{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/differentiated-uniformization-a-new-method","title":"Differentiated uniformization: A new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models","arxiv_id":"2112.10971","date":"2021-12-21","proceeding":null,"authors":["Kevin Rupp","Rudolf Schill","Jonas Süskind","Peter Georg","Maren Klever","Andreas Lösch","Lars Grasedyck","Tilo Wettig","Rainer Spang"],"abstract":"Motivation: We consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix $Q$ which depends on a parameter $\\theta$. Computing the probability distribution over states at time $t$ requires the matrix exponential $\\exp(tQ)$, and inferring $\\theta$ from data requires its derivative $\\partial\\exp\\!(tQ)/\\partial\\theta$. Both are challenging to compute when the state space and hence the size of $Q$ is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store $Q$. However, when $Q$ can be written as a sum of tensor products, computing $\\exp(tQ)$ becomes feasible by the uniformization method, which does not require explicit storage of $Q$. Results: Here we provide an analogous algorithm for computing $\\partial\\exp\\!(tQ)/\\partial\\theta$, the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that $Q$ can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Availability: Implementation and data are available at https://github.com/spang-lab/TenSIR.","url_abs":"https://arxiv.org/abs/2112.10971v1","url_pdf":"https://arxiv.org/pdf/2112.10971v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"differentiated-uniformization-a-new-method","repo_url":"https://github.com/spang-lab/TenSIR","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}