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Abstract Functional data analysis (FDA) is a fast-growing area of research and development in statistics. While most FDA literature imposes the classical$$\mathbb {L}^2$$ $L^2$Hilbert structure on function spaces, there is an emergent need for a different, shape-based approach for analyzing functional data. This paper reviews and develops fundamental geometrical concepts that help connect traditionally diverse fields of shape and functional analyses. It showcases that focusing on shapes is often more appropriate when structural features (number of peaks and valleys and their heights) carry salient information in data. It recaps recent mathematical representations and associated procedures for comparing, summarizing, and testing the shapes of functions. Specifically, it discusses three tasks: shape fitting, shape fPCA, and shape regression models. The latter refers to the models that separate the shapes of functions from their phases and use them individually in regression analysis. The ensuing results provide better interpretations and tend to preserve geometric structures. The paper also discusses an extension where the functions are not real-valued but manifold-valued. The article presents several examples of this shape-centric functional data analysis using simulated and real data. 

It is well-known that morphological features in the brain undergo changes due to traumatic events and associated disorders such as post-traumatic stress disorder (PTSD). However, existing approaches typically offer group-level comparisons, and there are limited predictive approaches for modeling behavioral outcomes based on brain shape features that can account for heterogeneity in PTSD, which is of paramount interest. We propose a comprehensive shape analysis framework representing brain sub-structures, such as the hippocampus, amygdala, and putamen, as parameterized surfaces and quantifying their shape differences using an elastic shape metric. Under this metric, we compute shape summaries (mean, covariance, PCA) of brain sub-structures and represent individual brain shapes by their principal scores under a shape-PCA basis. These representations are rich enough to allow visualizations of full 3D structures and help understand localized changes. In order to validate the elastic shape analysis, we use the principal components (PCs) to reconstruct the brain structures and perform further evaluation by performing a regression analysis to model PTSD and trauma severity using the brain shapes representedviaPCs and in conjunction with auxiliary exposure variables. We apply our method to data from the Grady Trauma Project (GTP), where the goal is to predict clinical measures of PTSD. The framework seamlessly integrates accurate morphological features and other clinical covariates to yield superior predictive performance when modeling PTSD outcomes. Compared to vertex-wise analysis and other widely applied shape analysis methods, the elastic shape analysis approach results in considerably higher reconstruction accuracy for the brain shape and reveals significantly greater predictive power. It also helps identify local deformations in brain shapes associated with PTSD severity.