Validity of covariance models for the analysis of geographical variation

Due to the availability of large molecular data sets, covariance models are increasingly used to describe the structure of genetic variation as an alternative to more heavily parameterized biological models. We focus here on a class of parametric covariance models that received sustained attention lately and show that the conditions under which they are valid mathematical models have been overlooked so far. We provide rigorous results for the construction of valid covariance models in this family. We also outline how to construct alternative covariance models for the analysis of geographical variation that are both mathematically well behaved and easily implementable. The full R code to reproduce the numerical analysis is available from: http://www2.imm.dtu.dk/~gigu/MEE_GSPB/covariance_validity.html.