Logistic Regression

参考网页：

http://ufldl.stanford.edu/tutorial/supervised/LogisticRegression/

在线性回归中，我们找到一个函数，对于输入x预测值为连续实值，但有时候我们想得到一个离散值，如在二分类问题中，我们想要给一个输入一个类别标签，在logistic regression中，我们想要找到另外一个函数来预测样本x属于类别1的概率。

　　

函数称为sigmoid或logistic函数，这也是logistic regression名字的由来 。

当时，样本属于类别1，反之属于类别0 。

目标函数：

　　

　　注：上式对于一个样本，只有一项为1

梯度：

　　

矩阵形式：

　　目标函数：

　　

　　梯度：

　　

　　注意：

1. m表示样本数，要除以样本数，如果不阶，那么会随着样本的数目学出不同的函数
2. 注意目标函数中的点乘

实验：

　　使用MNIST数据集中的标签为0,1的样本。图片大小为28*28

实验结果：

　　分类准确率为Accracy:100%

实验代码：

ex1b_logreg.m:

addpath ../common
addpath ../common/minFunc_2012/minFunc
addpath ../common/minFunc_2012/minFunc/compiled

% Load the MNIST data for this exercise.
% train.X and test.X will contain the training and testing images.
%   Each matrix has size [n,m] where:
%      m is the number of examples.
%      n is the number of pixels in each image.
% train.y and test.y will contain the corresponding labels (0 or 1).
binary_digits = true;
[train,test] = ex1_load_mnist(binary_digits);

% Add row of 1s to the dataset to act as an intercept term.
train.X = [ones(1,size(train.X,2)); train.X]; 
test.X = [ones(1,size(test.X,2)); test.X];

% Training set dimensions
m=size(train.X,2);
n=size(train.X,1);

% Train logistic regression classifier using minFunc
options = struct('MaxIter', 100);

% First, we initialize theta to some small random values.
theta = rand(n,1)*0.001;

% Call minFunc with the logistic_regression.m file as the objective function.
%
% TODO:  Implement batch logistic regression in the logistic_regression.m file!
%
tic;
theta=minFunc(@logistic_regression, theta, options, train.X, train.y);
fprintf('Optimization took %f seconds.\n', toc);

% Now, call minFunc again with logistic_regression_vec.m as objective.
%
% TODO:  Implement batch logistic regression in logistic_regression_vec.m using
% MATLAB's vectorization features to speed up your code.  Compare the running
% time for your logistic_regression.m and logistic_regression_vec.m implementations.
%
% Uncomment the lines below to run your vectorized code.
%theta = rand(n,1)*0.001;
%tic;
%theta=minFunc(@logistic_regression_vec, theta, options, train.X, train.y);
%fprintf('Optimization took %f seconds.\n', toc);

% Print out training accuracy.
tic;
accuracy = binary_classifier_accuracy(theta,train.X,train.y);
fprintf('Training accuracy: %2.1f%%\n', 100*accuracy);

% Print out accuracy on the test set.
accuracy = binary_classifier_accuracy(theta,test.X,test.y);
fprintf('Test accuracy: %2.1f%%\n', 100*accuracy);

 

logistic_regression.m

　　

function [f,g] = logistic_regression(theta, X,y)
  %
  % Arguments:
  %   theta - A column 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 label for each example.  y(j) is the j'th example's label.
  %

  m=size(X,2);
  
  % initialize objective value and gradient.
  f = 0;
  g = zeros(size(theta));


  %
  % TODO:  Compute the objective function by looping over the dataset and summing
  %        up the objective values for each example.  Store the result in 'f'.
  %
  % TODO:  Compute the gradient of the objective by looping over the dataset and summing
  %        up the gradients (df/dtheta) for each example. Store the result in 'g'.
  %
%%% YOUR CODE HERE %%%

% [n m] = size(X);
% theta : n * 1
% X : n * m
% y : 1 * m
%y_hat = sigmoid( theta' * X );

y_hat = sigmoid( theta' * X );  % 1*m
error = y .* log( y_hat ) + ( 1 - y ) .* log( 1 - y_hat ) ; % 1*1
error = y .* log( y_hat ) + ( 1 - y ) .* ( 1 -log( y_hat ) );
f = -1/m * sum( error(:) );
g = -1/m * X * ( y - y_hat )'; % n*1

end

 

参考资料：

http://ufldl.stanford.edu/tutorial/supervised/LogisticRegression/

http://ufldl.stanford.edu/tutorial/StarterCode/