Cancer cells acquire DNA mutations through many processes, such as environmental exposures and dysregulated repair mechanisms, and each process consistently produces distinct mutation types at characteristic frequencies, referred to as its signature. The usual approach for inferring these signatures is to decompose the matrix of mutation counts from a sample of tumors via non-negative matrix factorization (NMF). However, existing methods do not model the heterogeneity of mutation rates along the genome, which is driven in part by observed genomic features. In this paper, we introduce Poisson process factorization (PPF), which addresses this limitation by employing an inhomogeneous Poisson point process model to infer mutational signatures and their activities as they vary across the genome. PPF generalizes the baseline NMF model by representing a patient's exposure to each signature as a locus-specific function that depends on genomic covariates and patient-specific copy numbers via a log-linear model. We apply PPF to a sample of 113 breast tumors with $707{,}104$ mutations, using covariates representing histone modifications, cell replication timing, nucleosome positioning, and DNA methylation. Our analysis quantifies the joint effects of these features on the mutational processes in breast cancer, and estimates patient-specific activities of each signature as a function of genomic position.

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