SVM Classification with Cross Validation

Question

I am new to using Matlab and am trying to follow the example in the Bioinformatics Toolbox documentation (SVM Classification with Cross Validation) to handle a classification problem.

However, I am not able to understand Step 9, which says:

Set up a function that takes an input z=[rbf_sigma,boxconstraint], and returns the cross-validation value of exp(z).

The reason to take exp(z) is twofold: rbf_sigma and boxconstraint must be positive.

You should look at points spaced approximately exponentially apart. This function handle computes the cross validation at parameters

exp([rbf_sigma,boxconstraint]):
minfn = @(z)crossval('mcr',cdata,grp,'Predfun', ...
    @(xtrain,ytrain,xtest)crossfun(xtrain,ytrain,...
    xtest,exp(z(1)),exp(z(2))),'partition',c);

What is the function that I should be implementing here? Is it exp or minfn ? I will appreciate if you can give me the code for this section. Thanks.

Answer 1

I will like to know what does it mean when it says exp([rbf_sigma,boxconstraint])

1. rbf_sigma: The svm is using a gaussian kernel, the rbf_sigma set the standard deviation (~size) of the kernel. To understand how kernels work, the SVM is putting the kernel around every sample (so that you have a gaussian around every sample). Then the kernels are added up (sumed) for the samples of each category/type. At each point the type which sum is higher would be the "winner". For example if type A has a higher sum of these kernels at point X, then if you have a new datum to classify in point X, it will be classified as type A. (there are other configuration parameters that may change the actual threshold where a category is selected over another)

Fig. Analyze this figure from the webpage you gave us. You can see how by adding up the gaussian kernels on the red samples "sumA", and on the green samples "sumB"; it is logical that sumA>sumB in the center part of the figure. It is also logical that sumB>sumA in the outer part of the image.

2. boxconstraint: it is a cost/penalty over miss-classified data. During the training stage of the classifier, where you use the training data to adjust the SVM parameters, the training algorithm is using an error function to decide how to optimize the SVM parameters in an iterative fashion. The cost for a miss-classified sample is proportional to how far it is from the boundary where it would have been classified correctly. In the figure that I am attaching the boundary is the inner blue circumference.

Taking into account BGreene indications and from what I understand of the tutorial:

1. In the tutorial they advice to try values for rbf_sigma and boxconstraint that are exponentially apart. This means that you should compare values like {0.2, 2, 20, ...} (note that this is {2*10^(i-2), i=1,2,3,...}), and NOT like {0.2, 0.3, 0.4, 0.5} (which would be linearly apart). They advice this to try a wide range of values first. You can further optimize later FROM the first optimum that you obtained before.
2. The command "[searchmin fval] = fminsearch(minfn,randn(2,1),opts)" will give you back the optimum values for rbf_sigma and boxconstraint. Probably you have to use exp(z) because it affects how fminsearch increments the values of z(1) and z(2) during the search for the optimum value. I suppose that when you put exp(z(1)) in the definition of @minfn, then fminsearch will take 'exponentially' big steps.
3. In machine learning, always try to understand that there are three subsets in your data: training data, cross-validation data, and test data. The training set is used to optimize the parameters of the SVM classifier for EACH value of rbf_sigma and boxconstraint. Then the cross validation set is used to select the optimum value of the parameters rbf_sigma and boxconstraint. And finally the test data is used to obtain an idea of the performance of your classifier (the efficiency of the classifier is determined upon the test set).
4. So, if you start with 10000 samples you may divide the data for example as training(50%), cross-validation(25%), test(25%). So that you will sample randomly 5000 samples for the training set, then 2500 samples from the 5000 remaining samples for the cross-validation set, and the rest of samples (that is 2500) would be separated for the test set.

I hope that I could clarify your doubts. By the way, if you are interested in the optimization of the parameters of classifiers and machine learning algorithms I strongly suggest that you follow this free course -> www.ml-class.org (it is awesome, really).

Answer 2

You need to implement a function called crossfun (see example). The function handle minfn is passed to fminsearch to be minimized. exp([rbf_sigma,boxconstraint]) is the quantity being optimized to minimize classification error.

There are a number of functions nested within this function handle: - crossval is producing the classification error based on cross validation using partition c - crossfun - classifies data using an SVM - fminsearch - optimizes SVM hyperparameters to minimize classification error

Hope this helps