先貼一張圖解釋牛頓法原理：

      然後以一道問題為例：

      這個問題是《機器學習》周志華版本上的題目，給了西瓜的密度和含糖率資料，判斷西瓜的好壞。資料在程式碼裡。

      下面貼一下程式碼：

<span style="font-size:14px;">clear all;
close all;
clc;

density = [0.697, 0.774, 0.634, 0.608, 0.556, 0.403, 0.481, 0.437, 0.666, 0.243, 0.245, 0.343, 0.639, 0.657, 0.360, 0.593, 0.719];
density = density';
sugar = [0.460, 0.376, 0.264, 0.318, 0.215, 0.237, 0.149, 0.211, 0.091, 0.267, 0.057, 0.099, 0.161, 0.198, 0.370, 0.042, 0.103];
sugar = sugar';
y = [1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0];
y = y';

x = [ones(size(y,1),1) density  sugar];
[m,n] = size(x);

figure,
pos = find(y);
neg = find(y==0);
plot(x(pos,2),x(pos,3),'o');
hold on
plot(x(neg,2),x(neg,3),'*');
xlabel('density'),ylabel('sugar');


theta = zeros(n,1);
MaxIter = 10;
J=zeros(MaxIter,1);

for i=1:MaxIter
    z = x*theta;
    h = sigmoid(z);
    
    grad = 1./m*x'*(h-y);
    H = (1/m).*x' * diag(h) * diag(1-h) * x;
    theta = theta-H\grad;
     
    J(i) = 1/m*sum(-y.*log(h)-(1-y).*(log(1-h)));
end

hold on 
plot_x = [min(x(:,2)),max(x(:,2))];
plot_y = (-1/theta(3))*(theta(1)+theta(2)*plot_x);
plot(plot_x,plot_y);

figure,
plot(0:MaxIter-1,J,'o--', 'MarkerFaceColor', 'r', 'MarkerSize', 8)
xlabel('Iteration'); ylabel('J')</span>
 

對訓練資料的分類結果：

      損失函式隨迭代次數的變化，可以看出，下降非常快！