Functional CAR models for large spatially correlated functional datasets

A canonical polyadic tensor basis for fast Bayesian estimation of multi-subject brain activation patterns

Task-evoked functional magnetic resonance imaging studies, such as the Human Connectome Project (HCP), are a powerful tool for exploring how brain activity is influenced by cognitive tasks like memory retention, decision-making, and language processing. A fast Bayesian function-on-scalar model is proposed for estimating population-level activation maps linked to the working memory task. The model is based on the canonical polyadic (CP) tensor decomposition of coefficient maps obtained for each subject. This decomposition effectively yields a tensor basis capable of extracting both common features and subject-specific features from the coefficient maps. These subject-specific features, in turn, are modeled as a function of covariates of interest using a Bayesian model that accounts for the correlation of the CP-extracted features. The dimensionality reduction achieved with the tensor basis allows for a fast MCMC estimation of population-level activation maps. This model is applied to one hundred unrelated subjects from the HCP dataset, yielding significant insights into brain signatures associated with working memory.

Functional Bayesian networks for discovering causality from multivariate functional data

Multivariate functional data arise in a wide range of applications. One fundamental task is to understand the causal relationships among these functional objects of interest. In this paper, we develop a novel Bayesian network (BN) model for multivariate functional data where conditional independencies and causal structure are encoded by a directed acyclic graph. Specifically, we allow the functional objects to deviate from Gaussian processes, which is the key to unique causal structure identification even when the functions are measured with noises. A fully Bayesian framework is designed to infer the functional BN model with natural uncertainty quantification through posterior summaries. Simulation studies and real data examples demonstrate the practical utility of the proposed model.

Modeling post-holiday surge in COVID-19 cases in Pennsylvania counties

COVID-19 arrived in the United States in early 2020, with cases quickly being reported in many states including Pennsylvania. Many statistical models have been proposed to understand the trends of the COVID-19 pandemic and factors associated with increasing cases. While Poisson regression is a natural choice to model case counts, this approach fails to account for correlation due to spatial locations. Being a contagious disease and often spreading through community infections, the number of COVID-19 cases are inevitably spatially correlated as locations neighboring counties with a high COVID-19 case count are more likely to have a high case count. In this analysis, we combine generalized estimating equations (GEEs) for Poisson regression, a popular method for analyzing correlated data, with a semivariogram to model daily COVID-19 case counts in 67 Pennsylvania counties between March 20, 2020 to January 23, 2021 in order to study infection dynamics during the beginning of the pandemic. We use a semivariogram that describes the spatial correlation as a function of the distance between two counties as the working correlation. We further incorporate a zero-inflated model in our spatial GEE to accommodate excess zeros in reported cases due to logistical challenges associated with disease monitoring. By modeling time-varying holiday covariates, we estimated the effect of holiday timing on case count. Our analysis showed that the incidence rate ratio was significantly greater than one, 6-8 days after a holiday suggesting a surge in COVID-19 cases approximately one week after a holiday.

Fast Multilevel Functional Principal Component Analysis

We introduce fast multilevel functional principal component analysis (fast MFPCA), which scales up to high dimensional functional data measured at multiple visits. The new approach is orders of magnitude faster than and achieves comparable estimation accuracy with the original MFPCA (Di et al., 2009). Methods are motivated by the National Health and Nutritional Examination Survey (NHANES), which contains minute-level physical activity information of more than 10000 participants over multiple days and 1440 observations per day. While MFPCA takes more than five days to analyze these data, fast MFPCA takes less than five minutes. A theoretical study of the proposed method is also provided. The associated function mfpca.face() is available in the R package refund.

Ultra-Fast Approximate Inference Using Variational Functional Mixed Models

While Bayesian functional mixed models have been shown effective to model functional data with various complex structures, their application to extremely high-dimensional data is limited due to computational challenges involved in posterior sampling. We introduce a new computational framework that enables ultra-fast approximate inference for high-dimensional data in functional form. This framework adopts parsimonious basis to represent functional observations, which facilitates efficient compression and parallel computing in basis space. Instead of performing expensive Markov chain Monte Carlo sampling, we approximate the posterior distribution using variational Bayes and adopt a fast iterative algorithm to estimate parameters of the approximate distribution. Our approach facilitates a fast multiple testing procedure in basis space, which can be used to identify significant local regions that reflect differences across groups of samples. We perform two simulation studies to assess the performance of approximate inference, and demonstrate applications of the proposed approach by using a proteomic mass spectrometry dataset and a brain imaging dataset. Supplementary materials are available online.

Fixed-effects inference and tests of correlation for longitudinal functional data

We propose an inferential framework for fixed effects in longitudinal functional models and introduce tests for the correlation structures induced by the longitudinal sampling procedure. The framework provides a natural extension of standard longitudinal correlation models for scalar observations to functional observations. Using simulation studies, we compare fixed effects estimation under correctly and incorrectly specified correlation structures and also test the longitudinal correlation structure. Finally, we apply the proposed methods to a longitudinal functional dataset on physical activity. The computer code for the proposed method is available at https://github.com/rli20ST758/FILF.

From multivariate to functional data analysis: fundamentals, recent developments, and emerging areas

Functional data analysis (FDA), which is a branch of statistics on modeling infinite dimensional random vectors resided in functional spaces, has become a major research area for Journal of Multivariate Analysis. We review some fundamental concepts of FDA, their origins and connections from multivariate analysis, and some of its recent developments, including multi-level functional data analysis, high-dimensional functional regression, and dependent functional data analysis. We also discuss the impact of these new methodology developments on genetics, plant science, wearable device data analysis, image data analysis, and business analytics. Two real data examples are provided to motivate our discussions.

Digital twin-driven intelligence disaster prevention and mitigation for infrastructure: advances, challenges, and opportunities

Natural hazards, which have the potential to cause catastrophic damage and loss to infrastructure, have increased significantly in recent decades. Thus, the construction demand for disaster prevention and mitigation for infrastructure (DPMI) systems is increasing. Many studies have applied intelligence technologies to solve key aspects of infrastructure, such as design, construction, disaster prevention and mitigation, and rescue and recovery; however, systematic construction is still lacking. Digital twin (DT) is one of the most promising technologies for multi-stage management which has great potential to solve the above challenges. This paper initially puts forward a scientific concept, in which DT drives the construction of intelligent disaster prevention and mitigation for infrastructure (IDPMI) systematically. To begin with, a scientific review of DT and IDPMI is performed, where the development of DT is summarized and a DT-based life cycle of infrastructures is defined. In addition, the intelligence technologies used in disaster management are key reviewed and their relative merits are illustrated. Furthermore, the development and technical feasibility of DT-driven IDPMI are illustrated by reviewing the relevant practice of DT in infrastructure. In conclusion, a scientific framework of DT-IDPMI is programmed, which not only provides some guidance for the deep integration between DT and IDPMI but also identifies the challenges that inspire the professional community to advance these techniques to address them in future research.

Fast Univariate Inference for Longitudinal Functional Models

We propose fast univariate inferential approaches for longitudinal Gaussian and non-Gaussian functional data. The approach consists of three steps: (1) fit massively univariate pointwise mixed effects models; (2) apply any smoother along the functional domain; and (3) obtain joint confidence bands using analytic approaches for Gaussian data or a bootstrap of study participants for non-Gaussian data. Methods are motivated by two applications: (1) Diffusion Tensor Imaging (DTI) measured at multiple visits along the corpus callosum of multiple sclerosis (MS) patients; and (2) physical activity data measured by body-worn accelerometers for multiple days. An extensive simulation study indicates that model fitting and inference are accurate and much faster than existing approaches. Moreover, the proposed approach was the only one that was computationally feasible for the physical activity data application. Methods are accompanied by R software, though the method is "read-and-use", as it can be implemented by any analyst who is familiar with mixed effects model software.

A Spatio-temporal Model for Detecting the Effect of Cocaine Use Disorder on Functional Connectivity

Drug addiction can lead to many health-related problems and social concerns. Researchers are interested in the association between long-term drug usage and abnormal functional connectivity. Functional connectivity obtained from functional magnetic resonance imaging data promotes a variety of fundamental understandings in such association. Due to the complex correlation structure and large dimensionality, the modeling and analysis of the functional connectivity from neuroimage are challenging. By proposing a spatio-temporal model for multi-subject neuroimage data, we incorporate voxel-level spatio-temporal dependencies of whole-brain measurements to improve the accuracy of statistical inference. To tackle large-scale spatio-temporal neuroimage data, we develop a computational efficient algorithm to estimate the parameters. Our method is used to first identify functional connectivity, and then detect the effect of cocaine use disorder (CUD) on functional connectivity between different brain regions. The functional connectivity identified by our spatio-temporal model matches existing studies on brain networks, and further indicates that CUD may alter the functional connectivity in the medial orbitofrontal cortex subregions and the supplementary motor areas.

Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency

We consider spatially dependent functional data collected under a geostatistics setting, where locations are sampled from a spatial point process. The functional response is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect. Observations on each function are made on discrete time points and contaminated with measurement errors. Under the assumption of spatial stationarity and isotropy, we propose a tensor product spline estimator for the spatio-temporal covariance function. When a coregionalization covariance structure is further assumed, we propose a new functional principal component analysis method that borrows information from neighboring functions. The proposed method also generates nonparametric estimators for the spatial covariance functions, which can be used for functional kriging. Under a unified framework for sparse and dense functional data, infill and increasing domain asymptotic paradigms, we develop the asymptotic convergence rates for the proposed estimators. Advantages of the proposed approach are demonstrated through simulation studies and two real data applications representing sparse and dense functional data, respectively.

Multilevel modeling of spatially nested functional data: Spatiotemporal patterns of hospitalization rates in the US dialysis population

End-stage renal disease patients on dialysis experience frequent hospitalizations. In addition to known temporal patterns of hospitalizations over the life span on dialysis, where poor outcomes are typically exacerbated during the first year on dialysis, variations in hospitalizations among dialysis facilities across the US contribute to spatial variation. Utilizing national data from the United States Renal Data System (USRDS), we propose a novel multilevel spatiotemporal functional model to study spatiotemporal patterns of hospitalization rates among dialysis facilities. Hospitalization rates of dialysis facilities are considered as spatially nested functional data (FD) with longitudinal hospitalizations nested in dialysis facilities and dialysis facilities nested in geographic regions. A multilevel Karhunen-Loéve expansion is utilized to model the two-level (facility and region) FD, where spatial correlations are induced among region-specific principal component scores accounting for regional variation. A new efficient algorithm based on functional principal component analysis and Markov Chain Monte Carlo is proposed for estimation and inference. We report a novel application using USRDS data to characterize spatiotemporal patterns of hospitalization rates for over 400 health service areas across the US and over the posttransition time on dialysis. Finite sample performance of the proposed method is studied through simulations.

Bayesian analysis of longitudinal and multidimensional functional data

Multi-dimensional functional data arises in numerous modern scientific experimental and observational studies. In this article, we focus on longitudinal functional data, a structured form of multidimensional functional data. Operating within a longitudinal functional framework we aim to capture low dimensional interpretable features. We propose a computationally efficient nonparametric Bayesian method to simultaneously smooth observed data, estimate conditional functional means and functional covariance surfaces. Statistical inference is based on Monte Carlo samples from the posterior measure through adaptive blocked Gibbs sampling. Several operative characteristics associated with the proposed modeling framework are assessed comparatively in a simulated environment. We illustrate the application of our work in two case studies. The first case study involves age-specific fertility collected over time for various countries. The second case study is an implicit learning experiment in children with autism spectrum disorder.

Quantile Function on Scalar Regression Analysis for Distributional Data

Radiomics involves the study of tumor images to identify quantitative markers explaining cancer heterogeneity. The predominant approach is to extract hundreds to thousands of image features, including histogram features comprised of summaries of the marginal distribution of pixel intensities, which leads to multiple testing problems and can miss out on insights not contained in the selected features. In this paper, we present methods to model the entire marginal distribution of pixel intensities via the quantile function as functional data, regressed on a set of demographic, clinical, and genetic predictors to investigate their effects of imaging-based cancer heterogeneity. We call this approach quantile functional regression, regressing subject-specific marginal distributions across repeated measurements on a set of covariates, allowing us to assess which covariates are associated with the distribution in a global sense, as well as to identify distributional features characterizing these differences, including mean, variance, skewness, heavy-tailedness, and various upper and lower quantiles. To account for smoothness in the quantile functions, account for intrafunctional correlation, and gain statistical power, we introduce custom basis functions we call quantlets that are sparse, regularized, near-lossless, and empirically defined, adapting to the features of a given data set and containing a Gaussian subspace so non-Gaussianness can be assessed. We fit this model using a Bayesian framework that uses nonlinear shrinkage of quantlet coefficients to regularize the functional regression coefficients and provides fully Bayesian inference after fitting a Markov chain Monte Carlo. We demonstrate the benefit of the basis space modeling through simulation studies, and apply the method to Magnetic resonance imaging (MRI) based radiomic dataset from Glioblastoma Multiforme to relate imaging-based quantile functions to various demographic, clinical, and genetic predictors, finding specific differences in tumor pixel intensity distribution between males and females and between tumors with and without DDIT3 mutations.

Predictive functional ANOVA models for longitudinal analysis of mandibular shape changes

In this paper, we introduce a Bayesian statistical model for the analysis of functional data observed at several time points. Examples of such data include the Michigan growth study where we wish to characterize the shape changes of human mandible profiles. The form of the mandible is often used by clinicians as an aid in predicting the mandibular growth. However, whereas many studies have demonstrated the changes in size that may occur during the period of pubertal growth spurt, shape changes have been less well investigated. Considering a group of subjects presenting normal occlusion, in this paper we thus describe a Bayesian functional ANOVA model that provides information about where and when the shape changes of the mandible occur during different stages of development. The model is developed by defining the notion of predictive process models for Gaussian process (GP) distributions used as priors over the random functional effects. We show that the predictive approach is computationally appealing and that it is useful to analyze multivariate functional data with unequally spaced observations that differ among subjects and times. Graphical posterior summaries show that our model is able to provide a biological interpretation of the morphometric findings and that they comprehensively describe the shape changes of the human mandible profiles. Compared with classical cephalometric analysis, this paper represents a significant methodological advance for the study of mandibular shape changes in two dimensions.

Robust and Gaussian spatial functional regression models for analysis of event-related potentials

Event-related potentials (ERPs) summarize electrophysiological brain response to specific stimuli. They can be considered as correlated functions of time with both spatial correlation across electrodes and nested correlations within subjects. Commonly used analytical methods for ERPs often focus on pre-determined extracted components and/or ignore the correlation among electrodes or subjects, which can miss important insights, and tend to be sensitive to outlying subjects, time points or electrodes. Motivated by ERP data in a smoking cessation study, we introduce a Bayesian spatial functional regression framework that models the entire ERPs as spatially correlated functional responses and the stimulus types as covariates. This novel framework relies on mixed models to characterize the effects of stimuli while simultaneously accounting for the multilevel correlation structure. The spatial correlation among the ERP profiles is captured through basis-space Matérn assumptions that allow either separable or nonseparable spatial correlations over time. We induce both adaptive regularization over time and spatial smoothness across electrodes via a correlated normal-exponential-gamma (CNEG) prior on the fixed effect coefficient functions. Our proposed framework includes both Gaussian models as well as robust models using heavier-tailed distributions to make the regression automatically robust to outliers. We introduce predictive methods to select among Gaussian vs. robust models and models with separable vs. non-separable spatiotemporal correlation structures. Our proposed analysis produces global tests for stimuli effects across entire time (or time-frequency) and electrode domains, plus multiplicity-adjusted pointwise inference based on experiment-wise error rate or false discovery rate to flag spatiotemporal (or spatio-temporal-frequency) regions that characterize stimuli differences, and can also produce inference for any prespecified waveform components. Our analysis of the smoking cessation ERP data set reveals numerous effects across different types of visual stimuli.

Bayesian Semiparametric Functional Mixed Models for Serially Correlated Functional Data, with Application to Glaucoma Data

Glaucoma, a leading cause of blindness, is characterized by optic nerve damage related to intraocular pressure (IOP), but its full etiology is unknown. Researchers at UAB have devised a custom device to measure scleral strain continuously around the eye under fixed levels of IOP, which here is used to assess how strain varies around the posterior pole, with IOP, and across glaucoma risk factors such as age. The hypothesis is that scleral strain decreases with age, which could alter biomechanics of the optic nerve head and cause damage that could eventually lead to glaucoma. To evaluate this hypothesis, we adapted Bayesian Functional Mixed Models to model these complex data consisting of correlated functions on spherical scleral surface, with nonparametric age effects allowed to vary in magnitude and smoothness across the scleral surface, multi-level random effect functions to capture within-subject correlation, and functional growth curve terms to capture serial correlation across IOPs that can vary around the scleral surface. Our method yields fully Bayesian inference on the scleral surface or any aggregation or transformation thereof, and reveals interesting insights into the biomechanical etiology of glaucoma. The general modeling framework described is very flexible and applicable to many complex, high-dimensional functional data.

Simple fixed-effects inference for complex functional models

We propose simple inferential approaches for the fixed effects in complex functional mixed effects models. We estimate the fixed effects under the independence of functional residuals assumption and then bootstrap independent units (e.g. subjects) to conduct inference on the fixed effects parameters. Simulations show excellent coverage probability of the confidence intervals and size of tests for the fixed effects model parameters. Methods are motivated by and applied to the Baltimore Longitudinal Study of Aging, though they are applicable to other studies that collect correlated functional data.

Statistical Contributions to Bioinformatics: Design, Modeling, Structure Learning, and Integration

The advent of high-throughput multi-platform genomics technologies providing whole-genome molecular summaries of biological samples has revolutionalized biomedical research. These technologiees yield highly structured big data, whose analysis poses significant quantitative challenges. The field of Bioinformatics has emerged to deal with these challenges, and is comprised of many quantitative and biological scientists working together to effectively process these data and extract the treasure trove of information they contain. Statisticians, with their deep understanding of variability and uncertainty quantification, play a key role in these efforts. In this article, we attempt to summarize some of the key contributions of statisticians to bioinformatics, focusing on four areas: (1) experimental design and reproducibility, (2) preprocessing and feature extraction, (3) unified modeling, and (4) structure learning and integration. In each of these areas, we highlight some key contributions and try to elucidate the key statistical principles underlying these methods and approaches. Our goals are to demonstrate major ways in which statisticians have contributed to bioinformatics, encourage statisticians to get involved early in methods development as new technologies emerge, and to stimulate future methodological work based on the statistical principles elucidated in this article and utilizing all availble information to uncover new biological insights.

Comparison and Contrast of Two General Functional Regression Modeling Frameworks

In this article, Greven and Scheipl describe an impressively general framework for performing functional regression that builds upon the generalized additive modeling framework. Over the past number of years, my collaborators and I have also been developing a general framework for functional regression, functional mixed models, which shares many similarities with this framework, but has many differences as well. In this discussion, I compare and contrast these two frameworks, to hopefully illuminate characteristics of each, highlighting their respecitve strengths and weaknesses, and providing recommendations regarding the settings in which each approach might be preferable.

Edgewise and subgraph-level tests for brain networks

Resting-state functional magnetic resonance image is a useful technique for investigating brain functional connectivity at rest. In this work, we develop flexible regression models and methods for determining differences in resting-state functional connectivity as a function of age, gender, drug intervention, or neuropsychiatric disorders. We propose two complementary methods for identifying changes of edges and subgraphs. (i) For detecting changes of edges, we select the optimal model at each edge and then conduct contrast tests to identify the effects of the important variables while controlling the familywise error rate. (ii) We adopt the network-based statistics method to improve power by incorporating the graph topological structure. Both methods have wide applications for low signal-to-noise ratio data. We propose stability criteria for the choice of threshold in the network-based statistics procedure and utilize efficient massive parallel procedure to speed up the estimation and inference procedure. Results from our simulation studies show that the thresholds chosen by the proposed stability criteria outperform the Bonferroni threshold. To demonstrate applicability, we use both methods in the context of the Oxytocin and Aging Study to determine effects of age, gender, and drug treatment on resting-state functional connectivity, as well as in the context of the Autism Brain Imaging Data Exchange Study to determine effects of autism spectrum disorder on functional connectivity at rest. Copyright © 2016 John Wiley & Sons, Ltd.