Introduction
用one-vs-all logistic regression和neural networks识别手写阿拉伯数字。
1 Multi-class Classification
多分类问题是二分类问题的扩展。在二分类问题中，一类为正项，一类为负项，线性回归所拟合的这条decision boundary，在使用sigmoid函数之后，代入函数算出预测值y代表为正项的概率。
一个疑问，如果把正负项的值对换，再学习出来的theta会不变吗，但是decision boundary两边的值变了啊？
在1v1中，我们只做了一个分类器，在one-vs-all中，可以做n个（=类数）分类器，每个分类器负责该类的预测，该类为正项，其他类为负项。在predict的时候，计算这n个分类器的值，代表属于该类的概率，选择概率值最高的类。
1.1 Dataset
一共有5000个训练样本，每个训练样本是400维的列向量（20X20像素的 grayscale image），用矩阵 X 保存。样本的结果(label of training set)保存在向量 y 中，y 是一个5000行1列的列向量。比如 y = (1,2,3,4,5,6,7,8,9,10......)T，注意，由于Matlab下标是从1开始的，故用 10 表示数字 0
用load('ex3data1.mat');可以将数据加载到matlab。
%% Machine Learning Online Class - Exercise 3 | Part 1: One-vs-all
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% linear exercise. You will need to complete the following functions
% in this exericse:
%
% lrCostFunction.m (logistic regression cost function)
% oneVsAll.m
% predictOneVsAll.m
% predict.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%
%% Initialization
clear ; close all; clc
%% Setup the parameters you will use for this part of the exercise
input_layer_size = 400; % 20x20 Input Images of Digits
num_labels = 10; % 10 labels, from 1 to 10
% (note that we have mapped "0" to label 10)
%% =========== Part 1: Loading and Visualizing Data =============
% We start the exercise by first loading and visualizing the dataset.
% You will be working with a dataset that contains handwritten digits.
%
% Load Training Data
fprintf('Loading and Visualizing Data ...\n')
load('ex3data1.mat'); % training data stored in arrays X, y
m = size(X, 1);
1.2 Visualizing the data
报警了，不想看。
% Randomly select 100 data points to display
rand_indices = randperm(m);
sel = X(rand_indices(1:100), :);
displayData(sel);
fprintf('Program paused. Press enter to continue.\n');
pause;
function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
% [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
% stored in X in a nice grid. It returns the figure handle h and the
% displayed array if requested.
% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width)
example_width = round(sqrt(size(X, 2)));
end
% Gray Image
colormap(gray);
% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);
% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);
% Between images padding
pad = 1;
% Setup blank display
display_array = - ones(pad + display_rows * (example_height + pad), ...
pad + display_cols * (example_width + pad));
% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
for i = 1:display_cols
if curr_ex > m,
break;
end
% Copy the patch
% Get the max value of the patch
max_val = max(abs(X(curr_ex, :)));
display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
reshape(X(curr_ex, :), example_height, example_width) / max_val;
curr_ex = curr_ex + 1;
end
if curr_ex > m,
break;
end
end
% Display Image
h = imagesc(display_array, [-1 1]);
% Do not show axis
axis image off
drawnow;
end
1.3 Vectorizing Logistic Regression
熟悉的向量化，基本写三次就会了，写出routine了都。代价函数就是同一个套路。不懂得地方在于regularize。
后面加上一项lambda来regularize是为了防止过拟合，但是我忘记为什么这样可以防止过拟合了。
%% ============ Part 2a: Vectorize Logistic Regression ============
% In this part of the exercise, you will reuse your logistic regression
% code from the last exercise. You task here is to make sure that your
% regularized logistic regression implementation is vectorized. After
% that, you will implement one-vs-all classification for the handwritten
% digit dataset.
%
% Test case for lrCostFunction
fprintf('\nTesting lrCostFunction() with regularization');
theta_t = [-2; -1; 1; 2];
X_t = [ones(5,1) reshape(1:15,5,3)/10];
y_t = ([1;0;1;0;1] >= 0.5);
lambda_t = 3;
[J grad] = lrCostFunction(theta_t, X_t, y_t, lambda_t);
fprintf('\nCost: %f\n', J);
fprintf('Expected cost: 2.534819\n');
fprintf('Gradients:\n');
fprintf(' %f \n', grad);
fprintf('Expected gradients:\n');
fprintf(' 0.146561\n -0.548558\n 0.724722\n 1.398003\n');
fprintf('Program paused. Press enter to continue.\n');
pause;
1.3.1 Vectorizing the cost function
function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with
%regularization
% J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
% theta as the parameter for regularized logistic regression and the
% gradient of the cost w.r.t. to the parameters.
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
grad = zeros(size(theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
% You should set J to the cost.
% Compute the partial derivatives and set grad to the partial
% derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
% efficiently vectorized. For example, consider the computation
%
% sigmoid(X * theta)
%
% Each row of the resulting matrix will contain the value of the
% prediction for that example. You can make use of this to vectorize
% the cost function and gradient computations.
%
% Hint: When computing the gradient of the regularized cost function,
% there're many possible vectorized solutions, but one solution
% looks like:
% grad = (unregularized gradient for logistic regression)
% temp = theta;
% temp(1) = 0; % because we don't add anything for j = 0
% grad = grad + YOUR_CODE_HERE (using the temp variable)
%
%theta_s = [0; theta(2:end)];
J=(-y'*log(sigmoid(X*theta))-(1-y)'*log(1-sigmoid(X*theta)))/m+lambda * sum(theta(2:end).^2)/(2*m);
grad=(X'*(sigmoid(X*theta)-y))/m;
temp=theta;
temp(1)=0;
%grad=grad+lambda/m*sum(temp);
grad=grad+lambda/m*temp;
% =============================================================
grad = grad(:);
end
1.3.2 Vectorizing the gradient
要注意的一点是the extra bias unit 项不需要regularize，有好几种写法，都可以把bias项置为0。
1.3.3 Vectorizing regularized logistic regressionA
在写gradient的regularization的那一项不需要求和的。
1.4 One-vs-all Classification
num_labels 为分类器个数，共10个，每个分类器(模型)用来识别10个数字中的某一个。
我们一共有5000个样本，每个样本有400中特征变量，因此：模型参数θ 向量有401个元素。
all_theta是一个10*401的矩阵，每一行存储着一个分类器(模型)的模型参数θ 向量。
不知道怎么把theta行向量逐条依次插入到all_theta矩阵中。
一开始根本没有想到theta有四百个（大雾
没有见过这么多参数的线性方程，吓得我以为自己写错了什么。
应该是没错的。
1.4.1 One-vs-all Prediction
主要是matlab不太会使，对于每一条X，这里要找出算出10个分类器的10个假设函数的值中最大的一项，返回它的num_labels。
function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels
%are in the range 1..K, where K = size(all_theta, 1).
% p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
% for each example in the matrix X. Note that X contains the examples in
% rows. all_theta is a matrix where the i-th row is a trained logistic
% regression theta vector for the i-th class. You should set p to a vector
% of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
% for 4 examples)
m = size(X, 1);
num_labels = size(all_theta, 1);
% You need to return the following variables correctly
p = zeros(size(X, 1), 1);
% Add ones to the X data matrix
X = [ones(m, 1) X];
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned logistic regression parameters (one-vs-all).
% You should set p to a vector of predictions (from 1 to
% num_labels).
%
% Hint: This code can be done all vectorized using the max function.
% In particular, the max function can also return the index of the
% max element, for more information see 'help max'. If your examples
% are in rows, then, you can use max(A, [], 2) to obtain the max
% for each row.
%
[maxnum, p]=max(sigmoid(X*all_theta'),[],2);
% [maxnum,p]=max(A)：返回行向量maxnum和p，maxnum向量记录A的每列的最大值，p向量记录每列最大值的行号。
% =========================================================================
end
补充一下max函数的用法
求矩阵A的最大值的函数有3种调用格式，分别是：
(1) max(A)：返回一个行向量，向量的第i个元素是矩阵A的第i列上的最大值。
(2) [Y,U]=max(A)：返回行向量Y和U，Y向量记录A的每列的最大值，U向量记录每列最大值的行号。
(3) max(A,[],dim)：dim取1或2。dim取1时，该函数和max(A)完全相同；dim取2时，该函数返回一个列向量，其第i个元素是A矩阵的第i行上的最大值。
求最小值的函数是min，其用法和max完全相同。
C = max(A,[],dim)
返回A中有dim指定的维数范围中的最大值。比如C=max(A,[],2),在矩阵中，第1维度表示列，第2维度表示行，这个例子就是将第二维度，也就是行这个维度中，将同一行的不同列的最大值提取出来：
