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 datadata = load('ex1data2.txt');X = data(:, 1:2); %X : m*2y = data(:, 3);  % y : m*1m = length(y)%Scale features and set them to zero meanfprintf('Normalizing Features ...\n');[X mu sigma] = featureNormalize(X);% Add intercept term to XX = [ones(m, 1) X];fprintf('Running gradient descent ...\n');% Choose some alpha valuealpha = 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 graphfigure;plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2); xlabel('Number of iterations');ylabel('Cost J');% Display gradient descent's resultfprintf('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*2mu = zeros(size(X,2),1); sigma = zeros(size(X, 2),1); mu = mean(X)'; % m*1 ,对每列求meansigma = std(X)'; %m*1, 对每列求stdX_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 examplesJ_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);endend

其中computeCostMulti的代码如下

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

为方便查找，下附公式原图