Support Vector Machines for Binary Classification 2

Using Support Vector Machines

As with any supervised learning model, you first train a support vector machine, and then cross validate the classifier. Use the trained machine to classify (predict) new data. In addition, to obtain satisfactory predictive accuracy, you can use various SVM kernel functions, and you must tune the parameters of the kernel functions.

Training an SVM Classifier
Classifying New Data with an SVM Classifier
Tuning an SVM Classifier

Training an SVM Classifier

Train, and optionally cross validate, an SVM classifier using fitcsvm. The most common syntax is:

SVMModel = fitcsvm(X,Y,'KernelFunction','rbf',...
    'Standardize',true,'ClassNames',{'negClass','posClass'});

The inputs are:

X — Matrix of predictor data, where each row is one observation, and each column is one predictor.
Y — Array of class labels with each row corresponding to the value of the corresponding row in X. Y can be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors.
KernelFunction — The default value is 'linear' for two-class learning, which separates the data by a hyperplane. The value 'gaussian' (or 'rbf') is the default for one-class learning, and specifies to use the Gaussian (or radial basis function) kernel. An important step to successfully train an SVM classifier is to choose an appropriate kernel function.
Standardize — Flag indicating whether the software should standardize the predictors before training the classifier.
ClassNames — Distinguishes between the negative and positive classes, or specifies which classes to include in the data. The negative class is the first element (or row of a character array), e.g., 'negClass', and the positive class is the second element (or row of a character array), e.g., 'posClass'. ClassNames must be the same data type as Y. It is good practice to specify the class names, especially if you are comparing the performance of different classifiers.

The resulting, trained model (SVMModel) contains the optimized parameters from the SVM algorithm, enabling you to classify new data.

For more name-value pairs you can use to control the training, see the fitcsvm reference page.

Classifying New Data with an SVM Classifier

Classify new data using predict. The syntax for classifying new data using a trained SVM classifier (SVMModel) is:

[label,score] = predict(SVMModel,newX);

The resulting vector, label, represents the classification of each row in X. score is an n-by-2 matrix of soft scores. Each row corresponds to a row in X, which is a new observation. The first column contains the scores for the observations being classified in the negative class, and the second column contains the scores observations being classified in the positive class.

To estimate posterior probabilities rather than scores, first pass the trained SVM classifier (SVMModel) to fitPosterior, which fits a score-to-posterior-probability transformation function to the scores. The syntax is:

ScoreSVMModel = fitPosterior(SVMModel,X,Y);

The property ScoreTransform of the classifier ScoreSVMModel contains the optimal transformation function. Pass ScoreSVMModel to predict. Rather than returning the scores, the output argument score contains the posterior probabilities of an observation being classified in the negative (column 1 of score) or positive (column 2 of score) class.

Tuning an SVM Classifier

Use the 'OptimizeHyperparameters' name-value pair argument of fitcsvm to find parameter values that minimize the cross-validation loss. The eligible parameters are 'BoxConstraint', 'KernelFunction', 'KernelScale', 'PolynomialOrder', and 'Standardize'. For an example, see Optimize Classifier Fit Using Bayesian Optimization. Alternatively, you can use the bayesopt function, as shown in Optimize Cross-Validated Classifier Using bayesopt. The bayesopt function allows more flexibility to customize optimization. You can use the bayesopt function to optimize any parameters, including parameters that are not eligible to optimize when you use the fitcsvm function.

You can also try tuning parameters of your classifier manually according to this scheme:

Pass the data to fitcsvm, and set the name-value pair argument 'KernelScale','auto'. Suppose that the trained SVM model is called SVMModel. The software uses a heuristic procedure to select the kernel scale. The heuristic procedure uses subsampling. Therefore, to reproduce results, set a random number seed using rng before training the classifier.
Cross validate the classifier by passing it to crossval. By default, the software conducts 10-fold cross validation.
Pass the cross-validated SVM model to kfoldLoss to estimate and retain the classification error.
Retrain the SVM classifier, but adjust the 'KernelScale' and 'BoxConstraint' name-value pair arguments.
- BoxConstraint — One strategy is to try a geometric sequence of the box constraint parameter. For example, take 11 values, from 1e-5 to 1e5 by a factor of 10. Increasing BoxConstraint might decrease the number of support vectors, but also might increase training time.
- KernelScale — One strategy is to try a geometric sequence of the RBF sigma parameter scaled at the original kernel scale. Do this by:
  1. Retrieving the original kernel scale, e.g., ks, using dot notation: ks = SVMModel.KernelParameters.Scale.
  2. Use as new kernel scales factors of the original. For example, multiply ks by the 11 values 1e-5 to 1e5, increasing by a factor of 10.

Choose the model that yields the lowest classification error. You might want to further refine your parameters to obtain better accuracy. Start with your initial parameters and perform another cross-validation step, this time using a factor of 1.2.

Train SVM Classifiers Using a Gaussian Kernel

This example shows how to generate a nonlinear classifier with Gaussian kernel function. First, generate one class of points inside the unit disk in two dimensions, and another class of points in the annulus from radius 1 to radius 2. Then, generates a classifier based on the data with the Gaussian radial basis function kernel. The default linear classifier is obviously unsuitable for this problem, since the model is circularly symmetric. Set the box constraint parameter to Inf to make a strict classification, meaning no misclassified training points. Other kernel functions might not work with this strict box constraint, since they might be unable to provide a strict classification. Even though the rbf classifier can separate the classes, the result can be overtrained.

Generate 100 points uniformly distributed in the unit disk. To do so, generate a radius r as the square root of a uniform random variable, generate an angle t uniformly in (0, 2π), and put the point at (r cos(t), r sin(t)).

rng(1); % For reproducibility
r = sqrt(rand(100,1)); % Radius
t = 2*pi*rand(100,1);  % Angle
data1 = [r.*cos(t), r.*sin(t)]; % Points

Generate 100 points uniformly distributed in the annulus. The radius is again proportional to a square root, this time a square root of the uniform distribution from 1 through 4.

r2 = sqrt(3*rand(100,1)+1); % Radius
t2 = 2*pi*rand(100,1);      % Angle
data2 = [r2.*cos(t2), r2.*sin(t2)]; % points

Plot the points, and plot circles of radii 1 and 2 for comparison.

figure;
plot(data1(:,1),data1(:,2),'r.','MarkerSize',15)
hold on
plot(data2(:,1),data2(:,2),'b.','MarkerSize',15)
ezpolar(@(x)1);ezpolar(@(x)2);
axis equal
hold off

Figure contains an axes object. The axes object contains 6 objects of type line, text.

Put the data in one matrix, and make a vector of classifications.

data3 = [data1;data2];
theclass = ones(200,1);
theclass(1:100) = -1;

Train an SVM classifier with KernelFunction set to 'rbf' and BoxConstraint set to Inf. Plot the decision boundary and flag the support vectors.

%Train the SVM Classifier
cl = fitcsvm(data3,theclass,'KernelFunction','rbf',...
    'BoxConstraint',Inf,'ClassNames',[-1,1]);

% Predict scores over the grid
d = 0.02;
[x1Grid,x2Grid] = meshgrid(min(data3(:,1)):d:max(data3(:,1)),...
    min(data3(:,2)):d:max(data3(:,2)));
xGrid = [x1Grid(:),x2Grid(:)];
[~,scores] = predict(cl,xGrid);

% Plot the data and the decision boundary
figure;
h(1:2) = gscatter(data3(:,1),data3(:,2),theclass,'rb','.');
hold on
ezpolar(@(x)1);
h(3) = plot(data3(cl.IsSupportVector,1),data3(cl.IsSupportVector,2),'ko');
contour(x1Grid,x2Grid,reshape(scores(:,2),size(x1Grid)),[0 0],'k');
legend(h,{'-1','+1','Support Vectors'});
axis equal
hold off

Figure contains an axes object. The axes object contains 6 objects of type line, text, contour. These objects represent -1, +1, Support Vectors.

fitcsvm generates a classifier that is close to a circle of radius 1. The difference is due to the random training data.

Training with the default parameters makes a more nearly circular classification boundary, but one that misclassifies some training data. Also, the default value of BoxConstraint is 1, and, therefore, there are more support vectors.

cl2 = fitcsvm(data3,theclass,'KernelFunction','rbf');
[~,scores2] = predict(cl2,xGrid);

figure;
h(1:2) = gscatter(data3(:,1),data3(:,2),theclass,'rb','.');
hold on
ezpolar(@(x)1);
h(3) = plot(data3(cl2.IsSupportVector,1),data3(cl2.IsSupportVector,2),'ko');
contour(x1Grid,x2Grid,reshape(scores2(:,2),size(x1Grid)),[0 0],'k');
legend(h,{'-1','+1','Support Vectors'});
axis equal
hold off

Figure contains an axes object. The axes object contains 6 objects of type line, text, contour. These objects represent -1, +1, Support Vectors.

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