In recent years, in many application fields, extracting information from data in the form of functions is of most interest rather than investigating traditional multivariate vectors. Often these functions have complex spatial dependences that need to be accounted for using appropriate statistical analysis. Spatial Functional Statistics presents a fruitful analytics framework for this analysis. The definition of a distance measure between spatially dependent functional data is critical for many functional data analysis tasks such as clustering and classification. For this reason, and based on the specific characteristics of functional data, several distance measures have been proposed in the last few years. In this work we develop a weighted $L^2$ distance for spatially dependent functional data, with an optimized weight function. Assuming a penalized basis representation for the functional data, we consider weight functions depending also on the spatial location in two different situations: a classical georeferenced spatial structure and a connected network one. The performance of the proposed distances are compared using standard metrics applied to both real and simulated data analysis.

近年来，在许多应用领域中，最感兴趣的是从函数形式的数据中提取信息，而不是研究传统的多元向量。通常，这些功能具有复杂的空间依赖性，需要使用适当的统计分析来解决。空间功能统计为该分析提供了一个富有成果的分析框架。空间相关功能数据之间的距离度量的定义对于许多功能数据分析任务（例如聚类和分类）至关重要。因此，根据功能数据的特定特性，最近几年已经提出了几种距离测量方法。在这项工作中，我们制定了加权${\mathrm{大号}}^2$具有优化权重函数的空间相关功能数据的距离。假设功能数据的惩罚基础表示，我们考虑权重函数还取决于两种不同情况下的空间位置：经典地理参考空间结构和连接网络。使用适用于实际和模拟数据分析的标准指标比较建议距离的性能。