∂J(θ)∂θj=∑ix(i)j(hθ(x(i))−y(i))

%

%This exercise uses a data from the UCI repository:

% Bache, K. & Lichman, M. (2013). UCI Machine Learning Repository

% http://archive.ics.uci.edu/ml

% Irvine, CA: University of California, School of Information and Computer Science.

%

%Data created by:

% Harrison, D. and Rubinfeld, D.L.

% ''Hedonic prices and the demand for clean air''

% J. Environ. Economics & Management, vol.5, 81-102, 1978.

%

addpath ../common

addpath ../common/minFunc_2012/minFunc

addpath ../common/minFunc_2012/minFunc/compiled

% Load housing data from file.

data = load('housing.data');  % housing data  506x14

data=data'; % put examples in columns  14x506  一般这里将每一个样本放在每一列

% Include a row of 1s as an additional intercept feature.

data = [ ones(1,size(data,2)); data ];  % 15x506    添加intercept term 

% Shuffle examples. 乱序 目的在于之后可以随机选取training set和test sets

data = data(:, randperm(size(data,2))); %randperm(n)用于随机生成1到n的排列

% Split into train and test sets

% The last row of 'data' is the median home price.

train.X = data(1:end-1,1:400);   %选择前400个样本来训练，后面的样本来做測试

train.y = data(end,1:400);

test.X = data(1:end-1,401:end);

test.y = data(end,401:end);

m=size(train.X,2);  %训练样本数量

n=size(train.X,1);  %每一个样本的变量个数

% Initialize the coefficient vector theta to random values.

theta = rand(n,1); %随机生成初始theta 每一个值在(0,1)之间

% Run the minFunc optimizer with linear_regression.m as the objective.

%

% TODO:  Implement the linear regression objective and gradient computations

% in linear_regression.m

%

tic; %Start a stopwatch timer. 開始计时

options = struct('MaxIter', 200);

theta = minFunc(@linear_regression, theta, options, train.X, train.y);

fprintf('Optimization took %f seconds.\n', toc); %toc Read the stopwatch timer

% Run minFunc with linear_regression_vec.m as the objective.

%

% TODO:  Implement linear regression in linear_regression_vec.m

% using MATLAB's vectorization features to speed up your code.

% Compare the running time for your linear_regression.m and

% linear_regression_vec.m implementations.

%

% Uncomment the lines below to run your vectorized code.

%Re-initialize parameters

%theta = rand(n,1);

%tic;

%theta = minFunc(@linear_regression_vec, theta, options, train.X, train.y);

%fprintf('Optimization took %f seconds.\n', toc);

% Plot predicted prices and actual prices from training set.

actual_prices = train.y;

predicted_prices = theta'*train.X;

% Print out root-mean-squared (RMS) training error.平方根误差

train_rms=sqrt(mean((predicted_prices - actual_prices).^2));

fprintf('RMS training error: %f\n', train_rms);

% Print out test RMS error

actual_prices = test.y;

predicted_prices = theta'*test.X;

test_rms=sqrt(mean((predicted_prices - actual_prices).^2));

fprintf('RMS testing error: %f\n', test_rms);

% Plot predictions on test data.

plot_prices=true;

if (plot_prices)

  [actual_prices,I] = sort(actual_prices); %从小到大排序价格。I为index

  predicted_prices=predicted_prices(I);

  plot(actual_prices, 'rx');

  hold on;

  plot(predicted_prices,'bx');

  legend('Actual Price', 'Predicted Price');

  xlabel('House #');

  ylabel('House price ($1000s)');

end

function [f,g] = linear_regression(theta, X,y)

  %

  % Arguments:

  %   theta - A vector containing the parameter values to optimize.

  %   X - The examples stored in a matrix.

  %       X(i,j) is the i'th coordinate of the j'th example.

  %   y - The target value for each example.  y(j) is the target for example j.

  %

  m=size(X,2);

  n=size(X,1);

  f=0;

  g=zeros(size(theta));

  %

  % TODO:  Compute the linear regression objective by looping over the examples in X.

  %        Store the objective function value in 'f'.

  %

  % TODO:  Compute the gradient of the objective with respect to theta by looping over

  %        the examples in X and adding up the gradient for each example.  Store the

  %        computed gradient in 'g'.

%%% YOUR CODE HERE %%%

% Step 1 : Compute f cost function

for i = 1:m

    f = f + (theta' * X(:,i) - y(i))^2;

end

f = 1/2*f;

% Step 2: Compute gradient 

for j = 1:n

    for i = 1:m

        g(j) = g(j) + X(j,i)*(theta' * X(:,i) - y(i));

    end

end

3.3 Result

Optimization took 3.374166 seconds.
RMS training error: 4.679871
RMS testing error: 4.865463