【邏輯回歸函式模型】
1、訓練資料繪圖
cd d:\study\ai\data\ex2
data = load('ex2data1.txt');
x = data(:, [1, 2]);
y = data(:, 3);
pos = find(y==1); % y取1的所有行
plot(x(pos, 1), x(pos, 2), 'k+','linewidth', 2, 'markersize', 7);
hold on;
neg = find(y == 0); % y取0的所有行
plot(x(neg, 1), x(neg, 2), 'ko', 'marke***cecolor', 'y', 'markersize', 7);
xlabel('exam 1 score');
ylabel('exam 2 score');
% specified in plot order
legend('admitted', 'not admitted');
2、計算損失函式和梯度
[m, n] = size(x);
x = [ones(m, 1) x];
initial_theta = zeros(n + 1, 1);
test_theta = [-24; 0.2; 0.2];
[cost, grad] = costfunction(test_theta, x, y);
fprintf('\ncost at test theta: %f\n', cost);
fprintf('gradient at test theta: \n');
fprintf(' %f \n', grad);
【正則化】
1、訓練資料繪圖
cd d:\study\ai\data\ex2
data = load('ex2data2.txt');
x = data(:, [1, 2]);
y = data(:, 3);
pos = find(y==1); % y取1的所有行
plot(x(pos, 1), x(pos, 2), 'k+','linewidth', 2, 'markersize', 7);
hold on;
neg = find(y == 0); % y取0的所有行
plot(x(neg, 1), x(neg, 2), 'ko', 'marke***cecolor', 'y', 'markersize', 7);
xlabel('exam 1 score');
ylabel('exam 2 score');
% specified in plot order
legend('admitted', 'not admitted');
2、正則化計算
% 多項式特徵對映
x = mapfeature(x(:,1), x(:,2));
initial_theta = zeros(size(x, 2), 1);
[cost, grad] = costfunctionreg(initial_theta, x, y, 1);
fprintf('cost at initial theta (zeros): %f\n', cost);
fprintf(' %f \n', grad);
test_theta = ones(size(x,2),1);
[cost, grad] = costfunctionreg(test_theta, x, y, 10);
fprintf('\ncost at test theta (with lambda = 10): %f\n', cost);
fprintf(' %f \n', grad);
【工具函式】
% logistic函式
function g = sigmoid(z)
g = zeros(size(z));
g = 1./(1 + exp(-z));
end;
% 損失函式和梯度
function [j, grad] = costfunction(theta, x, y)
m = length(y);
j = 0;
grad = zeros(size(theta));
% j(θ) = −y * log(g(x)) − (1 − y) * log(1 − g(x))
j = (-y'*log(sigmoid(x*theta)) - (1-y)'*(log(1-sigmoid(x*theta))))/m;
% ▽j(θ)= x′*(g(θ)-y)/m
grad = x'*(sigmoid(x*theta) - y)/m;
end;
% 正則約束損失函式和梯度
function [j, grad] = costfunctionreg(theta, x, y, lambda)
m = length(y);
grad = zeros(size(theta));
j = 1/m * (-y' * log(sigmoid(x*theta)) - (1 - y') * log(1 - sigmoid(x * theta))) + lambda/2/m*sum(theta(2:end).^2);
grad(1,:) = 1/m * (x(:, 1)' * (sigmoid(x*theta) - y));
grad(2:size(theta), :) = 1/m * (x(:, 2:size(theta))' * (sigmoid(x*theta) - y)) + lambda/m*theta(2:size(theta), :);
end;
% **函式
function p = predict(theta, x)
m = size(x, 1); % number of training examples
p = zeros(m, 1);
p = sigmoid(x * theta)>=0.5;
end;
% 多項式特徵對映
function out = mapfeature(x1, x2)
degree = 6;
out = ones(size(x1(:,1)));
for i = 1:degree
for j = 0:i
out(:, end+1) = (x1.^(i-j)).*(x2.^j);
endend
end;
% 決策邊界繪圖
function plotdecisionboundary(theta, x, y)
pos = find(y==1); % y取1的所有行
plot(x(pos, 2), x(pos, 3), 'k+','linewidth', 2, 'markersize', 7);
hold on;
neg = find(y == 0); % y取0的所有行
plot(x(neg, 2), x(neg, 3), 'ko', 'marke***cecolor', 'y', 'markersize', 7);
xlabel('exam 1 score');
ylabel('exam 2 score');
% specified in plot order
legend('admitted', 'not admitted');
hold on
% 線性一階決策邊界直接繪製曲線圖
if size(x, 2) <= 3
plot_x = [min(x(:,2))-2, max(x(:,2))+2];
plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));
plot(plot_x, plot_y)
legend('admitted', 'not admitted', 'decision boundary')
axis([30, 100, 30, 100])
else % 高階決策邊界通過等高線繪製z=0切線圖
u = linspace(-1, 1.5, 50);
v = linspace(-1, 1.5, 50);
z = zeros(length(u), length(v));
for i = 1:length(u)
for j = 1:length(v)
% 多項式特徵對映的結構和x完全一樣
z(i,j) = mapfeature(u(i), v(j))*theta;
endend
z = z';
contour(u, v, z, [0, 0], 'linewidth', 2)
endhold off
end;
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