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Matrix-variate time series are now common in economic, medical, environmental, and atmospheric sciences, typically associated with large matrix dimensions. We introduce a structured autoregressive ...

In this article, we consider testing independence between two spatial Gaussian random fields evaluated, respectively, at p and q locations with sample size n, where both p and q are allowed to be larger than n. We impose no spatial stationarity and no parametric structure for the two random fields. Our approach is based on canonical correlation analysis (CCA). But instead of applying CCA directly to the two random fields, which is not feasible for high-dimensional testing considered, we adopt a dimension-reduction approach using a special class of multiresolution spline basis functions. These functions are ordered in terms of their degrees of smoothness. By projecting the data to the function space spanned by a few leading basis functions, the spatial variation of the data can be effectively preserved. The test statistic is constructed from the first sample canonical correlation coefficient in the projected space and is shown to have an asymptotic Tracy–Widom distribution under the null hypothesis. Our proposed method automatically detects the signal between the two random fields and is designed to handle irregularly spaced data directly. In addition, we show that our test is consistent under mild conditions and provide three simulation experiments to demonstrate its powers. Moreover, we apply our method to investigate whether the precipitation in continental East Africa is related to the sea surface temperature (SST) in the Indian Ocean and whether the precipitation in west Australia is related to the SST in the North Atlantic Ocean.

We consider estimation of intensity function for spatial point processes based on area-aggregated count data. Given a spatial point pattern of events, we can estimate its intensity function using the kernel method. However, when data are given in a spatially aggregated form with only the numbers of events in geographical subregions available, traditional methods developed for point patterns become infeasible. In this research, a modified kernel method is proposed to estimate the intensity function based on aggregated count data. Some numerical examples are provided to demonstrate the effectiveness of the proposed method.

The Taiwan air quality monitoring network (TAQMN) and the AirBox network both monitor PM

Rosset and Tibshirani are to be congratulated for addressing an important yet under-studied topic of modern statistics, so-called Random-X prediction in regression analysis. In statistics, Random-X...

Motivated by the need to analyze readily available data collected in space and time, especially in environmental sciences, we propose a parsimonious spatiotemporal model for time series data on a spatial grid. In essence, our model is a vector autoregressive model that utilizes the spatial structure to achieve parsimony of autoregressive matrices at two levels. The first level ensures the sparsity of the autoregressive matrices using a lagged-neighborhood scheme. The second level performs a spatial clustering of the nonzero autoregressive coefficients such that within some subregions, nearby locations share the same autoregressive coefficients while across different subregions the coefficients may have distinct values. The model parameters are estimated using the penalized maximum likelihood with an adaptive fused Lasso penalty. The estimation procedure can be tailored to accommodate the need and prior knowledge of a modeler. Performance of the proposed estimation algorithm is examined in a simulation study. Our method gives reliable estimation results that are interpretable and especially useful to identify geographical subregions, within each of which, the time series have similar dynamical behavior with homogeneous autoregressive coefficients. We apply our model to a wind speed time series dataset generated from a climate model over Saudi Arabia to illustrate its power in explaining the dynamics by the spatially structured coefficients. Moreover, the estimated model can be useful for building stochastic weather generators as an approximation of the computationally expensive climate model.

Layered manufacturing techniques have been successfully employed to construct scanned objects from three-dimensional medical image data sets. The printed physical models are useful tools for anatomical exploration, surgical planning, teaching, and related medical applications. Before fabricating scanned objects, we have to first build watertight geometrical representations of the target objects from medical image data sets. Many algorithms had been developed to fulfill this duty. However, some of these methods require extra efforts to resolve ambiguity problems and to fix broken surfaces. Other methods cannot generate legitimate models for layered manufacturing. To alleviate these problems, this article presents a modeling procedure to efficiently create geometrical representations of objects from computerized tomography scan and magnetic resonance imaging data sets. The proposed procedure extracts the iso-surface of the target object from the input data set at the first step. Then it converts the iso-surface into a three-dimensional image and filters this three-dimensional image using morphological operators to remove dangling parts and noises. At the next step, a distance field is computed in the three-dimensional image space to approximate the surface of the target object. Then the proposed procedure smooths the distance field to soothe sharp corners and edges of the target object. Finally, a boundary representation is built from the distance field to model the target object. Compared with conventional modeling techniques, the proposed method possesses the following advantages: (1) it reduces human efforts involved in the geometrical modeling process. (2) It can construct both solid and hollow models for the target object, and wall thickness of the hollow models is adjustable. (3) The resultant boundary representation guarantees to form a watertight solid geometry, which is printable using three-dimensional printers. (4) The proposed procedure allows users to tune the precision of the geometrical model to compromise with the available computational resources.

There are a number of ways to test for the absence/presence of a spatial signal in a completely observed fine-resolution image. One of these is a powerful nonparametric procedure called Enhanced False Discovery Rate (EFDR). A drawback of EFDR is that it requires the data to be defined on regular pixels in a rectangular spatial domain. Here, we develop an EFDR procedure for possibly incomplete data defined on irregular small areas. Motivated by statistical learning, we use conditional simulation (CS) to condition on the available data and simulate the full rectangular image at its finest resolution many times (M, say). EFDR is then applied to each of these simulations resulting in M estimates of the signal and M statistically dependent p-values. Averaging over these estimates yields a single, combined estimate of a possible signal, but inference is needed to determine whether there really is a signal present. We test the original null hypothesis of no signal by combining the M p-values into a single p-value using copulas and a composite likelihood. If the null hypothesis of no signal is rejected, we use the combined estimate. We call this new procedure EFDR-CS and, to demonstrate its effectiveness, we show results from a simulation study; an experiment where we introduce aggregation and incompleteness into temperature-change data in the Asia-Pacific; and an application to total-column carbon dioxide from satellite remote sensing data over a region of the Middle East, Afghanistan, and the western part of Pakistan., 45 pages, 23 figures, 2 tables

Partially ordered Markov models (POMMs) are Markov random fields (MRPs) with neigh­ borhood structures derivable from an associated partially ordered set. The most attractive feature of POMMs is that their joint distributions can be written in closed and product form. Therefore, simulation and maximum likelihood estimation for the models is quite straightfor­ ward, which is not the case in general for MRP models. In practice, one often has to modify the likelihood to account for edge components; the resulting composite likelihood for POMMs is similarly straightforward to maximize. In this article, we use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for POMMs. One of our results establishes that, under regularity conditions that include Dobrushin's condition for spatial mixing, the maximum composite likelihood estimator is consistent, asymptotically normal, and also asymptotically efficient. AMS 1991 subject classifications: Primary 62M40; secondary 62F12.

This article proposes a Rapid Prototyping (RP) procedure for manufacturing models of organs and tissues from medical imaging data. This RP procedure constitutes four stages: The key task of the first stage is to build the preliminary geometrical model of the target structure. To accomplish this goal, we extract the iso-surface of the target structure at first. Then we discretize the iso-surface into voxels and perform morphological operations upon the iso-surface to remove dangling parts and to fill small holes and gaps. At the second stage, a distance field is generated in the volume space, which encloses the iso-surface. Then a Laplacian filtering is performed to alleviate sharp geometrical features in the distance field. At the third stage, we extract a new iso-surface from the distance field to form the outer surface of the target structure. This iso-surface is the offset surface of the preliminary iso-surface. It forms a hollow solid model of the target object. At the final stage, this solid model is converted into G-codes and is manufactured by using a 3D printer.

We consider a stochastic process model with time trend and measurement error. We establish consistency and derive the limiting distributions of the maximum likelihood (ML) estimators of the covariance function parameters under a general asymptotic framework, including both the fixed domain and the increasing domain frameworks, even when the time trend model is misspecified or its complexity increases with the sample size. In particular, the convergence rates of the ML estimators are thoroughly characterized in terms of the growing rate of the domain and the degree of model misspecification/complexity., Published at http://dx.doi.org/10.3150/15-BEJ740 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

In layered manufacturing (LM), slicers are employed to convert input geometric models into G-codes. Conventional slicers accept only surface models as input data. Thus, volumetric models have to be converted into polygonal representations to fit the data format of the slicers. This results in extra computational costs and geometric errors. In this article, we present an efficient slicer aiming to generate G-codes for volumetric models. At first, our slicer computes the printing direction by exploring the inertia tensor of the input model to enhance the stability of the printed part and to decrease the build time. Then, it detects and classifies overhangs in the input model and generates necessary support structures by using a pattern-based method. Thirdly, the proposed slicer divides the input model into the skin and internal regions and cuts the model into 2D images. Subsequently, these images are transformed into toolpaths by utilizing texture mapping and graph traversal methods. Finally, the resultant toolpaths are smoothed to reduce staircases and encoded into G-codes. Test results verify that the proposed slicer produces decent G-codes for volumetric models. Scanned objects hidden in volume data can be directly manufactured without generating intermediate polygonal representations. LM processes become more efficient.

The spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging (FRK). However, the model depends on a class of basis functions, which if not selected properly, may result in unstable or undesirable results. Additionally, the maximum likelihood (ML) estimates of the model parameters are commonly computed using an expectation- maximization (EM) algorithm, which further limits its applicability when a large number of basis functions are required. In this research, we propose a class of basis functions extracted from thin-plate splines. The functions are ordered in terms of their degrees of smoothness with higher-order functions corresponding to larger-scale features and lower-order ones corresponding to smaller-scale details, leading to a parsimonious representation of a (nonstationary) spatial covariance function with the number of basis functions playing the role of spatial resolution. The proposed class of basis functions avoids the difficult knot-allocation or scale-selection problem. In addition, we show that ML estimates of the random effects covariance matrix can be expressed in simple closed forms, and hence the resulting FRK can accommodate a much larger number of basis functions without numerical difficulties. Finally, we propose to select the number of basis functions using Akaike's information criterion, which also possesses a simple closed-form expression. The whole procedure, involving no additional tuning parameter, is efficient to compute, easy to program, automatic to implement, and applicable to massive amounts of spatial data even when they are sparsely and irregularly located. Proofs of the theorems and an R package autoFRK are provided in supplementary materials available online.

In this article, we propose a regression method for simultaneous supervised clustering and feature selection over a given undirected graph, where homogeneous groups or clusters are estimated as well as informative predictors, with each predictor corresponding to one node in the graph and a connecting path indicating a priori possible grouping among the corresponding predictors. The method seeks a parsimonious model with high predictive power through identifying and collapsing homogeneous groups of regression coefficients. To address computational challenges, we present an efficient algorithm integrating the augmented Lagrange multipliers, coordinate descent and difference convex methods. We prove that the proposed method not only identifies the true homogeneous groups and informative features consistently but also leads to accurate parameter estimation. A gene network dataset is analysed to demonstrate that the method can make a difference by exploring dependency structures among the genes. Copyright 2012, Oxford University Press.

We develop a new approach for modeling non-stationary spatial–temporal processes on the basis of data sampled at fixed locations over time. The approach applies a basis function formulation and a constrained penalized least squares method recently proposed for estimating non-stationary spatial-only covariance functions. In this article, we further incorporate the temporal dependence into this framework and model the spatial–temporal process as the sum of a spatial–temporal stationary process and a linear combination of known basis functions with temporal dependent coefficients. A group lasso penalty is devised to select the basis functions and estimate the parameters simultaneously. In addition, a blockwise coordinate descent algorithm is applied for implementation. This algorithm computes the constrained penalized least squares solutions along a regularization path very rapidly. The resulting dynamic model has a state-space form, thereby the optimal spatial–temporal predictions can be computed efficiently using the Kalman filter. Moreover, the methodology is applied to a wind speed data set observed at the western Pacific Ocean for illustration. Copyright © 2011 John Wiley & Sons, Ltd.

Variable selection in geostatistical regression is an important problem, but has not been well studied in the literature. In this paper, we focus on spatial prediction and consider a class of conditional information criteria indexed by a penalty parameter. Instead of applying a fixed criterion, which leads to an unstable predictor in the sense that it is discontinuous with respect to the response variables due to that a small change in the response may cause a different model to be selected, we further stabilize the predictor by local model averaging, resulting in a predictor that is not only continuous but also differentiable even after plugging-in estimated model parameters. Then Stein’s unbiased risk estimate is applied to select the penalty parameter, leading to a data-dependent penalty that is adaptive to the underlying model. Some numerical experiments show superiority of the proposed model averaging method over some commonly used variable selection methods. In addition, the proposed method is applied to a mercury data set for lakes in Maine.

Spline smoothing is a popular technique for curve fitting, in which selection of the smoothing parameter is crucial. Many methods such as Mallows’ Cp, generalized maximum likelihood (GML), and the extended exponential (EE) criterion have been proposed to select this parameter. Although Cp is shown to be asymptotically optimal, it is usually outperformed by other selection criteria for small to moderate sample sizes due to its high variability. On the other hand, GML and EE are more stable than Cp, but they do not possess the same asymptotic optimality as Cp. Instead of selecting this smoothing parameter directly using Cp, we propose to select among a small class of selection criteria based on Stein's unbiased risk estimate (SURE). Due to the selection effect, the spline estimate obtained from a criterion in this class is nonlinear. Thus, the effective degrees of freedom in SURE contains an adjustment term in addition to the trace of the smoothing matrix, which cannot be ignored in small to moderate sample sizes. The resulting criterion, which we call adaptive Cp, is shown to have an analytic expression, and hence can be efficiently computed. Moreover, adaptive Cp is not only demonstrated to be superior and more stable than commonly used selection criteria in a simulation study, but also shown to possess the same asymptotic optimality as Cp.

SummarySpatial linear models are popular for the analysis of data on a spatial lattice, but statistical techniques for selection of covariates and a neighbourhood structure are limited. Here we develop new methodology for simultaneous model selection and parameter estimation via penalized maximum likelihood under a spatial adaptive lasso. A computationally efficient algorithm is devised for obtaining approximate penalized maximum likelihood estimates. Asymptotic properties of penalized maximum likelihood estimates and their approximations are established. A simulation study shows that the method proposed has sound finite sample properties and, for illustration, we analyse an ecological data set in western Canada.

Heterogeneous thyroid nodules have distinct components and vague boundaries in ultrasound (US) images. It is difficult for radiologists and physicians to manually draw the complete shape of a nodule, or distinguish what kind of components a nodule has. Hence, this article presents an automatic process for nodule segmentation and component classification. A decision-tree algorithm is used to segment the possible nodular area. A refinement process is then applied to recover the nodular shape. Finally, a hierarchical method based on support vector machines (SVMs) is used to identify the components in the nodular lesion. Experimental results of the proposed approach were compared with those of other methods.

Geographic information systems (GIS) organize spatial data in multiple two-dimensional arrays called layers. In many applications, a response of interest is observed on a set of sites in the landscape, and it is of interest to build a regression model from the GIS layers to predict the response at unsampled sites. Model selection in this context then consists not only of selecting appropriate layers, but also of choosing appropriate neighborhoods within those layers. We formalize this problem as a linear model and propose the use of Lasso to simultaneously select variables, choose neighborhoods, and estimate parameters. Spatially dependent errors are accounted for using generalized least squares and spatial smoothness in selected coefficients is incorporated through use of a priori spatial covariance structure. This leads to a modification of the Lasso procedure, called spatial Lasso. The spatial Lasso can be implemented by a fast algorithm and it performs well in numerical examples, including an applicat...

Extracting grouping structure or identifying homogenous subgroups of predictors in regression is crucial for high-dimensional data analysis. A low-dimensional structure in particular-grouping, when captured in a regression model, enables to enhance predictive performance and to facilitate a model's interpretability Grouping pursuit extracts homogenous subgroups of predictors most responsible for outcomes of a response. This is the case in gene network analysis, where grouping reveals gene functionalities with regard to progression of a disease. To address challenges in grouping pursuit, we introduce a novel homotopy method for computing an entire solution surface through regularization involving a piecewise linear penalty. This nonconvex and overcomplete penalty permits adaptive grouping and nearly unbiased estimation, which is treated with a novel concept of grouped subdifferentials and difference convex programming for efficient computation. Finally, the proposed method not only achieves high performance as suggested by numerical analysis, but also has the desired optimality with regard to grouping pursuit and prediction as showed by our theoretical results.

We propose a method for estimating nonstationary spatial covariance functions by representing a spatial process as a linear combination of some local basis functions with uncorrelated random coefficients and some stationary processes, based on spatial data sampled in space with repeated measurements. By incorporating a large collection of local basis functions with various scales at various locations and stationary processes with various degrees of smoothness, the model is flexible enough to represent a wide variety of nonstationary spatial features. The covariance estimation and model selection are formulated as a regression problem with the sample covariances as the response and the covariances corresponding to the local basis functions and the stationary processes as the predictors. A constrained least squares approach is applied to select appropriate basis functions and stationary processes as well as estimate parameters simultaneously. In addition, a constrained generalized least squares approach is ...

In many fields of science, predicting variables of interest over a study region based on noisy data observed at some locations is an important problem. Two popular methods for the problem are kriging and smoothing splines. The former assumes that the underlying process is stochastic, whereas the latter assumes it is purely deterministic. Kriging performs better than smoothing splines in some situations, but is outperformed by smoothing splines in others. However, little is known regarding selecting between kriging and smoothing splines. In addition, how to perform variable selection in a geostatistical model has not been well studied. In this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on (approximately) unbiased estimation of mean squared prediction errors using a data perturbation technique. The proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors, and is shown to be optimal in terms of two mean sq...

An intensive simulation study to compare the spatio-temporal prediction performances among various space-time models is presented. The models having separable spatio-temporal covariance functions and nonseparable ones, under various scenarios, are also considered. The computational performance among the various selected models are compared. The issue of how to select an appropriate space-time model by accounting for the tradeoff between goodness-of-fit and model complexity is addressed. Performances of the two commonly used model-selection criteria, Akaike information criterion and Bayesian information criterion are examined. Furthermore, a practical application based on the statistical analysis of surface shortwave radiation budget (SSRB) data is presented.

Data from remote-sensing platforms play an important role in monitoring environmental processes, such as the distribution of stratospheric ozone. Remote-sense data are typically spatial, temporal, and massive. Existing prediction methods such as kriging are computationally infeasible. The multi-resolution spatial model (MRSM) captures nonstationary spatial dependence and produces fast optimal estimates using a change-of-resolution Kalman filter. However, past data can provide valuable information about the current status of the process being investigated. In this article, we incorporate the temporal dependence into the process by developing a dynamic MRSM. An application of the dynamic MRSM to a month of daily total column ozone data is presented, and on a given day the results of posterior inference are compared to those for the spatial-only MRSM. It is apparent that there are advantages to using the dynamic MRSM in regions where data are missing, such as when a whole swath of satellite data is missing.

An autologistic regression model consists of a logistic regression of a response variable on explanatory variables and an autoregression on responses at neighboring locations on a lattice. It is a Markov random field with pairwise spatial dependence and is a popular tool for modeling spatial binary responses. In this article, we add a temporal component to the autologistic regression model for spatial-temporal binary data. The spatial-temporal autologistic regression model captures the relationship between a binary response and potential explanatory variables, and adjusts for both spatial dependence and temporal dependence simultaneously by a space-time Markov random field. We estimate the model parameters by maximum pseudo-likelihood and obtain optimal prediction of future responses on the lattice by a Gibbs sampler. For illustration, the method is applied to study the outbreaks of southern pine bettle in North Carolina. We also discuss the generality of our approach for modeling other types of spatial-temporal lattice data.

This article considers a class of multiresolution tree-structured models that are spatially shifted versions of each other and proposes a new spatial-prediction method that averages over the optimal spatial predictors produced from members of this class of models. As a consequence, the resulting predicted surface is smooth, even when the predictors generated separately from individual multiresolution tree-structured models are not. We call the new predictor the multiresolution spatial (MURS) predictor and develop a computationally efficient algorithm for it. The algorithm can handle massive datasets even when some observations are missing. Moreover, the MURS predictor can be shown to be the minimum mean squared error predictor for a large class of covariance functions. A simulation example for massive datasets shows that the MURS method consistently outperforms two commonly used filtering methods. Total column ozone data remotely sensed from a satellite are analyzed using the new methodology.

Typical modeling strategies involve model selection, which has a significant effect on inference of estimated parameters. Common practice is to use a selected model ignoring uncertainty introduced by the process of model selection. This could yield overoptimistic inferences, resulting in false discovery. In this article we develop a general methodology via optimal approximation for estimating the mean and variance of complex statistics that involve the process of model selection. This allows us to make approximately unbiased inferences, taking into account the selection process. We examine the operating characteristics of the proposed methodology via asymptotic analyses and simulations. These results show that the proposed methodology yields correct inferences and outperforms common alternatives.

In many scientific and engineering problems, selecting the optimal model from a large pool of candidate models is important, particularly in data mining. In the literature, model assessment in the context of nonnormal distributions has not yet received a lot of attention. Indeed, many existing model selection criteria such as the Bayes information criterion and Cp may not be suitable for a situation in which the conditional mean and variance of the response are dependent, such as in generalized linear model regression. In this article we propose a new adaptive model selection criterion and construct an approximately unbiased Kullback–Leibler loss estimator for model assessment in the context of exponential family distributions. This permits comparing any arbitrary complex modeling procedures. Our proposal uses a concept called generalized degrees of freedom that generalizes the concept originally proposed for the normal distribution. The proposed procedure is implemented for the binomial and Poisson distr...