Curve Registration and Human Connectome Data Public Deposited

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  • March 21, 2019
  • Yu, Qunqun
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
  • This thesis consists of three main parts: the usefulness of principal nested spheres for time warped functional data analysis, asymptotic study of the Fisher-Rao approach to time warped curve registration, the Joint and Individual Variation Explained method for Human Connectome Data. There are often two important types of variation in functional data: the horizontal (or phase) variation and the vertical (or amplitude) variation. These two types of variation have been appropriately separated and modeled through a domain warping method (or curve registration) based on the Fisher-Rao metric. The first part focuses on the analysis of the horizontal variation, captured by the domain warping functions. The square-root velocity function representation transforms the manifold of the warping functions to a Hilbert sphere. Motivated by recent results on manifold analogs of principal component analysis, we analyze the horizontal variation via a Principal Nested Spheres approach. Compared with earlier approaches, such as approximating tangent plane principal component analysis, this is seen to be an efficient and interpretable approach to decompose the horizontal variation in both simulated and real data examples. The mathematical underpinnings of the Fisher-Rao curve registration are studied by a consistency result for a signal that is observed under random warps, scaling and vertical translation. The signal estimator in the Fisher-Rao curve registration is known to be consistent. The second part of this dissertation studies more asymptotic properties. The ultimate goal is to compare available methods using rates of convergence. A challenging part is that closed form solutions on the surface of the sphere are generally not available. We study a simple case where the warps are piecewise linear warping functions. Points on the unit circle can represent each warp and we find the explicit solution and study the asymptotic properties of the signal estimation. A class of metrics that share some good properties of the Fisher-Rao metric is also studied. A major goal in neuroscience is to understand the neural pathways underlying human behavior. We introduce the recently developed Joint and Individual Variation Explained (JIVE) method to the neuroscience community to simultaneously analyze imaging and behavioral data from the Human Connectome Project. Motivated by recent computational and theoretical improvements in the JIVE approach, we simultaneously explore the joint and individual variation between and within imaging and behavioral data. In particular, we demonstrate that JIVE is an effective and efficient approach for integrating task fMRI and behavioral variables using three examples: one example where task variation is strong, one where task variation is weak and a reference case where the behavior is not directly related to the image. These examples are provided to visualize the different levels of signal found in the joint variation including working memory regions in the image data and accuracy and response time from the in-task behavioral variables. Joint analysis provides insights not available from conventional single block decomposition methods such as Singular Value Decomposition. Additionally, the joint variation estimated by JIVE appears to more clearly identify the working memory regions than Partial Least Squares (PLS), while Canonical Correlation Analysis (CCA) gives grossly overfit results. The individual variation in JIVE captures the behavior unrelated signals such as a background activation that is spatially homogeneous and activation in the default mode network. The information revealed by this individual variation is not examined in traditional methods such as CCA and PLS. We suggest that JIVE can be used as an alternative to PLS and CCA to improve estimation of the signal common to two or more datasets and reveal novel insights into the signal unique to each dataset.
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Rights statement
  • In Copyright
  • Zhang, Kai
  • Haaland, Perry
  • Hannig, Jan
  • Marron, James Stephen
  • Budhiraja, Amarjit
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2017

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