## Drifting Markov processes via generalized linear models

Drifting Markov processes via generalized linear models

Jeudi 8 juillet 2021, 11:15 à 12:15

Salle des séminaires M.0.1

Mircea Dumitru

LMRS

The drifting Markov models estimate the transition matrices in non-homogeneous sequences by assuming a link between them and a set of support point matrices. Different models are issued from the choice of the model’s basis functions, which is always limited by the stochasticity constraint and the possible constraints coming from the assumption over the support point matrices. When the support point matrices are assumed to belong to the sequence of the transition matrices, the model has a clear interpretation but has more restrictions in terms of possible choices for the basis functions. More precisely, in this case, the models are restricted to only two non zero weights - which si a significant limitation -, or to particular class of polynomials that require some extra processing of the estimates, since the model doesnt guarantee stochastic estimates. Furthermore, it requires the optimization involved in the estimation to be performed under constraints. When this assumption is dropped i.e. the support point matrices are still stochastic but without being necessarily in the sequence themselves, mixture models can be considered, i.e. finite support probability mass functions as basis functions. In particular, the Binomial and BetaBinomial mixture models are considered.  Finally, when the assumptions on the support point matrices are dropped completely and the support point matrices are free parameters, the model’s parameters interpretation is lost but a generalized linear model via a link function is possible, assuring the stochasticity of the transition matrices in a natural way and allowing any choice for the basis function and also allowing the optimization during estimation to be performed without constraints (which becomes particularly important for settings requiring a higher number of support point matrices).

In this presentation, firstly, the limitations of linear models under the assumption of support point matrices in the sequence of transition matrices are discussed and the results corresponding to suitable basis functions under this assumption are presented (polynomial basis functions – considered in [Vergne2008], but not optimized under stochasticity constraints – and finite element basis functions). Secondly, extensions of this model are introduced, by dropping the assumption of support point matrices in the sequence of the transition matrices, and considering finite support probability mass functions as basis functions. Then, by dropping the stochasticity assumption for the support point matrices (hence free parameters) and considering a generalized linear model assuring the stochasticity via a softmax link function – corresponding to a logistic model.