无MCMC的大规模空间变系数模型深度生成推断的不确定性量化

中文摘要 AI 辅助

变系数（VC）回归模型已成为空间数据分析中不可或缺的工具，在捕捉预测变量对响应的复杂非线性关系及空间变化效应方面提供了无与伦比的灵活性。尽管分层贝叶斯方法为VC建模中的不确定性量化提供了严格的概率框架，但其在大规模空间数据集上的实际应用仍受到马尔可夫链蒙特卡洛（MCMC）算法可扩展性限制的严重阻碍。为此，过去十年中，开发更高效的分层贝叶斯VC模型取得了实质性进展，主要通过用计算高效的随机替代模型替代传统高斯过程（GP）来估计未知系数函数。本文介绍了一种根本不同的方法：地质统计变分自编码器（GeoVAE），这是一种专门为联合估计多个空间变化系数函数而构建的分层深度生成框架。GeoVAE在两个方面区别于经典的基于GP的贝叶斯模型和标准变分自编码器（VAE）。首先，它为每个系数函数构建一个系数特定的自编码器，使每个系数能够捕捉其自身的空间分辨率和平滑度。其次，一个分层合成层将这些自编码器的信息整合到一个共享自编码器中，显式建模由共享空间域产生的跨系数依赖性。这种设计使GeoVAE能够以随样本量有利扩展的计算成本恢复复杂的空间模式，无需MCMC或显式GP协方差表示。我们描述了GeoVAE相对于分层贝叶斯空间模型的优势和局限性，展示了其在大规模空间分析中的强大潜力。

英文摘要

Varying coefficient (VC) regression models have become indispensable tools in spatial data analysis, providing unmatched flexibility in capturing complex, nonlinear relationships and spatially-varying effects of predictors on responses. Although hierarchical Bayesian approaches offer a rigorous probabilistic framework for uncertainty quantification in VC modeling, their practical application to large-scale spatial datasets remains severely hindered by the scalability limitations of Markov chain Monte Carlo (MCMC) algorithms. In response, the past decade has seen substantial advances in developing more efficient hierarchical Bayesian VC models, primarily through substituting traditional Gaussian processes (GP) with computationally efficient stochastic surrogates for estimating unknown coefficient functions. This article introduces a fundamentally different approach: the Geostatistical Variational Auto-Encoder (GeoVAE), a hierarchical deep generative framework built specifically for joint estimation of multiple spatially varying coefficient functions. GeoVAE departs from both classical GP-based Bayesian models and standard variational auto-encoders (VAEs) in two key ways. First, it constructs a coefficient-specific auto-encoder for each coefficient function, allowing each to capture its own spatial resolution and smoothness. Second, a hierarchical synthesis layer integrates information across these auto-encoders to a shared auto-encoder, explicitly modeling cross-coefficient dependencies arising from the shared spatial domain. This design enables GeoVAE to recover complex spatial patterns at computational cost that scales favorably with sample size, without MCMC or explicit GP covariance representations. We characterize the advantages and limitations of GeoVAE relative to hierarchical Bayesian spatial models, demonstrating its strong potential for large-scale spatial analysis.

发表机构

• University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
• University of California, San Francisco（加州大学旧金山分校）
• Texas A&M University（德克萨斯农工大学）

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