A2 Copula-Driven Spatial Bayesian Neural Network For Modeling Non-Gaussian Dependence: A Simulation Study

In this paper, we introduce the A2 Copula Spatial Bayesian Neural Network (A2-SBNN), a predictive spatial model designed to map coordinates to continuous fields while capturing both typical spatial patterns and extreme dependencies. By embedding the dual-tail novel Archimedean copula viz. A2 directly into the network's weight initialization, A2-SBNN naturally models complex spatial relationships, including rare co-movements in the data. The model is trained through a calibration-driven process combining Wasserstein loss, moment matching, and correlation penalties to refine predictions and manage uncertainty. Simulation results show that A2-SBNN consistently delivers high accuracy across a wide range of dependency strengths, offering a new, effective solution for spatial data modeling beyond traditional Gaussian-based approaches.

Theoretical Foundations of the Deep Copula Classifier: A Generative Approach to Modeling Dependent Features

Traditional classifiers often assume feature independence or rely on overly simplistic relationships, leading to poor performance in settings where real-world dependencies matter. We introduce the Deep Copula Classifier (DCC), a generative model that separates the learning of each feature's marginal distribution from the modeling of their joint dependence structure via neural network-parameterized copulas. For each class, lightweight neural networks are used to flexibly and adaptively capture feature interactions, making DCC particularly effective when classification is driven by complex dependencies. We establish that DCC converges to the Bayes-optimal classifier under standard conditions and provide explicit convergence rates of O(n^{-r/(2r + d)}) for r-smooth copula densities. Beyond theoretical guarantees, we outline several practical extensions, including high-dimensional scalability through vine and factor copula architectures, semi-supervised learning via entropy regularization, and online adaptation using streaming gradient methods. By unifying statistical rigor with the representational power of neural networks, DCC offers a mathematically grounded and interpretable framework for dependency-aware classification.

IGNIS: A Neural Network Framework for Robust Parameter Estimation in Archimedean Copulas

Parameter estimation for Archimedean copulas remains a challenging problem, particularly for the recently developed A1 and A2 families that exhibit complex dependency structures. Traditional methods, such as the Method of Moments (MoM), Maximum Likelihood Estimation (MLE), and Maximum Pseudo-Likelihood (MPL), often struggle due to issues of non-monotonic relationship with dependency measures such as Kendall's tau (as in the case of A1) and numerical instability. In this paper, we present the IGNIS Network, a novel, unified neural framework that learns a direct mapping from observable dependency measures to copula parameters, thereby overcoming the limitations of classical approaches. Our approach is trained on simulated data spanning five Archimedean copula families including Clayton, Gumbel, Frank, A1, and A2, ensuring its general applicability across the entire family. Extensive simulation studies demonstrate that the IGNIS Network reduces estimation errors compared to MoM, while inherently enforcing parameter constraints through theory-guided post-processing. We further validate the practical utility of our method on diverse real-world datasets, including financial returns (AAPL-MSFT), healthcare metrics (CDC Diabetes indicators), and environmental measurements (PM2.5 air quality). Our results underscore the transformative potential of neural methods for robust and accurate dependence modeling in modern applications.