机器学习－－线性回归的实践

1.鉴于之前提到的房价的问题，使用线性回归该如何解决呢？

首先我们假设有如下的数据方便计算机进行学习

面积	卧室	价格
2140	3	400
1600	3	330
2400	3	369
1416	2	232
...	...	...

根据之前的演算过程（使房价与面积和卧室数目线性相关）：

hθ(x)=θ0 +θ1x1 +θ2x2 

θ为计算时的权重，x1为房间面积，x2为我是数目。
为了降低计算的模糊程度，将hθ(x)变成h(x)来进行计算，这时计算公式为：

n为学习次数。

2. 有了相关数据之后就要开始训练算法了（ex1a_linreg.m）

<span style="font-size:24px;">%
%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    增加intercept term 

% Shuffle examples. 乱序 目的在于之后能够随机选取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，后面的样本来做测试
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. 开始计时
options = struct('MaxIter', 200);
theta = minFunc(@linear_regression, theta, options, train.X, train.y);
fprintf('Optimization took %f seconds.
', 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.
', 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
', 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
', 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
  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</span>

3.为了保证运算的准确性我们需要对cost function函数进行运行，得到最小成本函数（本次练习时我添加的代码）

    % Step 1 : 计算成本函数
    for i = 1:m  
        f = f + (theta' * X(:,i) - y(i))^2;  
    end  
    f = 1/2*f;  

    % Step 2:计算gradient并储存在g中   

    for j = 1:n  
        for i = 1:m  
            g(j) = g(j) + X(j,i)*(theta' * X(:,i) - y(i));  
    end