监督学习-逻辑回归及编程作业（二）

 一、过拟合问题

欠拟合、恰当拟合、过拟合（代价函数约为0，泛化太差）

样本数量少，而样本特征很多，容易出现过拟合问题，如何解决？

1.利用一些算法自动舍弃一部分特征；

2.正则化，保留所有特征，减小 θ 量级。

二、线性回归的正则化

三、逻辑回归的正则化

编程作业

1.plotDate.m  

在第一部分作业中，无法完成在这部分的绘图，修改了代码。

function plotData(X, y)
%PLOTDATA Plots the data points X and y into a new figure 
%   PLOTDATA(x,y) plots the data points with + for the positive examples
%   and o for the negative examples. X is assumed to be a Mx2 matrix.

% Create New Figure
figure; hold on;

% ====================== YOUR CODE HERE ======================
% Instructions: Plot the positive and negative examples on a
%               2D plot, using the option 'k+' for the positive
%               examples and 'ko' for the negative examples.
%
n0 = 1;
n1 = 1;
for i=1:length(y),
	if y(i)==0,
		matrix0(n0,:) = X(i,:);
		n0 = n0 + 1;
	end;
	if y(i)==1,
		matrix1(n1,:) = X(i,:);
		n1 = n1 + 1;
	end;
end;
 
plot(matrix0(:,1),matrix0(:,2),'ko', 'MarkerFaceColor', 'y', ...
 'MarkerSize', 7);   
plot(matrix1(:,1),matrix1(:,2),'k+','LineWidth', 2, ...
'MarkerSize', 7);

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



hold off;

end

2.costFunctionReg.m

正则化逻辑回归的代价函数

下降梯度

 在这部分需要注意的是，求代价函数的公式中的正则化项中的 θ 是不包括 θ0 的（约定俗成），在因为在MATLAB中下标从1开始，故指的是θ1.

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

h = sigmoid(X*theta);
[J, grad] = costFunction(theta,X,y);
J = J + lambda*(theta'*theta-theta(1).^2)/2/m;

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

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

end