【Exercise 5 Regularized Linear Regression and Bias v.s. Variance】
【代碼】
ex5.m
-> 加載數據（上升曲線）可視化
-> 實現成本 實現梯度
不進行特徵匹配：（只用原始的一維x）
-> 訓練，繪製結果
-> 畫學習曲線：樣本個數-誤差關係
進行特徵匹配（p維x，有高次特徵）
-> 特徵匹配
-> 特徵匹配產生了“多個”特徵，故歸一化、並輸出均值標準差、
cv集 測試集需要做同樣處理，但是歸一化要減去訓練集的均值、除以訓練集的標準差
-> 訓練，繪製結果
-> 畫學習曲線：樣本個數-誤差關係
-> 誤差-λ曲線
不計分部分:
-> 選取cv集上表現最優的λ，利用這個值訓練，計算測試集誤差
-> 作圖顯示訓練集、預測、測試集關係
note：原始數據沒有x_0，不論匹配不匹配特徵，學習前都要加全1列（預測前也要）
%% 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.
% 計時開始（畫平滑學習曲線耗時較大）
tic
% 原始取0，平滑時爲更好效果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
% 計時結束
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+梯度函數、option送入求解器。
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【基礎】
for循環，從只用1個樣本、只用2個樣本到全部m個樣本，各情況下分別訓練，再計算訓練集誤差、cv集誤差。
計算誤差可以列公式，也可以調用已經實現好的linearRegCostFunction函數。
注意：1、列公式或調用函數時，有關位置要改成前i個而非全部m個。尤其注意1/2m。
2、cv集上的誤差總是對全體cv集來說，不受i的影響
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，根據定義，誤差值沒有正則項；
% 外層傳進來的lambda是用來訓練參數theta的
end
% -------------------------------------------------------------
% =========================================================================
end
learningCurve.m【平滑】
原始版本簡單地取前i個樣本；這裏對每個i重複進行若干次計算，每次計算重新隨機取i個樣本，最後重複計算的誤差取平均，作爲這個i值的誤差。
第一層循環同前爲1到m。第二層循環爲重複計算。隨機取i個樣本：
random_index=randperm(m);
random_index=random_index(1:i);
Xtrain=X(random_index,:);
ytrain=y(random_index);
代碼內部設置了變量smooth，手動修改其爲0時，實現功能與基礎版本相同；修改爲1時，作平滑的學習曲線，無法Coursera通過在線測試。
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 ----------------------
%說明：以下爲繪製平滑learning curve有關變量，爲使其它部分能正常通過在線測
% 試，僅採用手動修改參數方法簡單演示。
% - smooth變量設置爲0，即不進行平滑，可正常提交通過測試；
% - smooth變量設置爲1，即進行平滑。lambda_smooth設置正則化係數。
% repeat_times爲重複計算次數，_temp變量存儲重複計算誤差數據。
% （ex5.m中傳入的lambda應做相應修改，以得到更好效果）
% - 繪製多項式的平滑learning curve需要接近6分鐘
smooth=0;
repeat_times=50;
error_train_temp=zeros(repeat_times,1);
error_val_temp=zeros(repeat_times,1);
%-------------------------------------------------------------------------
%設置smooth=0,不進行平滑，可通過在線測試
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，根據定義，誤差值沒有正則項；
% 外層傳進來的lambda是用來訓練參數theta的
end
%設置smooth=1,進行平滑，
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
取平均達到平滑效果：
polyFeature.m
特徵映射feature mapping
原本每個樣本只有一個特徵x，現加入其冪形成更多特徵，可以用來擬合更復雜的曲線
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
% =========================================================================
endfeatureNormalize.m
特徵縮放、均值歸一
利用bsfunc函數可以自動維度擴展
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
-> 預測曲線是連續函數，而不是散點，故生成密集、等距離x作爲橫軸作圖向量
-> 對該向量進行預處理（特徵匹配 特徵縮放 均值歸一化 加全1列）
-> 預測θTx、畫圖
note：預測前也要預處理
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，分別訓練，求取誤差
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訓練集、預測、測試集關係
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