theta] = trainLinearReg(Xtrain, ytrain, lambda);
            [error_train_temp(j),~] = linearRegCostFunction(Xtrain, ytrain, theta, 0);
            [error_val_temp(j),~] = linearRegCostFunction(Xval, yval, theta, 0); 
        end
        error_train(i)=mean(error_train_temp);
        error_val(i)=mean(error_val_temp);
    end
end


% -------------------------------------------------------------


% =========================================================================


end

取平均达到平滑效果:

Coursera-Machine Learning-Andrew Ng-Programming Exercise 5_第2张图片

Coursera-Machine Learning-Andrew Ng-Programming Exercise 5_第3张图片


polyFeature.m

特征映射feature mapping

Coursera-Machine Learning-Andrew Ng-Programming Exercise 5_第4张图片

原本每个样本只有一个特征x,现加入其幂形成更多特征,可以用来拟合更复杂的曲线

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.
%
% 
    for i=1:p
        X_poly(:,i)= X.^i;
    end
% =========================================================================

end


featureNormalize.m

特征缩放、均值归一

利用bsfunc函数可以自动维度扩展

function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X 
%   FEATURENORMALIZE(X) returns a normalized version of X where
%   the mean value of each feature is 0 and the standard deviation
%   is 1. This is often a good preprocessing step to do when
%   working with learning algorithms.

mu = mean(X);
X_norm = bsxfun(@minus, X, mu);

sigma = std(X_norm);
X_norm = bsxfun(@rdivide, X_norm, sigma);

% ============================================================

end

polyFit.m

-> 预测曲线是连续函数,而不是散点,故生成密集、等距离x作为横轴作图向量
-> 对该向量进行预处理(特征匹配 特征缩放 均值归一化 加全1列)
-> 预测θTx、画图
note:预测前也要预处理

function plotFit(min_x, max_x, mu, sigma, theta, p)
%PLOTFIT Plots a learned polynomial regression fit over an existing figure.
%Also works with linear regression.
%   PLOTFIT(min_x, max_x, mu, sigma, theta, p) plots the learned polynomial
%   fit with power p and feature normalization (mu, sigma).

% Hold on to the current figure
hold on;

% We plot a range slightly bigger than the min and max values to get
% an idea of how the fit will vary outside the range of the data points
x = (min_x - 15: 0.05 : max_x + 25)';

% Map the X values 
X_poly = polyFeatures(x, p);
X_poly = bsxfun(@minus, X_poly, mu);
X_poly = bsxfun(@rdivide, X_poly, sigma);

% Add ones
X_poly = [ones(size(x, 1), 1) X_poly];

% Plot
plot(x, X_poly * theta, '--', 'LineWidth', 2)

% Hold off to the current figure
hold off

end

validationCurve.m

一系列λ值,分别训练,求取误差

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
%
%
       for i = 1:length(lambda_vec)
           lambda = lambda_vec(i);
           [theta] = trainLinearReg(X, y, lambda);
           error_train(i) = 1/2/size(X,1)*sum((X*theta-y).^2);
           error_val(i) = 1/2/size(Xval,1)*sum((Xval*theta-yval).^2);         
       end
% =========================================================================

end

训练集、预测、测试集关系


Coursera-Machine Learning-Andrew Ng-Programming Exercise 5_第5张图片


2-27

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