The distribution of singletons suggest an underlying nonhomogeneous Poisson process, where the rate of incidence varies across the genome. In other areas of research, it has been shown that the waiting times between events arising from other nonhomogeneous Poisson processes, such as volcano eruptions or extreme weather events, can be accurately modelled as a mixture of exponential distributions98,99. Taking a similar approach, we model the distribution of inter-singleton distances across all Si singletons in individual i as a mixture of K exponential component distributions (fk(di;θi,k)), given by:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({d}_{i};{\lambda }_{i},\,{\theta }_{i})=\mathop{\sum }\limits_{k=1}^{K}{\lambda }_{i,k}\,{f}_{k}({d}_{i};{\theta }_{i,k})$$\end{document}f(di;λi,θi)=∑k=1Kλi,kfk(di;θi,k)where θi,1 < θi,2 < … < θi,K and λi,k = Si,k/Si is the proportion of singletons arising from component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}k, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sum }_{k=1}^{K}{\lambda }_{i,k}=1$$\end{document}∑k=1Kλi,k=1.
