The International Committee on Taxonomy of Viruses authorizes and organizes the taxonomic classification of viruses. Thus far, the detailed classifications for all viruses are neither complete nor free from dispute. For example, the current missing label rates in GenBank are 12.1% for family label and 30.0% for genus label. Using the proposed Natural Vector representation, all 2,044 single-segment referenced viral genomes in GenBank can be embedded in R12. Unlike other approaches, this allows us to determine phylogenetic relations for all viruses at any level (e.g., Baltimore class, family, subfamily, genus, and species) in real time. Additionally, the proposed graphical representation for virus phylogeny provides a visualization of the distribution of viruses in R12. Unlike the commonly used tree visualization methods which suffer from uniqueness and existence problems, our representation always exists and is unique. This approach is successfully used to predict and correct viral classification information, as well as to identify viral origins; e.g. a recent public health threat, the West Nile virus, is closer to the Japanese encephalitis antigenic complex based on our visualization. Based on cross-validation results, the accuracy rates of our predictions are as high as 98.2% for Baltimore class labels, 96.6% for family labels, 99.7% for subfamily labels and 97.2% for genus labels.