一、Logistic回归——分类

对于分类问题，采用线性回归是不合理的。

1.假设函数（logistic函数/Sigmoid函数）：

注：假设函数 h 的值，看作结果为y=1的概率估计。决策界限可以看作是h=0.5的线。

2.代价函数

3.高级优化fminunc

在上文优化过程中需要提供α值，而高级优化α是自动选择。

优化结果

二、Logistic回归——多元分类（一对多种类别）

三、编程作业

1.sigmoid.m写假设函数

function g = sigmoid(z)
%SIGMOID Compute sigmoid function
%   g = SIGMOID(z) computes the sigmoid of z.

% You need to return the following variables correctly 
g = zeros(size(z));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the sigmoid of each value of z (z can be a matrix,
%               vector or scalar).


g = 1./(1+ exp(-z));


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

end

2.plotDate.m数据可视化

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.
%
axis([30 100 30 100]);
pos = find( y==1 );
neg = find( y==0 );
plot(X(pos, 1), X(pos, 2), 'k+','LineWidth', 2, ...
'MarkerSize', 7);
plot(X(neg, 1), X(neg, 2), 'ko', 'MarkerFaceColor', 'y', ...
'MarkerSize', 7);

3.costFunction.m写代价函数和梯度

function [J, grad] = costFunction(theta, X, y)
%COSTFUNCTION Compute cost and gradient for logistic regression
%   J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the
%   parameter for 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
%
% Note: grad should have the same dimensions as theta
%

h =sigmoid(X*theta);
costfun = y.*log(h)+(1-y).*log(1-h);
J = -1/m*sum(costfun);
grad = X'*(h-y)/m;

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

end

4.fminunc高级优化

命令行：

% Set options for fminunc
options = optimset('GradObj', 'on', 'MaxIter', 400);
% Run fminunc to obtain the optimal theta
% This function will return theta and the cost
[theta, cost] = ...
fminunc(@(t)(costFunction(t, X, y)), initial theta, options);

5.predict.m

对每个样本预测分类结果（根据假设函数），将分类结果存到向量v中，与实际的分类结果y比较，得到正确率。

function p = predict(theta, X)
%PREDICT Predict whether the label is 0 or 1 using learned logistic 
%regression parameters theta
%   p = PREDICT(theta, X) computes the predictions for X using a 
%   threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)

m = size(X, 1); % Number of training examples

% You need to return the following variables correctly
p = zeros(m, 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned logistic regression parameters. 
%               You should set p to a vector of 0's and 1's
%

h = sigmoid(X*theta);
h(h>=0.5)=1;
h(h<0.5)=0;
p = h;

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


end

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