Functional Data Analysis represents a field of growing interest in statistics. Despite several studies have been proposed leading to fundamental results, the problem of obtaining valid and efficient prediction sets has not been thoroughly covered. Indeed, the great majority of methods currently in the literature rely on strong distributional assumptions (e.g, Gaussianity), dimension reduction techniques and/or asymptotic arguments. In this work, we propose a new nonparametric approach in the field of Conformal Prediction based on a new family of nonconformity measures inducing conformal predictors able to create closed-form finite-sample valid or exact prediction sets under very minimal distributional assumptions. In addition, our proposal ensures that the prediction sets obtained are bands, an essential feature in the functional setting that allows the visualization and interpretation of such sets. The procedure is also fast, scalable, does not rely on functional dimension reduction techniques and allows the user to select different nonconformity measures depending on the problem at hand always obtaining valid bands. Within this family of measures, we propose also a specific measure leading to prediction bands asymptotically no less efficient than those with constant width.
翻译:功能数据分析代表了对统计越来越感兴趣的一个领域。尽管提出了若干项研究,提出了一系列基本结果,但获得有效、高效的预测组的问题尚未完全涵盖。事实上,文献中目前绝大多数方法都依赖于强有力的分布假设(例如高森度)、减少尺寸技术和/或零时论。在这项工作中,我们提议在非正式预测领域采取新的非对称方法,其基础是新的不兼容措施,促使符合要求的预测器在极小的分布假设下创建封闭式有限抽样有效或精确的预测组。此外,我们的提议确保获得的预测组是频谱,这是功能环境中允许直观化和解释这类组合的一个基本特征。该程序也是快速的、可伸缩缩的,并不依赖功能减少技术,而且允许用户根据手头的问题选择不同的不兼容性措施,总是得到有效的波段。在这个措施组中,我们还提议采取一项具体措施,导致预测频段的效率不低于具有恒度的宽度的频段。