Coursera机器学习-Week 3-编程作业:Logistic Regression

感觉自己做的好慢，不懂的也太多，读英文文档障碍太多了……不过，这个必须慢慢克服，感觉自己在逐渐步入正轨。

1.1 plotData.m

完成这个函数，依照给定的数据进行描点。这个函数文档中已经给出了正确答案……所以这里是不用提交的。

function plotData(X, y)
%PLOTDATA Plots the data points X and y into a new figure 
%   PLOTDATA(x,y) plots the data points with + for the positive examples
%   and o for the negative examples. X is assumed to be a Mx2 matrix.

% Create New Figure
figure; hold on;

% ====================== YOUR CODE HERE ======================
% Instructions: Plot the positive and negative examples on a
%               2D plot, using the option 'k+' for the positive
%               examples and 'ko' for the negative examples.
%

% 根据 y 进行分组
y1 = find(y == 1);
y0 = find(y == 0);

% 描点
plot(X(y1, 1), X(y1, 2), 'k+', 'LineWidth', 2, 'MarkerSize', 7);
plot(X(y0, 1), X(y0, 2), 'ko', 'MarkerFaceColor', 'y', 'MarkerSize', 7);

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

hold off;

end

1.2.1 sigmoid.m

实现函数 hθ(x)=g(θTx) 中的 g(z) ，公式如下：

g (z) = 1 1 + e - z

，这里强调

z z 有可能是矩阵，所以需要考虑到矩阵的运算。

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

g = 1 ./ (ones(size(z)) + e .^ (-z));

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

end

1.2.2 costFunction.m

根据代价和梯度公式分别求代价和梯度，代价公式：

J (θ) = \frac{1}{m} <munderover> \sum i = 1 m </munderover> [- y^{(i)} l o g (h_{θ} (x (i)) - (1 - y^{(i)}) l o g (1 - h_{θ} (x^{(i)}))]

梯度公式：

\partial J ( θ ) \partial θ j = 1 m \sum i = 1 m (h θ (x (i)) - y (i)) x (i) j

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
%

h_theta = sigmoid(X * theta); % X: m * n theta: n * 1 h_theta: m * 1

J = (-y' * log(h_theta) - (1 - y)' * log(1 - h_theta)) / m; % y: m * 1
grad = X' * (h_theta - y) / m;

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

end

1.2.4 predict.m

预测函数，返回判断结果为 0 or 1 。

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
%

p = sigmoid(X * theta) > 0.5; % X: m * n theta: n * 1

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


end

2.3 costFunctionReg.m

和 costFunction.m 相似，不过加入了正则化处理，公式有所变化。代价公式：

J (θ) = 1 m \sum i = 1 m [- y (i) l o g (h θ (x (i)) - (1 - y (i)) l o g (1 - h θ (x (i)))] + 1 2 m \sum j = 1 n θ 2 j

梯度公式：

\partial J ( θ ) \partial θ j = 1 m \sum i = 1 m (h θ (x (i)) - y (i)) x (i) j, f o r j = 0 \partial J ( θ ) \partial θ j = (1 m \sum i = 1 m (h θ (x (i)) - y (i)) x (i) j) + λ m θ j, f o r j \geq 1

function [J, grad] = costFunctionReg(theta, X, y, lambda)
%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
%   J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using
%   theta as the parameter for regularized 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

h_theta = sigmoid(X * theta); % X: m * n theta: n * 1 h_theta: m * 1

J = (-y' * log(h_theta) - (1 - y)' * log(1 - h_theta)) / m + ...
    lambda * (theta' * theta - (theta(1, 1))^2) / (2 * m); % y: m * 1
theta(1) = 0;
grad = (X' * (h_theta - y) + theta * lambda) / m;

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

end

结果

最后应该有的实验结果在文档中都有给出，所以这里就不截图了。这次编程作业一共是两个部分，前者是逻辑回归算法，后者是在此基础上进行了正则化处理，不过两部分的数据是不一样的，刚好适应两种算法。

代码亲测，都是可以通过测试的。