Introduction
用one-vs-all logistic regression和neural networks識(shí)別手寫阿拉伯?dāng)?shù)字。
1 Multi-class Classification
多分類問題是二分類問題的擴(kuò)展。在二分類問題中，一類為正項(xiàng)，一類為負(fù)項(xiàng)，線性回歸所擬合的這條decision boundary，在使用sigmoid函數(shù)之后，代入函數(shù)算出預(yù)測(cè)值y代表為正項(xiàng)的概率。
一個(gè)疑問，如果把正負(fù)項(xiàng)的值對(duì)換，再學(xué)習(xí)出來的theta會(huì)不變嗎，但是decision boundary兩邊的值變了啊？
在1v1中，我們只做了一個(gè)分類器，在one-vs-all中，可以做n個(gè)（=類數(shù)）分類器，每個(gè)分類器負(fù)責(zé)該類的預(yù)測(cè)，該類為正項(xiàng)，其他類為負(fù)項(xiàng)。在predict的時(shí)候，計(jì)算這n個(gè)分類器的值，代表屬于該類的概率，選擇概率值最高的類。
1.1 Dataset
一共有5000個(gè)訓(xùn)練樣本，每個(gè)訓(xùn)練樣本是400維的列向量（20X20像素的 grayscale image），用矩陣 X 保存。樣本的結(jié)果(label of training set)保存在向量 y 中，y 是一個(gè)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下標(biāo)是從1開始的，故用 10 表示數(shù)字 0
用load('ex3data1.mat');可以將數(shù)據(jù)加載到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
報(bào)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，基本寫三次就會(huì)了，寫出routine了都。代價(jià)函數(shù)就是同一個(gè)套路。不懂得地方在于regularize。
后面加上一項(xiàng)lambda來regularize是為了防止過擬合，但是我忘記為什么這樣可以防止過擬合了。
%% ============ 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
要注意的一點(diǎn)是the extra bias unit 項(xiàng)不需要regularize，有好幾種寫法，都可以把bias項(xiàng)置為0。
1.3.3 Vectorizing regularized logistic regressionA
在寫gradient的regularization的那一項(xiàng)不需要求和的。
1.4 One-vs-all Classification
num_labels 為分類器個(gè)數(shù)，共10個(gè)，每個(gè)分類器(模型)用來識(shí)別10個(gè)數(shù)字中的某一個(gè)。
我們一共有5000個(gè)樣本，每個(gè)樣本有400中特征變量，因此：模型參數(shù)θ 向量有401個(gè)元素。
all_theta是一個(gè)10*401的矩陣，每一行存儲(chǔ)著一個(gè)分類器(模型)的模型參數(shù)θ 向量。
不知道怎么把theta行向量逐條依次插入到all_theta矩陣中。
一開始根本沒有想到theta有四百個(gè)（大霧
沒有見過這么多參數(shù)的線性方程，嚇得我以為自己寫錯(cuò)了什么。
應(yīng)該是沒錯(cuò)的。
1.4.1 One-vs-all Prediction
主要是matlab不太會(huì)使，對(duì)于每一條X，這里要找出算出10個(gè)分類器的10個(gè)假設(shè)函數(shù)的值中最大的一項(xiàng)，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向量記錄A的每列的最大值，p向量記錄每列最大值的行號(hào)。
% =========================================================================
end
補(bǔ)充一下max函數(shù)的用法
求矩陣A的最大值的函數(shù)有3種調(diào)用格式，分別是：
(1) max(A)：返回一個(gè)行向量，向量的第i個(gè)元素是矩陣A的第i列上的最大值。
(2) [Y,U]=max(A)：返回行向量Y和U，Y向量記錄A的每列的最大值，U向量記錄每列最大值的行號(hào)。
(3) max(A,[],dim)：dim取1或2。dim取1時(shí)，該函數(shù)和max(A)完全相同；dim取2時(shí)，該函數(shù)返回一個(gè)列向量，其第i個(gè)元素是A矩陣的第i行上的最大值。
求最小值的函數(shù)是min，其用法和max完全相同。
C = max(A,[],dim)
返回A中有dim指定的維數(shù)范圍中的最大值。比如C=max(A,[],2),在矩陣中，第1維度表示列，第2維度表示行，這個(gè)例子就是將第二維度，也就是行這個(gè)維度中，將同一行的不同列的最大值提取出來：
