Stanford机器学习课程的主页是： http://cs229.stanford.edu/
主讲人Andrew Ng是机器学习界的大牛，创办最大的公开课网站coursera，前段时间还听说加入了百度。他讲的机器学习课程可谓每个学计算机的人必看。整个课程的大纲大致如下：
- Introduction (1 class) Basic concepts.
- Supervised learning. (6 classes) Supervised learning setup. LMS. Logistic regression. Perceptron. Exponential family. Generative learning algorithms. Gaussian discriminant analysis. Naive Bayes. Support vector machines. Model selection and feature selection. Ensemble methods: Bagging, boosting, ECOC.
- Learning theory. (3 classes) Bias/variance tradeoff. Union and Chernoff/Hoeffding bounds. VC dimension. Worst case (online) learning. Advice on using learning algorithms.
- Unsupervised learning. (5 classes) Clustering. K-means. EM. Mixture of Gaussians. Factor analysis. PCA. MDS. pPCA. Independent components analysis (ICA).
- Reinforcement learning and control. (4 classes) MDPs. Bellman equations. Value iteration. Policy iteration. Linear quadratic regulation (LQR). LQG. Q-learning. Value function approximation. Policy search. Reinforce. POMDPs.
本笔记主要是关于Linear Regression和Logistic Regression部分的学习实践记录。
举了一个房价预测的例子，
| Area(feet^2) | #bedrooms | Price(1000$) |
|---|---|---|
| 2014 | 3 | 400 |
| 1600 | 3 | 330 |
| 2400 | 3 | 369 |
| 1416 | 2 | 232 |
| 3000 | 4 | 540 |
| 3670 | 4 | 620 |
| 4500 | 5 | 800 |
Assume：房价与“面积和卧室数量”是线性关系，用线性关系进行放假预测。因而给出线性模型，
上述
解
Gradient descent algorithm执行过程中又分两种方法：batch gradient descent和stochastic gradient descent。batch gradient descent每次更新
-
batch gradient descent
-
stochastic gradient descent
for i=1 to m
两者只不过一个将样本放在了for循环上，一者放在了\sum上求和。事实证明，只要选择合适的学习率
关于收敛条件，Andrew Ng没有在课程中提到更多，我给出两种收敛准则：
- J小于一定值收敛
- 两次迭代之间的J差值，即|J-J_pre|<一定值则收敛
下面是使用batch gradient descent拟合上面房价问题的例子，
clear all;
clc
%% 原数据
x = [2014, 1600, 2400, 1416, 3000, 3670, 4500;...
3,3,3,2,4,4,5;];
y = [400 330 369 232 540 620 800];
error = Inf;
threshold = 4300;
alpha = 10^(-10);
x = [zeros(1,size(x,2)); x]; % x0 = 0，拟合常数项
theta = [0;0;0]; % 常数项为0
J = 1/2*sum((y-theta'*x).^2);
costs = [];
while error > threshold
tmp = y-theta'*x;
theta(1) = theta(1) + alpha*sum(tmp.*x(1,:));
theta(2) = theta(2) + alpha*sum(tmp.*x(2,:));
theta(3) = theta(3) + alpha*sum(tmp.*x(3,:));
% J_last = J;
J = 1/2*sum((y-theta'*x).^2);
% error = abs(J-J_last);
error = J;
costs =[costs, error];
end
%% 绘制
figure,
subplot(211);
plot3(x(2,:),x(3,:),y, '*');
grid on;
xlabel('Area');
ylabel('#bedrooms');
zlabel('Price(1000$)');
hold on;
H = theta'*x;
plot3(x(2,:),x(3,:),H,'r*');
hold on
hx(1,:) = zeros(1,20);
hx(2,:) = 1000:200:4800;
hx(3,:) = 1:20;
[X,Y] = meshgrid(hx(2,:), hx(3,:));
H = theta(2:3)'*[X(:)'; Y(:)'];
H = reshape(H,[20,20]);
mesh(X,Y,H);
legend('原数据', '对原数据的拟合', '拟合平面');
subplot(212);
plot(costs, '.-');
grid on
title('error=J(\theta)的迭代收敛过程');
xlabel('迭代次数');
ylabel('J(\theta)');拟合及收敛过程如下：
不管是梯度下降，还是线性回归模型，都是工具！！分析结果更重要，从上面的拟合平面可以看到，影响房价的主要因素是面积而非卧室数量。
很多情况下，模型并不是线性的，比如股票随时间的涨跌过程。这种情况下，
- 建立一个非线性的模型，比如指数或者其它的符合数据变化的模型
- 局部线性模型，对每段数据进行局部建立线性模型。Andrew Ng课堂上讲解了Locally Weighted Linear Regression，即局部加权的线性模型
其中权值的一种好的选择方式是：
Linear Regression解决的是连续的预测和拟合问题，而Logistic Regression解决的是离散的分类问题。两种方式，但本质殊途同归，两者都可以算是指数函数族的特例。
在分类问题中，y取值在{0,1}之间，因此，上述的Liear Regression显然不适应。修改模型如下
该模型称为Logistic函数或Sigmoid函数。为什么选择该函数，我们看看这个函数的图形就知道了，
Sigmoid函数范围在[0,1]之间，参数
假设
对m个训练样本，使其似然函数最大，则有
同样的可以用梯度下降法求解上述的最大值问题，只要将最大值求解转化为求最小值，则迭代公式一模一样，
最后的梯度下降方式和Linear Regression一致。我做了个例子（数据集链接），下面是Logistic的Matlab代码，
function Logistic
clear all;
close all
clc
data = load('LogisticInput.txt');
x = data(:,1:2);
y = data(:,3);
% Plot Original Data
figure,
positive = find(y==1);
negtive = find(y==0);
hold on
plot(x(positive,1), x(positive,2), 'k+', 'LineWidth',2, 'MarkerSize', 7);
plot(x(negtive,1), x(negtive,2), 'bo', 'LineWidth',2, 'MarkerSize', 7);
% Compute Likelihood(Cost) Function
[m,n] = size(x);
x = [ones(m,1) x];
theta = zeros(n+1, 1);
[cost, grad] = cost_func(theta, x, y);
threshold = 0.1;
alpha = 10^(-1);
costs = [];
while cost > threshold
theta = theta + alpha * grad;
[cost, grad] = cost_func(theta, x, y);
costs = [costs cost];
end
% Plot Decision Boundary
hold on
plot_x = [min(x(:,2))-2,max(x(:,2))+2];
plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));
plot(plot_x, plot_y, 'r-');
legend('Positive', 'Negtive', 'Decision Boundary')
xlabel('Feature Dim1');
ylabel('Feature Dim2');
title('Classifaction Using Logistic Regression');
% Plot Costs Iteration
figure,
plot(costs, '*');
title('Cost Function Iteration');
xlabel('Iterations');
ylabel('Cost Fucntion Value');
end
function g=sigmoid(z)
g = 1.0 ./ (1.0+exp(-z));
end
function [J,grad] = cost_func(theta, X, y)
% Computer Likelihood Function and Gradient
m = length(y); % training examples
hx = sigmoid(X*theta);
J = (1./m)*sum(-y.*log(hx)-(1.0-y).*log(1.0-hx));
grad = (1./m) .* X' * (y-hx);
end
判决边界(Decesion Boundary)的计算是令h(x)=0.5得到的。当输入新的数据，计算h(x)：h(x)>0.5为正样本所属的类，h(x)< 0.5 为负样本所属的类。
这部分在Andrew Ng课堂上没有讲，参考了网络上的资料。
上面的数据可以通过直线进行划分，但实际中存在那么种情况，无法直接使用直线判决边界(看后面的例子)。
为解决上述问题，必须将特征映射到高维，然后通过非直线判决界面进行划分。特征映射的方法将已有的特征进行多项式组合，形成更多特征，
上面将二维特征映射到了2阶（还可以映射到更高阶），这便于形成非线性的判决边界。
但还存在问题，尽管上面方法便于对非线性的数据进行划分，但也容易由于高维特性造成过拟合。因此，引入泛化项应对过拟合问题。似然函数添加泛化项后变成，
此时梯度下降算法发生改变，
最后来个例子，样本数据链接，对应的含泛化项和特征映射的matlab分类代码如下：
function LogisticEx2
clear all;
close all
clc
data = load('ex2data2.txt');
x = data(:,1:2);
y = data(:,3);
% Plot Original Data
figure,
positive = find(y==1);
negtive = find(y==0);
subplot(1,2,1);
hold on
plot(x(positive,1), x(positive,2), 'k+', 'LineWidth',2, 'MarkerSize', 7);
plot(x(negtive,1), x(negtive,2), 'bo', 'LineWidth',2, 'MarkerSize', 7);
% Compute Likelihood(Cost) Function
[m,n] = size(x);
x = mapFeature(x);
theta = zeros(size(x,2), 1);
lambda = 1;
[cost, grad] = cost_func(theta, x, y, lambda);
threshold = 0.53;
alpha = 10^(-1);
costs = [];
while cost > threshold
theta = theta + alpha * grad;
[cost, grad] = cost_func(theta, x, y, lambda);
costs = [costs cost];
end
% Plot Decision Boundary
hold on
plotDecisionBoundary(theta, x, y);
legend('Positive', 'Negtive', 'Decision Boundary')
xlabel('Feature Dim1');
ylabel('Feature Dim2');
title('Classifaction Using Logistic Regression');
% Plot Costs Iteration
% figure,
subplot(1,2,2);plot(costs, '*');
title('Cost Function Iteration');
xlabel('Iterations');
ylabel('Cost Fucntion Value');
end
function f=mapFeature(x)
% Map features to high dimension
degree = 6;
f = ones(size(x(:,1)));
for i = 1:degree
for j = 0:i
f(:, end+1) = (x(:,1).^(i-j)).*(x(:,2).^j);
end
end
end
function g=sigmoid(z)
g = 1.0 ./ (1.0+exp(-z));
end
function [J,grad] = cost_func(theta, X, y, lambda)
% Computer Likelihood Function and Gradient
m = length(y); % training examples
hx = sigmoid(X*theta);
J = (1./m)*sum(-y.*log(hx)-(1.0-y).*log(1.0-hx)) + (lambda./(2*m)*norm(theta(2:end))^2);
regularize = (lambda/m).*theta;
regularize(1) = 0;
grad = (1./m) .* X' * (y-hx) - regularize;
end
function plotDecisionBoundary(theta, X, y)
%PLOTDECISIONBOUNDARY Plots the data points X and y into a new figure with
%the decision boundary defined by theta
% PLOTDECISIONBOUNDARY(theta, X,y) plots the data points with + for the
% positive examples and o for the negative examples. X is assumed to be
% a either
% 1) Mx3 matrix, where the first column is an all-ones column for the
% intercept.
% 2) MxN, N>3 matrix, where the first column is all-ones
% Plot Data
% plotData(X(:,2:3), y);
hold on
if size(X, 2) <= 3
% Only need 2 points to define a line, so choose two endpoints
plot_x = [min(X(:,2))-2, max(X(:,2))+2];
% Calculate the decision boundary line
plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));
% Plot, and adjust axes for better viewing
plot(plot_x, plot_y)
% Legend, specific for the exercise
legend('Admitted', 'Not admitted', 'Decision Boundary')
axis([30, 100, 30, 100])
else
% Here is the grid range
u = linspace(-1, 1.5, 50);
v = linspace(-1, 1.5, 50);
z = zeros(length(u), length(v));
% Evaluate z = theta*x over the grid
for i = 1:length(u)
for j = 1:length(v)
z(i,j) = mapFeature([u(i), v(j)])*theta;
end
end
z = z'; % important to transpose z before calling contour
% Plot z = 0
% Notice you need to specify the range [0, 0]
contour(u, v, z, [0, 0], 'LineWidth', 2)
end
end
我们再回过头来看Logistic问题：对于非线性的问题，只不过使用了一个叫Sigmoid的非线性映射成一个线性问题。那么，除了Sigmoid函数，还有其它的函数可用吗？Andrew Ng老师还讲了指数函数族。