机器学习week2 ex1 review

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机器学习week2 ex1 review

 

  • 这周的作业主要关于线性回归

 


1. Linear regression with one variable

1.1 Plotting the Data

  • 通过已有的城市人口和盈利的数据,来预测在一个新城市营业的收入。
    文件ex1data1.txt包含了以下数据:

    第一列是城市人口数据,第二列是盈利金额(负数代表亏损)。

  • 可以首先通过绘图来直观感受。
    这里我们可以使用散点图scatter plot )。
    通过 Octave/MATLAB 实现。
    数据存储在ex1data1.txt中,如下图所示:
    image_1bto7dffa1k4b7vnvs2nl7u6mm.png-34.9kB
    文件ex1.mex1data1.txt中的所有数据载入data:

    data = load(\'ex1data1.txt\');      % read comma separated data
    

Octave/MATLAB 会根据数据文件中的逗号来分隔数据。这里的data以矩阵形式存储。因为有两组数据,所以是两列。
然后分别将人口和盈利存入 Xy 中。

X = data(:,1); y = data(:,2);
m = length(y);

Xdata的第一列,y 取第二列。mtraining example 的数量。因为 Xy 大小相同,用哪个都无所谓。


下一步调用 PlotData 函数绘制散点图。
我们需要首先把这个函数补充完整。
这是文件中包含的原始版本:

function plotData
%PLOTDATA Plots the data points x and y into a new figure 
%   PLOTDATA(x,y) plots the data points and gives the figure axes labels of
%   population and profit.

figure; % open a new figure window

% ====================== YOUR CODE HERE ======================
% Instructions: Plot the training data into a figure using the 
%               "figure" and "plot" commands. Set the axes labels using
%               the "xlabel" and "ylabel" commands. Assume the 
%               population and revenue data have been passed in
%               as the x and y arguments of this function.
%
% Hint: You can use the \'rx\' option with plot to have the markers
%       appear as red crosses. Furthermore, you can make the
%       markers larger by using plot(..., \'rx\', \'MarkerSize\', 10);





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

end

我们可以看到他要求我们绘制散点图,并给横纵坐标加上 population 和 profit 的标签。
Hint 提示我们这样做:

  • 使用figure打开一个图形窗口。
  • 使用plot函数绘图。用xlabelylabel分别设置横纵坐标的标签。rx将散点设置为红十字形(red cross)。通过\'Markersize\',10\'设置散点大小。

代码如下:

plot(x, y, \'rx\', \'Markersize\', 10);           % plot the data
ylabel(\'Profit in $10,000s\');                 % set the y-axis label
xlabel(\'Population of City in 10,000s\');      % set the x-axis label

这时再运行ex1.m,就可以得到如下图形:
Figure 1: Scatter plot of training data

1.2 Gradient Descent

  • 通过梯度下降算法计算线性回归参数 \\theta

1.2.1 Update Equations



线性回归的目标就是使代价函数最小
J(\\theta) = 
\\displaystyle{\\frac {1} {2m}\\sum_{i=1}^{m}}(h_\\theta(x^{(i)})-y^{(i)})^2
其中
h_\\theta(x) = \\theta^Tx = \\theta_0+\\theta_1x_1


使用批量梯度下降法batch gradient descent ) 以使 J(\\theta) 最小。
\\theta_j := \\theta_j -\\alpha \\displaystyle{\\frac{1}{m}\\sum_{i=1}^{m}}(h_\\theta(x^{(i)})-y^{(i)})x_j^{(i)}
( simultaneously update \\theta_j for all j )

1.2.2 Implementation

目前的 X 是一个列向量,每一行存储一个 training example,即 x_1
因此在脚本文件ex1.m中,为了处理 \\theta_0 , 给每一行增加一个 x_0 = 1

X = [ones(m,1) data(:,1)];   % Add a column of ones to x

因为只有一个变量 x_1 影响盈利,
\\theta = \\left[ \\begin{matrix} \\theta _{0}\\\\\\theta _{1}\\end{matrix} \\right]
初始化 \\theta\\left[ \\begin{matrix} 0 \\\\ 0 \\end{matrix} \\right]

theta = zeros(2,1);  % initialize fitting parameters

设置迭代次数和 \\alpha 的值:

iterations = 1500;
alpha = 0.01;

1.2.3 Computing the cost J(\\theta)

根据上述公式,完成computeCost以计算代价函数 J(\\theta)

function J = computeCost(X, y, theta)
%COMPUTECOST Compute cost for linear regression
%   J = COMPUTECOST(X, y, theta) computes the cost of using theta as the
%   parameter for linear regression to fit the data points in X and y

% Initialize some useful values
m = length(y); % number of training examples
prediction = X * theta;
sqError = (prediction - y).^2;

% You need to return the following variables correctly 
J = 0;

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
%               You should set J to the cost.

J = 1/(2 * m) * sum(sqError);



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

end

我们来看一下脚本文件ex1.m中这一部分的测试代码

fprintf(\'\\nTesting the cost function ...\\n\')
% compute and display initial cost
J = computeCost(X, y, theta);
fprintf(\'With theta = [0 ; 0]\\nCost computed = %f\\n\', J);
fprintf(\'Expected cost value (approx) 32.07\\n\');

% further testing of the cost function
J = computeCost(X, y, [-1 ; 2]);
fprintf(\'\\nWith theta = [-1 ; 2]\\nCost computed = %f\\n\', J);
fprintf(\'Expected cost value (approx) 54.24\\n\');

fprintf(\'Program paused. Press enter to continue.\\n\');
pause;

它对两组数据进行了测试。一组是我们之前初始化后的\\theta = \\left[ \\begin{matrix} 0 \\\\ 0 \\end{matrix} \\right] , 另一组是\\left[ \\begin{matrix} -1 \\\\ 2 \\end{matrix} \\right]
如果你的computeCost.m计算正确的话,输出的两个答案应该是32.072734和54.242455。

1.2.4 Gradient descent

  • 根据之前的公式
    \\theta_j := \\theta_j -\\alpha \\displaystyle{\\frac{1}{m}\\sum_{i=1}^{m}}(h_\\theta(x^{(i)})-y^{(i)})x_j^{(i)}
    ( simultaneously update \\theta_j for all j )

补充gradientDescent.m的代码。如下:

%GRADIENTDESCENT Performs gradient descent to learn theta
%   theta = GRADIENTDESCENT(X, y, theta, alpha, num_iters) updates theta by 
%   taking num_iters gradient steps with learning rate alpha

% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);
temp = theta;
n = length(theta);

for iter = 1:num_iters

    % ====================== YOUR CODE HERE ======================
    % Instructions: Perform a single gradient step on the parameter vector
    %               theta. 
    %
    % Hint: While debugging, it can be useful to print out the values
    %       of the cost function (computeCost) and gradient here.
    %
    for j = 1:n
        temp(j) = theta(j) - 1/m*alpha*sum((X*theta-y).*X(:,j));
    end;
    theta = temp;






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

    % Save the cost J in every iteration    
    J_history(iter) = computeCost(X, y, theta);

end

end

事实上这已经解决了多变量的线性回归问题,尽管这里只用处理 n=2 的情况。

再来看一下脚本文件里这一部分的内容:

fprintf(\'\\nRunning Gradient Descent ...\\n\')
% run gradient descent
theta = gradientDescent(X, y, theta, alpha, iterations);

% print theta to screen
fprintf(\'Theta found by gradient descent:\\n\');
fprintf(\'%f\\n\', theta);
fprintf(\'Expected theta values (approx)\\n\');
fprintf(\' -3.6303\\n  1.1664\\n\\n\');

% Plot the linear fit
hold on; % keep previous plot visible
plot(X(:,2), X*theta, \'-\')
legend(\'Training data\', \'Linear regression\')
hold off  % do not overlay any more plots on this figure

% Predict values for population sizes of 35,000 and 70,000
predict1 = [1, 3.5] *theta;
fprintf(\'For population = 35,000, we predict a profit of %f\\n\',...
    predict1*10000);
predict2 = [1, 7] * theta;
fprintf(\'For population = 70,000, we predict a profit of %f\\n\',...
    predict2*10000);

fprintf(\'Program paused. Press enter to continue.\\n\');
pause;

使用梯度下降法后输出计算得到的 \\theta:
image_1btongdv2afu1dae13bv1e3i1mimm.png-7.7kB
与正确情况吻合。

之后绘制图线。对之前的散点图使用hold on,保留图形。
再绘制经过梯度下降后得到的 Xh_\\theta(x) = X\\theta 的图线。
得到如下图形:
image_1bton9ngt29919hc1duq3uh9339.png-36kB
再对population = 35000 和 70000的情况进行估计,输出这两种情况下的估计值:
image_1btoni94o1rg9d14107ktcu1oat13.png-8.5kB

1.3 Visualizing J(\\theta)

脚本文件ex1.m提供了对 J(\\theta) 可视化的部分。

fprintf(\'Visualizing J(theta_0, theta_1) ...\\n\')

% Grid over which we will calculate J
theta0_vals = linspace(-10, 10, 100);
theta1_vals = linspace(-1, 4, 100);

函数 linspace(BASE,LIMIT,N=100) 返回一个从BASE到LIMIT的等间距分布的行向量;如果BASE和LIMIT是列向量的话,返回一个矩阵。不输入N的时候默认为100。

% initialize J_vals to a matrix of 0 s
J_vals = zeros(length(theta0_vals), length(theta1_vals));

% Fill out J_vals
for i = 1:length(theta0_vals)
    for j = 1:length(theta1_vals)
      t = [theta0_vals(i); theta1_vals(j)];
      J_vals(i,j) = computeCost(X, y, t);
    end
end

\\theta_0\\theta_1 平面上的点求出其代价函数值。
绘制曲面图:

% Because of the way meshgrids work in the surf command, we need to
% transpose J_vals before calling surf, or else the axes will be flipped
J_vals = J_vals\';
% Surface plot
figure;
surf(theta0_vals, theta1_vals, J_vals)
xlabel(\'\\theta_0\'); ylabel(\'\\theta_1\');

image_1btorckqkshi1l161shbka43263a.png-179.6kB
绘制等值线图:

% Contour plot
figure;
% Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100
contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20))
xlabel(\'\\theta_0\'); ylabel(\'\\theta_1\');
hold on;
plot(theta(1), theta(2), \'rx\', \'MarkerSize\', 10, \'LineWidth\', 2);

image_1btordvej9j1e3sdun1lvcnm33t.png-53.8kB
从图中可以看出最小值所在的位置。


2. Linear regression with multiple variables

这一次需要处理多变量的线性回归。比如预测房价,需要考虑的因素可能就包括房子的大小、卧室的数量。
在文件ex1data2.txt中有多变量的 training example。 如下图所示:
image_1btorn0cm5f9vrqa3i1tjr1n374a.png-40.5kB
共有三列,第一列是房子面积(单位:平方英尺),第二列是卧室数量,第三列是房价。

2.1 Feature normalization

脚本文件ex1_multi中首先展示部分数据:

%% Load Data 
data = load(\'ex1data2.txt\');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);

% Print out some data points
fprintf(\'First 10 examples from the dataset: \\n\');
fprintf(\' x = [%.0f %.0f], y = %.0f \\n\', [X(1:10,:) y(1:10,:)]\');

fprintf(\'Program paused. Press enter to continue.\\n\');
pause;

image_1btos2cq66mm8d91d00oa21v14n.png-12.3kB
通过观察数据可以发现,第一列数据大小比第二列数据高三个数量级,需要进行标准化Normalization
标准化包括如下步骤:

  • 减去平均值
  • 除以标准差(因为大部分数据会落在平均值\\pm标准差的范围内),也可以直接选择用max-min来代替

代码如下:

function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X 
%   FEATURENORMALIZE(X) returns a normalized version of X where
%   the mean value of each feature is 0 and the standard deviation
%   is 1. This is often a good preprocessing step to do when
%   working with learning algorithms.

% You need to set these values correctly
X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));

% ====================== YOUR CODE HERE ======================
% Instructions: First, for each feature dimension, compute the mean
%               of the feature and subtract it from the dataset,
%               storing the mean value in mu. Next, compute the 
%               standard deviation of each feature and divide
%               each feature by it\'s standard deviation, storing
%               the standard deviation in sigma. 
%
%               Note that X is a matrix where each column is a 
%               feature and each row is an example. You need 
%               to perform the normalization separately for 
%               each feature. 
%
% Hint: You might find the \'mean\' and \'std\' functions useful.
% 

mu = mean(X);
sigma = std(X);
for i = 1:size(X,1)
   X_norm(i,:) = (X(i,:)-mu)./sigma;
end;









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

end

meanstd 分别用来计算向量的平均值和标准差,如果是对象是矩阵的话,默认计算每列的平均值和标准差,然后返回一个行向量。

2.2 Gradient descent

这一部分(包括梯度下降和代价函数),我们在单变量的时候处理的时候已经可以用于多变量了。略去。
值得一提的是,在计算代价函数的时候,用如下方法计算是很有效的:
image_1btot5mnqv7tmgp1tce1dhr1cn754.png-56.1kB

2.2.1 Selecting learning rates

可以通过修改ex1_multi.m中的learning rate来直观感受其作用。
其中有如下代码:

% Init Theta and Run Gradient Descent 
theta = zeros(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\');

这段代码的功能是画出 J(\\theta) 随迭代次数的变化情况。
numel函数返回对象的元素个数。
J_history内存储了每次迭代后的代价函数值。在gradientDescentMulti.m中,我们每一次循环中有这样的步骤来计算J_history:

    % Save the cost J in every iteration    
    J_history(iter) = computeCostMulti(X, y, theta);

设置learning rate \\alpha 为 0.01,0.03,0.1,1,1.5 画出的图像依次如下所示:

image_1btp4lfg21t7j7fpej21qiq19985h.png-25.4kB
image_1btp4ncpa1cfg1248kei1eec1d195u.png-27.2kB
image_1btp4prvh161v7o9f1e1flsral7b.png-25.3kB
image_1btp4r2nk11181hmjh85f6p13hq7o.png-22kB
image_1btp4vmu3p6rhbi7lrqbkllq85.png-22.4kB
可以注意到,起初 \\alpha 设置得很小的时候,下降非常缓慢;\\alpha 适当增大之后,下降速度变快;而 \\alpha 过大时,图线不降反升。

用梯度下降法在ex1_multi中计算1650平方英尺,3间卧室的房子的价格:

testify = [1,1650, 3];
price = (testify - [0 mu]) ./ [1 sigma] * theta;

需要记得,在使用时要先进行normalization。
输出的结果是:
image_1btp6hacsggttin1uvvq4d1io92.png-10.8kB

2.3 Normal equations

\\theta = (X^TX)^{-1}X^T\\vec {y}

代码相当简单:

function [theta] = normalEqn(X, y)
%NORMALEQN Computes the closed-form solution to linear regression 
%   NORMALEQN(X,y) computes the closed-form solution to linear 
%   regression using the normal equations.

theta = zeros(size(X, 2), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the code to compute the closed form solution
%               to linear regression and put the result in theta.
%

% ---------------------- Sample Solution ----------------------


theta = pinv(X\' * X) * X\' * y;

% -------------------------------------------------------------

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

end

用normal Equation在ex1_multi中计算1650平方英尺,3间卧室的房子的价格:

price = testify * theta;

normal equation 不需要进行normalization。
输出的结果是:
image_1btp6hv3r1uecu3s1sg1osjfdu9f.png-11.1kB

与之前用梯度下降法求出的结果吻合得相当精确。