吴恩达 coursera ML 第九课总结+作业答案
前言
吴恩达的课程堪称经典，有必要总结一下。
学以致用，以学促用，通过笔记总结，巩固学习成果，复习新学的概念。
目录
文章目录
正文
本节主要介绍在机器学习系统设计的实践过程中如何进行调节。在决定采取措施前，首先必须判断问题的症结。
机器学习诊断，判断一项措施可以在多大程度上提升你的性能。
首先评估你的假设。
将你手上的数据集划分成训练集和测试集。
通过训练集选出最优的参数，通过测试集验证模型的泛化性能。
逻辑回归时的相应手续。
过拟合问题，模型总是会更加符合训练集表现出来的样子。
测试不同的超参数，选择适宜的模型。
将数据集划分为，训练集，交叉验证集和测试集。
选择测试集表现最好的模型。
测试集上的表现近似为泛化误差。
问题判断是方差还是误差
欠拟合，刚好，以及过拟合的情况。
误差函数的表现以及训练的情况。
根据误差图，判断问题情况的展示。
正则化以及方差和偏差的关系
正则化对于模型拟合情况的影响。
选择合适的正则化参数。
在对进行测试的时候，选择方法。
偏差和方差是正则化参数的函数。
学习曲线
高偏差时的学习曲线
高方差时的学习曲线。
作业答案
仅作参考，切勿抄袭。
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
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
[theta] = trainLinearReg(X, y, lambda_vec(3));
[error_test,~]=linearRegCostFunction(Xtest, ytest, theta, lambda_vec(3));
fprintf('Program paused. Press enter to continue.\n');
pause;
fprintf('theta\terror_test\tlambda')
fprintf('%f\t%f\t%f',theta,error_test,lambda_vec(3));
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
% 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=J+sum((X*theta-y).^2);
J=J+lambda*sum(theta(2:end).^2);
J=J/2/m;
grad=(X*theta-y)'*X;
theta(1)=0;
grad=grad+lambda*theta';
grad=grad/m;
% =========================================================================
grad = grad(:);
end
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);
% 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
% % 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)
[theta] = trainLinearReg(X(1:i,:), y(1:i), 1);
[error_train(i),~]=linearRegCostFunction(X(1:i,:), y(1:i), theta, 0);
[error_val(i),~]=linearRegCostFunction(Xval, yval, theta, 0);
end
% -------------------------------------------------------------
% =========================================================================
end
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);
% % 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)
% ....
%
lambda = lambda_vec(i);
[theta] = trainLinearReg(X, y, lambda);
[error_train(i),~]=linearRegCostFunction(X, y, theta, 0);
[error_val(i),~]=linearRegCostFunction(Xval, yval, theta, 0);
end
% =========================================================================
end