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Accurately estimating data density is crucial for making informed decisions and modeling in various fields. This paper presents a novel nonparametric density estimation procedure that utilizes bivariate penalized spline smoothing over triangulation for data scattered over irregular spatial domains. The approach is likelihood-based with a regularization term that addresses the roughness of the logarithm of density based on a second-order differential operator. The proposed method offers greater efficiency and flexibility in estimating density over complex domains and has been theoretically supported by establishing the asymptotic convergence rate under mild natural conditions. Through extensive simulation studies and a real-world application that analyzes motor vehicle theft data from Portland City, Oregon, we demonstrate the advantages of the proposed method over existing techniques detailed in the literature.

Computed tomography (CT) is an imaging technique that uses X-ray projections from multiple rotation angles to create detailed cross-sectional images, widely used in industrial inspection and medical diagnostics. Reducing the projection data in CT scans is often necessary to decrease radiation exposure, scanning time, and computational costs. However, this reduction makes accurate image reconstruction challenging due to the incomplete sinogram. Existing RGB inpainting models struggle with severe feature overlap, while current sinogram-specific models fail to employ efficient feature extraction methods that account for the physical principles underlying the sinogram generation process. To tackle these challenges, we introduce the Frequency Convolution Diffusion Model (FCDM), a novel diffusion-based inpainting framework tailored for sinogram data. FCDM leverages frequency-domain convolutions to capture global and fine-grained structural features, effectively disentangling overlapping components across projection angles. Additionally, we propose a custom loss function that incorporates unique sinogram properties of total absorption consistency and frequency-domain consistency. Extensive experiments on synthetic and real-world datasets demonstrate that FCDM significantly outperforms existing methods, achieving SSIM over 0.95 and PSNR above 30 dB, with improvements of up to 33% in SSIM and 29% in PSNR compared to baselines.

Surface-based data is commonly observed in diverse practical applications spanning various fields. In this paper, we introduce a novel nonparametric method to discover the underlying signals from data distributed on complex surface-based domains. Our approach involves a penalized spline estimator defined on a triangulation of surface patches, which enables effective signal extraction and recovery. The proposed method offers several advantages over existing methods, including superior handling of "leakage" or "boundary effects" over complex domains, enhanced computational efficiency, and potential applications in analyzing sparse and irregularly distributed data on complex objects. We provide rigorous theoretical guarantees for the proposed method, including convergence rates of the estimator in both the $L_2$ and supremum norms, as well as the asymptotic normality of the estimator. We also demonstrate that the convergence rates achieved by our estimation method are optimal within the framework of nonparametric estimation. Furthermore, we introduce a bootstrap method to quantify the uncertainty associated with the proposed estimators accurately. The superior performance of the proposed method is demonstrated through simulation experiments and data applications on cortical surface functional magnetic resonance imaging data and oceanic near-surface atmospheric data., Comment: 56 pages, 16 figures

Over the past two decades, we have seen an exponentially increased amount of point clouds collected with irregular shapes in various areas. Motivated by the importance of solid modeling for point clouds, we develop a novel and efficient smoothing tool based on multivariate splines over the tetrahedral partitions to extract the underlying signal and build up a 3D solid model from the point cloud. The proposed smoothing method can denoise or deblur the point cloud effectively and provide a multi-resolution reconstruction of the actual signal. In addition, it can handle sparse and irregularly distributed point clouds and recover the underlying trajectory. The proposed smoothing and interpolation method also provides a natural way of numerosity data reduction. Furthermore, we establish the theoretical guarantees of the proposed method. Specifically, we derive the convergence rate and asymptotic normality of the proposed estimator and illustrate that the convergence rate achieves the optimal nonparametric convergence rate. Through extensive simulation studies and a real data example, we demonstrate the superiority of the proposed method over traditional smoothing methods in terms of estimation accuracy and efficiency of data reduction., Comment: 64 pages, 16 figures

Motivated by recent work involving the analysis of biomedical imaging data, we present a novel procedure for constructing simultaneous confidence corridors for the mean of imaging data. We propose to use flexible bivariate splines over triangulations to handle irregular domain of the images that is common in brain imaging studies and in other biomedical imaging applications. The proposed spline estimators of the mean functions are shown to be consistent and asymptotically normal under some regularity conditions. We also provide a computationally efficient estimator of the covariance function and derive its uniform consistency. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions from two populations of imaging data. Through Monte Carlo simulation studies we examine the finite-sample performance of the proposed method. Finally, the proposed method is applied to analyze brain Positron Emission Tomography (PET) data in two different studies. One dataset used in preparation of this article was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database.

Motivated by recent data analyses in biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose using flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images, as well as other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-n consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals and data-driven simultaneous confidence corridors for the coefficient functions are constructed. Our method can simultaneously estimate and make inferences on the coefficient functions while incorporating spatial heterogeneity and spatial correlation. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method, which is then applied to the spatially normalized positron emission tomography data of the Alzheimer's Disease Neuroimaging Initiative.

In this paper, we focus on the variable selection techniques for a class of semiparametric spatial regression models which allow one to study the effects of explanatory variables in the presence of the spatial information. The spatial smoothing problem in the nonparametric part is tackled by means of bivariate splines over triangulation, which is able to deal efficiently with data distributed over irregularly shaped regions. In addition, we develop a unified procedure for variable selection to identify significant covariates under a double penalization framework, and we show that the penalized estimators enjoy the "oracle" property. The proposed method can simultaneously identify non-zero spatially distributed covariates and solve the problem of "leakage" across complex domains of the functional spatial component. To estimate the standard deviations of the proposed estimators for the coefficients, a sandwich formula is developed as well. In the end, Monte Carlo simulation examples and a real data example are provided to illustrate the proposed methodology. All technical proofs are given in the supplementary materials.

Over the past few months, the outbreak of Coronavirus disease (COVID-19) has been expanding over the world. A reliable and accurate dataset of the cases is vital for scientists to conduct related research and for policy-makers to make better decisions. We collect the United States COVID-19 daily reported data from four open sources: the New York Times, the COVID-19 Data Repository by Johns Hopkins University, the COVID Tracking Project at the Atlantic, and the USAFacts, then compare the similarities and differences among them. To obtain reliable data for further analysis, we first examine the cyclical pattern and the following anomalies, which frequently occur in the reported cases: (1) the order dependencies violation, (2) the point or period anomalies, and (3) the issue of reporting delay. To address these detected issues, we propose the corresponding repairing methods and procedures if corrections are necessary. In addition, we integrate the COVID-19 reported cases with the county-level auxiliary information of the local features from official sources, such as health infrastructure, demographic, socioeconomic, and environmental information, which are also essential for understanding the spread of the virus., Comment: 34 pages, 9 figures

Epidemic modeling is an essential tool to understand the spread of the novel coronavirus and ultimately assist in disease prevention, policymaking, and resource allocation. In this article, we establish a state of the art interface between classic mathematical and statistical models and propose a novel space-time epidemic modeling framework to study the spatial-temporal pattern in the spread of infectious disease. We propose a quasi-likelihood approach via the penalized spline approximation and alternatively reweighted least-squares technique to estimate the model. Furthermore, we provide a short-term and long-term county-level prediction of the infected/death count for the U.S. by accounting for the control measures, health service resources, and other local features. Utilizing spatiotemporal analysis, our proposed model enhances the dynamics of the epidemiological mechanism and dissects the spatiotemporal structure of the spreading disease. To assess the uncertainty associated with the prediction, we develop a projection band based on the envelope of the bootstrap forecast paths. The performance of the proposed method is evaluated by a simulation study. We apply the proposed method to model and forecast the spread of COVID-19 at both county and state levels in the United States., Comment: 37 pages, 6 figures, 8 tables