Network data are increasingly available in various research fields, motivating statistical analysis for populations of networks where a network as a whole is viewed as a data point. Due to the non-Euclidean nature of networks, basic statistical tools available for scalar and vector data are no longer applicable when one aims to relate networks as outcomes to Euclidean covariates, while the study of how a network changes in dependence on covariates is often of paramount interest. This motivates to extend the notion of regression to the case of responses that are network data. Here we propose to adopt conditional Fr\'{e}chet means implemented with both global least squares regression and local weighted least squares smoothing, extending the Fr\'{e}chet regression concept to networks that are quantified by their graph Laplacians. The challenge is to characterize the space of graph Laplacians so as to justify the application of Fr\'{e}chet regression. This characterization then leads to asymptotic rates of convergence for the corresponding M-estimators by applying empirical process methods. We demonstrate the usefulness and good practical performance of the proposed framework with simulations and with network data arising from resting-state fMRI in neuroimaging, as well as New York taxi records.
翻译:各个研究领域都越来越多地提供网络数据,为整个网络被视为数据点的网络人口提供统计分析。由于网络的非欧化性质,在将网络与欧化共变结果相联系时,如果旨在将网络与欧化共变结果联系起来,而研究网络依赖共变物系变化对共变物的依赖性变化如何经常引起极大的兴趣。这促使将回归概念扩大到网络数据答复的回归情况。我们建议采用有条件的Fr\'{{e}chet,这意味着既在全球最小正方回归和局部加权最小平方均匀的情况下实施有条件的Fr\{e}chelect 回归概念,又将其扩展至其图 Laplacians量化的网络时,不再适用这些基本统计工具。挑战在于如何描述拉平面图的空间,从而证明应用Fr\'{{e}chetchregrel 回归值的正当性。然后通过应用实证过程方法,使相应的M-Sertaminators 实现一致。我们展示了拟议框架的有用性和良好实际表现,作为模拟记录,并借助网络进行模拟和记录。