Spatial Regression With Partial Differential Equation Regularisation
本文综述了一类分析复杂二维空间和函数型数据的新方法，该方法在回归中引入偏微分方程正则化项，结合统计学、应用数学和工程技术，能处理经典方法不适用的数据结构，并通过血流速度和脑功能成像数据展示其应用。
Summary This work gives an overview of an innovative class of methods for the analysis of spatial and of functional data observed over complicated two‐dimensional domains. This class is based on regression with regularising terms involving partial differential equations. The associated estimation problems are solved resorting to advanced numerical analysis techniques. The synergical interplay of approaches from statistics, applied mathematics and engineering endows the methods with important advantages with respect to the available techniques, and makes them able to accurately deal with data structures for which the classical techniques are unfit. Spatial regression with differential regularisation is illustrated via applications to the analysis of eco‐colour doppler measurements of blood‐flow velocity, and to functional magnetic resonance imaging signals associated with neural connectivity in the cerebral cortex.