Inflammatory Bowel Disease (IBD), including Crohn's Disease (CD) and Ulcerative Colitis (UC), presents significant public health challenges due to its complex etiology. Motivated by the IBD study of the Integrative Human Microbiome Project, our objective is to identify microbial pathways that distinguish between CD, UC and non-IBD over time. Most current research relies on simplistic analyses that examine one variable or time point at a time, or address binary classification problems, limiting our understanding of the dynamic interactions within the microbiome over time. To address these limitations, we develop a novel functional data analysis approach for discriminant analysis of multivariate functional data that can effectively handle multiple high-dimensional predictors, sparse time points, and categorical outcomes. Our method seeks linear combinations of functions (i.e., discriminant functions) that maximize separation between two or more groups over time. We impose a sparsity-inducing penalty when estimating the discriminant functions, allowing us to identify relevant discriminating variables over time. Applications of our method to the motivating data identified microbial features related to mucin degradation, amino acid metabolism, and peptidoglycan recognition, which are implicated in the progression and development of IBD. Furthermore, our method highlighted the role of multiple vitamin B deficiencies in the context of IBD. By moving beyond traditional analytical frameworks, our innovative approach holds the potential for uncovering clinically meaningful discoveries in IBD research.

翻译：炎症性肠病（IBD），包括克罗恩病（CD）和溃疡性结肠炎（UC），因其复杂的病因学构成了重大的公共卫生挑战。受整合人类微生物组计划中IBD研究的启发，我们的目标是识别能够随时间推移区分CD、UC和非IBD的微生物通路。当前大多数研究依赖于简化的分析方法，这些方法一次只检查一个变量或时间点，或处理二元分类问题，限制了我们理解微生物组内随时间变化的动态相互作用。为解决这些局限性，我们开发了一种新颖的函数数据分析方法，用于多元函数数据的判别分析，该方法能够有效处理多个高维预测因子、稀疏时间点和分类结果。我们的方法寻求函数的线性组合（即判别函数），以最大化两个或更多组别随时间变化的分离度。在估计判别函数时，我们施加了稀疏诱导惩罚，使我们能够识别随时间变化的相关判别变量。将我们的方法应用于激励数据，识别出了与粘蛋白降解、氨基酸代谢和肽聚糖识别相关的微生物特征，这些特征与IBD的进展和发展有关。此外，我们的方法强调了多种维生素B缺乏在IBD背景下的作用。通过超越传统的分析框架，我们的创新方法有望在IBD研究中揭示具有临床意义的发现。