Machine Learning：Regression with multi variables

学习NG的Machine Learning教程，先关推导及代码。由于在matleb或Octave中需要矩阵或向量，经常搞混淆，因此自己推导，并把向量的形式写出来了，主要包括cost function及gradient descent 见下图。

图中可见公式推导，及向量化表达形式的cost function（J）.

图中为参数更新的向量化表达方式(其中有一处写错了，不想改了。。。)

图中为feature scaling的推导，及向量化表示

下面regression with multi variable的代码

%loading data data = load('ex1data2.txt'); X = data(:, 1:2); %X : m*2 y = data(:, 3); % y : m*1 m = length(y) %Scale features and set them to zero mean fprintf('Normalizing Features ...\n'); [X mu sigma] = featureNormalize(X); % Add intercept term to X X = [ones(m, 1) X]; fprintf('Running gradient descent ...\n'); % Choose some alpha value alpha = 0.05; num_iters = 400; % Init Theta and Run Gradient Descent theta = zeros(3, 1); % theta:3*1 [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters); % Plot the convergence graph figure; plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2); xlabel('Number of iterations'); ylabel('Cost J'); % Display gradient descent's result fprintf('Theta computed from gradient descent: \n'); fprintf(' %f \n', theta); fprintf('\n');

其中featureNormalize代码如下

function [X_norm, mu, sigma] = featureNormalize(X) X_norm = X; % X: m*2 mu = zeros(size(X,2),1); sigma = zeros(size(X, 2),1); mu = mean(X)'; % m*1 ,对每列求mean sigma = std(X)'; %m*1, 对每列求std X_norm = (X .- mu') ./ sigma'; end

其中gradientDescentMulti的代码如下

function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters) m = length(y); % number of training examples J_history = zeros(num_iters, 1); for iter = 1:num_iters error = X*theta - y; % m*1 theta = theta - alpha/m*(X'*error); J_history(iter) = computeCostMulti(X, y, theta); end end

其中computeCostMulti的代码如下

function J = computeCostMulti(X, y, theta) m = length(y); % number of training examples J = 0; error = X*theta - y; %m*1 J = 1/(2*m)*sum(error .^ 2); end

为方便查找，下附公式原图