SVDD Procedure

The most elemental form of SVDD is a normal data description. The SVDD model for normal data description builds a minimum-radius hypersphere around the data. PROC SVDD does not support this description because it is very simplistic and often leads to higher rates of misclassification. PROC SVDD supports the flexible data description (as described in the section Flexible Data Description). SVDD formulation can be expressed in either of the following forms:

Primal Form

The objective function in primal form is

subject to

where for i=1, …, n represents the training data, R is the radius and represents the decision variable, is the slack for each variable, a is the center, is the penalty constant that controls the trade-off between the volume and the errors, and f is the expected outlier fraction.

Dual Form

The dual formulation is obtained from primal formulation by using the method of Lagrange multipliers. The objective function is

subject to

where are the Lagrange multipliers and is the penalty constant.

Duality Information

The solution to the dual formulation of SVDD provides values of Lagrange multipliers for all training data observations . Based on the value of , the position of an observation in the context of the data description can be determined as follows:

Inside position:

Boundary position:

Outside position:

The center of the data description a can be obtained by using the values of for all n observations in the training data as follows:

Center position:

The circular data boundary can include a significant amount of space in which training observations are very sparsely distributed. Scoring with this model can increase the probability of false positives. Hence, instead of a circular shape, a compact bounded outline around the data is often desired. Such an outline should approximate the shape of the one-class training data. Such a data boundary is possible with the use of kernel functions.

The support vector data description is made flexible by replacing the inner product present in the objective function of (Dual Form) with a suitable kernel function . The radial basis kernel function is defined as

where x and y are any two observations from the training data and s is the bandwidth parameter. Using a kernel function, the objective function is

subject to

The results for the center, inside, boundary, and outside positions, which are described in the section Duality Information, hold when the kernel function is used in the mathematical formulation.

The threshold is calculated as

using any , where is the set of support vectors for which .

For each observation z in the scoring data set, the distance is calculated as follows:

Observations in the scoring data set for which are designated as outliers.

Figure 7 illustrates key SVDD statistics. The unfilled markers indicate the bivariate training data set. The filled green marker indicates the center a. The filled blue markers are support vectors, which define the data boundary with a threshold value of 0.66. The plot shows contour lines for distance values from 0.76 to 1.06. The red markers indicate the outlier observations from the scoring data; their distance values are greater than the threshold value. The plot illustrates that as points move away from the training data, their distance value increases. It can be noted that the contours corresponding to different distance values follow the natural geometry of the training data.