邏輯迴歸(logistic regression)

基本概念

邏輯迴歸是一種概率型非線性迴歸模型。雖然名字裏面有迴歸，但是他其實是一種分類的方法，通常是用來研究在某些條件下某個結果會不會發生，例如：已知病人身體裏的腫瘤的情況，然後判斷這個腫瘤是良性還是惡性。

邏輯迴歸與線性迴歸

邏輯迴歸同線性迴歸一樣都需要有一個假設函數hθ(x) ,代價函數J(θ) ,在大體上基本相同。
在線性迴歸中：

hθ(x)=θ0+θ1∗x1+...+..θn∗xn

但是我們引入了一個Sigmoid 函數，將結果給映射到區間(0,1) .
Sigmoid 函數：

π(x)=11+e−x

因此在邏輯迴歸中的假設函數就是

hθ(x)=π(θτx)=11+e−θτx

π(x) 的定義域爲(−∞,+∞) ,值域爲(0,1) 。因此就將最後的結果映射到了(0,1) 中。

P(y=1|x,θ)=π(θτx)P(y=0|x,θ)=1−π(θτx)

當hθ(x)≥0.5 時,y=1
當hθ(x)<0.5 時,y=0

代價函數J(θ)

在求解參數θ 的時候我們用用極大似然估計來求解。設pi=P(yi=1|xi;θ) 表示在給定條件下yi=1 的概率，則pi=P(yi=0|xi;θ)=1−pi ,所以可以得到一個觀測值的概率P(yi)=pyii(1−pi)1−yi ,各樣本間相互獨立就可以得到似然函數爲：

L(θ)=Πmi=1[hθ(xi)]yi[1−hθ(xi)]1−yi

目標就是求這個函數的值最大的時候的參數θ ,推導過程如下：

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)))

我們要求得θ ，使得L(θ) 最大，那麼我們的代價函數就可以這樣表示：

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

爲了求得θ 我們可以用梯度下降法:
對J(θ) 求偏導：

∂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應該爲：

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