New Challenges, Models and Methods for Functional Data Analysis

我的研究项目将通过解决遗传学和公共卫生等不同学科的应用，开发新的统计方法。******我们的第一个研究主题是功能数据分析（FDA），这是一个分析曲线、图像或任何多维函数的新兴领域。功能数据的常见来源包括来自可穿戴设备的健身数据、与空气污染有关的数据、纵向研究、时间过程基因表达数据和脑成像数据。主要的挑战是处理功能数据的规模和维度的增加，同时保持FDA模型的灵活性和可解释性之间的平衡。******我们将为高维函数/标量变量开发通用和有效的FDA方法。本研究项目将为各种大型FDA模型提供基础，并在大数据分析领域产生广泛影响。例如，我们将为各种健康结果制定空气污染指数。我们还将研究大脑不同部位之间信号的相关性。******我们的第二个研究主题侧重于动态模型的统计推断。动力学模型用微分方程（DEs）来描述动力系统的变化率。它们被广泛用于理解生物和物理等领域的复杂系统。虽然DE模型的参数通常有科学的解释，但它们的值往往是未知的，难以估计。******我们将开发有效的方法，从实际数据中为DE模型提供准确和稳健的参数估计，并将这些DE参数与高维函数和标量预测器联系起来。本研究项目将填补DE建模与真实数据分析之间的空白，使DE模型在实际应用中大众化、现实性。例如，我提出对大型网络的动力学机制进行建模，不仅提供了网络的连接结构，还提供了网络的调节机制。该方法可用于构造时变有向神经网络和社会网络。我还建议使用偏微分方程从地震记录数据中提取信息。这种方法将使用地震记录数据生成地下地质结构的高清图像。这项技术对加拿大的石油工业非常有用。******这项拟议的研究不仅有助于培养高素质的人才，而且还会导致我们开发的用户友好的软件，作为解决研究问题的办法。该软件将向公众开放；我们预计，其他国家对它的使用将对研究和工业产生广泛的影响。此外，拟议的研究可能会通过在功能数据分析和动态模型统计推断方面取得突破性进展来影响统计领域。**************

My research program will develop novel statistical methodologies by addressing applications in various disciplines such as genetics and public health. ******Our first research theme is functional data analysis (FDA), a growing area for analyzing curves, images, or any multidimensional functions. Common sources of functional data include fitness data from wearable devices, air pollution-related data, longitudinal studies, time-course gene expression data, and brain imaging data. The main challenge is dealing with the increasing scale and dimension of functional data while maintaining a balance between the flexibility and interpretability of FDA models. ******We will develop general and efficient FDA methods for high-dimensional functional/scalar variables. This research program will provide the foundation for various large-scale FDA models and have a broad impact in big data analysis. For instance, we will develop air pollution indices for various health outcomes. We will also study the correlation of brain signals among different brain locations. ******Our second research theme focuses on statistical inference for dynamical models. Dynamical models describe the rate of change of a dynamical system by using differential equations (DEs). They are widely used to understand complex systems in areas including biology and physics. While the parameters of DE models usually have scientific interpretations, their values are often unknown and are difficult to estimate. ******We will develop efficient methods that provide accurate and robust parameter estimates for DE models from real data and to link these DE parameters to high-dimensional functional and scalar predictors. This research program will fill the gap between DE modeling and real data analysis and make DE models popular and realistic in practical applications. For example, I propose to model the dynamical mechanism of large-scale networks, which provides not only the connection structure but also the regulation mechanism of networks. The proposed method can be applied to construct time-varying directed neural networks and social networks. I also propose to use partial differential equations to extract information from the seismogram data. This method will use seismogram data to produce high-definition images of subsurface geologic structures. This technology will be extremely useful for Canada's oil industry.******Not only will the proposed research support the training of highly qualified personnel, but it will also result in user-friendly software that we develop as solutions to the research problems tackled. This software will be available to the general public; we anticipate that its use by others will have a broad impact in research and industry. Furthermore, the proposed research is likely to impact the field of statistics through enabling groundbreaking advances in functional data analysis and statistical inference for dynamical models.**************