Factorial latent dynamic models trained on Markovian simulations of biological processes using scRNAseq. data.

With a transition probability matrix $T$ over observed states $O$ and assuming Markovian dynamics, $P(o \mid i) = P(o \mid o_{i-1})$ For iteration $i$, $P(o \mid i) = P(o \mid i=0) \cdot T^i$ The animation overlays $P(i \mid o)$ on a 2D UMAP embedding of the data (Cerletti et. al. 2020) Since we are interested in modelling the dynamics in a smaller latent state space, we factorise the MSM simulation, $P(o \mid i) = \sum\limits_{s \in S} P(o \mid s,i) P(s \mid i)$ Assuming Markovian dynamics in the latent space aswell, $P(o \mid i) = \sum\limits_{s_{i} \in S} P(o \mid s_{i}) \sum\limits_{s_{i-1} \in S} P(s_{i} \mid s_{i-1})$ Multiple independent chains in a common latent space can be modelled using conditional latent TPMs (Ghahramani & Jordan 1997), $P(o \mid i) = \sum\limits_{s_{i} \in S} P(o \mid s_{i}) \sum\limits_{l \in L} P(l) \sum\limits_{s_{i-1} \in S} P(s_{i} \mid s_{i-1}, l)$

Citation

Claassen, M., & Gupta, R. (2023). Factorial state-space modelling for kinetic clustering and lineage inference. https://doi.org/10.1101/2023.08.21.554135

Notebooks

Demonstration notebooks can be found here.