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.

中文翻译：

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带偏微分方程正则化的空间回归

这项工作概述了一类创新的方法，用于分析在复杂的二维域上观察到的空间和功能数据。此类基于回归，其中正则化项涉及偏微分方程。借助先进的数值分析技术解决了相关的估计问题。统计学、应用数学和工程学方法的协同相互作用赋予了这些方法在可用技术方面的重要优势，并使它们能够准确地处理经典技术不适合的数据结构。通过对血流速度的生态彩色多普勒测量分析的应用，说明了具有微分正则化的空间回归，