Logistic迴歸

代價函式

Logistic迴歸是分類演算法，它的輸出值在0和1之間。

h(x)的作用是，對於給定的輸入變數，根據選擇的引數計算輸出變數等於1的可能性（estimated probablity）即h(x)=P(y=1|x;θ)

梯度下降

練習2

資料集

x1,x2 : score on two exams
y : admissions decision  

視覺化資料集

function plotData(X, y)
	figure; hold on;
	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);
	hold off;
end
 

find()函式的基本功能是返回向量或者矩陣中不為0的元素的位置索引。
如果需要找到其中滿足一定條件的元素，find(X==4)找到矩陣X中值為4的資料的索引

sigmoid函式

function g = sigmoid(z)
	g = zeros(size(z));
	g = 1./(1+exp.^(-z));
end

代價函式及梯度下降

function [J, grad] = costFunction(theta, X, y)
	m = length(y); 
	J = 0;
	grad = zeros(size(theta));
	h = sigmoid(X*theta);
	J = (1/m)*sum((-y)*log(h)-(1-y)*log(1-h));
	for j = 1 : size(X,2)
		grad(j) = (1/m)*sum((h-y).*X(:,j));
end

 

使用fminunc學習引數

%  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);

評估

function p = predict(theta, X)
	m = size(X, 1); 
	p = zeros(m, 1);
	p = sigmoid(X*theta)>=0.5;
end