特征空间映射

1. 问题

• 简单的0,1分类 – 即标签$y =$ {$0,1$}
• 特征值：$x = [x_1, x_2]$二维

数据离散点如图：

2.解答

• 数据是二维的，因此如果利用Logistics Regression 的到的$\theta$只有三个数，所以分类超平面是二维坐标下的直线
• 由数据分布图可以知道分类超平面应该是一个二次曲线，所以这里利用多项式核函数：$K = (<x_1, x_2> + R)^d$，令R= 1， d = 2.可以得出二维数据映射到五维时，二次曲线成为五维空间中的直线，因此我们将二维数据$x = [x_1, x_2]$ 扩展到五维$x = [x_1, x_1^2, x_2, x_2^2, x_1x_2]$
• 因此超平面表达式如下：
  $\theta_1 + \theta_2*x_1 + \theta_3*x_1^2 + \theta_4*x_2 + \theta_5*x_2^2 + \theta_6*x_1x_2 = 0$
• $\theta$和$J$的计算方式依然不变，这里采用fminunc函数计算最优解

3.效果如下

　　　最后结果：
　　

theta =

    5.1555
    3.2317
  -11.9929
    4.1505
  -11.7849
   -7.4954

>> cost

cost =

    0.3481

图形如下：

代码

1. Logistics_Regression

%%  part0： 准备
data = load(‘ex2data2.txt‘);
x = data(:,[1,2]);
y = data(:,3);
pos = find(y==1);
neg = find(y==0);

x1 = x(:,1);
x2 = x(:,2);
plot(x(pos,1),x(pos,2),‘r*‘,x(neg,1),x(neg,2),‘co‘);
m = size(y,1);
X = zeros(m,6);
X(:,1) = ones(m,1);
X(:,2) = x(:,1);
X(:,3) = x(:,1).*x(:,1);
X(:,4) = x(:,2);
X(:,5) = x(:,2).*x(:,2);
X(:,6) = x(:,1).*x(:,2);

pause;


%% part1: GradientDecent and compute cost of J
[m,n] = size(X);
theta = zeros(6,1);
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)(computeCost(X,y,t)), theta, options);
display(theta);
ezplot(‘5.15 + 3.2317*XX  - 11.99*XX*XX + 4.15*YY  - 11.78*YY*YY  - 7.50*XX*YY‘,[-1.0,1.5,-0.8,1.2]);
hold on
plot(x(pos,1),x(pos,2),‘r*‘,x(neg,1),x(neg,2),‘co‘);
pause;

2.computeCost

function [J,grad] = computeCost(x, y, theta)

%% compute cost: J

m = size(x,1);
grad = zeros(size(theta));
hx = sigmoid(x * theta);  
 J = (1.0/m) * sum(-y .* log(hx) - (1.0 - y) .* log(1.0 - hx));  
 grad = (1.0/m) .* x‘ * (hx - y);
end

3.sigmoid

function g = sigmoid(z)

%% SIGMOID Compute sigmoid functoon

g = zeros(size(z));
g = 1.0 ./ (1.0 + exp(-z));

end