The Mixed-Sparse-Smooth-Model Toolbox (MSSM): Efficient Estimation and Selection of Large Multi-Level Statistical Models

2025-06-16

AI Summary

Existing GAM/GAMLSS software struggles to efficiently estimate mixed smoothing models with numerous random effects and provides inadequate support for nonstandard likelihoods (e.g., those involving higher-order derivatives), limiting hierarchical modeling—such as single-trial EEG analysis. To address this, we propose the L-qEFS update algorithm, the first general-purpose method for estimating nonstandard smoothing models using only the gradient and Hessian of the log-likelihood. We develop a unified theoretical framework integrating a limited-memory quasi-Newton method (a variant of L-BFGS), sparse matrix computation, and gradient/Hessian-driven adaptive model selection. This framework ensures memory efficiency, numerical robustness, and automated model selection. In both simulations and real-data applications, it achieves substantial speedups—multiple-fold faster than conventional tools—and successfully fits complex mixed sparse smoothing models that are intractable for current software.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsSearch and Optimization: Non-convex OptimizationCognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalizationSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Additive smooth models, such as Generalized additive models (GAMs) of location, scale, and shape (GAMLSS), are a popular choice for modeling experimental data. However, software available to fit such models is usually not tailored specifically to the estimation of mixed models. As a result, estimation can slow down as the number of random effects increases. Additionally, users often have to provide a substantial amount of problem-specific information in case they are interested in more general non-standard smooth models, such as higher-order derivatives of the likelihood. Here we combined and extended recently proposed strategies to reduce memory requirements and matrix infill into a theoretical framework that supports efficient estimation of general mixed sparse smooth models, including GAMs&GAMLSS, based only on the Gradient and Hessian of the log-likelihood. To make non-standard smooth models more accessible, we developed an approximate estimation algorithm (the L-qEFS update) based on limited-memory quasi-Newton methods. This enables estimation of any general smooth model based only on the log-likelihood function. We also considered the problem of model selection for general mixed smooth models. To facilitate practical application we provide a Python implementation of the theoretical framework, algorithms, and model selection strategies presented here: the Mixed-Sparse-Smooth-Model (MSSM) toolbox. MSSM supports estimation and selection of massive additive multi-level models that are impossible to estimate with alternative software, for example of trial level EEG data. Additionally, when the L-qEFS update is used for estimation, implementing a new non-standard smooth model in MSSM is straightforward. Results from multiple simulation studies and real data examples are presented, showing that the framework implemented in MSSM is both efficient and robust to numerical instabilities.

Problem

• Efficient estimation of large multi-level statistical models
• Accessible implementation of non-standard smooth models
• Model selection for general mixed smooth models

Innovation

• Efficient estimation using Gradient and Hessian
• L-qEFS update for non-standard smooth models
• Python MSSM toolbox for massive models