1.正规化的线性回归
（1）代价函数
（2）梯度
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. % %求h（θ） h = X * theta; J = 1/2/m *((h-y)'*(h-y)) + lambda/2/m*(theta(2:end,:)'*theta(2:end,:)); grad(1,1) = X(:,1)'*(h-y)/m; grad(2:end,1) = X(:,2:end)'*(h-y)/m +lambda/m * theta(2:end,1); % ========================================================================= grad = grad(:); end
用fmincg最优的theta来拟合线性回归，画出线性回归函数（在这里是低维度的可以画出来）
2.偏差与方差
（1）求训练样本的误差代价：
（2）交叉样本集
Jcv
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 ----------------------
%进行训练的时候，对训练样本i个进行训练得到theta值，再求J
for i = 1:m
theta = 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
% -------------------------------------------------------------
% =========================================================================
end
学习曲线如下：
3.多项式回归
（1) 上面学习曲线可以看出来高偏差，欠拟合。采用增加特性来拟合，即多项式如下：
polyFeatures.m
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
(2) 画出学习曲线
（2）可以看出出现了高方差，过拟合。选择一个好的正则化参数lambda。
利用交叉验证集来选择合适的lambda，选择最小的Jcv对应的lambda。(在这里求代价误差的时候就不用加正则化项了)
trainLinearReg.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);
theta = trainLinearReg(X, y, lambda); %10x1选择最优的theta
error_train(i,1) = linearRegCostFunction(X, y, theta, 0);
error_val(i,1) = linearRegCostFunction(Xval, yval, theta, 0);
end
% =========================================================================
end
（3）计算测试集代价误差3.8599，（根据上面得到的最优的λ= 3）
（4）画出学习曲线
