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Support Vector Machine Regression

 

Support Vector Machines are very specific class of algorithms, characterized by usage of kernels, absence of local minima, sparseness of the solution and capacity control obtained by acting on the margin, or on number of support vectors, etc.

They were invented by Vladimir Vapnik and his co-workers, and first introduced at the Computational Learning Theory (COLT) 1992 conference with the paper. All these nice features however were already present in machine learning since 1960’s: large margin hyper planes usage of kernels, geometrical interpretation of kernels as inner products in a feature space. Similar optimization techniques were used in pattern recognition and sparsness techniques were widely discussed. Usage of slack variables to overcome noise in the data and non - separability was also introduced in 1960s. However it was not until 1992 that all these features were put together to form the maximal margin classifier, the basic Support Vector Machine, and not until 1995 that the soft margin version was introduced.

 

Support Vector Machine can be applied not only to classification problems but also to the case of regression. Still it contains all the main features that characterize maximum margin algorithm: a non-linear function is leaned by linear learning machine mapping into high dimensional kernel induced feature space. The capacity of the system is controlled by parameters that do not depend on the dimensionality of feature space.

 

In the same way as with classification approach there is motivation to seek and optimize the generalization bounds given for regression. They relied on defining the loss function that ignores errors, which are situated within the certain distance of the true value. This type of function is often called – epsilon intensive – loss function. The figure below shows an example of one-dimensional linear regression function with – epsilon intensive – band. The variables measure the cost of the errors on the training points. These are zero for all points that are inside the band.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Figure 1 One-dimensional linear regression with epsilon intensive band.


Another picture shows a similar situation but for non-linear regression case.


 

 


Figure 2. Non-linear regression function.

 

One of the most important ideas in Support Vector Classification and Regression cases is that presenting the solution by means of small subset of training points gives enormous computational advantages. Using the epsilon intensive loss function we ensure existence of the global minimum and at the same time optimization of reliable generalization bound.

Figure 3. Detailed picture of epsilon band with slack variables and selected data points

 

In SVM regression, the inputis first mapped onto a m-dimensional feature space using some fixed (nonlinear) mapping, and then a linear model is constructed in this feature space. Using mathematical notation, the linear model (in the feature space) is given by

                                                                                       

where denotes a set of nonlinear transformations, and b is the “bias” term. Often the data are assumed to be zero mean (this can be achieved by preprocessing), so the bias term is dropped.

 

 Figure 4. Non-linear mapping of input examples into high dimensional feature space. (Classification case, however the same stands for regression as well).

 

The quality of estimation is measured by the loss function. SVM regression uses a new type of loss function called -insensitive loss function proposed by Vapnik:

                              

The empirical risk is:

                                                                       

SVM regression performs linear regression in the high-dimension feature space using -insensitive loss and, at the same time, tries to reduce model complexity by minimizing . This can be described by introducing (non-negative) slack variables , to measure the deviation of training samples outside -insensitive zone. Thus SVM regression is formulated as minimization of the following functional:

min  

s.t.                                                                                

This optimization problem can transformed into the dual problem and its solution is given by 

    s.t. ,,                        

where is the number of Support Vectors (SVs) and the kernel function

                                                                              

It is well known that SVM generalization performance (estimation accuracy) depends on a good setting of meta-parameters parameters C,  and the kernel parameters. The problem of optimal parameter selection is further complicated by the fact that SVM model complexity (and hence its generalization performance) depends on all three parameters. Existing software implementations of SVM regression usually treat SVM meta-parameters as user-defined inputs. Selecting a particular kernel type and kernel function parameters is usually based on application-domain knowledge and also should reflect distribution of input (x) values of the training data.

 

Figure 5. Performance of Support Vector Machine in regression case. The epsilon boundaries are given with the green lines. Blue points represent data instances.

 

Parameter C determines the trade off between the model complexity (flatness) and the degree to which deviations larger than  are tolerated in optimization formulation for example, if C is too large (infinity), then the objective is to minimize the empirical risk only, without regard to model complexity part in the optimization formulation.

 

Parameter controls the width of the -insensitive zone, used to fit the training data. The value of can affect the number of support vectors used to construct the regression function. The bigger , the fewer support vectors are selected. On the other hand, bigger -values results in more ‘flat’ estimates. Hence, both C and -values affect model complexity (but in a different way).

 

 

 

Figure 6. Performance of Support Vector Machine in regression case. The epsilon boundaries are given with the green lines. Blue points represent data instances.

(Additional instances are introduced in this case and after being supplied to the model are inside epsilon band. The regression function has changed. However this makes some points that were initially inside the interval to be misclassified.)

 

 

 

Figure 7. Performance of Support Vector Machine in regression case. The epsilon boundaries are given with the green lines. Blue points represent data instances.

(Introducing new data instances that are located inside the epsilon band, do not influence the structure of the model. It can be seen that regression function has not changed at all.)

 

 

One of the advantages of Support Vector Machine, and Support Vector Regression as the part of it, is that it can be used to avoid difficulties of using linear functions in the high dimensional feature space and optimization problem is transformed into dual convex quadratic programmes. In regression case the loss function is used to penalize errors that are grater than threshold - . Such loss functions usually lead to the sparse representation of the decision rule, giving significant algorithmic and representational advantages.

 

 

References

 

O. Chapelle and V. Vapnik, Model Selection for Support Vector Machines. In Advances in Neural Information Processing Systems, Vol 12, (1999)

V. Cherkassky and Y. Ma, Selecting of the Loss Function for Robust Linear Regression, Neural computation 2002.

V. Cherkassky and F. Mulier, Learning from Data: Concepts, Theory, and Methods. (John Wiley & Sons, 1998)

Y.B. Dibike, S. Velickov, D.P. Solomatine. Support vector machines: review and applications in civil engineering. In: AI methods in Civil Engineering Applications (O. Schleider, A. Zijderveld, eds). Cottbus, 2000, p.45-58

Y.B. Dibike, S. Velickov, D.P. Solomatine. and M. B. Abbott Model induction with support vector machines: introduction and applications. Journal of Computing in Civil Engineering, American Society of Civil Engineers (ASCE), vol. 15, No.3, 2001, pp. 208-216

A. Smola and B. Schölkopf. A Tutorial on Support Vector Regression. NeuroCOLT Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK, 1998

T. Yoshioka and S. Ishii, Fast Gaussian process regression using representative data,
International Joint Conference on Neural Networks 1, 132-137, 2001.

 

Useful links:

1.      Kernels-Machines.org Ultimate resource in machine learning.

2.      Intelligent Data Analysis. GMD.

3.      Isabelle Guyon's SVM application list. Berkeley.

4.      COLT Group. Royal Holloway College.

5.      Machine Learning Group. ANU Canberra.

6.      Gaussian Processes Homepage. Toronto.

7.      Support Vector Machines. Royal Holloway College.

8.      SVM Book Site. Nello Cristianini and John Shawe-Taylor.

9.      Learning Theory Website. COLT.