## 2 Materials and Methods  
Our experimental analysis is focused on public available data of The Cancer Genome Atlas (TCGA) [15] program hosted on the Genomic Data Commons (GDC) portal [[https://portal.gdc.cancer.gov/](https://portal.gdc.cancer.gov/)]([https://portal.gdc.cancer.gov/](https://portal.gdc.cancer.gov/))[14]. This project aims to catalog the genetic mutations responsible for cancer, applying NGS techniques to improve the ability to diagnose, treat, and prevent cancer through a better understanding of the genetic basis of this disease. TCGA contains the genomic characterization and analysis of 33 types of tumors, and patient samples are processed through different types of NGS techniques, such as DNA sequencing, gene expression profiling, and DNA methylation. Additionally, TCGA collects and analyzes high quality cancer samples and makes the following data available to the research community: (i) clinical information on the patients participating in the program; (ii) samples metadata (for example, the weight of the sample, etc.); and, (iii) histopathological images of portions of the sample.  
In this work, we focus on DNA methylation (DM) experiments, which describe the amount of methylated molecules for each of the known CpG regions on bisulfite-treated DNA sequences. If a methylation occurs on a DNA region, this causes the interruption of the transcriptional process of the promoter gene, where the involved region falls in. The amount of methylated molecules detected by the sequencing machine are represented by the beta value, [37]. Which is computed according to the Formula (1), where \(Meth_{n}\) and \(Unmeth_{n}\) are both intensities of the \(n\)-th methylated and unmethylated allele, respectively. The \(\epsilon\) parameter acts as a modulator which is required when both the \(Meth_{n}\) and \(Unmeth_{n}\) are very low. The beta value is therefore a continuous variable in the (0, 1) range, where 1 represents full methylation and 0 no methylation at all.  
\[\beta_{n}=\frac{max(Meth_{n},0)}{max(Meth_{n},0)+max(Unmeth_{n},0)+\epsilon} \tag{1}\]  
The relevance of DNA methylation data has been widely discussed in the literature, and it has been proven that their processing can discriminate the diseased (tumor) from the healthy (also called normal) samples in case of cancer. For more details, we point the reader to Section 1.  
The analysis of DNA methylation data is challenging, because of the huge amount of CpG sites processed by the sequencing machines, i.e., up to 485,577 CpG sites, which represent the features in a supervised machine learning problem. Canonical classification approaches are not designed to manage hundred of thousands of features and most of the time fail in producing results, because of the required amount of computational resources. For this reason, we redesigned a classification procedure previously applied mainly on language detection and speech recognition problems. This method exploits high-dimensional numerical vectors (thus, hyperdimensional--HD) to represent information in the considered dataset. In particular, the atomic numerical values (or properly encoded categorical values) are represented as HD vectors, and the observations are also HD vectors built by combining their values--vector representations. In the context of DNA methylation data of cancer, we create two HD vectors representing the tumor and normal classes by collapsing all of the HD vector representations of the experiments considered during a first training phase. Thus, the class prediction of every sample is performed by computing the inner product between their HD representations and the classes HD vectors.  
From a theoretical perspective, the whole procedure is as simple as powerful, and it can be summarised in Figure 1, and thoroughly explained with the steps that are described in the next subsections.