CAREER: New Topics in Functional Data Analysis

泛函数据是泛函空间中的随机向量，通常在离散点上观测，测量时带有误差。功能数据还与其他类型的相关数据（如空间数据）密切相关，因为它们都可以被建模为随机过程。在本研究中，研究者将通过提出一种新的功能数据方法来模拟疾病监测应用中的时空点模式数据，从而扩大功能数据分析的应用范围，其中时空随机效应被建模为潜在的功能过程。受结肠癌实验和高血压研究中的科学问题的启发，研究者提出了新的降维方法，该方法广泛适用于具有功能预测因子的半参数回归问题。研究者提出了基于样条近似和粗糙度惩罚的新估计程序，并将研究与这些方法相关的模型选择和推理问题。研究者还建议通过协方差模型的功能分析对纵向临床试验数据进行建模，其中不同的治疗效果用时间上的非参数函数表示。提出的非参数假设检验可用于检测治疗效果。具体而言，研究者将研究受试者内相关性对测试有效性的影响。提出的疾病监测数据的功能数据方法将有助于建立疾病发生与某些环境变量（如污染水平）之间的关系模型，估计疾病发病率的时间趋势，并预测以潜在随机效应为代表的未知危险因素。研究结果将有助于疾病控制机构和地方官员更好地了解疾病风险，并制定更好的公共卫生政策，如排放或水质控制政策。所提出的降维方法为结肠癌发生和高血压研究中的半参数回归问题提供了急需的统计工具。提出的协方差模型功能分析的非参数假设检验程序回答了临床试验中的一个基本问题，即比较不同治疗方法的有效性。研究者将为科研人员提供免费且用户友好的软件，并将其研究活动与研究生教育相结合。为了进一步传播研究成果，激发新的想法，研究者将开设一门新的功能数据分析课程，并组织一次研究研讨会。

Functional data are random vectors in a functional space, which are usually observed on discrete points and measured with error. Functional data are also deeply connected to other types of correlated data, such as spatial data, since they can both be modeled as stochastic processes. In the proposed research, the investigator will broaden the applications of functional data analysis by proposing a new functional data approach to model spatio-temporal point pattern data from disease surveillance applications, where the spatio-temporal random effects are modeled as latent functional processes. Motivated by scientific problems in colon carcinogenesis experiments and hypertension studies, the investigator proposes new dimension reduction methods, which are widely applicable to semiparametric regression problems with functional predictors. The investigator proposes new estimation procedures based on spline approximation and roughness penalties, and will also investigate the model selection and inference problems related to these methods. The investigator also proposes to model longitudinal clinical trial data by the functional analysis of covariance models, where different treatment effects are represented by nonparametric functions in time. The proposed nonparametric hypothesis test can be used to detect the treatment effects. Specifically, the investigator will study the effect of the within-subject correlation on the power of the test.The proposed functional data approach to disease surveillance data will help to model the relationship between disease occurrence and some environmental variables (such as pollution level), estimate the time trend in the disease rate, and predict the unknown risk factors represented by latent random effects. The results will help disease control agencies and local officials to gain better understanding of the disease risk and develop better public health policies, such as emission or water quality control policies. The proposed dimension reduction methods provide the much needed statistical tools in the semiparametric regression problems in colon carcinogenesis and hypertension studies. The proposed nonparametric hypothesis testing procedure for functional analysis of covariance models answers a fundamental question in clinical trials, which is to compare the effectiveness of different treatments. The investigator will provide free and user-friendly software to scientific researchers and incorporate his research activity with graduate education. To further disseminate the research results and motivate new ideas, the investigator will develop a new course on functional data analysis and organize a research workshop.