Scale-Invariant Correspondence Analysis of Compositional Data

Correspondence analysis is a dimension reduction technique for visualizing a non-negative matrix N=(nij) of size I×J, particularly contingency tables or compositional datasets, but it depends on the row and column marginals of N. Three complementary transformations of the data T(N)=(T(ainijbj)) render CA of T(N) invariant for any ai>0 and bj>0: first, Greenacre’s scale-invariant approach, valid for positive data; second, Goodman’s marginal-free correspondence analysis, valid for positive or moderately sparse data; third, correspondence analysis of the sign-transformed matrix, sign(N)=(sign(nij)), valid for sparse or extremely sparse data. We demonstrate these three methods on four real-world datasets with varying levels of sparsity to compare their exploratory performance.