斯坦福大学(吴恩达) 机器学习课后习题详解 第六周 编程题 正则化线性回归以及方差与偏差

作业下载地址：https://download.youkuaiyun.com/download/wwangfabei1989/10303134

1. 正则化线性回归代价函数 linearRegCostFunction

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)+ lambda*sum((theta(2:end).^2)) )


grad=(1/m)*(X'*(X*theta-y));%all 此处注意 没有 使用求和，因为 矩阵相乘时，已经进行了求和
grad(2:end) += (lambda / m) * theta(2:end)


















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


grad = grad(:);


end

2. 学习曲线 learningCurve

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
    %计算theta值，找到拟合参数
   theta=trainLinearReg(X(1:i, :),y(1:i),lambda); 
   %计算 训练集 的 cost 
   error_train(i)=linearRegCostFunction(X(1:i, :),y(1:i),theta,0)
   %计算交叉验证集 的 cost
   error_val(i)=linearRegCostFunction(Xval,yval,theta,0)
  
  




end   








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


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


end

3. 多项式特征polyFeatures

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 %循环列
  X_poly(:,i)=X.^i;
  
end  








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


end

4. 验证曲线validationCurve

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:size(lambda_vec, 1)
    theta = trainLinearReg(X, y, lambda_vec(i));
    error_train(i) = linearRegCostFunction(X, y, theta, 0);
    error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
end
















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


end