Aggregate PCoA Chart

The principal coordinates analysis (PCoA) chart in the aggregate report is generated using classical multidimensional scaling (MDS) on normalized classification vectors for each sample. An overview of the steps of the algorithm is presented in this section.

Normalized Classification Matrix

A normalized classification matrix is created for a range of taxonomic levels. The first matrix includes only kingdom classifications, the second includes kingdom and phylum, and so on, until the full range of taxonomic levels is considered in the species-level classification matrix. These ranges correspond to the user-selectable levels in the aggregate report.

The set of classifications present within the current range of taxonomic levels across all samples is collected. Then a label vector is created by placing each unique classification at a unique index in the vector. Classifications for each sample within the current taxonomic range are collected, and unclassified classifications are discarded. A vector is created for each sample, which is the projection of the sample classifications within the current taxonomic range onto the label vector of all non-empty classifications at each index. Each sample vector is then L-1 normalized by multiplying every index by the inverse of the sum of the sample vector.

The resulting vectors form a classification matrix, in which each row represents a unique non-empty classification present within the current taxonomic range. Each column represents one sample L-1 normalized projection of non-empty classifications onto this space.

Pearson Correlation Distance Matrix

Pearson correlation is calculated for each pair of L-1 normalized sample classification vectors. A distance matrix is then calculated as 1 – r, where r is the Pearson correlation measure between two samples normalized classification vectors.

Classical MDS

Classical MDS is performed on the distance matrix output from the previous step. The MDS is implemented as described in mathpsy.uni-tuebingen.de/wickelmaier/pubs/Wickelmaier2003SQRU.pdf.