Line 8: Line 8:
 
*Background in Linear Classifiers  
 
*Background in Linear Classifiers  
 
*Support Vector Machine  
 
*Support Vector Machine  
*Effect of Kernal functions on SVM  
+
*Effect of Kernal Functions on SVM  
*Effect of Kernal parameters on SVM<br>  
+
*Effect of Kernal Parameters on SVM<br>  
 
*References<br>
 
*References<br>
  
Line 36: Line 36:
 
<br>&nbsp; &nbsp;The above approach is equivalent to finding a solution for  
 
<br>&nbsp; &nbsp;The above approach is equivalent to finding a solution for  
 
<center><math>\textbf{Y}\textbf{c} = \begin{bmatrix} b_1\\b_2\\...\\b_n\end{bmatrix}</math>&nbsp;,</center>  
 
<center><math>\textbf{Y}\textbf{c} = \begin{bmatrix} b_1\\b_2\\...\\b_n\end{bmatrix}</math>&nbsp;,</center>  
where <math>\textbf{Y} =\begin{bmatrix} y_1^T \\ y_2^T \\ ... \\ y_n^T \end{bmatrix}</math>&nbsp;.
+
where <math>\textbf{Y} =\begin{bmatrix} y_1^T \\ y_2^T \\ ... \\ y_n^T \end{bmatrix}</math>&nbsp;.  
  
 
<br>&nbsp; &nbsp;In most cases when n&gt;p, it is always impossible to find a solution for <math>\textbf{c}</math>. An alternative approach is to find c that minimize a criterion function <math>J(\textbf{c})</math>. There are variant forms of criterion functions. For example, we can try to minimize the error vector between <math>\textbf{c}\cdot\textbf{y}</math> and <span class="texhtml">''b''</span>, hence the criterion function can be defined as  
 
<br>&nbsp; &nbsp;In most cases when n&gt;p, it is always impossible to find a solution for <math>\textbf{c}</math>. An alternative approach is to find c that minimize a criterion function <math>J(\textbf{c})</math>. There are variant forms of criterion functions. For example, we can try to minimize the error vector between <math>\textbf{c}\cdot\textbf{y}</math> and <span class="texhtml">''b''</span>, hence the criterion function can be defined as  
Line 46: Line 46:
 
otherwise, the solution is defined generally by  
 
otherwise, the solution is defined generally by  
  
 +
<br>
 +
<center><math>\lim_{\epsilon \to 0}(\textbf{Y}^T\textbf{Y}+\epsilon\textbf{I})^{-1}\textbf{Y}^Tb</math>.</center>
 +
<br>
  
<center><math>\lim_{\epsilon \to 0}(\textbf{Y}^T\textbf{Y}+\epsilon\textbf{I})^{-1}\textbf{Y}^Tb</math>.</center>
+
<br> &nbsp; &nbsp;This solution is a MSE solution to <math>\textbf{Y}\textbf{c} = \textbf{b}</math>&nbsp;and it always exists.  
  
 +
<br>
  
 +
= Support Vector Machine  =
  
 +
&nbsp; &nbsp;Support vector machines are an example of a linear two-class classifier. For a given hyperlane we denote by <math>\textbf{y}_1,\textbf{y}_2</math> the closest point to the hyperpalne among the positive (negative). The distance from the two points to the hyperplane is <math>g(\textbf{y}_1)/||c||</math> and <math>g(\textbf{y}_2)/||c||</math>. The margin is defined as the region between the two points. In SVM, the hyperplane is chosen so that the margin is maximized, i.e. we want to maximize 1/||c||, which is equivalent to minimizing <span class="texhtml"> |  | ''c'' |  | <sup>2</sup></span>. This leads to the following optimization problem:
  
&nbsp; &nbsp;This solution is a MSE solution to <math>\textbf{Y}\textbf{c} = \textbf{b}</math>&nbsp;and it always exists.  
+
<br>
 +
<center><math>arg \min \limits_{c,b} \quad\quad \frac{1}{2}||c||^2</math>,</center>
 +
<br>
 +
<center></center> <center><math>subject \quad to: \quad w_i(\textbf{c}\textbf{y}_i+b) \geqslant 1, \quad i = 1,...,n</math>.</center>
 +
<br>
  
 +
<br> &nbsp; &nbsp;A more general classifier that allows misclassfication would be
  
 +
<br>
 +
<center><math>arg \min \limits_{c,b} \quad\quad \frac{1}{2}||c||^2 +C\sum_{i=1}^{n} \xi_i</math>,</center> <center><math>subject \quad to: \quad w_i(\textbf{c}\textbf{y}_i+b) \geqslant 1-\xi_i, \xi\geqslant 0 \quad i = 1,...,n</math>.</center>
 +
<br>
  
= Support Vector Machine =
+
<br> where ξi ≥ 0 are slack variables that allow an example to be in the margin (0 ≤ ξi ≤ 1, also called a margin error) or to be misclassified (ξi &gt; 1). The constant C &gt; 0 sets the relative importance of maximizing the margin and minimizing the amount of slack. Using the method of Lagrange multipliers, we can obtain the dual formulation which is expressed in terms of variables αi:
 +
<center><math>arg \max \limits_{\alpha} \quad\quad \sum_{i=1}^{n}\alpha_i-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n} w_iw_j\alpha_i\alpha_j\textbf{y}_i^T\textbf{y}_j</math></center> <center><math>subject \quad to: \sum_{i=1}^nw_i\alpha_i = 0, C\geqslant \alpha_i \geqslant 0</math></center>
 +
<br>&nbsp; &nbsp;In many applications the data is not linearly separable, in this case 'kernel trick' is applied to map the data into high-dimensional feature spaces. We first map the input space to a feature space using <math>\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^m</math> Then the discriminant function is then
 +
<center><math>f(\textbf{y}) = \textbf{c}^T\varphi(\textbf{y})) + b.</math></center>
 +
&nbsp; &nbsp;Suppose <math>\textbf{c} = \sum_{i=1}^{n}\alpha_i\textbf{y}_i</math>, then the discriminant function in the new feature space takes the form
  
&nbsp; &nbsp;Support vector machines are an example of a linear two-class classifier. For a given hyperlane we denote by <math>\textbf{y}_1,\textbf{y}_2</math> the closest point to the hyperpalne among the positive (negative). The distance from the two points to the hyperplane is <math>g(\textbf{y}_1)/||c||</math> and <math>g(\textbf{y}_2)/||c||</math>. The margin is defined as the region between the two points. In SVM, the hyperplane is chosen so that the margin is maximized, i.e. we want to maximize 1/||c||, which is equivalent to minimizing <span class="texhtml"> |  | ''c'' |  | <sup>2</sup></span>. This leads to the following optimization problem:
+
<br>
 +
<center><math>f(\textbf{y}) = \sum_{i=1}^{n}\alpha_i\varphi(\textbf{y}_i)^T\varphi(\textbf{y})+b</math>.</center>  
 +
<br>  
  
 +
<br> &nbsp; &nbsp;By defining a kernel function as a mapping function
 +
<center><math>k: \mathbb{R}^n \times \mathbb{R}^n to \mathbb{R}</math></center>
 +
that satisfies
 +
<center><math>k(\textbf{y}_i,\textbf{y}_j) = \varphi(\textbf{y}_i)^T\varphi(\textbf{y}_j)</math>&nbsp;,</center>
 +
the discriminant function in terms of the kernel function is
 +
<center><math>f(\textbf{y}) = \sum_{i=1}^n\alpha_i k(\textbf{y},\textbf{y}_i)+b.</math></center>
 +
&nbsp; &nbsp;The resulting algorithm is formally similar, except that every dot product is replaced by a nonlinear kernel function.
  
<center><math>arg \min \limits_{c,b} \quad\quad \frac{1}{2}||c||^2</math>,</center>
+
<br>  
  
 +
= Effect of Kernel Functions on SVM  =
  
 +
There are several kernel functions we can choose from. Some common kernels include:
  
<center></center> <center><math>subject \quad to: \quad w_i(\textbf{c}\textbf{y}_i+b) \geqslant 1, \quad i = 1,...,n</math>.</center>
+
Linear: <math>k(\textbf{x}_i,\textbf{x}_j)=\textbf{x}_i\cdot\textbf{x}_j</math>  
  
 +
Polynomial:&nbsp;<math>k(\textbf{x}_i,\textbf{x}_j) = (\textbf{x}_i\cdot\textbf{x}_j+1)^d</math>
  
 +
Gaussian radial basis function:&nbsp;<math>k(\textbf{x}_i,\textbf{x}_j) = exp(-\gamma||\text{x}_i-\textbf{x}_j||^2)</math>.
  
 +
<br> In the section we give examples of using SVM to classify the Ripley data set based on the above kernel functions. The classfications are illustrated in the Fig.1~3.&nbsp;The mis-classification rate are as follows:
  
&nbsp; &nbsp;A more general classifier that allows misclassfication would be
+
Linear: &nbsp;0.1488<br>Polynomial: &nbsp;0.0744<br>Gaussian radial basis function: 0.0651<br>
 +
<center>[[Image:LinearKernel.png|600px]]</center> <center>Fig 1. SVM classification using linear kernel function</center> <center>[[Image:Polynomial.png|600px]]</center> <center>Fig 2. SVM classification using polynomial kernel function</center> <center>[[Image:Rbf.png|600px]]</center> <center>Fig 3. SVM classification using Gaussian radial basis function</center>
 +
<br>
  
 +
= Effect of Kernal Parameters on SVM =
  
<center><math>arg \min \limits_{c,b} \quad\quad \frac{1}{2}||c||^2 +C\sum_{i=1}^{n} \xi_i</math>,</center> <center><math>subject \quad to: \quad w_i(\textbf{c}\textbf{y}_i+b) \geqslant 1-\xi_i, \xi\geqslant 0 \quad i = 1,...,n</math>.</center>
+
The effect of degree of polynomial kernel function on the classification is shown in Fig. 4. Figure 5 shows the&nbsp;mis-classification rate as a function of the degree.&nbsp;We can see that higher degree polynomial kernels allow a more flexible decision boundary and higher accuracy.  
  
  
  
  
where ξi ≥ 0 are slack variables that allow an example to be in the margin (0 ≤ ξi ≤ 1, also called a margin error) or to be misclassified (ξi &gt; 1). The constant C &gt; 0 sets the relative importance of maximizing the margin and minimizing the amount of slack. Using the method of Lagrange multipliers, we can obtain the dual formulation which is expressed in terms of variables αi:  
+
<center>[[Image:DegreePoly2.png|210px]] [[Image:DegreePoly7.png|210px]][[Image:DegreePoly12.png|210px]]</center> <center>Fig 4. SVM classification using polynomial kernel function of degree 2(left), 7(middle), and 12(right)</center>
<center><math>arg \max \limits_{\alpha} \quad\quad \sum_{i=1}^{n}\alpha_i-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n} w_iw_j\alpha_i\alpha_j\textbf{y}_i^T\textbf{y}_j</math></center> <center><math>subject \quad to: \sum_{i=1}^nw_i\alpha_i = 0, C\geqslant \alpha_i \geqslant 0</math></center>
+
<br>&nbsp; &nbsp;In many applications the data is not linearly separable, in this case 'kernel trick' is applied to map the data into high-dimensional feature spaces. We first map the input space to a feature space using <math>\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^m</math> Then the discriminant function is then
+
<center><math>f(\textbf{y}) = \textbf{c}^T\varphi(\textbf{y})) + b.</math></center>  
+
&nbsp; &nbsp;Suppose <math>\textbf{c} = \sum_{i=1}^{n}\alpha_i\textbf{y}_i</math>, then the discriminant function in the new feature space takes the form
+
  
  
<center><math>f(\textbf{y}) = \sum_{i=1}^{n}\alpha_i\varphi(\textbf{y}_i)^T\varphi(\textbf{y})+b</math>.</center>
 
  
  
  
 +
<center></center> <center>[[Image:DegreeOfpolynomial.png|500px]]</center> <center>Fig 5. The effect of the degree of a polynomial kernel</center>
  
&nbsp; &nbsp;By defining a kernel function as a mapping function
 
<center><math>k: \mathbb{R}^n \times \mathbb{R}^n to \mathbb{R}</math></center>
 
that satisfies
 
<center><math>k(\textbf{y}_i,\textbf{y}_j) = \varphi(\textbf{y}_i)^T\varphi(\textbf{y}_j)</math>&nbsp;,</center>
 
the discriminant function in terms of the kernel function is
 
<center><math>f(\textbf{y}) = \sum_{i=1}^n\alpha_i k(\textbf{y},\textbf{y}_i)+b.</math></center>
 
&nbsp; &nbsp;The resulting algorithm is formally similar, except that every dot product is replaced by a nonlinear kernel function.
 
  
  
 +
<center></center> <center></center>
  
= Effect of Kernel Functions on SVM =
 
  
There are several kernel functions we can choose from. Some common kernels include:
 
  
Linear: <math>k(\textbf{x}_i,\textbf{x}_j)=\textbf{x}_i\cdot\textbf{x}_j</math>
 
  
Polynomial:&nbsp;<math>k(\textbf{x}_i,\textbf{x}_j) = (\textbf{x}_i\cdot\textbf{x}_j+1)^d</math>
 
  
Gaussian radial basis function:&nbsp;<math>k(\textbf{x}_i,\textbf{x}_j) = exp(-\gamma||\text{x}_i-\textbf{x}_j||^2)</math>.
 
 
<br> In the section we give examples of using SVM to classify the Ripley data set based on the above kernel functions. The classfications are illustrated in the Fig.1~3.&nbsp;The misclassification rate are as follows:
 
 
Linear: &nbsp;0.1488<br>Polynomial: &nbsp;0.0744<br>Gaussian radial basis function: 0.0651<br>
 
  
<center>[[Image:LinearKernel.png|600px]]</center> <center>Fig 1. SVM classification using linear kernel function</center> <center>[[Image:Polynomial.png|600px]]</center> <center>Fig 2. SVM classification using polynomial kernel function</center> <center>[[Image:Rbf.png|600px]]</center> <center>Fig 3. SVM classification using Gaussian radial basis function</center>
 
  
  
  
The effect of degree of polynomial kernel function on the classification
+
<center>[[Image:Gamma1RBF.png|300px]] [[Image:Gamma2RBF.png|300px]][[Image:Gamma3RBF.png|300px]][[Image:Gamma4RBF.png|300px]][[Image:Gamma5RBF.png|300px]]</center> <center>Fig 6. SVM classification using RBF kernel function with gamma =&nbsp;0.001, 0.01, 0.1, 1, and 10</center>
<center>[[Image:DegreePoly2.png|210px]] [[Image:DegreePoly7.png|210px]][[Image:DegreePoly12.png|210px]]</center> <center>Fig 4. SVM classification using polynomial kernel function of degree 2(left), 7(middle), and 12(right)</center> <center>[[Image:DegreeOfpolynomial.png|500px]]</center> <center>Fig 5. SVM classification using linear kernel function</center> <center>[[Image:Gamma1RBF.png|300px]] [[Image:Gamma2RBF.png|300px]][[Image:Gamma3RBF.png|300px]][[Image:Gamma4RBF.png|300px]][[Image:Gamma5RBF.png|300px]]</center> <center>Fig 6. SVM classification using polynomial kernel function of degree 2(left), 7(middle), and 12(right)</center>
+

Revision as of 20:59, 4 May 2014

'Support Vector Machine and its Applications in Classification Problems
A slecture by Xing Liu Partially based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.


Outline of the slecture

  • Background in Linear Classifiers
  • Support Vector Machine
  • Effect of Kernal Functions on SVM
  • Effect of Kernal Parameters on SVM
  • References

 Background in Linear Classifiers 

   In this section, we will introduce the framework and basic idea of linear classification problem.    

   In a linear classification problem, the feature space can be divided into different regions by hyperplanes. In this lecture, we will take a two-catagory case to illustrate. Given training samples

$ \textbf{y}_1,\textbf{y}_2,...\textbf{y}_n \in \mathbb{R}^p $ , each $ \textbf{y}_i $ is a p-dimensional vector and belongs to either class w1 or w2. The goal is to find the maximum-margin hyperplane that separate the points in the feature space that belong to class w1 from those that belong to class w2. The discriminate function can be written as

$ g(\textbf{y}) = \textbf{c}\cdot\textbf{y} $


   We want to find $ \textbf{c}\in\mathbb{R}^{n+1} $ so that a testing data point $ \textbf{y}_i $ is labelled

$ w_1 ~~ if ~ \textbf{c}\cdot\textbf{y}>0 $ ,
$ w_2 ~~ if ~ \textbf{c}\cdot \textbf{y} < 0 $ .


   We can apply a trick here to replace all <math>\textbf{y} 's in class w2 by $ -\textbf{y} $, then the above task is equivalent to looking for $ \textbf{c} $ so that 

                                                   $ \textbf{c}\cdot \textbf{y}>0, \forall \textbf{y} \in ~new~ sample ~space $ .


   You might have already observed the ambiguity of c in the above discussion: if c separates data, $ \lambda \textbf{c} $ also separates the data. One solution might be set $ |\textbf{c}|=1 $. Another solution is to introduce a bias denoted b, and ask

$ \textbf{c}\cdot\textbf{y}\ge b > 0, \forall \textbf{y} $


   In this scenario, the hyperplane is defined by $ \{\textbf{y}: f(\textbf{y})=\textbf{c}\cdot \textbf{y} - \textbf{b}=0\} $ and it divides the space into two, the sign of the discriminant function $ f(\textbf{y}) = \textbf{c}\cdot \textbf{y} - \textbf{b} $ denotes the side of the hyperplane a testing point is on. We notice that he decision boundary by this hyperplane is linear, hence the classifier is called a linear classifier. $ \textbf{c} $ is the normal of the plane lying on the positive side of every hyperplane. $ \frac{b_i}{||c||} $ is the distance from each point $ \textbf{y}_i $ to the hyperplane. We notice that he decision boundary by this hyperplane is linear, hence the classifier is called a linear classifier.


   The above approach is equivalent to finding a solution for

$ \textbf{Y}\textbf{c} = \begin{bmatrix} b_1\\b_2\\...\\b_n\end{bmatrix} $ ,

where $ \textbf{Y} =\begin{bmatrix} y_1^T \\ y_2^T \\ ... \\ y_n^T \end{bmatrix} $ .


   In most cases when n>p, it is always impossible to find a solution for $ \textbf{c} $. An alternative approach is to find c that minimize a criterion function $ J(\textbf{c}) $. There are variant forms of criterion functions. For example, we can try to minimize the error vector between $ \textbf{c}\cdot\textbf{y} $ and b, hence the criterion function can be defined as

$ J(\textbf{c}) = ||\textbf{Y}\textbf{c}-\textbf{b}||^2 $.


   The solution to the above problem is

$ \textbf{c} = (\textbf{Y}^T\textbf{Y})^{-1}\textbf{Y}^T\textbf{b} $     if $ det(\textbf{Y}^T\textbf{Y})\neq 0 $,

otherwise, the solution is defined generally by


$ \lim_{\epsilon \to 0}(\textbf{Y}^T\textbf{Y}+\epsilon\textbf{I})^{-1}\textbf{Y}^Tb $.



   This solution is a MSE solution to $ \textbf{Y}\textbf{c} = \textbf{b} $ and it always exists.


Support Vector Machine

   Support vector machines are an example of a linear two-class classifier. For a given hyperlane we denote by $ \textbf{y}_1,\textbf{y}_2 $ the closest point to the hyperpalne among the positive (negative). The distance from the two points to the hyperplane is $ g(\textbf{y}_1)/||c|| $ and $ g(\textbf{y}_2)/||c|| $. The margin is defined as the region between the two points. In SVM, the hyperplane is chosen so that the margin is maximized, i.e. we want to maximize 1/||c||, which is equivalent to minimizing | | c | | 2. This leads to the following optimization problem:


$ arg \min \limits_{c,b} \quad\quad \frac{1}{2}||c||^2 $,


$ subject \quad to: \quad w_i(\textbf{c}\textbf{y}_i+b) \geqslant 1, \quad i = 1,...,n $.



   A more general classifier that allows misclassfication would be


$ arg \min \limits_{c,b} \quad\quad \frac{1}{2}||c||^2 +C\sum_{i=1}^{n} \xi_i $,
$ subject \quad to: \quad w_i(\textbf{c}\textbf{y}_i+b) \geqslant 1-\xi_i, \xi\geqslant 0 \quad i = 1,...,n $.



where ξi ≥ 0 are slack variables that allow an example to be in the margin (0 ≤ ξi ≤ 1, also called a margin error) or to be misclassified (ξi > 1). The constant C > 0 sets the relative importance of maximizing the margin and minimizing the amount of slack. Using the method of Lagrange multipliers, we can obtain the dual formulation which is expressed in terms of variables αi:

$ arg \max \limits_{\alpha} \quad\quad \sum_{i=1}^{n}\alpha_i-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n} w_iw_j\alpha_i\alpha_j\textbf{y}_i^T\textbf{y}_j $
$ subject \quad to: \sum_{i=1}^nw_i\alpha_i = 0, C\geqslant \alpha_i \geqslant 0 $


   In many applications the data is not linearly separable, in this case 'kernel trick' is applied to map the data into high-dimensional feature spaces. We first map the input space to a feature space using $ \varphi: \mathbb{R}^n \rightarrow \mathbb{R}^m $ Then the discriminant function is then

$ f(\textbf{y}) = \textbf{c}^T\varphi(\textbf{y})) + b. $

   Suppose $ \textbf{c} = \sum_{i=1}^{n}\alpha_i\textbf{y}_i $, then the discriminant function in the new feature space takes the form


$ f(\textbf{y}) = \sum_{i=1}^{n}\alpha_i\varphi(\textbf{y}_i)^T\varphi(\textbf{y})+b $.



   By defining a kernel function as a mapping function

$ k: \mathbb{R}^n \times \mathbb{R}^n to \mathbb{R} $

that satisfies

$ k(\textbf{y}_i,\textbf{y}_j) = \varphi(\textbf{y}_i)^T\varphi(\textbf{y}_j) $ ,

the discriminant function in terms of the kernel function is

$ f(\textbf{y}) = \sum_{i=1}^n\alpha_i k(\textbf{y},\textbf{y}_i)+b. $

   The resulting algorithm is formally similar, except that every dot product is replaced by a nonlinear kernel function.


Effect of Kernel Functions on SVM

There are several kernel functions we can choose from. Some common kernels include:

Linear: $ k(\textbf{x}_i,\textbf{x}_j)=\textbf{x}_i\cdot\textbf{x}_j $

Polynomial: $ k(\textbf{x}_i,\textbf{x}_j) = (\textbf{x}_i\cdot\textbf{x}_j+1)^d $

Gaussian radial basis function: $ k(\textbf{x}_i,\textbf{x}_j) = exp(-\gamma||\text{x}_i-\textbf{x}_j||^2) $.


In the section we give examples of using SVM to classify the Ripley data set based on the above kernel functions. The classfications are illustrated in the Fig.1~3. The mis-classification rate are as follows:

Linear:  0.1488
Polynomial:  0.0744
Gaussian radial basis function: 0.0651

LinearKernel.png
Fig 1. SVM classification using linear kernel function
Polynomial.png
Fig 2. SVM classification using polynomial kernel function
Rbf.png
Fig 3. SVM classification using Gaussian radial basis function


Effect of Kernal Parameters on SVM

The effect of degree of polynomial kernel function on the classification is shown in Fig. 4. Figure 5 shows the mis-classification rate as a function of the degree. We can see that higher degree polynomial kernels allow a more flexible decision boundary and higher accuracy.



DegreePoly2.png DegreePoly7.pngDegreePoly12.png
Fig 4. SVM classification using polynomial kernel function of degree 2(left), 7(middle), and 12(right)



DegreeOfpolynomial.png
Fig 5. The effect of the degree of a polynomial kernel






Gamma1RBF.png Gamma2RBF.pngGamma3RBF.pngGamma4RBF.pngGamma5RBF.png
Fig 6. SVM classification using RBF kernel function with gamma = 0.001, 0.01, 0.1, 1, and 10

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