On the Non-uniqueness of Representations of Coxian Phase-Type Distributions

Abstract

Parameter estimation in Coxian phase-type models can be challenging due to their non-unique representation leading to a multi-modal likelihood. Since each representation corresponds to a different underlying data-generating mechanism, it is of interest to identify those supported by given data (i.e., find all likelihood modes). The standard approach is to simply refit using various initial values, but this has no guarantee of working. Thus, we develop new properties specific to this class of models, and employ these to determine all the equivalent model representations. The proposed approach only requires fitting the model once, and is guaranteed to find all representations.

An Alternative Formulation of Coxian Phase-type Distributions with Covariates: Application to Emergency Department Length of Stay

“…These optimisations methods are numerical and require initiation from a variety of initial values. For more details see Rizk et al (2019) and references therein.…”

“…Notwithstanding the usefulness of the CPH model in this context, the density function is complicated by the appearance of the matrix exponential, the likelihood surface is multimodal (cf. Rizk et al (2019) for further details), and, consequently, parameter estimation can be quite computationally intensive. Including covariates into all model parameters increases the model dimensionality further, and can lead to infeasibly large computational times.…”

“…However, a limitation of the models considered by those authors is that covariate influence was not considered. The inclusion of covariates even in the basic Coxian model is already computationally challenging as discussed in (Rizk et al, 2019). The compartmental model of course suffers from the same issues (but is even more complex still), and, perhaps, this is the reason that covariates have not been considered previously in the literature.…”

An alternative formulation of Coxian phase‐type distributions with covariates: Application to emergency department length of stay

“…Notwithstanding the usefulness of the CPH model in this context, the density function is complicated by the appearance of the matrix exponential, the likelihood surface is multimodal (see Rizk et al 16 for further details), and, consequently, parameter estimation can be quite computationally intensive. Including covariates into all model parameters increases the model dimensionality further, and can lead to infeasibly large computational times.…”

“…However, a limitation of the models considered by those authors is that covariate influence was not considered. The inclusion of covariates even in the basic Coxian model is already computationally challenging as discussed in Rizk et al 16 The compartmental model of course suffers from the same issues (but is even more complex still), and, perhaps, this is the reason that covariates have not been considered previously in the literature.…”