Classification and logistic regression

logistic 回归

1.问题：

在上面讨论回归问题时。讨论的结果都是连续类型。但假设要求做分类呢？即讨论结果为离散型的值。

2.解答：

• 假设：
  
  当中：
  
  g(z)的图形例如以下：
  
  
  
  由此可知：当hθ(x)<0.5时我们能够觉得为0，反之为1，这样就变成离散型的数据了。

• 推导迭代式：

  • 利用概率论进行推导，找出样本服从的分布类型，利用最大似然法求出对应的θ
  • 
  • 因此：
    
    

• 结果：

• 注意：这里的迭代式增量迭代法

Newton迭代法：

1.问题：

上述迭代法，收敛速度非常慢，在利用最大似然法求解的时候能够运用Newton迭代法，即θ := θ−f(θ)f′(θ)

2.解答：

• 推导：

  • Newton迭代法是求θ，且f(θ)=0。刚好：l′(θ)=0
  • 所以能够将Newton迭代法改写成：

• 定义：

  • 当中：l′(θ) =
  • 
    
    因此：H矩阵就是l′′(θ)，即H−1 = 1/l′′(θ)
  • 所以：

• 应用：

  • 特征值比較少的情况，否则H−1的计算量是非常大的

Logistic 0、1分类：

1.自己设定迭代次数

　　自己编写对应的循环，给出迭代次数以及下降坡度alpha，进行增量梯度下降。

主要函数及功能：

• Logistic_Regression 相当于主函数
• gradientDecent 梯度下降更新θ函数
• computeCost 计算损失J函数

Logistic_Regression

%%  part0： 准备
data = load('ex2data1.txt');
x = data(:,[1,2]);
y = data(:,3);
pos = find(y==1);
neg = find(y==0);

x1 = x(:,1);
x2 = x(:,2);
plot(x(pos,1),x(pos,2),'r*',x(neg,1),x(neg,2),'co');
pause;

%% part1: GradientDecent and compute cost of J
[m,n] = size(x);
x = [ones(m,1),x];
theta = zeros(3,1);
J = computeCost(x,y,theta);

theta = gradientDecent(x, y, theta);
X = 25:100;
Y = ( -theta(1,1) - theta(3,1)*X)/theta(2,1);
plot(x(pos,2),x(pos,3),'r*',x(neg,2),x(neg,3),'co', X, Y, 'b');
pause;

gradientDecent

function theta = gradientDecent(x, y, theta)

%% compute GradientDecent 更新theta,利用的是增量梯度下降
m = size(x,1);
alph = 0.001;
for iter = 1:150000
    for j = 1:3
        dec = 0;
        for i = 1:m
            dec = dec + (y(i) - sigmoid(x(i,:)*theta))*x(i,j);
        end
        theta(j,1) = theta(j,1) + dec*alph/m;
    end
end
end

sigmoid

function g = sigmoid(z)

%% SIGMOID Compute sigmoid functoon

g = 1/(1+exp(-z));

end

computeCost

function J = computeCost(x, y, theta)

%% compute cost: J

m = size(x,1);
J = 0;
for i = 1:m
   J =  J + y(i)*log(sigmoid(x(i,:)*theta)) + (1 - y(i))*log(1 - sigmoid(x(i,:)*theta));
end
J = (-1/m)*J;
end

结果例如以下：

2. 利用fminunc函数：

　　给出损失J的计算方式和θ的计算方式。然后调用fminunc函数计算出最优解

主要函数及功能：

• Logistics_Regression 相当于主函数
• computeCost给出J和θ的计算方式
• sigmoid函数

Logistics_Regression

%%  part0： 准备
data = load('ex2data1.txt');
x = data(:,[1,2]);
y = data(:,3);
pos = find(y==1);
neg = find(y==0);

x1 = x(:,1);
x2 = x(:,2);
plot(x(pos,1),x(pos,2),'r*',x(neg,1),x(neg,2),'co');
pause;

%% part1: GradientDecent and compute cost of J
[m,n] = size(x);
x = [ones(m,1),x];
theta = zeros(3,1);
options = optimset('GradObj', 'on', 'MaxIter', 400);

%  Run fminunc to obtain the optimal theta
%  This function will return theta and the cost
[theta, cost] = ...
    fminunc(@(t)(computeCost(x,y,t)), theta, options);
X = 25:100;
Y = ( -theta(1,1) - theta(3,1)*X)/theta(2,1);
plot(x(pos,2),x(pos,3),'r*',x(neg,2),x(neg,3),'co', X, Y, 'b');
pause;

sigmoid

function g = sigmoid(z)

%% SIGMOID Compute sigmoid functoon

g = zeros(size(z));
g = 1.0 ./ (1.0 + exp(-z));

end

computeCost

function [J,grad] = computeCost(x, y, theta)

%% compute cost: J

m = size(x,1);
grad = zeros(size(theta));
hx = sigmoid(x * theta);
 J = (1.0/m) * sum(-y .* log(hx) - (1.0 - y) .* log(1.0 - hx));
 grad = (1.0/m) .* x' * (hx - y);
end

结果

Logistic multi_class

1.条件

• 自己做的数据：

1,5,1
1,6,1
1.5,3.5,1
2.5,3.5,1
2,6,1
3,7,1
4,6,1
3.5,4.5,1
2,4,1
2,5,1
4,4,1
5,5,1
6,4,1
5,3,1
4,2,1
4,3,2
5,3,2
5,2,2
5,1.5,2
7,1.5,2
5,2.5,2
6,2.5,2
5.5,2.5,2
5,1,2
6,2,2
6,3,2
5,4,2
7,5,2
7,2,2
8,1,2
8,3,2
7,4,3
7,5,3
8.5,5.5,3
9,4,3
8,5.5,3
8,4.5,3
9.5,5.5,3
8,4.5,3
8.5,4.5,3
7,6,3
6,5,3
9,5,3
9,6,3
8,6,3
8,7,3
10,6,3
10,4,3

• 数据离散图：

  

2.算法推到

• 花费J :
  
  

• 更新θ：
  
  

• 算法思路（这个算法也叫one_vs_all）：

  

  假设样本分成K类，。那我们训练K组θ，依次考虑每一类样本，然后把其他的全部样本当做一类样本，这样就把这类样本和其他分开了。我们把考虑的那类样本的y值改为1，其他为0。这样就得到K组θ值。

3.代码实现：

这里採用fminuc函数实现

1.函数级功能简单介绍：

• Logistic_Regression : 相当于主函数
• oneVsAll: 写成一个循环，依次计算出K组θ。利用fminunc调用计算函数
• computeCost：当中主要写J&θ更新函数

2.代码：

• Logistic_Regerssion:

%%  part0： 准备
data = load('data.txt');
x = data(:,[1,2]);
y = data(:,3);
y1 = find(y==1);
y2 = find(y==2);
y3 = find(y==3);

plot(x(y1,1),x(y1,2),'r*',x(y2,1),x(y2,2),'c+',x(y3,1),x(y3,2),'bo');
pause;

%% part1: GradientDecent and compute cost of J
[m,n] = size(x);
x = [ones(m,1),x];
theta = zeros(3,3);

%  Run fminunc to obtain the optimal theta
%  This function will return theta and the cost 

[thetas,cost]= one_vs_all(x,y,theta);
X = 1:10;
Y1 = -(thetas(1,1) + thetas(2,1)*X)/thetas(3,1);
Y2 = -(thetas(1,2) + thetas(2,2)*X)/thetas(3,2);
Y3 = -(thetas(1,3) + thetas(2,3)*X)/thetas(3,3);
plot(x(y1,2),x(y1,3),'r*',x(y2,2),x(y2,3),'c+',x(y3,2),x(y3,3),'bo');
hold on
plot(X,Y1,'r',X,Y2,'g',X,Y3,'c');

• one_vs_all:

function [theta,cost] = one_vs_all(x, y, theta)

%% compute cost: J

options = optimset('GradObj', 'on', 'MaxIter', 400);
n = size(x,2);
cost = zeros(n,1);
num_labels = 3;
for i = 1:num_labels
    L = logical(y==i);
    [theta(:,i), cost(i,1)] = ...
    fminunc(@(t)(computeCost(x,L,t)), theta(:,i), options);
end

• computeCost:

function [J,grad] = computeCost(x, y, thetas)

%% compute cost: J

m = size(x,1);
grad = zeros(size(thetas));
hx = sigmoid(x * thetas);
 J = (1.0/m) * sum(-y .* log(hx) - (1 - y) .* log(1 - hx));
 grad = (1.0/m) .* x' * (hx - y);
end

3.效果：

• θ & J cost：

thetas =

    6.3988    5.1407  -24.4266
   -2.0773    0.2173    2.1641
    0.9857   -1.9490    2.2038

>> cost

cost =

    0.1715
    0.2876
    0.1031

• 图形显示：
  
  

• 注意三条线组成的三角形。。这个地方的点不属于不论什么类别。

补充：

1.regularized Logistic Regerssion

• regularized 和 普通的Logistics没有太大的差别，仅仅是在J的计算和θ更新中加上了曾经的结果。

2.one_vs_all:

1.简单介绍：

• 事实上one_vs_all另一种算法，把θ当做单隐层前馈神经网络进行计算。比方说我们有K类样本，第一类样本我们能够看成[1,0,0,0...]共k个数，，然后依次。，第i个为1则代表第i类样本。计算方式和上面multi_class一样。

• 前馈神经网络模型例如以下：
  
  

2.代码：

• 函数介绍：

  • one_vs_all:相当于主函数。
  • IrCostFunction:花费J和θ更新
  • myPredict:统计训练误差

• 数据 和 训练得到的θ：
  
  点击这儿下载

• 训练结果：

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

Training Set Accuracy: 100.000000

• one_vs_all:

function [all_theta,cost] = oneVsAll(X, y, num_labels)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta
%corresponds to the classifier for label i
%   [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
%   logisitc regression classifiers and returns each of these classifiers
%   in a matrix all_theta, where the i-th row of all_theta corresponds
%   to the classifier for label i

% Some useful variables

m = size(X, 1);
n = size(X, 2);

% You need to return the following variables correctly
all_theta = zeros(n+1,num_labels);

% Add ones to the X data matrix
X = [ones(m, 1),X];

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
%               logistic regression classifiers with regularization
%               parameter lambda.
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell use
%       whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
%       function. It is okay to use a for-loop (for c = 1:num_labels) to
%       loop over the different classes.
%
%       fmincg works similarly to fminunc, but is more efficient when we
%       are dealing with large number of parameters.
%
% Example Code for fmincg:
%
%     % Set Initial theta
%     initial_theta = zeros(n + 1, 1);
%
%     % Set options for fminunc
%     options = optimset('GradObj', 'on', 'MaxIter', 50);
%
%     % Run fmincg to obtain the optimal theta
%     % This function will return theta and the cost
%     [theta] = ...
%         fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
%                 initial_theta, options);
%

cost = zeros(num_labels,1);
options = optimset('GradObj', 'on', 'MaxIter', 50);
for i =1:num_labels
    L = logical(y==i);
     [all_theta(:,i),cost(i,1)] = ...
       fminunc (@(t)(lrCostFunction(t, X, L)),all_theta(:,i), options);
end

myPredict(all_theta,X,y);

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

end

• IrCostFunction:

function [J,grad] = lrCostFunction(thetas,x, y)
%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

%单独调试该函数时用的代码
%x = [ones(m,1),x];
%theta = zeros(size(x,2),1);
%y = logical(y==1);

% ====================== 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)
%

grad = zeros(size(thetas));
hx = sigmoid(x * thetas);
J = (1.0/m) * sum(-y .* log(hx) - (1 - y) .* log(1 - hx));
grad = (1.0/m) .* x' * (hx - y);
% ================================================x=============

end

• myPredict:

function p = myPredict(Theta1,X,y)
%PREDICT Predict the label of an input given a trained neural network
%   p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
%   trained weights of a neural network (Theta1, Theta2)

% Useful values
m = size(X, 1);
num_labels = 10;

% You need to return the following variables correctly
p = zeros(size(X, 1), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned neural network. You should set p to a
%               vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. 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.
%

z_2 = X*Theta1;
a_2 = sigmoid(z_2);
for i = 1:m
    for j = 1:num_labels
        if a_2(i,j) >= 0.5
            p(i,1) = j;
            break;
        end
    end
end
  fprintf('\nTraining Set Accuracy: %f\n', mean(double(p == y)) * 100);

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

end

与本博客相关知识链接：

• 由Logistics Regression multi_class中的one_vs _all算法 ——> 双隐层前馈神经网络 ：BP神经网络
• 由Logistic Regerssion —–> SVM : 特征空间映射
• Logistic Regerssion 的理论解释： 概率论解释