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.
%reg = (lambda / (2*m)) * ( ( theta( 2:length(theta) ) )' * theta(2:length(theta)) );
J = sum((X*theta-y).^2)/(2*m) + reg;
grad_tmp = X'*(X*theta-y) / m;
grad = [ grad_tmp(1:1); grad_tmp(2:end) + (lambda/m)*theta(2:end) ];% =========================================================================grad = grad(:);end

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

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:mtheta = trainLinearReg(X(1:i, :), y(1:i), lambda);error_train(i) = linearRegCostFunction(X(1:i, :), y(1:i), theta, 0);error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);% -------------------------------------------------------------% =========================================================================end

for i = 1:pX_poly(:,i) = X.^i;
end

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

Concretely, you will use a cross validation set to evaluate how good each lambda value is. After selecting the best lambda value using the cross validation set, we can then evaluate the model on the test set to estimate how well the model will perform on actual unseen data.

for i = 1:length(lambda_vec)theta = trainLinearReg(X,y,lambda_vec(i));%对于每个lambda,训练出模型参数theta%compute jcv and jval without regularization,causse last arguments(lambda) is zero error_train(i) = linearRegCostFunction(X, y, theta, 0);%计算训练误差error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);%计算交叉验证误差
end

lambda     Train Error   Validation Error0.000000    0.173616    22.0666020.001000    0.156653    18.5976380.003000    0.190298    19.9815030.010000    0.221975    16.9690870.030000    0.281852    12.8290030.100000    0.459318    7.5870130.300000    0.921760    4.6368331.000000    2.076188    4.2606253.000000    4.901351    3.82290710.000000   16.092213   9.945508