## 3 Methods and Materials
### Support Vector Domain Description for Confidence Volumes  
A method that has been proposed for determining the support of a set of points is Support Vector Domain Description (SVDD), which is different to the two-class Support Vector Classifier. SVDD is a one-class procedure which aims to separate one single class of objects from the subset different from them [23, 24]. It has been widely used in anomaly detection, as well as to determine the support of a given distribution, which is the main use in the present work.  
In our case, let us denote by \(\{\mathbf{t}_{i}\}_{i=1}^{N}\) a dataset from a _class of interest_. Our purpose is to build the _hypersphere_ with the lowest possible volume in a high-dimensional feature space,denoted as \(\mathcal{H}\), whose radius is \(R>0\), and it is centered at point \(\mathbf{a}\in\mathcal{H}\), in such a way that this geometrical entity conveys inside most of said set of objects. A nonlinear mapping is used to map the original data \(\mathbf{t}_{i}\) to the \(\mathcal{H}\) space, given by \(\boldsymbol{\phi}(\mathbf{t}_{i})\). Note also that, in this case, we are interested in projecting vectors \(\mathbf{t}_{i}\) corresponding to indicator vectors for each category in the MCA decomposition.  
Not only the set of interest can contain outliers, but also we are interested in leaving out of the domain region some portion of the data, so that we can interpret the result as a confidence domain, similar to the concept of confidence interval in statistics. The SVDD algorithm considers this according to the inclusion of _slack variables_\(\xi_{i}\geq 0\) accounting for the distance to the hypersphere of some of the observations, and the problem statement is then  
\[\min_{R,\mathbf{a}}\left\{R^{2}+C\sum_{i=1}^{N}\xi_{i}\right\} \tag{8}\]  
where constrained to  
\[\|\boldsymbol{\phi}(\mathbf{t}_{i})-\mathbf{a}\|^{2}\leq R^{2}+\xi_{i},\xi_{i }\geq 0,\quad\forall i=1,\ldots,N \tag{9}\]  
where parameter \(C\) represents a trade-off between the hypersphere volume and the committed errors. It is common to use the so-called reject parameter, defined as \(\
u=1/(NC)\), which represents an upper bound of the number of outliers or samples outside the domain, so that it provides a convenient way to control for the confidence required to the volumes provided by the SVDD in our problem.  
It can be shown [23; 24] that the dual functional is a constrained quadratic form and it yields a set of Lagrange multipliers (\(\alpha_{i}\)) in terms of the constraints represented by each observation. When we adjust \(C\) properly, most of \(\alpha_{i}\) are nulls and they give a sparse solution. The unbounded support vectors are those observations \(\mathbf{t}_{i}\) fulfilling \(0\leqslant\alpha_{i}<C\), whereas the bounded support vectors exactly fulfill \(\alpha_{i}=C\), and they can be considered either outliers or tails of a multidimensional distribution. The utility of these Lagrange multipliers is to obtain the distance of a test point to the hypersphere center,  
\[R(\mathbf{t}_{*})=K(\mathbf{t}_{*},\mathbf{t}_{*})-2\sum_{i=1}^{N}K(\mathbf{t }_{i},\mathbf{t}_{*})+\sum_{i,j=1}^{N}K(\mathbf{t}_{i},\mathbf{t}_{j}), \tag{10}\]  
where to be compared with radius \(R\). We can calculate the boundary of the domain and represent projections of each separate feature on the MCA subspace, which provides as with a way to estimate the statistical uncertainty of our estimations in the MCA method. When using a Gaussian kernel method, the kernel width has to be preset in a similar way as in the case of Parzen windows.