  Somatic mutations in cancer can be viewed as a mixture distribution of
several mutational signatures, which can be inferred using non-negative matrix
factorization (NMF). Mutational signatures have previously been parametrized
using either simple mono-nucleotide interaction models or general
tri-nucleotide interaction models. We describe a flexible and novel framework
for identifying biologically plausible parametrizations of mutational
signatures, and in particular for estimating di-nucleotide interaction models.
The estimation procedure is based on the expectation--maximization (EM)
algorithm and regression in the log-linear quasi--Poisson model. We show that
di-nucleotide interaction signatures are statistically stable and sufficiently
complex to fit the mutational patterns. Di-nucleotide interaction signatures
often strike the right balance between appropriately fitting the data and
avoiding over-fitting. They provide a better fit to data and are biologically
more plausible than mono-nucleotide interaction signatures, and the
parametrization is more stable than the parameter-rich tri-nucleotide
interaction signatures. We illustrate our framework on three data sets of
somatic mutation counts from cancer patients.
