Spatial Stochastic Process Research Articles

The Many Forms of Co-kriging: A Diversity of Multivariate Spatial Estimators

AbstractIn this expository review paper, we show that co-kriging, a widely used geostatistical multivariate optimal linear estimator, has a diverse range of extensions that we have collected and illustrated to show the potential of this spatial interpolator. In the context of spatial stochastic processes, this paper covers scenarios including increasing the spatial resolution of a spatial variable (downscaling), solving inverse problems, estimating directional derivatives, and spatial interpolation taking boundary conditions into account. All these spatial interpolators are optimal linear estimators in the sense of being unbiased and minimising the variance of the estimation error.

Time-dependent performance assessment of mountain tunnels considering the hazards associated with squeezing soft rock and nonuniform steel corrosion in RC lining structure

Mountain tunnels buried in squeezing soft rock inevitably undergo a rapid degradation with time due to creep behaviors of surrounding rock. The performance of tunnel would become more exacerbated when steel corrosion occurs in lining structures. High failure risk of tunnel lining within a short time after construction would significantly shorten its service life. Since the properties of rock mass in space and the corrosion of reinforcements in space–time exhibit a high uncertainty in practices, it is necessary to perform a reliable performance assessment of tunnels by incorporating the concept of random field-based reliability analysis. In this study, a novel approach for reliability assessment of mountain tunnels under the hazards associated with creep behavior and non-uniform steel corrosion is presented. A creep constitutive model of rock is introduced firstly; Random field-based modeling for the spatial-variant properties of rock mass is proposed; and a spatio-temporal approach for describing the spatial variability and stochastic process of steel corrosion is established. Numerical simulation for the deteriorated mountain tunnel is implemented by integrating the FLAC3D with MATLAB. In an illustrative example, both deterministic and probabilistic analysis on structural performance were conducted. The time-dependent deformation, axial forces and bending moments of lining structure are investigated.

Periodicity and Variability in Daily Activity Satisfaction: Toward a Space-Time Modeling of Subjective Well-Being

Understanding how geographical environments influence peoples’ subjective experiences of daily activities is of great potential for improving subjective well-being (SWB), a subject that is presently limited by a lack of available data and proper statistical methods. Focusing on Beijing and using a unique data set that linked residents’ seven-day mobility trajectories at a fine spatiotemporal resolution to their complete activity participation and momentary well-being, this article investigates the temporal dynamics of and geographical contextual effects on SWB of daily activities. We developed a unified spatial multilevel stochastic process model to simultaneously capture periodicity, stochastic dynamics, and individual heterogeneity effects. Results show that momentary SWB has a twenty-four-hour periodicity and evolves stochastically around individual equilibrium states depending on key life-circumstance variables. Migrants and low-income residents tend to have lower equilibrium states than their counterparts. Real-time air pollution exposure significantly lowers daily activity satisfaction levels, and an inverted-U-shaped relationship exists between city vibrancy and satisfaction.

Sedimentary, littoral and planktic diatoms show different diversity patterns and assembly mechanisms in mountain lakes of the northern European Alps

This study analysed diversity patterns of sedimentary, littoral and planktic diatoms in 43 mountain lakes in the northern European Alps and identified processes that contribute to these patterns. Linear regression models showed a significant increase of sedimentary α-diversity with lake area and conductivity and a negative trend with increasing elevation, whilst the littoral diatom α-diversity increased significantly with conductivity and lake water temperature. Planktic diatom α-diversity significantly decreased with lake area and depth. August water temperature, total phosphorus, conductivity and lake depth explained a significant part of the variation and were significantly correlated with pairwise β-diversities in the data sets, but spatial and shared effects of space and environment were more important for planktic and littoral diatoms. A null model approach based on assemblages’ dissimilarities revealed that the structure of littoral and planktic assemblages was predominantly stochastic. In contrast, sedimentary diatoms were formed by both deterministic and stochastic processes. Abundant and widespread species contributed a large part to the assemblage β-diversity. The results point to a stronger role of niche assembly in sedimentary than for littoral and planktic diatoms. Dispersal limitation, in turn, is likely to contribute to the spatial patterns and stochastic assembly processes observed for littoral and planktic diatoms.

Distribution-free P-box processes based on translation theory: Definition and simulation

Typically, non-deterministic models of spatial or time dependent uncertainty are modelled using the well-established random field framework. However, while tailored for exactly these types of time and spatial variations, stochastic processes and random fields currently have only limited success in industrial engineering practice. This is mainly caused by its computational burden, which renders the analysis of industrially sized problems very challenging, even when resorting to highly efficient random field analysis methods such as EOLE. Apart from that, also the methodological complexity, high information demand and rather indirect control of the spatial (or time) variation has limited its cost–benefit potential for potential end-users. This data requirement was recently relaxed by some of the authors with the introduction of imprecise random fields, but so far the method is only applicable to parametric p-box valued stochastic processes and random fields. This paper extends these concepts by expanding the framework towards distribution-free p-boxes. The main challenges addressed in this contribution are related to both the non-Gaussianity of realisations of the imprecise random field in between the p-box bounds, as well as maintaining the imposed auto-correlation structure while sampling from the p-box. Two case studies involving a dynamical model of a car suspension and the settlement of an embankment are included to illustrate the presented concepts.

Correlation of variations in species abundance of Atlantic forests regenerating on abandoned pastures with different environmental and spatial variables

The conversion of native forests into pastures is still a common practice in Brazil. Abandoned pastures have great potential for natural regeneration and therefore could play an important role in meeting the enormous demand for forest restoration. Few studies, however, have investigated the extent to which spatially-structured environmental variables and community structure are correlated with the variation in species abundance of regenerating forests on abandoned pastures. Therefore, we aimed to determine whether environmental and spatial variables were capable of explaining the variation in abundance of woody species on abandoned pastures in the subtropical Atlantic Forest. We systematically distributed 45 sample plots with size and inclusion criteria that changed according to the vegetation layer in three different abandoned pastures. In general, most of the variation in species abundance that our models were able to explain was correlated with spatially-unstructured physical-chemical soil properties. A smaller part of the variation was correlated with spatially-structured soil variables and topography-related variables. An even smaller portion of the variation was spatially-structured but was not correlated with spatially-structured environmental variables. Therefore, our results suggest that the variation in species abundance of regenerating subtropical Atlantic forests on abandoned pastures is more closely related to niche-based processes mediated by environmental variables than to stochastic spatial processes.

Responses of a soil mollusk community to an extreme cold event

Soil communities are impacted by increasingly frequent and intense climate extremes. To accurately predict the effects of these events on soil communities and effectively design recovery and management programs for soil biodiversity, it is necessary to understand the spatiotemporal patterns and mechanisms underlying the soil community responses. Here, we investigated a mangrove soil macrofaunal (mollusk) community before and after an extreme cold event (ECE) to determine the effects of the ECE on the spatial pattern, temporal succession, and deterministic and stochastic processes affecting the soil metacommunity. We used distance-decay and time-decay analyses to characterize the spatial pattern and temporal succession of soil mollusk community dissimilarity (e.g., species identity, abundance, and biomass) across a reginal scale (~350 km scale). We used null models to compare the relative importance of deterministic and stochastic processes in shaping the regional soil mollusk community assembly. The ECE resulted in a reorganization of the soil mollusk metacommunity reflected by the mortality and extinction of several original species, colonization by new opportunistic species, and a significant reduction in mollusk abundance and biomass. The ECE increased divergent temporal succession but decreased spatial segregation in the soil mollusks, consistent with changes in the community dissimilarity of species abundance, richness, and biomass. The ECE increased the numbers of non-random segregated and random structures of soil mollusks while decreased the number of non-random aggregated communities. Notably, non-random segregated communities were the most strongly affected. Thus, the response of the soil mollusk metacommunity to the ECE was mediated by an increase in limiting similarity processes and stochastic processes and a decrease in limiting dissimilarity processes. The dominant limiting similarity processes involved species competition and/or dispersal limitation. More broadly, our study indicated that alterations in the spatiotemporal dynamics of soil fauna in response to climate change should be studied at the metacommunity scale. These spatiotemporal dynamics should be further tested in an ecological processes framework to reliably evaluate soil community responses to climate change and to devise management strategies for global soils.

Reduced-bias estimation of spatial autoregressive models with incompletely geocoded data

The application of spatial Cliff–Ord models requires information about spatial coordinates of statistical units to be reliable, which is usually the case in the context of areal data. With micro-geographic point-level data, however, such information is inevitably affected by locational errors, that can be generated intentionally by the data producer for privacy protection or can be due to inaccuracy of the geocoding procedures. This unfortunate circumstance can potentially limit the use of the spatial autoregressive modelling framework for the analysis of micro data, as the presence of locational errors may have a non-negligible impact on the estimates of model parameters. This contribution aims at developing a strategy to reduce the bias and produce more reliable inference for spatial models with location errors. The proposed estimation strategy models both the spatial stochastic process and the coarsening mechanism by means of a marked point process. The model is fitted through the maximisation of a doubly-marginalised likelihood function of the marked point process, which cleans out the effects of coarsening. The validity of the proposed approach is assessed by means of a Monte Carlo simulation study under different real-case scenarios, whereas it is applied to real data on house prices.

Shape-constrained Gaussian process regression for surface reconstruction and multimodal, non-rigid image registration

We present a new statistical framework for landmark ?>curve-based image registration and surface reconstruction. The proposed method first elastically aligns geometric features (continuous, parameterized curves) to compute local deformations, and then uses a Gaussian random field model to estimate the full deformation vector field as a spatial stochastic process on the entire surface or image domain. The statistical estimation is performed using two different methods: maximum likelihood and Bayesian inference via Markov Chain Monte Carlo sampling. The resulting deformations accurately match corresponding curve regions while also being sufficiently smooth over the entire domain. We present several qualitative and quantitative evaluations of the proposed method on both synthetic and real data. We apply our approach to two different tasks on real data: (1) multimodal medical image registration, and (2) anatomical and pottery surface reconstruction.

Geostatistical modeling of positive-definite matrices: An application to diffusion tensor imaging.

Geostatistical modeling for continuous point-referenced data has extensively been applied to neuroimaging because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging technique characterizing the brain's anatomical structure, produces a positive-definite (p.d.) matrix for each voxel. Currently, only a few geostatistical models for p.d. matrices have been proposed because introducing spatial dependence among p.d. matrices properly is challenging. In this paper, we use the spatial Wishart process, a spatial stochastic process (random field), where each p.d. matrix-variate random variable marginally follows a Wishart distribution, and spatial dependence between random matrices is induced by latent Gaussian processes. This process is valid on an uncountable collection of spatial locations and is almost-surely continuous, leading to a reasonable way of modeling spatial dependence. Motivated by a DTI data set of cocaine users, we propose a spatial matrix-variate regression model based on the spatial Wishart process. A problematic issue is that the spatial Wishart process has no closed-form density function. Hence, we propose an approximation method to obtain a feasible Cholesky decomposition model, which we show to be asymptotically equivalent to the spatial Wishart process model. A local likelihood approximation method is also applied to achieve fast computation. The simulation studies and real data application demonstrate that the Cholesky decomposition process model produces reliable inference and improved performance, compared to othermethods.

Estimation the Variogram Function Indicator which represent the Transmissivity Coefficient in the groundwater

The problem tackled in this paper is the estimation of variogram function Indicator of spatial stochastic process for the Levels of groundwater, by the method of weighted Least squares. This methods is well known in regression analysis in estimating the coefficient of ression model. After defining the indicator variable the parameters of Indicator variogram estimated based on mean squares error. The final formula of weighted least squares estimator can be not be solved exactly, then through the use of iterative Newten - Raphson algorithm and for some iterations the convergence of solution is obtained with certain termination criterion or number of repeats (that used in this paper).

Analysis of the atmospheric macro-physical using spatial methods

The central western area of Venezuela has an unequal distribution of precipitation. Due to its agricultural importance, is necessary to plan water accounting and this requires a evaluation of spatial and temporal variability of precipitation and an estimate of local geophysical effect from the relief. In this research we use an iterative computationally lattice approach to perform a confirmatory analysis of the variability and the spatial correlation structure in monthly precipitation stations. Spatial correlograms and pooled empirical semivariogram were applied to evaluate the most appropriate spatial weighting matrix to estimate the Moran’s I. The altitude effect over monthly rainfall was estimated through spatial regression algorithm which determine the predominant spatial process in each slice. A homogeneous spatial stochastic process with positive spatial autocorrelation is evidenced. There is a trend towards a higher frequency of spatial error and spatial auto-regressive processes between the months of June and August whilst there are not dominant process between October and December. This response is caused by the dynamics of the intertropical convergence zone, which generates a seasonal effect on precipitation. These estimations allows decision-making in modeling and will lead to an improvement for analysis and forecasting in areas strongly affected by climate change and water stress.

Estimating Exceedance Probabilities of Railway Bridge Vibrations in the Presence of Random Rail Irregularities

This contribution addresses the estimation of exceedance probabilities of the dynamic random response of railway bridges subjected to high-speed trains in the presence of random rail irregularities. The random nature of the irregular rail track is described by a spatial ergodic stochastic process, and consequently the dynamic bridge response becomes a stochastic process in time with generally unknown distributions. Using numerical simulation methods, the response thresholds for bridge deflection and acceleration are estimated to obtain small exceedance probabilities. Combining these limits with the response at perfect rail geometry provides an estimate of the dynamic response amplification due to random rail irregularities. This is in line with the semi-probabilistic safety concept of modern civil engineering, where critical response thresholds for structures are associated with small exceedance probabilities. It is shown that modeling the maximum bridge deflection as a normally distributed random variable with parameters fitted to the results of a Monte Carlo simulation with small sample size is a computationally efficient approach for estimating the amplified deflection. In contrast, the random maximum bridge acceleration is better captured by a lognormal distribution. As an efficient alternative, the subset simulation method provides accurate predictions for very small exceedance probabilities. If the amplitudes of the rail irregularities at discrete spatial coordinates along the rail axis are considered as random variables, the stability of subset simulation increases.

Prediction of tensile response of UHPC with aligned and ZnPh treated steel fibers based on a spatial stochastic process

This paper proposed a new method for predicting tensile response of UHPC based on a spatial stochastic strength distribution model. The steel fibers in UHPC were aligned by a developed device and chemically treated by ZnPh. A direct tension test was conducted to obtain the tensile response of UHPC with treated fibers. Image analysis was used to obtain the frequency of overall fiber orientation distribution. The spatial stochastic distribution model was proposed for cross sectional strength of UHPC based on the Ornstein-Uhlenbeck stochastic process. The model introduced some assumptions such as the correlation coefficient between the cracking strength and fiber bridging strength. Parametric analysis was conducted to obtain the crucial stochastic parameters in the model. Combining with the single fiber pullout model, tensile behavior of a single crack, and minimum crack spacing, the model predicted the strain hardening and multiple cracking behaviors with precise accuracy, as compared to experimental results.

Measuring, mapping, and uncertainty quantification in the space-time cube

The space-time cube is not a cube of course, but the idea of one is useful. Its base is a spatial domain, $$D_t$$ , and the “cube” is traced out by a process of spatial domains, $$\{D_t:t\ge 0\}$$ . Now fill the cube with a spatio-temporal stochastic process $$\{Y_t(\mathbf{s} ):\mathbf{s} \in D_t,t\ge 0\}$$ . Assume that $$\{D_t\}$$ is fixed and known (but clearly it too could be stochastic). Slicing the cube laterally for a fixed $$t_0$$ generates a spatial stochastic process $$\{Y_{t_0}(\mathbf{s} ):\mathbf{s} \in D_{t_0}\}$$ . Slicing the cube longitudinally for a fixed $$\mathbf{s} _0$$ generates a temporal process $$\{Y_t(\mathbf{s} _0):t\ge 0\}$$ that, after dicing, yields a time series, $$\{Y_0(\mathbf{s} _0),Y_1(\mathbf{s} _0),\ldots \}$$ . These are the main highways that traverse the cube but other, less-traveled paths, can be taken. In this paper, we discuss spatio-temporal data and processes whose domain is the space-time cube, and we incorporate them into hierarchical statistical models for spatio-temporal data.

Aridity Trends in Central America: A Spatial Correlation Analysis

Trend analyses are common in several types of climate change studies. In many cases, finding evidence that the trends are different from zero in hydroclimate variables is of particular interest. However, when estimating the confidence interval of a set of hydroclimate stations or gridded data the spatial correlation between can affect the significance assessment using for example traditional non-parametric and parametric methods. For this reason, Monte Carlo simulations are needed in order to generate maps of corrected trend significance. In this article, we determined the significance of trends in aridity, modeled runoff using the Variable Infiltration Capacity Macroscale Hydrological model, Hagreaves potential evapotranspiration (PET) and near-surface temperature in Central America. Linear-regression models were fitted considering that the predictor variable is the time variable (years from 1970 to 1999) and predictand variable corresponds to each of the previously mentioned hydroclimate variables. In order to establish if the temporal trends were significantly different from zero, a Mann Kendall and a Monte Carlo test were used. The spatial correlation was calculated first to correct the variance of each trend. It was assumed in this case that the trends form a spatial stochastic process that can be modeled as such. Results show that the analysis considering the spatial correlation proposed here can be used for identifying those extreme trends. However, a set of variables with strong spatial correlation such as temperature can have robust and widespread significant trends assuming independence, but the vast majority of the stations can still fail the Monte Carlo test. We must be vigilant of the statistically robust changes in key primary parameters such as temperature and precipitation, which are the driving sources of hydrological alterations that may affect social and environmental systems in the future.

Prediction of Stochastic Spatial Processes Applied to the Water of Wells

Prediction of Stochastic Spatial Processes Applied to the Water of Wells

Uncovering the relative influences of space and environment in shaping the biogeographic patterns of mangrove mollusk diversity

AbstractDetermining the relative importance of ecological processes regulating the biogeographic patterns of marine species, especially with respect to α- and β-diversity in multi-habitat communities, is a central goal in marine ecology. We explored the relative contribution of spatial (stochastic processes) and environmental factors (deterministic processes) to the biogeographic patterns of the α- and β-diversity of mangrove mollusks. A total of 16 mangrove areas were sampled in southeast coast China from 18°N to 28°N latitude. The highest mean α-diversity was found at 20°N and that of β-diversity was at 21°N. Both spatial and environmental factors had significant effects on the α- and β-diversity patterns. The environments had greater effects than the spaces on shaping the α-diversity pattern, while the spaces were relatively more important in governing the β-diversity patterns than the environments. Our results suggest that the α-diversity pattern was mainly controlled by deterministic processes (environmental filtering), while β-diversity was primarily shaped by stochastic processes (dispersal-related), although both processes had significant impacts on α- and β-diversity patterns. Identifying the ecological variables and mechanisms that drive variations in α- and β-diversity may help guide the conservation for biodiversity in endangered mangrove ecosystems under anthropogenic and global changes.

Hybrid simulation scheme for volatility modulated moving average fields

We develop a simulation scheme for a class of spatial stochastic processes called volatility modulated moving averages. A characteristic feature of this model is that the behaviour of the moving average kernel at zero governs the roughness of realisations, whereas its behaviour away from zero determines the global properties of the process, such as long range dependence. Our simulation scheme takes this into account and approximates the moving average kernel by a power function around zero and by a step function elsewhere. For this type of approach, Bennedsen et al. (2017), who considered an analogous model in one dimension, coined the term hybrid simulation scheme. We derive the asymptotic mean square error of the simulation scheme and compare it in a simulation study with several other simulation techniques and exemplify its favourable performance in a simulation study.

Spatial-Temporal Modelling of Temperature for Pricing Temperature Index Insurance

This paper discusses the pricing methodology of the temperature index insurance based on spatial temporal modelling of temperature. The crucial problem here is the location of the potential insurance buyer relative to the station where index is calculated. Since the observed temperatures at particular station are not always correlated to the temperature where the insurance holder lives, it is important to consider spatial issues in the pricing methodology. Thus, we model the temperature using spatial temporal stochastic processes and employ the universal Kriging method to predict the future temperature at some specific locations. Based on temperature index, we may then price the temperature insurance. We illustrate the pricing methodology using 20 years data from five stations in Malaysia. The findings are important for the development of weather index insurance.