Backscoring in principal coordinates analysis

Principal coordinates analysis refers to the low-dimensional projection of data obtained from distance matrix based methods such as multidimensional scaling. Principal components analysis also produces a low-dimensional projection of data and has the convenience of explicit mappings to and from the data space and the projected score space being readily available. The map from data to score is called called out-of-sample embedding. We call the map from score to data, backscoring. We discuss how these mappings may be obtained for a principal coordinates analysis and demonstrate applications for orientation, shape, functional and mixed data. The application to functional data shows how both phase and amplitude variation can be described together. Backscoring is helpful for interpreting the meaning of scores and in simulating new data. Data and R code necessary to reproduce the results are provided as supplemental materials.