Non-stationary Covariance Structure Research Articles

Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations.

Many techniques have been proposed to model space-varying observation processes with a nonstationary spatial covariance structure and/or anisotropy, usually on a geostatistical framework. Nevertheless, there is an increasing interest in point process applications, and methodologies that take nonstationarity into account are welcomed. In this sense, this work proposes an extension of a class of spatial Cox process using spatial deformation. The proposed method enables the deformation behavior to be data-driven, through a multivariate latent Gaussian process. Inference leads to intractable posterior distributions that are approximated via MCMC. The convergence of algorithms based on the Metropolis-Hastings steps proved to be slow, and the computational efficiency of the Bayesian updating scheme was improved by adopting Hamiltonian Monte Carlo (HMC) methods. Our proposal was also compared against an alternative anisotropic formulation. Studies based on synthetic data provided empirical evidence of the benefit brought by the adoption of nonstationarity through our anisotropic structure. A real data application was conducted on the spatial spread of the Spodoptera frugiperda pest in a corn-producing agricultural area in southern Brazil. Once again, the proposed method demonstrated its benefit over alternatives.

Projection-based two-sample inference for sparsely observed multivariate functional data.

Modern longitudinal studies collect multiple outcomes as the primary endpoints to understand the complex dynamics of the diseases. Oftentimes, especially in clinical trials, the joint variation among the multidimensional responses plays a significant role in assessing the differential characteristics between two or more groups, rather than drawing inferences based on a single outcome. We develop a projection-based two-sample significance test to identify the population-level difference between the multivariate profiles observed under a sparse longitudinal design. The methodology is built upon widely adopted multivariate functional principal component analysis to reduce the dimension of the infinite-dimensional multi-modal functions while preserving the dynamic correlation between the components. The test applies to a wide class of (non-stationary) covariance structures of the response, and it detects a significant group difference based on a single p-value, thereby overcoming the issue of adjusting for multiple p-values that arise due to comparing the means in each of components separately. Finite-sample numerical studies demonstrate that the test maintains the type-I error, and is powerful to detect significant group differences, compared to the state-of-the-art testing procedures. The test is carried out on two significant longitudinal studies for Alzheimer's disease and Parkinson's disease (PD) patients, namely, TOMMORROW study of individuals at high risk of mild cognitive impairment to detect differences in the cognitive test scores between the pioglitazone and the placebo groups, and Azillect study to assess the efficacy of rasagiline as a potential treatment to slow down the progression of PD.

Bayesian regularization for a nonstationary Gaussian linear mixed effects model.

In omics experiments, estimation and variable selection can involve thousands of proteins/genes observed from a relatively small number of subjects. Many regression regularization procedures have been developed for estimation and variable selection in such high-dimensional problems. However, approaches have predominantly focused on linear regression models that ignore correlation arising from long sequences of repeated measurements on the outcome. Our work is motivated by the need to identify proteomic biomarkers that improve the prediction of rapid lung-function decline for individuals with cystic fibrosis (CF) lung disease. We extend four Bayesian penalized regression approaches for a Gaussian linear mixed effects model with nonstationary covariance structure to account for the complicated structure of longitudinal lung function data while simultaneously estimating unknown parameters and selecting important protein isoforms to improve predictive performance. Different types of shrinkage priors are evaluated to induce variable selection in a fully Bayesian framework. The approaches are studied with simulations. We apply the proposed method to real proteomics and lung-function outcome data from our motivating CF study, identifying a set of relevant clinical/demographic predictors and a proteomic biomarker for rapid decline of lung function. We also illustrate the methods on CD4 yeast cell-cycle genomic data, confirming that the proposed method identifies transcription factors that have been highlighted in the literature for their importance as cell cycle transcription factors.

Deep Compositional Spatial Models

Spatial processes with nonstationary and anisotropic covariance structure are often used when modeling, analyzing, and predicting complex environmental phenomena. Such processes may often be expressed as ones that have stationary and isotropic covariance structure on a warped spatial domain. However, the warping function is generally difficult to fit and not constrained to be injective, often resulting in “space-folding.” Here, we propose modeling an injective warping function through a composition of multiple elemental injective functions in a deep-learning framework. We consider two cases; first, when these functions are known up to some weights that need to be estimated, and, second, when the weights in each layer are random. Inspired by recent methodological and technological advances in deep learning and deep Gaussian processes, we employ approximate Bayesian methods to make inference with these models using graphics processing units. Through simulation studies in one and two dimensions we show that the deep compositional spatial models are quick to fit, and are able to provide better predictions and uncertainty quantification than other deep stochastic models of similar complexity. We also show their remarkable capacity to model nonstationary, anisotropic spatial data using radiances from the MODIS instrument aboard the Aqua satellite.

Statistical Interpolation of Spatially Varying but Sparsely Measured 3D Geo-Data Using Compressive Sensing and Variational Bayesian Inference

Real geo-data are three-dimensional (3D) and spatially varied, but measurements are often sparse due to time, resource, and/or technical constraints. In these cases, the quantities of interest at locations where measurements are missing must be interpolated from the available data. Several powerful methods have been developed to address this problem in real-world applications over the past several decades, such as two-point geo-statistical methods (e.g., kriging or Gaussian process regression, GPR) and multiple-point statistics (MPS). However, spatial interpolation remains challenging when the number of measurements is small because a suitable covariance function is difficult to select and the parameters are challenging to estimate from a small number of measurements. Note that a covariance function form and its parameters are key inputs for some methods (e.g., kriging or GPR). MPS is a non-parametric simulation method that combines training images as prior knowledge for sparse measurements. However, the selection of a suitable training image for continuous geo-quantities (e.g., soil or rock properties) faces certain difficulties and may become increasingly complicated when the geo-data to be interpolated are high-dimensional (e.g., 3D) and exhibit non-stationary (e.g., with unknown trends or non-stationary covariance structure) and/or anisotropic characteristics. This paper proposes a non-parametric approach that systematically combines compressive sensing and variational Bayesian inference for statistical interpolation of 3D geo-data. The method uses sparse measurements and their locations as the input and provides interpolated values at unsampled locations with quantified interpolation uncertainty as the output. The proposed method is illustrated using a series of numerical 3D examples, and the results indicate a reasonably good performance.

Modeling spatial data using local likelihood estimation and a Matérn to spatial autoregressive translation

Modeling data with nonstationary covariance structure is important to represent heterogeneity in geophysical and other environmental spatial processes. In this work, we investigate a two‐stage approach to modeling nonstationary covariances that is efficient for large data sets. First, maximum likelihood estimation is used in local, moving windows to infer spatially varying covariance parameters. These surfaces of covariance parameters are then encoded into a global covariance model specifying the second‐order structure for the complete spatial domain. From this second step, the resulting global model allows for efficient simulation and prediction. This work uses a nonstationary spatial autoregressive (SAR) model, related to Gaussian Markov random field methods, as the global model which is amenable to plug in local estimates and practical for large datasets. A simulation study is used to establish the accuracy of local Matérn parameter estimation as a reliable technique for small window sizes and a modest number of replicated fields. This modeling approach is implemented on a nonstationary climate model dataset with the goal of emulating the variation in the numerical model ensemble using a Gaussian process.

"ADAPTIVE PARAMETRIC MODEL FOR NONSTATIONARY SPATIAL COVARIANCE"

In modelling environment processes, multi-disciplinary methods are used to explain, explore and predict how the earth responds to natural human-induced environmental changes over time. Consequently, when analyzing spatial processes in environmental and ecological studies, the spatial parameters of interest are always heterogeneous. This often negates the stationarity assumption. In this article, we propose the adaptive parametric nonstationary covariance structure for spatial processes. The adaptive turning parameter for this model was also proposed for nonstationary processes. The flexibility and efficiency of the propose model was examined through simulation. A real life data was also use to examine the efficiency of the propose model. The results show that the propose model perform competitively with existing models.

Spatiotemporal signal detection using continuous shrinkage priors.

Periodontal disease (PD) is a chronic inflammatory disease that affects the gum tissue and bone supporting the teeth. Although tooth-site level PD progression is believed to be spatio-temporally referenced, the whole-mouth average periodontal pocket depth (PPD) has been commonly used as an indicator of the current/active status of PD. This leads to imminent loss of information, and imprecise parameter estimates. Despite availability of statistical methods that accommodates spatiotemporal information for responses collected at the tooth-site level, the enormity of longitudinal databases derived from oral health practice-based settings render them unscalable for application. To mitigate this, we introduce a Bayesian spatiotemporal model to detect problematic/diseased tooth-sites dynamically inside the mouth for any subject obtained from large databases. This is achieved via a spatial continuous sparsity-inducing shrinkage prior on spatially varying linear-trend regression coefficients. A low-rank representation captures the nonstationary covariance structure of the PPD outcomes, and facilitates the relevant Markov chain Monte Carlo computing steps applicable to thousands of study subjects. Application of our method to both simulated data and to a rich database of electronic dental records from the HealthPartners Institute reveal improved prediction performances, compared with alternative models with usual Gaussian priors for regression parameters and conditionally autoregressive specification of the covariance structure.

Modelling Spatio-Temporal Variation in Sparse Rainfall Data Using a Hierarchical Bayesian Regression Model

Rainfall is a critical component of climate governing vegetation growth and production, forage availability and quality for herbivores. However, reliable rainfall measurements are not always available, making it necessary to predict rainfall values for particular locations through time. Predicting rainfall in space and time can be a complex and challenging task, especially where the rain gauge network is sparse and measurements are not recorded consistently for all rain gauges, leading to many missing values. Here, we develop a flexible Bayesian model for predicting rainfall in space and time and apply it to Narok County, situated in southwestern Kenya, using data collected at 23 rain gauges from 1965 to 2015. Narok County encompasses the Maasai Mara ecosystem, the northern-most section of the Mara–Serengeti ecosystem, famous for its diverse and abundant large mammal populations and spectacular migration of enormous herds of wildebeest, zebra and Thomson’s gazelle. The model incorporates geographical and meteorological predictor variables, including elevation, distance to Lake Victoria and minimum temperature. Salient features of our model are the use of non-stationary covariance structures and the facility to handle excess zeros and many missing observations. We assess the efficiency of the model by comparing it empirically with the established Gaussian process, Kriging, simple linear and Bayesian linear models. We use the model to predict total monthly rainfall and its standard error for all 5 $$\times $$ 5 km grid cells in Narok County for each year from 1965 to 2015. Using the Monte Carlo integration method, we estimate seasonal and annual rainfall and their standard errors for 29 sub-regions of Narok for each of the 51 years spanning 1965–2015. The non-stationary model can handle data from a sparse network of observations with many missing values and performs at least as well as or better than four established and widely used models on the Narok rainfall data set. The model produces rainfall predictions consistent with expectation and in good agreement with blended station and satellite rainfall values. The predictions are precise enough for most practical purposes. The model is very general and applicable to other variables besides rainfall. Supplementary materials accompanying this paper appear online

Locally stationary spatio-temporal interpolation of Argo profiling float data.

Argo floats measure seawater temperature and salinity in the upper 2000 m of the global ocean. Statistical analysis of the resulting spatio-temporal dataset is challenging owing to its non-stationary structure and large size. We propose mapping these data using locally stationary Gaussian process regression where covariance parameter estimation and spatio-temporal prediction are carried out in a moving-window fashion. This yields computationally tractable non-stationary anomaly fields without the need to explicitly model the non-stationary covariance structure. We also investigate Student t-distributed fine-scale variation as a means to account for non-Gaussian heavy tails in ocean temperature data. Cross-validation studies comparing the proposed approach with the existing state of the art demonstrate clear improvements in point predictions and show that accounting for the non-stationarity and non-Gaussianity is crucial for obtaining well-calibrated uncertainties. This approach also provides data-driven local estimates of the spatial and temporal dependence scales for the global ocean, which are of scientific interest in their own right.

Locally stationary spatio-temporal processes

This paper proposes a locally stationary spatio-temporal process to analyze the motivating example of US precipitation data, which is a huge data set composed of monthly observations of precipitation on thousands of monitoring points scattered irregularly all over US continent. Allowing the parameters of continuous autoregressive and moving average (CARMA) random fields by Brockwell and Matsuda (J R Stat Soc Ser B 79(3):833–857, 2017) to be dependent spatially, we generalize locally stationary time series by Dahlhaus (Ann Stat 25:1–37, 1997) to spatio-temporal processes that are locally stationary in space. We develop Whittle likelihood estimation for the spatially dependent parameters and derive the asymptotic properties rigorously. We demonstrate that the spatio-temporal models actually work to account for nonstationary spatial covariance structures in US precipitation data.

Prediction of weakly locally stationary processes by auto-regression

In this contribution we introduce weakly locally stationary time series through the local approximation of the non-stationary covariance structure by a stationary one. This allows us to define autoregression coefficients in a non-stationary context, which, in the particular case of a locally stationary Time Varying Autoregressive (TVAR) process, coincide with the generating coefficients. We provide and study an estimator of the time varying autoregression coefficients in a general setting. The proposed estimator of these coefficients enjoys an optimal minimax convergence rate under limited smoothness conditions. In a second step, using a bias reduction technique, we derive a minimax-rate estimator for arbitrarily smooth time-evolving coefficients, which outperforms the previous one for large data sets. In turn, for TVAR processes, the predictor derived from the estimator exhibits an optimal minimax prediction rate.

Accounting for covariate information in the scale component of spatio-temporal mixing models

Spatio-temporal processes in the environmental science are usually assumed to follow a Gaussian process, possibly after some transformation. Gaussian processes might not be appropriate to handle the presence of outlying observations. Our proposal is based on the idea of modelling the process as a scale mixture between a Gaussian and log-Gaussian process. And the novelty is to allow the scale process to vary as a function of covariates. The resultant model has a nonstationary covariance structure in space. Moreover, the resultant kurtosis varies with location, allowing the time series at each location to have different distributions with different tail behaviour. Inference procedure is performed under the Bayesian framework. The analysis of an artificial dataset illustrates how this proposal is able to capture heterogeneity in space caused by dependence on some spatial covariate or by a transformation of the process of interest. Furthermore, an application to maximum temperature data observed in the Spanish Basque country illustrates the effects of altitude in the variability of the process and how our proposed model identifies this dependence through parameters which can be interpreted as regression coefficients in the variance model.

Modeling nonstationarity in space and time.

We propose to model a spatio-temporal random field that has nonstationary covariance structure in both space and time domains by applying the concept of the dimension expansion method in Bornn et al. (2012). Simulations are conducted for both separable and nonseparable space-time covariance models, and the model is also illustrated with a streamflow dataset. Both simulation and data analyses show that modeling nonstationarity in both space and time can improve the predictive performance over stationary covariance models or models that are nonstationary in space but stationary in time.

Smoothing and Mean-Covariance Estimation of Functional Data with a Bayesian Hierarchical Model.

Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean-covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean-covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.

Modeling nonstationary covariance function with convolution on sphere

The wide use of satellite-based instruments provides measurements in climatology on a global scale, which often have nonstationary covariance structure. The issue of modeling a spatial random fields on sphere which is stationary across longitudes is addressed with a kernel convolution approach. The observed random field is generated by convolving a latent uncorrelated random field with a class of Matérn type kernel functions. By allowing the parameters in the kernel functions to vary with locations, it is possible to generate a flexible class of covariance functions and capture the nonstationary properties. Since the corresponding covariance functions generally do not have a closed form, numerical evaluations are necessary and a pre-computation table is used to speed up the computation. For regular grid data on sphere, the circulant block property of the covariance matrix enables us to use Fast Fourier Transform (FFT) to get its determinant and inverse matrix efficiently. The proposed approach is applied to the Total Ozone Mapping Spectrometer (TOMS) data for illustration.

Functional CAR Models for Large Spatially Correlated Functional Datasets

ABSTRACTWe develop a functional conditional autoregressive (CAR) model for spatially correlated data for which functions are collected on areal units of a lattice. Our model performs functional response regression while accounting for spatial correlations with potentially nonseparable and nonstationary covariance structure, in both the space and functional domains. We show theoretically that our construction leads to a CAR model at each functional location, with spatial covariance parameters varying and borrowing strength across the functional domain. Using basis transformation strategies, the nonseparable spatial-functional model is computationally scalable to enormous functional datasets, generalizable to different basis functions, and can be used on functions defined on higher dimensional domains such as images. Through simulation studies, we demonstrate that accounting for the spatial correlation in our modeling leads to improved functional regression performance. Applied to a high-throughput spatially correlated copy number dataset, the model identifies genetic markers not identified by comparable methods that ignore spatial correlations. Supplementary materials for this article are available online.

The impact of covariance misspecification in group-based trajectory models for longitudinal data with non-stationary covariance structure.

One purpose of a longitudinal study is to gain a better understanding of how an outcome of interest changes among a given population over time. In what follows, a trajectory will be taken to mean the series of measurements of the outcome variable for an individual. Group-based trajectory modelling methods seek to identify subgroups of trajectories within a population, such that trajectories that are grouped together are more similar to each other than to trajectories in distinct groups. Group-based trajectory models generally assume a certain structure in the covariances between measurements, for example conditional independence, homogeneous variance between groups or stationary variance over time. Violations of these assumptions could be expected to result in poor model performance. We used simulation to investigate the effect of covariance misspecification on misclassification of trajectories in commonly used models under a range of scenarios. To do this we defined a measure of performance relative to the ideal Bayesian correct classification rate. We found that the more complex models generally performed better over a range of scenarios. In particular, incorrectly specified covariance matrices could significantly bias the results but using models with a correct but more complicated than necessary covariance matrix incurred little cost.

Spacebender

This paper presents an algorithm that deforms geographical space such that local spatial variation is more regular in the transformed space than in the original space. Geographical-space deformation may be useful in situations where there is an indication that the assumption of stationarity of the variance is unrealistic. In this paper we present an algorithm, Spacebender, a variance equalising algorithm that can be used for generating the non-affine transform between geographic space and the transformed space. This transformed space, which takes care of the non-stationary spatial covariance structure in the geographic space, is the one in which regularly spaced points would be optimal for sampling. This transformed space is a kind of enviro-geographic space as opposed to a simple geographic one. Two examples are given; one is based on a digital elevation model, and the other used a data from a soil gamma radiometric potassium survey.

Reduced-Order Dynamic Stall Modeling with Swept Flow Effects Using a Surrogate-Based Recurrence Framework

The extension of the surrogate-based recurrence framework approach to account for time-varying swept flow effects under dynamic stall conditions is described. Using full-order solutions generated by the OVERFLOW computational fluid dynamics code, the surrogate-based recurrence framework reduced-order modeling approach is shown to effectively mimic full-order solutions of unsteady lift, moment, and drag under dynamic stall conditions while maintaining the computational efficiency associated with semiempirical dynamic stall models. This level of functionality represents a new capability for rotary-wing aeroelasticity applications. Furthermore, a generalized kriging formulation based on nonstationary Gaussian process modeling is implemented in a tractable manner by locally optimizing the high-dimensional likelihood function in the vicinity of the stationary solution. The resulting nonstationary covariance structures are shown to significantly improve the accuracy of the surrogate-based recurrence framework predicted moment stall characteristics compared to a stationary model. It is shown that the nonstationary surrogate-based recurrence framework approach is better able to adapt to abrupt changes in airload behavior caused by the underlying dynamic stall vortex dynamics. The results indicate that the surrogate-based recurrence framework approach based on nonstationary Gaussian process models is a promising alternative to widely used semiempirical rotorcraft dynamic stall models that cannot account for the effects of time-varying velocity components associated with forward flight.