Nonlinear Models for Functional Data Analysis

函数数据分析的非线性方法导致灵活和通用的统计模型，推理和分析方法的数据，包括随机函数的样本。这些数据积累在时间动态现象的研究中，如电力消耗曲线或生物轨迹，以及跨科学的大量纵向研究中。在过去的几年中，函数数据分析的方法得到了迅速的发展，并越来越多地被视为分析时间动态现象的必要条件。到目前为止，线性模型的功能数据已经比较好的调查，无论是在理论和实践方面，和统计工具，是基于这些方法的数据分析。然而，线性函数模型的类别相当狭窄，往往不足以用于实际数据分析。相比之下，对更一般和更灵活的非线性方法知之甚少。本研究试图通过发展一类非线性泛函方法来弥补这种情况。这种模型的潜在应用价值很高，特别是在观察时间或空间上重复函数的情况下，或者希望研究包括函数分量作为预测因子或响应的回归关系的情况下。非线性函数方法包括通过非线性分量或通过混合模型来表示轨迹的样本。这些方法对于随机时间规整发挥作用的应用程序以及在函数回归设置中构建分位数非常有用。所提出的方法和相关的软件提供了一个合理的灵活性和结构约束之间的平衡。调查员和他的研究小组开发新的方法，旨在对重复观察到的趋势和轨迹进行统计分析。由于新的复杂传感器、测量系统以及人们越来越广泛地认识到，对时间趋势及其模式的深入理解和解释往往是更好地做出个人和社会决策的关键，这些数据越来越普遍。这种新方法有助于深入了解时间依赖性过程的动态，如人类增长，高速公路交通模式的特征，或比较各国和历年的生命表。提出的非线性方法，这样的功能数据导致改进和更紧凑的描述和更好的预测结果，所观察到的时间趋势。

Nonlinear methods for Functional Data Analysis lead to flexible and versatile statistical models, inference and analysis methods for data that include samples of random functions. Such data accrue in the study of time-dynamic phenomena such as electricity consumption curves or biological trajectories and also in a large number of longitudinal studies across the sciences. Methods for functional data analysis have been rapidly evolving over the past few years and are increasingly viewed as essential for the analysis of time-dynamic phenomena. To date, linear models for functional data have been relatively well investigated, both in terms of theoretical and practical aspects, and statistical tools that are based on these methods are available for data analysis. However, the class of linear functional models is quite narrow and often not adequate in practical data analysis. In contrast, relatively little is known about more general and more flexible nonlinear approaches. This research seeks to remedy this situation by developing a class of nonlinear functional methods. The potential value of such models for applications is high, especially in scenarios where one observes repeated functions in time or space, or where one wishes to study regression relations that include functional components as predictors or responses. Nonlinear functional methodology includes representations of samples of trajectories by means of nonlinear components or through mixture models. These approaches are useful for applications where random time warping plays a role, and for the construction of quantiles in functional regression settings. The proposed methodology and associated software provides a sensible balance between increased flexibility and structural constraints.The investigator and his research group develop new methods aimed at the statistical analysis of repeatedly observed trends and trajectories. Such data are increasingly common due to new sophisticated sensors, measurement systems and the widening recognition that a deep understanding and interpretation of time trends and their patterns is often key to better individual and societal decision making. This new methodology is useful to gain insights into the dynamics of time-dependent processes such as human growth, characteristics of freeway traffic patterns, or the comparison of lifetables across countries and calendar years. The proposed nonlinear approaches to such functional data lead to improved and more compact descriptions and to better predictions of outcomes that are related to observed time trends.