A scalable approach for continuous time Markov models with covariates

Existing methods for fitting continuous time Markov models (CTMM) in the presence of covariates suffer from scalability issues due to high computational cost of matrix exponentials calculated for each observation. In this paper we propose an optimization technique for CTMM which uses a stochastic gradient descent algorithm combined with differentiation of the matrix exponential using a Pade approximation which we show then makes fitting large scale data feasible. We compare two methods for computing standard errors, one using Pade expansion, and the other using power series expansion of the matrix exponential. Through simulations we find improved performance relative to existing CTMM methods, and we demonstrate the method on the large-scale multiple sclerosis NO.MS dataset.