arXiv subjects

Xiongtao Dai

Publications and source records attributed to Xiongtao Dai.

16 recordsLinked to original sources

arXiv2026

Wild bootstrap for mean response inference in functional linear regression models

Functional regressors complicate inference in linear regression problems so that the bootstrap can play a useful role in quantifying uncertainty and calibrating intervals. The best bootstrap in practice, though, can depend on factors in the data as well as computational considerations and existing bootstraps can have limitations: residual bootstrap is computationally fast and simple but may fail when the errors are heterogeneous, while paired bootstrap applies more generally in functional linear regression at a cost of much higher computation. To bridge this gap, we develop a wild bootstrap method for functional linear regression, which is akin to a modified version of residual bootstrap but designed to have a wide scope of application like paired bootstrap, including to heteroscedastic errors. Its theoretical consistency is established and numerical studies suggest that wild bootstrap can provide accurate and computationally fast inference. Importantly, we also suggest a practical and effective approach of selecting truncation levels, specifically designed for mean response inference problems. The proposed bootstrap in functional linear regression is further illustrated through a weather data example, and an accompanying R package BTSinFLRM provides numerical implementations.

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arXiv2026

Spatial Prediction of Local Soil Erosion Distribution in the Wasserstein Space

Obtaining precise erosion measurements requires costly fieldwork, making it infeasible to directly survey large domains such as a province or river basin. To extend fieldwork results across such extensive domains, we propose a novel spatial prediction method that treats local erosion distributions as objects in the Wasserstein space. These distributions are mapped into square-integrable trajectories and represented via basis expansion, forming a multivariate random field that captures spatial dependence. By applying local regression and Kriging in this representation, our approach flexibly models and predicts erosion distributions at arbitrary locations. This framework improves prediction for functionals of the distribution, such as the mean and exceedance probabilities. Simulation studies demonstrate that the proposed method outperforms a misspecified parametric alternative and existing Fréchet regression approaches. We illustrate the approach with a detailed erosion analysis in Shaanxi province, China, where local measurements from surveyed watersheds are extended to predict erosion distributions across the entire province using covariates such as land use and elevation.

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arXiv2026

Generalized linear models with spatial dependence and a functional covariate

We extend generalized functional linear models under independence to a situation in which a functional covariate is related to a scalar response variable that exhibits spatial dependence-a complex yet prevalent phenomenon. For estimation, we apply basis expansion and truncation for dimension reduction of the covariate process followed by a composite likelihood estimating equation to handle the spatial dependency. We establish asymptotic results for the proposed model under a repeating lattice asymptotic context, allowing us to construct a confidence interval for the spatial dependence parameter and a confidence band for the regression parameter function. A binary conditionals model with functional covariates is presented as a concrete illustration and is used in simulation studies to verify the applicability of the asymptotic inferential results. We apply the proposed model to a problem in which the objective is to relate annual corn yield in counties of states in the Midwestern United States to daily maximum temperatures from April to September in those same geographic regions. The extension to an expanding lattice context is further discussed in the supplement.

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arXiv2026

Conditional regularized halfspace depth for sparse functional data and its applications

Many functional datasets are observed sparsely and irregularly. Ordering such data is challenging because only limited information is available from each observation, while the underlying trajectories remain infinite-dimensional. This paper develops a novel depth notion for sparse functional data, called the conditional regularized halfspace depth (CRHD). CRHD is defined as the infimum of conditional halfspace probabilities of the underlying trajectory given the observed sparse measurements, thereby enabling depth evaluation directly at sparse observations without requiring trajectory reconstruction. We study several basic theoretical properties of CRHD that clarify its behavior as a depth measure. The proposed depth is applicable even to extremely sparsely observed functional data, overcoming key limitations of existing sparse functional depths that often rely on reconstructed curves. In addition, CRHD induces meaningful rankings for complex functional data. Its numerical performance is demonstrated through rank-based tests, and its practical utility is illustrated using an infant growth dataset.

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arXiv2026

Predictive Distributions and the Transition from Sparse to Dense Functional Data

A representation of Gaussian distributed sparsely sampled longitudinal data in terms of predictive distributions for their functional principal component scores (FPCs) maps available data for each subject to a multivariate Gaussian predictive distribution. Of special interest is the case where the number of observations per subject increases in the transition from sparse (longitudinal) to dense (functional) sampling of underlying stochastic processes. We study the convergence of the predicted scores given noisy longitudinal observations towards the true but unobservable FPCs, and under Gaussianity demonstrate the shrinkage of the entire predictive distribution towards a point mass located at the true FPCs and also extensions to the shrinkage of functional $K$-truncated predictive distributions when the truncation point $K=K(n)$ diverges with sample size $n$. To address the problem of non-consistency of point predictions, we construct predictive distributions aimed at predicting outcomes for the case of sparsely sampled longitudinal predictors in functional linear models and derive asymptotic rates of convergence for the $2$-Wasserstein metric between true and estimated predictive distributions. Predictive distributions are illustrated for longitudinal data from the Baltimore Longitudinal Study of Aging.

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arXiv2024

Exploring Spatial Generalized Functional Linear Models: A Comparative Simulation Study and Analysis of COVID-19

Implementation of spatial generalized linear models with a functional covariate can be accomplished through the use of a truncated basis expansion of the covariate process. In practice, one must select a truncation level for use. We compare five criteria for the selection of an appropriate truncation level, including AIC and BIC based on a log composite likelihood, a fraction of variance explained criterion, a fitted mean squared error, and a prediction error with one standard error rule. Based on the use of extensive simulation studies, we propose that BIC constitutes a reasonable default criterion for the selection of the truncation level for use in a spatial functional generalized linear model. In addition, we demonstrate that the spatial model with a functional covariate outperforms other models when the data contain spatial structure and response variables are in fact influenced by a functional covariate process. We apply the spatial functional generalized linear model to a problem in which the objective is to relate COVID-19 vaccination rates in counties of states in the Midwestern United States to the number of new cases from previous weeks in those same geographic regions.

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arXiv2024

Regularized Halfspace Depth for Functional Data

Data depth is a powerful nonparametric tool originally proposed to rank multivariate data from center outward. In this context, one of the most archetypical depth notions is Tukey's halfspace depth. In the last few decades notions of depth have also been proposed for functional data. However, Tukey's depth cannot be extended to handle functional data because of its degeneracy. Here, we propose a new halfspace depth for functional data which avoids degeneracy by regularization. The halfspace projection directions are constrained to have a small reproducing kernel Hilbert space norm. Desirable theoretical properties of the proposed depth, such as isometry invariance, maximality at center, monotonicity relative to a deepest point, upper semi-continuity, and consistency are established. Moreover, the regularized halfspace depth can rank functional data with varying emphasis in shape or magnitude, depending on the regularization. A new outlier detection approach is also proposed, which is capable of detecting both shape and magnitude outliers. It is applicable to trajectories in $L^2$, a very general space of functions that include non-smooth trajectories. Based on extensive numerical studies, our methods are shown to perform well in terms of detecting outliers of different types. Three real data examples showcase the proposed depth notion.

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arXiv2024

Bootstrap inference in functional linear regression models with scalar response under heteroscedasticity

Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean estimates have complicated sampling distributions due to the infinite dimensional regressors, where truncation bias and scaling issues are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated conditional mean under general dependent errors, and subsequently we develop a paired bootstrap method to provide better approximations of sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for centered projections and for conducting hypothesis tests for the multiple conditional means. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with a real data example.

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arXiv2021

Nonparametric Estimation of Repeated Densities with Heterogeneous Sample Sizes

We consider the estimation of densities in multiple subpopulations, where the available sample size in each subpopulation greatly varies. This problem occurs in epidemiology, for example, where different diseases may share similar pathogenic mechanism but differ in their prevalence. Without specifying a parametric form, our proposed method pools information from the population and estimate the density in each subpopulation in a data-driven fashion. Drawing from functional data analysis, low-dimensional approximating density families in the form of exponential families are constructed from the principal modes of variation in the log-densities. Subpopulation densities are subsequently fitted in the approximating families based on likelihood principles and shrinkage. The approximating families increase in their flexibility as the number of components increases and can approximate arbitrary infinite-dimensional densities. We also derive convergence results of the density estimates with discrete observations. The proposed methods are shown to be interpretable and efficient in simulation as well as applications to electronic medical record and rainfall data.

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arXiv2021

Tukey's Depth for Object Data

We develop a novel exploratory tool for non-Euclidean object data based on data depth, extending the celebrated Tukey's depth for Euclidean data. The proposed metric halfspace depth, applicable to data objects in a general metric space, assigns to data points depth values that characterize the centrality of these points with respect to the distribution and provides an interpretable center-outward ranking. Desirable theoretical properties that generalize standard depth properties postulated for Euclidean data are established for the metric halfspace depth. The depth median, defined as the deepest point, is shown to have high robustness as a location descriptor both in theory and in simulation. We propose an efficient algorithm to approximate the metric halfspace depth and illustrate its ability to adapt to the intrinsic data geometry. The metric halfspace depth was applied to an Alzheimer's disease study, revealing group differences in the brain connectivity, modeled as covariance matrices, for subjects in different stages of dementia. Based on phylogenetic trees of 7 pathogenic parasites, our proposed metric halfspace depth was also used to construct a meaningful consensus estimate of the evolutionary history and to identify potential outlier trees.

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arXiv2021

A Geospatial Functional Model For OCO-2 Data with Application on Imputation and Land Fraction Estimation

Data from NASA's Orbiting Carbon Observatory-2 (OCO-2) satellite is essential to many carbon management strategies. A retrieval algorithm is used to estimate CO2 concentration using the radiance data measured by OCO-2. However, due to factors such as cloud cover and cosmic rays, the spatial coverage of the retrieval algorithm is limited in some areas of critical importance for carbon cycle science. Mixed land/water pixels along the coastline are also not used in the retrieval processing due to the lack of valid ancillary variables including land fraction. We propose an approach to model spatial spectral data to solve these two problems by radiance imputation and land fraction estimation. The spectral observations are modeled as spatially indexed functional data with footprint-specific parameters and are reduced to much lower dimensions by functional principal component analysis. The principal component scores are modeled as random fields to account for the spatial dependence, and the missing spectral observations are imputed by kriging the principal component scores. The proposed method is shown to impute spectral radiance with high accuracy for observations over the Pacific Ocean. An unmixing approach based on this model provides much more accurate land fraction estimates in our validation study along Greece coastlines.

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arXiv2021

Statistical Inference on the Hilbert Sphere with Application to Random Densities

The infinite-dimensional Hilbert sphere $S^\infty$ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on $S^\infty$. To break a path for sound statistical inference, we derive properties of the Fréchet mean on $S^\infty$ by establishing its existence and uniqueness as well as a root-$n$ central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on $S^\infty$. Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan modeled as densities, of which the square roots are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.

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arXiv2018

Modeling Longitudinal Data on Riemannian Manifolds

When considering functional principal component analysis for sparsely observed longitudinal data that take values on a nonlinear manifold, a major challenge is how to handle the sparse and irregular observations that are commonly encountered in longitudinal studies. Addressing this challenge, we provide theory and implementations for a manifold version of the principal analysis by conditional expectation (PACE) procedure that produces representations intrinsic to the manifold, extending a well-established version of functional principal component analysis targeting sparsely sampled longitudinal data in linear spaces. Key steps are local linear smoothing methods for the estimation of a Fréchet mean curve, mapping the observed manifold-valued longitudinal data to tangent spaces around the estimated mean curve, and applying smoothing methods to obtain the covariance structure of the mapped data. Dimension reduction is achieved via representations based on the first few leading principal components. A finitely truncated representation of the original manifold-valued data is then obtained by mapping these tangent space representations to the manifold. We show that the proposed estimates of mean curve and covariance structure achieve state-of-the-art convergence rates. For longitudinal emotional well-being data for unemployed workers as an example of time-dynamic compositional data that are located on a sphere, we demonstrate that our methods lead to interpretable eigenfunctions and principal component scores. In a second example, we analyze the body shapes of wallabies by mapping the relative size of their body parts onto a spherical pre-shape space. Compared to standard functional principal component analysis, which is based on Euclidean geometry, the proposed approach leads to improved trajectory recovery for sparsely sampled data on nonlinear manifolds.

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arXiv2017

Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fréchet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores.

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arXiv2017

Derivative Principal Component Analysis for Representing the Time Dynamics of Longitudinal and Functional Data

We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Loève expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other components of the model are derived under a unified scheme for sparse or dense observations and mild conditions. We compare the proposed representations for derivatives with alternative approaches in simulation settings and also in a wallaby growth curve application. It emerges that representations using the proposed derivative principal component analysis recover the underlying derivatives more accurately compared to principal component analysis-based approaches especially in settings where the functional data are represented with only a very small number of components or are densely sampled. In a second wheat spectra classification example, derivative principal component scores were found to be more predictive for the protein content of wheat than the conventional functional principal component scores.

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arXiv2016

Optimal Bayes Classifiers for Functional Data and Density Ratios

Bayes classifiers for functional data pose a challenge. This is because probability density functions do not exist for functional data. As a consequence, the classical Bayes classifier using density quotients needs to be modified. We propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified. The density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one-dimensional density estimates. This is an extension to functional data of some of the very earliest nonparametric Bayes classifiers that were based on simple density ratios in the one-dimensional case. By means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect Bayes classifiers for functional data can be avoided. We demonstrate that in the case of Gaussian functional data, the proposed functional Bayes classifier reduces to a functional version of the classical quadratic discriminant. A study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as "perfect classification". The proposed classifiers also perform favorably in finite sample applications, as we demonstrate in comparisons with other functional classifiers in simulations and various data applications, including wine spectral data, functional magnetic resonance imaging (fMRI) data for attention deficit hyperactivity disorder (ADHD) patients, and yeast gene expression data.

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