Nonstationary Spatial Processes Research Articles

Comparing two spatial variables with the probability of agreement.

Computing the agreement between 2 continuous sequences is of great interest in statistics when comparing 2 instruments or one instrument with a gold standard. The probability of agreement quantifies the similarity between 2 variables of interest, and it is useful for determining what constitutes a practically important difference. In this article, we introduce a generalization of the PA for the treatment of spatial variables. Our proposal makes the PA dependent on the spatial lag. We establish the conditions for which the PA decays as a function of the distance lag for isotropic stationary and nonstationary spatial processes. Estimation is addressed through a first-order approximation that guarantees the asymptotic normality of the sample version of the PA. The sensitivity of the PA with respect to the covariance parameters is studied for finite sample size. The new method is described and illustrated with real data involving autumnal changes in the green chromatic coordinate (Gcc), an index of "greenness" that captures the phenological stage of tree leaves, is associated with carbon flux from ecosystems, and is estimated from repeated images of forest canopies.

CoYangCZ: a new spatial interpolation method for nonstationary multivariate spatial processes

In multivariate spatial interpolation, the accuracy of a variable of interest can be improved using ancillary variables. Although geostatistical methods are widely used for multivariate spatial interpolation, these methods usually require second-order stationary assumption of spatial processes, which is difficult to satisfy in practice. We developed a new multivariate spatial interpolation method based on Yang-Chizhong filtering (CoYangCZ) to overcome this limitation. CoYangCZ does not solve the multivariate spatial interpolation problem from a purely statistical point of view but integrates geometry and statistics-based strategies. First, we used a weighted moving average method based on binomial coefficients (i.e. Yang-Chizhong filtering) to fit the spatial autocorrelation structure of each spatial variable from a geometric perspective. We then quantified the spatial autocorrelation of each spatial variable and the correlations between different spatial variables by analyzing the variances of different spatial variables. Finally, we obtain the best linear unbiased estimators at the unsampled locations. Experiments on air pollution and meteorological datasets show that CoYangCZ has a higher interpolation accuracy than cokriging, regression kriging, gradient plus-inverse distance squared, sequential Gaussian co-simulation, and the kriging convolutional network. CoYangCZ can adapt to second-order non-stationary spatial processes; therefore, it has a wider scope of application than purely statistical methods.

The Second Competition on Spatial Statistics for Large Datasets

In the last few decades, the size of spatial and spatio-temporal datasets in many research areas has rapidly increased with the development of data collection technologies. As a result, classical statistical methods in spatial statistics are facing computational challenges. For example, the kriging predictor in geostatistics becomes prohibitive on traditional hardware architectures for large datasets as it requires high computing power and memory footprint when dealing with large dense matrix operations. Over the years, various approximation methods have been proposed to address such computational issues, however, the community lacks a holistic process to assess their approximation efficiency. To provide a fair assessment, in 2021, we organized the first competition on spatial statistics for large datasets, generated by our ExaGeoStat software, and asked participants to report the results of estimation and prediction. Thanks to its widely acknowledged success and at the request of many participants, we organized the second competition in 2022 focusing on predictions for more complex spatial and spatio-temporal processes, including univariate nonstationary spatial processes, univariate stationary space-time processes, and bivariate stationary spatial processes. In this paper, we describe in detail the data generation procedure and make the valuable datasets publicly available for a wider adoption. Then, we review the submitted methods from fourteen teams worldwide, analyze the competition outcomes, and assess the performance of each team.

On Nonparametric Conditional Quantile Estimation for Non-stationary Random

A kernel conditional quantile estimate of a real-valued non-stationary spatial process is proposed for a prediction goal at a non-observed location of the underlying process. The originality is based on the ability to take into account some local spatial dependency. Large sample properties based on almost complete and \(L^q\)-consistencies of the estimator are established. A numerical study is given in order to illustrate the performance of our methodology.

Airbnb and crime in Barcelona (Spain): testing the relationship using a geographically weighted regression

ABSTRACT The existence of works proving the possible relationship empirically that Airbnb lodgings could have with crime in Spain is not known. This research analyzes the relationship between Airbnb lodgings and crimes against the properties and people in Barcelona’s neighbourhoods. To achieve this, we use an ordinary least squares regression model and a geographically weighted regression model. The results show a significant and positive relationship between the higher density of Airbnb lodgings and the higher crime rates in the neighbourhoods, especially of patrimonial nature. Divided by type of leased space, the Airbnb homes, in which the guest shares a room with other guests, show a higher relationship with crimes against property and people. The results of the local model show a spatial heterogeneity in all variables used, indicating the need to address non-stationary spatial processes that reveal hidden patterns. However, the only variable that shows statistically significant local variability is the total Airbnb lodgings variable. Finally, we discussed some unexpected results, proposing some future lines of research.

Durağan Olmayan Uzamsal Kovaryans Yapılar İçin Uyarlanabilir Modeller

Objective: In modeling 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 spatial domains, the spatial covariance of interest are always heterogeneous. However, this article proposed locally adaptive covariance for the spatial domain whose covariance is nonstationary in their spatial domain. The objectives of the study are to propose parametric, non-parametric and semi- parametric models for nonstationary spatial structure, continuous model for nonstationary spatial processes whose distance is far apart and to propose the adaptive weighting scheme approach that generates the optimal value for the nonparametric and semi-parametric models. Material and Methods: The spatial covariances are derived by applying the concept of adaptive weighting scheme approach on the covariance proposed in Nott and Dunsmuir (2002). Consequently, the local adaptive bandwidth for the nonstationary covariance was obtained for both the nonparametric and semi-parametric models. Simulations are conducted on the proposed model to examine the proposed model. Results and Conclusion: The results obtained are compared with existing models. The results indicate proposed spatial covariance are driven by the local bandwidths, penalty, weighted scheme, and tuning parameters. The adaptive models performed better in relation to existing covariances in terms of their mean square prediction errors (MSPE). The proposed models were further applied to real life Sulphate spatial data.

Sampling design optimisation for rainfall prediction using a non-stationary geostatistical model

The accuracy of spatial predictions of rainfall by merging rain-gauge and radar data is partly determined by the sampling design of the rain-gauge network. Optimising the locations of the rain-gauges may increase the accuracy of the predictions. Existing spatial sampling design optimisation methods are based on minimisation of the spatially averaged prediction error variance under the assumption of intrinsic stationarity. Over the past years, substantial progress has been made to deal with non-stationary spatial processes in kriging. Various well-documented geostatistical models relax the assumption of stationarity in the mean, while recent studies show the importance of considering non-stationarity in the variance for environmental processes occurring in complex landscapes. We optimised the sampling locations of rain-gauges using an extension of the Kriging with External Drift (KED) model for prediction of rainfall fields. The model incorporates both non-stationarity in the mean and in the variance, which are modelled as functions of external covariates such as radar imagery, distance to radar station and radar beam blockage. Spatial predictions are made repeatedly over time, each time recalibrating the model. The space-time averaged KED variance was minimised by Spatial Simulated Annealing (SSA). The methodology was tested using a case study predicting daily rainfall in the north of England for a one-year period. Results show that (i) the proposed non-stationary variance model outperforms the stationary variance model, and (ii) a small but significant decrease of the rainfall prediction error variance is obtained with the optimised rain-gauge network. In particular, it pays off to place rain-gauges at locations where the radar imagery is inaccurate, while keeping the distribution over the study area sufficiently uniform.

Bayesian geostatistical modelling for mapping schistosomiasis transmission

Progress has been made in mapping and predicting the risk of schistosomiasis using Bayesian geostatistical inference. Applications primarily focused on risk profiling of prevalence rather than infection intensity, although the latter is particularly important for morbidity control. In this review, the underlying assumptions used in a study mapping Schistosoma mansoni infection intensity in East Africa are examined. We argue that the assumption of stationarity needs to be relaxed, and that the negative binomial assumption might result in misleading inference because of a high number of excess zeros (individuals without an infection). We developed a Bayesian geostatistical zero-inflated (ZI) regression model that assumes a non-stationary spatial process. Our model is validated with a high-quality georeferenced database from western Côte d'Ivoire, consisting of demographic, environmental, parasitological and socio-economic data. Nearly 40% of the 3818 participating schoolchildren were infected with S. mansoni, and the mean egg count among infected children was 162 eggs per gram of stool (EPG), ranging between 24 and 6768 EPG. Compared to a negative binomial and ZI Poisson and negative binomial models, the Bayesian non-stationary ZI negative binomial model showed a better fit to the data. We conclude that geostatistical ZI models produce more accurate maps of helminth infection intensity than the spatial negative binomial ones.

An autoregressive point source model for spatial processes

We suggest a parametric modeling approach for nonstationary spatial processes driven by point sources. Baseline near-stationarity, which may be reasonable in the absence of a point source, is modeled using a conditional autoregressive (CAR) Markov random field. Variability due to the point source is captured by our proposed autoregressive point source (ARPS) model. Inference proceeds according to the Bayesian hierarchical paradigm, and is implemented using Markov chain Monte Carlo (MCMC) methods. The parametric approach allows a formal test of effectiveness of the point source. Application is made to a real dataset on electric potential measurements in a field containing a metal pole and the finding is that our approach captures the pole's impact on small-scale variability of the electric potential process.

Spatially-explicit risk profiling of Plasmodium falciparum infections at a small scale: a geostatistical modelling approach

BackgroundThere is a renewed political will and financial support to eradicate malaria. Spatially-explicit risk profiling will play an important role in this endeavour. Patterns of Plasmodium falciparum infection prevalence were examined among schoolchildren in a highly malaria-endemic area.MethodsA questionnaire was administered and finger prick blood samples collected from 3,962 children, aged six to 16 years, attending 55 schools in a rural part of western Côte d'Ivoire. Information was gathered from the questionnaire on children's socioeconomic status and the use of bed nets for the prevention of malaria. Blood samples were processed with standardized, quality-controlled methods for diagnosis of Plasmodium spp. infections. Environmental data were obtained from satellite images and digitized maps. Bayesian variogram models for spatially-explicit risk modelling of P. falciparum infection prevalence were employed, assuming for stationary and non-stationary spatial processes.FindingsThe overall prevalence of P. falciparum infection was 64.9%, ranging between 34.0% and 91.9% at the unit of the school. Risk factors for a P. falciparum infection included age, socioeconomic status, not sleeping under a bed net, distance to health care facilities and a number of environmental features (i.e. normalized difference vegetation index, rainfall and distance to rivers). After taking into account spatial correlation only age remained significant. Non-stationary models performed better than stationary models.ConclusionSpatial risk profiling of P. falciparum prevalence data provides a useful tool for targeting malaria control intervention, and hence will play a role in the quest of local elimination and ultimate eradication of the disease.

Diffusion-based spatial priors for functional magnetic resonance images

We recently outlined a Bayesian scheme for analyzing fMRI data using diffusion-based spatial priors [Harrison, L.M., Penny, W., Ashburner, J., Trujillo-Barreto, N., Friston, K.J., 2007. Diffusion-based spatial priors for imaging. NeuroImage 38, 677–695]. The current paper continues this theme, applying it to a single-subject functional magnetic resonance imaging (fMRI) study of the auditory system. We show that spatial priors on functional activations, based on diffusion, can be formulated in terms of the eigenmodes of a graph Laplacian. This allows one to discard eigenmodes with small eigenvalues, to provide a computationally efficient scheme. Furthermore, this formulation shows that diffusion-based priors are a generalization of conventional Laplacian priors [Penny, W.D., Trujillo-Barreto, N.J., Friston, K.J., 2005. Bayesian fMRI time series analysis with spatial priors. NeuroImage 24, 350–362]. Finally, we show how diffusion-based priors are a special case of Gaussian process models that can be inverted using classical covariance component estimation techniques like restricted maximum likelihood [Patterson, H.D., Thompson, R., 1974. Maximum likelihood estimation of components of variance. Paper presented at: 8th International Biometrics Conference (Constanta, Romania)]. The convention in SPM is to smooth data with a fixed isotropic Gaussian kernel before inverting a mass-univariate statistical model. This entails the strong assumption that data are generated smoothly throughout the brain. However, there is no way to determine if this assumption is supported by the data, because data are smoothed before statistical modeling. In contrast, if a spatial prior is used, smoothness is estimated given non-smoothed data. Explicit spatial priors enable formal model comparison of different prior assumptions, e.g., that data are generated from a stationary (i.e., fixed throughout the brain) or non-stationary spatial process. Indeed, for the auditory data we provide strong evidence for a non-stationary process, which concurs with a qualitative comparison of predicted activations at the boundary of functionally selective regions.

Variance modeling for nonstationary spatial processes with temporal replications

We have previously formulated a Bayesian approach to the Sampson and Guttorp model for the nonstationary correlation function r(x, x′) of a Gaussian spatial process [Damian et al., 2001]. This model assumes that the nonstationarity can be encoded through a bijective space deformation, f, that defines a new coordinate system in which the spatial correlation function can be considered isotropic, namely r(x, x′) = ρ(∥f(x) − f(x′)∥), where ρ belongs to a known parametric family. We extend this model to incorporate spatial heterogeneity in site‐specific temporal variances. In our Bayesian framework the variances are considered (hidden) realizations of another spatial process, which we model as log‐Gaussian, with correlation structure expressed in terms of the same spatial deformation function underlying that of the observed process. We demonstrate the method in simulations and in an application.

Interpolation of nonstationary air pollution processes: a spatial spectral approach

Spatial processes are important models for many environmental problems. Classical geostatistics and Fourier spectral methods are powerful tools for stuyding the spatial structure of stationary processes. However, it is widely recognized that in real applications spatial processes are rarely stationary and isotropic. Consequently, it is important to extend these spectral methods to processes that are nonstationary. In this work, we present some new spectral approaches and tools to estimate the spatial structure of a nonstationary process. More specifically, we propose an approach for the spectral analysis of nonstationary spatial processes that is based on the concept of spatial spectra, i.e., spectral functions that are space-dependent. This notion of spatial spectra generalizes the definition of spectra for stationary processes, and under certain conditions, the spatial spectrum at each Location can be estimated from a single realization of the spatial process.The motivation for this work is the modeling and prediction of ozone concentrations over different geopolitical boundaries for assessment of compliance with ambient air quality standards.

Spectral methods for nonstationary spatial processes

SUMMARY We propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process. We also study the asymptotic properties of the proposed estimators via shrinking asymptotics, assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound. With this type of asymptotic model we can uniquely determine the spectral density, avoiding the aliasing problem. We also present a new class of nonstationary processes, based on a convolution of local stationary processes. This model has the advantage that the model is simultaneously defined everywhere, unlike 'moving window' approaches, but it retains the attractive property that, locally in small regions, it behaves like a stationary spatial process. Applications include the spatial analysis and modelling of air pollution data provided by the US Environmental Protection Agency.

Ornstein–Uhlenbeck–Cauchy process

We combine earlier investigations of linear systems subject to Lévy fluctuations with recent attempts to give meaning to so-called Lévy flights in external force fields. We give a complete construction of the Ornstein–Uhlenbeck–Cauchy process as a fully computable paradigm example of Doob’s stable noise-supported Ornstein–Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, and relevant (pseudodifferential) evolution equations. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense.

Increment-Vector Methodology: Transforming Non-Stationary Series to Stationary Series

In time series analysis, it is well-known that the differencing operator ∇ d may transform a non-stationary series, {Z(t)} say, to a stationary one, {W(t)} = ∇ d Z(t)}; and there are many procedures for analysing and modelling {Z(t)} which exploit this transformation. Rather differently, Matheron (1973) introduced a set of measures on R n that transform an appropriate non-stationary spatial process to stationarity, and Cressie (1988) then suggested that specialized low-order analogues of these measures, called increment-vectors, be used in time series analysis. This paper develops a general theory of increment-vectors which provides a more powerful transformation tool than mere simple differencing. The methodology gives a handle on the second-moment structure and divergence behaviour of homogeneously non-stationary series which leads to many important applications such as determining the correct degree of differencing, forecasting and interpolation.

Increment-Vector Methodology: Transforming Non-Stationary Series to Stationary Series

In time series analysis, it is well-known that the differencing operator ∇d may transform a non-stationary series, {Z(t)} say, to a stationary one, {W(t)} = ∇dZ(t)}; and there are many procedures for analysing and modelling {Z(t)} which exploit this transformation. Rather differently, Matheron (1973) introduced a set of measures on Rn that transform an appropriate non-stationary spatial process to stationarity, and Cressie (1988) then suggested that specialized low-order analogues of these measures, called increment-vectors, be used in time series analysis. This paper develops a general theory of increment-vectors which provides a more powerful transformation tool than mere simple differencing. The methodology gives a handle on the second-moment structure and divergence behaviour of homogeneously non-stationary series which leads to many important applications such as determining the correct degree of differencing, forecasting and interpolation.

On second-order non-stationarity in two dimensionallattice processes

In this paper, we consider the problem of estimating the unknown parameters of non-stationary spatial process when only a single replication of the process is available. The approach suggested here to overcome the problem of estimating a number of parameters a lot larger than the number of observations, is based upon the general idea of resampling. A set of “moving samples” are considered in the study are and the unknown parameters are estimated within them. An application of the method to some agricultural field trials data is also considered in the paper.

Sull'analisi di serie spaziali non stazionarie. un metodo basato sul ricampionamento

In this paper we consider the problem of estimating the unknown parameters of non-stationary spatial process when only a single replication of the process is available. The approach suggested here, to overcome the problem of estimating a number of parameters a lot larger then the number of observation, is based upon the general idea of resampling. A set of moving subsample are considered in the study area and the unknown parameters are estimated within them. An application of the method to some agricultural field trials data is also considered in the paper.