Flexible specification of directional spatial processes for eigenvector maps

Understanding spatial structure is crucial when analysing species distributions, biodiversity and community dynamics. Spatial analysis using eigenvector maps provides a general framework for capturing spatial patterns in ecological data. Among these methods, asymmetric eigenvector maps (AEM) are particularly useful for capturing asymmetric spatial patterns caused by directional processes, such as transport by water flow. Spatial analysis using AEM is widely applied in ecological studies of aquatic environments. However, the interpretation of spatial eigenfunctions extracted by AEM and their underlying representation of spatial dependence remains unclear. This study clarifies the conceptual basis of AEM and proposes an extension, termed extended asymmetric eigenvector maps (XAEM), to better represent directional spatial dependence induced by transport processes. First, we demonstrate that the original AEM can be interpreted as representing spatial processes based on reachability. We then extend this framework by introducing a spatial lag structure to represent directional dependence with attenuation rather than assuming simple reachability, as in the original AEM. We conduct simulations to compare the performances of XAEM and AEM across a variety of spatial patterns. The simulation results show that the proposed model flexibly captures asymmetric spatial patterns across a wide range of scenarios, from deterministic unidirectional gradients to stochastic spatial structures, generated under varying attenuation parameters. To demonstrate its applicability to real‐world ecosystems, we apply the proposed method to environmental DNA (eDNA)–based coastal fish community data. The spatial eigenfunctions derived from the observed current flow using XAEM capture spatial patterns that closely match the water temperature distributions, whereas AEM did not detect this pattern. This highlights the potential of XAEM for analysing flow‐driven spatial structures in coastal environments. The proposed approach improves our understanding of asymmetric spatial patterns arising from the spatial dependence induced by transport processes, which may be overlooked by conventional methods.