逻辑回归(logistic regression)

基本概念

逻辑回归是一种概率型非线性回归模型。虽然名字里面有回归，但是他其实是一种分类的方法，通常是用来研究在某些条件下某个结果会不会发生，例如：已知病人身体里的肿瘤的情况，然后判断这个肿瘤是良性还是恶性。

逻辑回归与线性回归

逻辑回归同线性回归一样都需要有一个假设函数hθ(x),代价函数J(θ),在大体上基本相同。

在线性回归中：

hθ(x)=θ0+θ1∗x1+...+..θn∗xn

但是我们引入了一个Sigmoid函数，将结果给映射到区间(0,1).

Sigmoid函数：

π(x)=11+e−x

因此在逻辑回归中的假设函数就是

hθ(x)=π(θτx)=11+e−θτx

π(x)的定义域为(−∞,+∞),值域为(0,1)。因此就将最后的结果映射到了(0,1)中。

P(y=1|x,θ)=π(θτx)P(y=0|x,θ)=1−π(θτx)

当hθ(x)≥0.5时,y=1

当hθ(x)<0.5时,y=0

代价函数J(θ)

在求解参数θ的时候我们用用极大似然估计来求解。设pi=P(yi=1|xi;θ)表示在给定条件下yi=1的概率，则pi=P(yi=0|xi;θ)=1−pi,所以可以得到一个观测值的概率P(yi)=pyii(1−pi)1−yi,各样本间相互独立就可以得到似然函数为：

L(θ)=Πmi=1[hθ(xi)]yi[1−hθ(xi)]1−yi

目标就是求这个函数的值最大的时候的参数θ,推导过程如下：

lnL(θ)=Σmi=1[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]

lnL(θ)=Σmi=1(y(i)ln(11+e−x(i))+(1−y(i))ln(1−11+e−x(i)))

lnL(θ)=Σmi=1(y(i)ln(ex(i)ex(i)+1)+(1−y(i))ln(1ex(i)+1))

lnL(θ)=Σmi=1(y(i)∗lnex(i)−y(i)∗ln(1+ex(i))−(1−y(i))∗ln(1+ex(i)))

lnL(θ)=Σmi=1(x(i)y(i)−ln(1+ex(i)))

我们要求得θ，使得L(θ)最大，那么我们的代价函数就可以这样表示：

J(θ)=−1mlnL(θ)=−1mΣmi=1(x(i)y(i)−ln(1+ex(i)))

为了求得θ我们可以用梯度下降法:

对J(θ)求偏导：

∂J(θ)∂θj=−Σmi=1[y(i)−hθ(x(i))]∗x(i)j=Σmi=1[hθ(x(i))−y(i)]∗x(i)j

因此完整的梯度下降应该为：

θj=θj+α∂J(θ)∂θj=θj+αΣmi=1[hθ(x(i))−y(i)]∗x(i)j

Matlab code

costfunction

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
%

z = X*theta;
hx = 1 ./ (1 + exp(-z));
J = -1/m *sum([y'*log(hx) + (1-y)'*log(1-hx)]);
%J = 1/m * sum([-y' * log(hx) - (1 - y)' * log(1 - hx)]);
for j = 1:length(theta)
grad(j)=1/m*sum((hx-y)'*X(:,j));
end;

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

end

Sigmoid

function g = sigmoid(z)
%SIGMOID Compute sigmoid functoon
%   J = 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).

sz = size(z);

for i = 1:sz(1),
for j = 1:sz(2),
g(i,j) = 1./(1+exp(-z(i,j)));
end;
end;
% =============================================================

end

predict

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
%

pp = sigmoid(X*theta);

pos = find(pp>=0.5);

p(pos,1)=1;

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

end