| Autor: |
Rupp, Kevin, Schill, Rudolf, Süskind, Jonas, Georg, Peter, Klever, Maren, Lösch, Andreas, Grasedyck, Lars, Wettig, Tilo, Spang, Rainer |
| Rok vydání: |
2021 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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