Spatial, amplitude and phase variations in spatial functional data are confounded. Conclusions from the popular functional trace-variogram, which quantifies spatial variation, can be misleading when analyzing misaligned functional data with phase variation. To remedy this, we describe a framework that extends amplitude-phase separation methods in functional data to the spatial setting, with a view towards performing clustering and spatial prediction. We propose a decomposition of the trace-variogram into amplitude and phase components, and quantify how spatial correlations between functional observations manifest in their respective amplitude and phase. This enables us to generate separate amplitude and phase clustering methods for spatial functional data, and develop a novel spatial functional interpolant at unobserved locations based on combining separate amplitude and phase predictions. Through simulations and real data analyses, we demonstrate advantages of our approach when compared to standard ones that ignore phase variation, through more accurate predictions and more interpretable clustering results.

空间函数数据中的空间、幅度和相位变化是混杂的。在分析具有相位变化的未对准函数数据时，流行的函数迹变差图量化了空间变化，其结论可能会产生误导。为了解决这个问题，我们描述了一个框架，将功能数据中的幅度相位分离方法扩展到空间设置，以执行聚类和空间预测。我们建议将迹变差函数分解为幅度和相位分量，并量化功能观测值之间的空间相关性如何在各自的幅度和相位中体现。这使我们能够为空间函数数据生成单独的幅度和相位聚类方法，并基于组合单独的幅度和相位预测在未观测的位置开发一种新颖的空间函数插值。通过模拟和真实数据分析，我们通过更准确的预测和更可解释的聚类结果，展示了我们的方法与忽略相位变化的标准方法相比的优势。