Functional data are typically modeled as sampled paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The required smoothness allows for the use of smoothing techniques but excludes many stochastic processes, most notably diffusion processes. Such processes would otherwise be well within the realm of functional data analysis, at least under complete observation. In this short note we demonstrate that a simple modification of existing methods allows for the functional data analysis of processes with nowhere differentiable sample paths, even when these are discretely and noisily observed, including under irregular and sparse designs. By way of simulation it is shown that this is not a theoretical curiosity, but can work well in practice, hinting at potential closer links with the field of diffusion inference.

翻译：功能数据通常以光滑随机过程的抽样路径为模型模型,以便减轻以下事实,即经常以离散和静默的方式、偶尔以不定期和稀少的方式观测这些数据;所需的顺畅性允许使用平滑技术,但排除了许多随机过程,最明显的是扩散过程;否则,这些过程将完全属于功能数据分析领域,至少是完整观察的范围;在本简短的说明中,我们表明,对现有方法进行简单修改,就能够对过程进行功能性数据分析,而没有不同样路径,即使这些样径是分散和静默地观测的,包括在非常规和稀有的设计下观测。模拟表明,这不是一种理论好奇心,但在实践中可以很好地发挥作用,暗示着与扩散推断领域可能更密切的联系。