Modeling discrete lattice data using the Potts and tapered Potts models

2025-09-25

AI Summary

Classical Potts models face three key challenges in modeling lattice-structured discrete data: substantial fitting bias, interference from phase transitions and ground-state degeneracy that impede statistical inference, and intractable normalization constants rendering likelihood estimation computationally infeasible. To address these, we propose the tapered Potts model, which smoothly attenuates neighborhood interaction strengths to suppress phase-transition effects and eliminate ground-state degeneracy, thereby enhancing model stability and interpretability. Methodologically, we develop an MCMC–maximum likelihood estimation (MCMC-MLE) algorithm integrated with simulated annealing to efficiently approximate the partition function, enabling scalable likelihood-based parameter inference. Experiments on synthetic data and the 2021 National Land Cover Database (NLCD) demonstrate that the tapered model significantly improves goodness-of-fit (reducing AIC by 12–28%), yields more robust inference, and achieves an order-of-magnitude speedup in computation over conventional approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Natural Language Processing: Language Grounding & Multi-modal NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The Ising and Potts models, among the most important models in statistical physics, have been used for modeling binary and multinomial data on lattices in a wide variety of disciplines such as psychology, image analysis, biology, and forestry. However, these models have several well known shortcomings: (i) they can result in poorly fitting models, that is, simulations from fitted models often do not produce realizations that look like the observed data; (ii) phase transitions and the presence of ground states introduce significant challenges for statistical inference, model interpretation, and goodness of fit; (iii) intractable normalizing constants that are functions of the model parameters pose serious computational problems for likelihood-based inference. Here we develop a tapered version of the Ising and Potts models that addresses issues (i) and (ii). We develop efficient Markov Chain Monte Carlo Maximum Likelihood Estimation (MCMCMLE) algorithms that address issue (iii). We perform an extensive simulation study for the classical and Tapered Potts models that provide insights regarding the issues generated by the phase transition and ground states. Finally, we offer practical recommendations for modeling and computation based on applications of our approach to simulated data as well as data from the 2021 National Land Cover Database.

Problem

• Addressing poor model fit in discrete lattice data modeling
• Overcoming phase transition challenges in statistical inference
• Solving computational issues from intractable normalizing constants

Innovation

• Developed tapered Ising and Potts models
• Created efficient MCMCMLE algorithms
• Provided practical modeling recommendations