matlab(7) Regularized logistic regression : mapFeature(将feature增多) and costFunctionReg

Regularized logistic regression : mapFeature(将feature增多) and costFunctionReg

ex2_reg.m文件中的部分内容

%% =========== Part 1: Regularized Logistic Regression ============
% In this part, you are given a dataset with data points that are not
% linearly separable. However, you would still like to use logistic
% regression to classify the data points.
%
% To do so, you introduce more features to use -- in particular, you add
% polynomial features to our data matrix (similar to polynomial
% regression).
%

% Add Polynomial Features

% Note that mapFeature also adds a column of ones for us, so the intercept
% term is handled
X = mapFeature(X(:,1), X(:,2));  %调用下面的mapFeature.m文件中的mapFeature(X1,X2)函数

                                           %将只有x1,x2feature map成一个有28个feature的6次的多项式 ,这样就能画出更复杂的decision boundary,                                            但同时也有可能带来overfitting的结果(取决于λ的值)

                                            %  调用完后X变为118*28(118个example,28个属性，包括前面的1做为一列)的矩阵

% Initialize fitting parameters
initial_theta = zeros(size(X, 2), 1);    %initial_theta: 28*1

% Set regularization parameter lambda to 1  
lambda = 1;                            % λ=1;当λ=0时表示不正则化(No regularization ),这时会出现overfitting;当λ=100时会出现Too much regularization(Underfitting)

% Compute and display initial cost and gradient for regularized logistic
% regression
[cost, grad] = costFunctionReg(initial_theta, X, y, lambda);  %调用costFunctionReg.m文件中的costFunctionReg(theta, X, y, lambda)函数

fprintf('Cost at initial theta (zeros): %f ', cost);    %计算initial theta (zeros)时的cost 值

fprintf(' Program paused. Press enter to continue. ');
pause;

mapFeature.m文件

function out = mapFeature(X1, X2)
% MAPFEATURE Feature mapping function to polynomial features
%
% MAPFEATURE(X1, X2) maps the two input features
% to quadratic features used in the regularization exercise.
%
% Returns a new feature array with more features, comprising of
% X1, X2, X1.^2, X2.^2, X1*X2, X1*X2.^2, etc..
%
% Inputs X1, X2 must be the same size
%

degree = 6;       %map the features into all polynomial terms of x1 and x2 up to the sixth power

out = ones(size(X1(:,1)));
for i = 1:degree
for j = 0:i
out(:, end+1) = (X1.^(i-j)).*(X2.^j);
end
end

end

costFunctionReg.m文件

function [J, grad] = costFunctionReg(theta, X, y, lambda)
%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
% J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using
% theta as the parameter for regularized logistic regression and the
% gradient of the cost w.r.t. to the parameters.

% 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 of a particular choice of theta.
% You should set J to the cost.
% Compute the partial derivatives and set grad to the partial
% derivatives of the cost w.r.t. each parameter in theta

J = 1/m*(-1*y'*log(sigmoid(X*theta)) - (ones(1,m)-y')*log(ones(m,1)-sigmoid(X*theta)))...     
+ lambda/(2*m) * (theta(2:end,:))' * theta(2:end,:);     %Note that you should not regularize the parameter θ0.

                                                                

                                                             the regularized cost function,

                    

grad = 1/m * (X' * (sigmoid(X*theta) - y)) + (lambda/m)*theta;        % 
grad(1) = 1/m * (X(:,1))' * (sigmoid(X*theta) - y);                       %

                                                                                                % Note that you should not regularize the parameter θ0.

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

end