Generalised Linear Models Driven by Latent Processes: Asymptotic Theory and Applications

2026-02-18

AI Summary

This work proposes a latent-process-driven generalized linear model based on the two-parameter exponential family, overcoming the limitations of existing approaches that rely on the single-parameter exponential family assumption and thus struggle to jointly model count, binary, real-valued, and positive continuous time series. By introducing multiplicative latent process effects, the framework accommodates distributions such as Gamma, relaxes constraints on link functions, and establishes asymptotic normality for likelihood-based estimators under marginalization of the latent process. Key innovations include the first prediction and forecasting methodology for this class of models, method-of-moments estimation for unknown discrete parameters, and a corrected information matrix. Empirical validation on German measles incidence and paleoclimatic varve data demonstrates the model’s flexibility and practical utility in estimation, inference, and prediction.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This paper introduces a class of generalised linear models (GLMs) driven by latent processes for modelling count, real-valued, binary, and positive continuous time series. Extending earlier latent-process regression frameworks based on Poisson or one-parameter exponential family assumptions, we allow the conditional distribution of the response to belong to a bi-parameter exponential family, with the latent process entering the conditional mean multiplicatively. This formulation substantially broadens the scope of latent-process GLMs, for instance, it naturally accommodates gamma responses for positive continuous data, enables estimation of an unknown dispersion parameter via method of moments, and avoids restrictive conditions on link functions that arise under existing formulations. We establish the asymptotic normality of the GLM estimators obtained from the GLM likelihood that ignores the latent process, and we derive the correct information matrix for valid inference. In addition, we provide a principled approach to prediction and forecasting in GLMs driven by latent processes, a topic not previously addressed in the literature. We present two real data applications on measles infections in North Rhine-Westphalia (Germany) and paleoclimatic glacial varves, which highlight the practical advantages and enhanced flexibility of the proposed modelling framework.

Problem

• latent processes
• generalised linear models
• bi-parameter exponential family
• time series
• asymptotic inference

Innovation

• latent process
• generalised linear models
• biparameter exponential family
• asymptotic normality
• forecasting