Spatial models for areal data are often constructed such that all pairs of adjacent regions are assumed to have near-identical spatial autocorrelation. In practice, data can exhibit dependence structures more complicated than can be represented under this assumption. In this article we develop a new model for spatially correlated data observed on graphs, which can flexibly represented many types of spatial dependence patterns while retaining aspects of the original graph geometry. Our method implies an embedding of the graph into Euclidean space wherein covariance can be modeled using traditional covariance functions, such as those from the Mat\'{e}rn family. We parameterize our model using a class of graph metrics compatible with such covariance functions, and which characterize distance in terms of network flow, a property useful for understanding proximity in many ecological settings. By estimating the parameters underlying these metrics, we recover the "intrinsic distances" between graph nodes, which assist in the interpretation of the estimated covariance and allow us to better understand the relationship between the observed process and spatial domain. We compare our model to existing methods for spatially dependent graph data, primarily conditional autoregressive models and their variants, and illustrate advantages of our method over traditional approaches. We fit our model to bird abundance data for several species in North Carolina, and show how it provides insight into the interactions between species-specific spatial distributions and geography.

翻译：面向区域数据的空间模型通常构建为假设所有相邻区域对具有近乎相同的空间自相关性。实际上，数据可能展现出比此假设所能表征的更为复杂的依赖结构。本文针对在图上观测到的空间相关数据开发了一种新模型，该模型能够灵活表征多种类型的空间依赖模式，同时保留原始图几何的某些特征。我们的方法隐含着将图嵌入欧几里得空间，在该空间中可使用传统协方差函数（例如来自Matérn族的函数）对协方差进行建模。我们使用一类与此类协方差函数兼容的图度量来参数化模型，这类度量基于网络流来表征距离——这一特性对于理解许多生态场景中的邻近性非常有用。通过估计这些度量背后的参数，我们恢复了图节点间的“本征距离”，这有助于解释估计的协方差，并使我们能更好地理解观测过程与空间域之间的关系。我们将本模型与现有的空间依赖图数据处理方法（主要是条件自回归模型及其变体）进行比较，并阐明本方法相较于传统方法的优势。我们将模型应用于北卡罗来纳州多种鸟类的丰度数据，展示了其如何帮助理解物种特异性空间分布与地理环境之间的相互作用。