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Regularized Tensor Quantile Regression for Functional Data Analysis with Applications to Neuroimaging
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- Author / Creator
- Pietrosanu, Matthew
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With accelerating technological advancements, functional data analysis is of ever-growing importance in statistics, engineering, and medicine as the collection, storage, and analysis of data becomes ingrained in every aspect of modern life beyond any one field of study. From continuous biometric measurements to environmental monitoring to medical imaging, data collected across time, space, and mathematical manifolds in general are increasingly common. The advent of ``big data'' ensures that the rate, volume, and complexity of collected data will only continue to grow. Despite its richness, the high dimensionality of functional data is inherently challenging to incorporate into statistical models in both theory and practice.
This thesis is primarily concerned with incorporating multiple scalar-valued functional covariates, each with a high-dimensional domain but observed in a discrete, uniform grid, into functional quantile regression models for a scalar response. This type of functional data is ubiquitous in science and medicine as imaging data observed across spatiotemporal dimensions in a tensor format. The existing accommodation of functional and tensor-valued covariates in generalized linear models does not extend immediately to the quantile setting, although such a development would be useful in practice. Indeed, quantile models are well-known to be more efficient and robust for non-Gaussian, heavy-tailed error distributions or when outliers are present---typically the case with real-world data. Throughout this work, we emphasize direct regularization of tensor effects for more generalizable models and interpretable signal recovery for imaging data.
The Tucker tensor decomposition is the main tool used in this thesis: we assume a low-dimensional representation of a tensor with a particular polyadic structure to reduce the dimension of the parameter space and make model estimation feasible. We obtain this decomposition in a supervised manner, relying on partial quantile covariance between tensor covariates and the scalar response, similar to partial least squares in traditional linear regression. We estimate decomposition parameters and fit the proposed $q$-dimensional tensor quantile regression (\qdt{}) model by minimizing quantile loss. To address the non-convexity and non-differentiability of the loss in Tucker tensor decomposition parameters, we use a block relaxation technique and a continuously differentiable smoothing approximation of the quantile loss. After proposing a new algorithm and gradient-based implementation for models with one functional covariate, we extend our approach to multiple functional covariates and discuss simplifications exploiting the Tucker decomposition's nonsingular transformation indeterminacy. We consider convex penalty functions that, unlike previous approaches, directly regularize the estimated tensor effect through the assumed structure rather than only its decomposition parameters.
We establish theoretical properties for the proposed model, including global, local, and approximation convergence for the proposed algorithm and, using empirical process theory, asymptotic statistical results regarding estimator consistency and normality.
Finally, we demonstrate the performance of our model in simulated and real-world settings. Through a simulation study proposed in previous works that attempt to recover image signals of various geometric shape, we highlight the superiority of quantile-based methods for heavy-tailed error distributions. We examine the effect of tensor decomposition rank, quantile level, signal-to-noise ratio, and sample size on model estimates, further improving signal recovery by using a LASSO-type penalty. Second, we apply our methods to a real-world neuroimaging dataset from the Alzheimer's Disease Neuroimaging Initiative. Our model relates clinical covariates and and four functional covariates obtained from magnetic resonance imaging scans to mini-mental state examination score, a screening tool for Alzheimer's disease. After LASSO regularization leaves more be desired in estimate interpretability, we explore fused LASSO penalization to enforce estimate smoothness in a post-hoc analysis. Results show improvement that would not be possible with previous work through direct penalization of decomposition parameters.
The major work in this thesis fills the need for an extension of existing functional quantile methods to tensor and high-dimensional functional data. Our results furthermore address the practical issue of multiple functional covariates---typically ignored in other work---and demonstrate the utility of direct regularization in tensor effect interpretability for imaging data.
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- Graduation date
- Fall 2019
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- Type of Item
- Thesis
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- Degree
- Master of Science
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- License
- Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.