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 article, 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. This approach makes fitting large scale data feasible. We present two methods for computing standard errors, one novel approach using the 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 data set.

Automated summary

In this article, an optimization technique for fitting continuous time Markov models (CTMM) in the presence of covariates is proposed. This technique combines a stochastic gradient descent algorithm with differentiation of the matrix exponential using a Pade approximation, making it feasible to fit large scale data. Two methods for computing standard errors are presented, one utilizing the Pade expansion and the other using power series expansion of the matrix exponential. Simulation results show improved performance compared to existing CTMM methods, and the method is demonstrated on a large-scale multiple sclerosis NO.MS dataset.