Unobserved Locations Research Articles

Predicting measurements at unobserved locations in an electrical transmission system

Electrical transmission systems consist of a huge number of locations (nodes) with different types of measurements available. Our aim is to derive a subset of nodes which contains almost sufficient information to describe the whole energy network. We derive a parameter set which characterises every single measuring location or node, respectively. Via analysing the behaviour of each node with respect to its neighbours, we construct a feasible random field metamodel over the whole transmission system. The metamodel is used to smooth the measurements across the network. In the next step we work with a subset of locations to predict the unobserved ones. We derive different graph kernels to define the missing covariance matrix from the neighbourhood structures of the network. This results in a metamodel that is able to smooth observed and predict unobserved locations in a spatial domain with non-isotropic distance functions.

Integration of Diverse Data Sources for Spatial PM2.5 Data Interpolation

Heterogeneous data fusion from disparate geospatial sensors has drawn increasing attention in multimedia. Unfortunately, environmental sensors are usually sparsely and preferentially located, which restricts situation recognition of geographical regions and results in uncertainty in derived inferences. Spatial interpolation is an effective way to solve the problem of data sparsity, which demands the availability of related data sources. However, these data sources are usually in different resolutions, distributions, scales, and densities, which poses a major challenge in data integration. To address this problem, we present a novel spatial interpolation framework to incorporate diverse data sources and model the spatial processes explicitly at multiple resolutions. Spectral analysis is deployed to generate features at multiple spatial resolutions and to improve the interpolation accuracy at unobserved locations. A statistical operator based on the spatial Gaussian process is implemented and integrated into a geospatial situation recognition system, which can analyze heterogeneous spatio-temporal data streams derived from sensors. To verify the effectiveness and efficiency of the proposed framework, this framework is applied to the PM2.5 air pollution application. Experiments conducted in California, USA, demonstrate that the proposed method outperforms state-of-the-art approaches.

Estimation of intensity-duration-frequency relationships according to the property of scale invariance and regionalization analysis in a Mediterranean coastal area

Usually, networks of daily rainfall raingauges have a higher spatial cover than tippet bucket raingauges networks. Consequently, it would be of high interest to make use of daily rainfall information to asses IDF curves for unobserved locations. The present work proposes achieving this goal by using the assumption of simple scaling invariance. Indeed, series observed over sufficiently long periods for 10 tippet bucket raingauge, allowed us to test the hypothesis of simple scaling of annual maximum rainfall intensities in northern Tunisia. This assumption, combined with Gumbel model of maximum rainfall intensities allowed us to develop a methodology to estimate IDF curves from the daily rainfall totals. In fact, a regionalization formula which involves the percentile 90% of the annual maximum daily rainfall was developed and validated. This regionalization formula applied to daily data of 25 rainfall stations in the sub area of Tunis region, combined with the assumption of simple scaling has enabled us to develop Intensity Duration Area Frequency (IDAF) curves for Tunis area.

Using compute unified device architecture-enabled graphic processing unit to accelerate fast Fourier transform-based regression Kriging interpolation on a MODIS land surface temperature image

Kriging interpolation provides the best linear unbiased estimation for unobserved locations, but its heavy computation limits the manageable problem size in practice. To address this issue, an efficient interpolation procedure incorporating the fast Fourier transform (FFT) was developed. Extending this efficient approach, we propose an FFT-based parallel algorithm to accelerate regression Kriging interpolation on an NVIDIA® compute unified device architecture (CUDA)-enabled graphic processing unit (GPU). A high-performance cuFFT library in the CUDA toolkit was introduced to execute computation-intensive FFTs on the GPU, and three time-consuming processes were redesigned as kernel functions and executed on the CUDA cores. A MODIS land surface temperature 8-day image tile at a resolution of 1 km was resampled to create experimental datasets at eight different output resolutions. These datasets were used as the interpolation grids with different sizes in a comparative experiment. Experimental results show that speedup of the FFT-based regression Kriging interpolation accelerated by GPU can exceed 1000 when processing datasets with large grid sizes, as compared to the traditional Kriging interpolation running on the CPU. These results demonstrate that the combination of FFT methods and GPU-based parallel computing techniques greatly improves the computational performance without loss of precision.

Estimation of primaries by sparse inversion with scattering-based multiple predictions for data with large gaps

We have solved the estimation of primaries by sparse inversion problem for a seismic record with large near-offset gaps and other contiguous holes in the acquisition grid without relying on explicit reconstruction of the missing data. Eliminating the unknown data as an explicit inversion variable is desirable because it sidesteps possible issues arising from overfitting the primary model to the estimated data. Instead, we have simulated their multiple contributions by augmenting the forward prediction model for the total wavefield with a scattering series that mimics the action of the free surface reflector within the area of the unobserved trace locations. Each term in this scattering series involves convolution of the total predicted wavefield once more with the current estimated Green’s function for a medium without the free surface at these unobserved locations. It is important to note that our method cannot by itself mitigate regular undersampling issues that result in significant aliases when computing the multiple contributions, such as source-receiver sampling differences or crossline spacing issues in 3D acquisition. We have investigated algorithms that handle the nonlinearity in the modeling operator due to the scattering terms, and we also determined that just a few of the terms can be enough to satisfactorily mitigate the effects of near-offset data gaps during the inversion process. Numerical experiments on synthetic data found that the final derived method can significantly outperform explicit data reconstruction for large near-offset gaps, with a similar computational cost and better memory efficiency. We have also found on real data that our scheme outperforms the unmodified primary estimation method that uses an existing Radon-based interpolation of the near-offset gap.

Bayesian spatial binary classification

In analyses of spatially-referenced data, researchers often have one of two goals: to quantify relationships between a response variable and covariates while accounting for residual spatial dependence or to predict the value of a response variable at unobserved locations. In this second case, when the response variable is categorical, prediction can be viewed as a classification problem. Many classification methods either ignore response-variable/covariate relationships and rely only on spatially proximate observations for classification, or they ignore spatial dependence and use only the covariates for classification. The Bayesian spatial generalized linear (mixed) model offers a tool to accommodate both spatial and covariate sources of information in classification problems. In this paper, we formally define spatial classification rules based on these models. We also take a close look at two of these models that have been proposed in the literature, namely the probit versions of the spatial generalized linear model (SGLM) and the Bayesian spatial generalized linear mixed model (SGLMM). We describe the implications of the seemingly slight differences between these models for spatial classification and explore the issue of robustness to model misspecification through a simulation study. We also provide an overview of alternatives to the SGLM/SGLMM-based classifiers and illustrate the various methods using satellite-derived land cover data from Southeast Asia.

Information-Driven Adaptive Sampling Strategy for Mobile Robotic Wireless Sensor Network

This brief addresses the issue of monitoring physical spatial phenomena of interest using information collected by a resource-constrained network of mobile, wireless, and noisy sensors that can take discrete measurements as they navigate through the environment. We first propose an efficient novel optimality criterion for designing a sampling strategy to find the most informative locations in taking future observations to minimize the uncertainty at all unobserved locations of interest. This solution is proven to be within bounds. The computational complexity of this proposition is shown to be practically feasible. We then prove that under a certain condition of monotonicity property, the approximate entropy at resulting locations obtained by our proposed algorithm is within $1-({1}/{e})$ of the optimum, which is then utilized as a stopping criterion for the sampling algorithm. The criterion enables the prediction results to be within user-defined accuracies by controlling the number of mobile sensors. The effectiveness of the proposed method is illustrated using a prepublished data set.

Very short-term irradiance forecasting at unobserved locations using spatio-temporal kriging

Several variants of spatio-temporal kriging are used to perform very short-term solar irradiance forecasting by utilizing data from a sensor network. Kriging can produce forecasts not only at the locations of the irradiance monitoring stations, but also at locations where sensors are not installed. Leave-one-out cross-validation is used to test the kriging performance at unobserved locations. Kriging weights are determined either empirically or using a correlation function. Four parametric correlation functions (correlograms) are herein considered, namely, separable, fully symmetric, and two polynomial-adjusted correlation functions. A dense 1km×1.2km network of 17 stations located on Oahu island, Hawaii, is used in this paper.We find that kriging based on a polynomial-adjusted correlation function (the best among the parametric models) is able to obtain forecast skill up to 0.43 and 0.36 for observed and unobserved locations respectively, for a forecast horizon of 50s. It is also found that empirical kriging performs better than parametric models at small forecast horizons (such as 30s). However, it loses accuracy for forecast horizons longer than 100s.

A kriging procedure for processes indexed by graphs

We provide a new kriging procedure of processes on graphs. Based on the construction of Gaussian random processes indexed by graphs, we extend to this framework the usual linear prediction method for spatial random fields, known as kriging. We provide the expression of the estimator of such a random field at unobserved locations as well as a control for the prediction error.

Animal movement constraints improve resource selection inference in the presence of telemetry error.

Multiple factors complicate the analysis of animal telemetry location data. Recent advancements address issues such as temporal autocorrelation and telemetry measurement error, but additional challenges remain. Difficulties introduced by complicated error structures or barriers to animal movement can weaken inference. We propose an approach for obtaining resource selection inference from animal location data that accounts for complicated error structures, movement constraints, and temporally autocorrelated observations. We specify a model for telemetry data observed with error conditional on unobserved true locations that reflects prior knowledge about constraints in the animal movement process. The observed telemetry data are modeled using a flexible distribution that accommodates extreme errors and complicated error structures. Although constraints to movement are often viewed as a nuisance, we use constraints to simultaneously estimate and account for telemetry error. We apply the model to simulated data, showing that it outperforms common ad hoc approaches used when confronted with measurement error and movement constraints. We then apply our framework to an Argos satellite telemetry data set on harbor seals (Phoca vitulina) in the Gulf of Alaska, a species that is constrained to move within the marine environment and adjacent coastlines.

A spatial model to examine rainfall extremes in Colorado’s Front Range

Summary Between 9th and 16th September 2013, northeast Colorado received some of its most extreme rainfall on record. The event affected 6 major rivers and their tributaries and 14 counties, breaking observed records for accumulations from sub-daily through to annual total. NOAA’s rainfall atlases indicated that this event had an anticipated return period of 1000 years. We use the rainfall that led to the 2013 Colorado floods as a case study in order to explore how a large event can affect the generalized extreme value (GEV) parameter estimates often used by designers and planners. We employ daily rainfall observations, with at least 30 years of data, from stations across Colorado’s Front Range of the Rocky Mountains to develop a spatial statistical model for annual maximum daily rainfall. We produce estimates of relatively rare events such as the 1% Annual Exceedance Probability (AEP) level and of extremely rare events such as the return period associated with Boulder’s 2013 observation. To explore sensitivity, we compare estimates including and excluding data from 2013, and both using only individual station data and our model which borrows strength across multiple stations. We compute the uncertainty associated with all of our estimates, and find large uncertainties associated with extremely rare events. Our statistical model is a spatial hierarchical model and we employ a two-stage approach for inference which can be implemented by practitioners. Additionally, the spatial model allows us to interpolate spatially and estimate the GEV parameters at unobserved locations. A further development of the model makes use of an alternatively defined space in terms of elevation and a climate variable, rather than geographical space defined by longitude and latitude, which seems to better account for orographic effects. In addition to producing AEP level and return period estimates to the annual maximum data, we investigate sensitivity to the choice of block length. We find point estimates indicate the tail to be much heavier when a longer block length is used, but the uncertainty associated with this parameter is such that one cannot say the difference is significant. To describe the spatial extent of severe storms, we also investigate the amount of data dependence between station locations. We find evidence in the record for storms with large spatial extent, although an extremal dependence parameter estimate indicates that this dependence is relatively weak.

Optimal sensor configuration for flexible structures with multi-dimensional mode shapes

A framework for deciding the optimal sensor configuration is implemented for civil structures with multi-dimensional mode shapes, which enhances the applicability of structural health monitoring for existing structures. Optimal sensor placement (OSP) algorithms are used to determine the best sensor configuration for structures with a priori knowledge of modal information. The signal strength at each node is evaluated by effective independence and modified variance methods. Euclidean norm of signal strength indices associated with each node is used to expand OSP applicability into flexible structures. The number of sensors for each method is determined using the threshold for modal assurance criterion (MAC) between estimated (from a set of observations) and target mode shapes. Kriging is utilized to infer the modal estimates for unobserved locations with a weighted sum of known neighbors. A Kriging model can be expressed as a sum of linear regression and random error which is assumed as the realization of a stochastic process. This study presents the effects of Kriging parameters for the accurate estimation of mode shapes and the minimum number of sensors. The feasible ranges to satisfy MAC criteria are investigated and used to suggest the adequate searching bounds for associated parameters. The finite element model of a tall building is used to demonstrate the application of optimal sensor configuration. The dynamic modes of flexible structure at centroid are appropriately interpreted into the outermost sensor locations when OSP methods are implemented. Kriging is successfully used to interpolate the mode shapes from a set of sensors and to monitor structures associated with multi-dimensional mode shapes.

Assessment of spatial models for interpolation of elevation in Pakistan

Elevation has major impact on the climate change. Interpolation of elevation at any location in Pakistan may be useful for predicting environmental parameters such as precipitation, temperature, humidity and wind speed. The locations with low elevations are more effecting global warming as compared with locations at high elevation. Present study interpolates the amount of elevation at unobserved locations using: 1) model-based ordinary kriging; 2) model-based Bayesian kriging with constant trend; 3) model-based Bayesian kriging with varying trend; 4) spatial artificial neural network. Prediction maps of elevation for complete domain are estimated along with prediction standard deviation. The results of suggested methods are compared with means of leave one take others cross validation method. It is observed from cross validation method that model-based Bayesian kriging with constant trend performs better than the other methods of predicting the amount of elevation in Pakistan.

Spatial Interpolation of Sulfate Concentration in Groundwater Including Covariates Using Bayesian Hierarchical Models

Sulfate is a key parameter for water quality and is commonly used in manufacturing of fertilizers, soaps, glass, papers, and common household items. If sulfate quantity is more than a threshold, it is hazardous for health. In the present paper, we use Bayesian kriging with external drift and Gaussian spatial predictive process model to analyze the spatial behavior of response variable (Sulfate). Different informative and non-informative priors are utilized to estimate the correlation parameters. The performance of these models are compared by means of twofold cross validation with deviance information criterion, and root mean square prediction as criterion. In summary, the inclusion of covariates plays an important role in minimizing the mean square prediction error. Bayesian kriging with external drift performs better than Gaussian spatial predictive process. The predictive distribution of Bayesian kriging with external drift is also applicable for interpolation of sulfate concentration at unobserved locations.

Dynamic spatial regression models for space‐varying forest stand tables

Many forest management planning decisions are based on information about the number of trees by species and diameter per unit area. This information is commonly summarized in astand table, where a stand is defined as a group of forest trees of sufficiently uniform species composition, age, condition, or productivity to be considered a homogeneous unit for planning purposes. Typically, information used to construct stand tables is gleaned from observed subsets of the forest selected using a probability‐based sampling design. Such sampling campaigns are expensive, and hence, only a small number of sample units are typically observed. This data paucity means that stand tables can only be estimated for relatively large areal units. Contemporary forest management planning and spatially explicit ecosystem models require stand table input at higher spatial resolution than can be affordably provided using traditional approaches. We propose a dynamic multivariate Poisson spatial regression model that accommodates both spatial correlation between observed diameter distributions and also correlation between tree counts across diameter classes within each location. To improve fit and prediction at unobserved locations, diameter specific intensities can be estimated using auxiliary data such as management history or remotely sensed information. The proposed model is used to analyze a diverse forest inventory dataset collected on the United States Forest Service Penobscot Experimental Forest in Bradley, Maine. Results demonstrate that explicitly modeling the residual spatial structure via a multivariate Gaussian process and incorporating information about forest structure from Light Detection and Ranging (LiDAR) covariates improve model fit and can provide high spatial resolution stand table maps with associated estimates of uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.

Accounting for the space‐varying nature of the relationships between temporal community turnover and the environment

Non‐spatial regression models are rarely adequate for exploring ecological phenomena, especially in settings where the processes operate at large spatial scales and when model covariates do not explain all variation present in the outcome variable. Given the complexity of ecological processes, it is often unrealistic to assume a set of stationary regression coefficients can capture space‐varying and scale‐dependent relationships between covariates and an ecological response. Spatially‐varying coefficients (SVC) models fit within a Bayesian inferential framework provide a statistically robust method to explore potential space‐varying and scale‐dependent impacts of covariates. Our study objective was to assess the utility of SVC models for capturing non‐stationary relationships between temporal community dynamics in avian assemblages and variation in environmental factors. We also wanted to compare the inference drawn from SVC models to that obtained from space‐varying intercept models and also models that do not acknowledge any spatial structure beyond what is introduced by the covariates. Our analysis examines the temporal turnover, expressed as a proportion, of avian communities across New York State, USA. Given the expected outcome is non‐Gaussian, we detail a generalized linear model specification of the proposed model structures. Our results show the SVC model outperformed the spatially‐varying intercept and non‐spatial models in terms of model fit and model predictive inference. Further, by fitting these models within a Bayesian inferential framework, we were able to make inferences about the spatial impact of covariates and other process parameters, as well as obtain full posterior predictive inference about the rate of turnover at new, unobserved locations. We conclude that SVC models provide a flexible framework for exploring and accounting for non‐stationary mechanisms driving ecological patterns.

Integration of Satellite Imagery, Topography and Human Disturbance Factors Based on Canonical Correspondence Analysis Ordination for Mountain Vegetation Mapping: A Case Study in Yunnan, China

The integration between vegetation data, human disturbance factors, and geo-spatial data (Digital Elevation Model (DEM) and image data) is a particular challenge for vegetation mapping in mountainous areas. The present study aimed to incorporate the relationships between species distribution (or vegetation spatial distribution pattern) and topography and human disturbance factors with remote sensing data, to improve the accuracy of mountain vegetation maps. Two different mountainous areas located in Lancang (Mekong) watershed served as study sites. An Artificial Neural Network (ANN) architecture classification was used as image classification protocol. In addition, canonical correspondence analysis (CCA) ordination was applied to address the relationships between topography and human disturbance factors with the spatial distribution of vegetation patterns. We used ordinary kriging at unobserved locations to predict the CCA scores. The CCA ordination results showed that the vegetation spatial distribution patterns are strongly affected by topography and human disturbance factors. The overall accuracy of vegetation classification was significantly improved by incorporating DEM or four CCA axes as additional channels in both the northern and southern study areas. However, there was no significant difference between using DEM or four CCA axes as extra channels in the northern steep mountainous areas because of a strong redundancy between CCA axes and DEM data. In the southern lower mountainous areas, the accuracy was significantly higher using four CCA axes as extra bands, compared to using DEM as an extra band. In the southern study area, the variance of vegetation data explained by human disturbance factors was larger than the variance explained by topographic attributes.

SSN: AnRPackage for Spatial Statistical Modeling on Stream Networks

The SSN package for R provides a set of functions for modeling stream network data. The package can import geographic information systems data or simulate new data as a ‘SpatialStreamNetwork’, a new object class that builds on the spatial sp classes. Functions are provided that fit spatial linear models (SLMs) for the ‘SpatialStreamNetwork’ object. The covariance matrix of the SLMs use distance metrics and geostatistical models that are unique to stream networks; these models account for the distances and topological configuration of stream networks, including the volume and direction of flowing water. In addition, traditional models that use Euclidean distance and simple random effects are included, along with Poisson and binomial families, for a generalized linear mixed model framework. Plotting and diagnostic functions are provided. Prediction (kriging) can be performed for missing data or for a separate set of unobserved locations, or block prediction (block kriging) can be used over sets of stream segments. This article summarizes the SSN package for importing, simulating, and modeling of stream network data, including diagnostics and prediction.

Spatial mixed effects model for compositional data with applications to coal geology

We analyze data on the geochemical make-up of coal samples throughout the state of Illinois. The goal is to estimate the geochemical properties at unobserved locations over a specified region. Multivariate spatial modeling requires characterization of both spatial and cross-spatial covariances. Reduced rank spatial models are popular in analyzing large spatial datasets. We develop a multivariate spatial mixed effects model for log-normal processes and show how to implement with compositional data to predict on point locations, as well as the average prediction over a finite area. We use log-normal kriging for the components of compositional data, and show how to obtain estimates and measures of precision in isometric log-ratio coordinates.

A dynamic nonstationary spatio-temporal model for short term prediction of precipitation

Precipitation is a complex physical process that varies in space and time. Predictions and interpolations at unobserved times and/or locations help to solve important problems in many areas. In this paper, we present a hierarchical Bayesian model for spatio-temporal data and apply it to obtain short term predictions of rainfall. The model incorporates physical knowledge about the underlying processes that determine rainfall, such as advection, diffusion and convection. It is based on a temporal autoregressive convolution with spatially colored and temporally white innovations. By linking the advection parameter of the convolution kernel to an external wind vector, the model is temporally nonstationary. Further, it allows for nonseparable and anisotropic covariance structures. With the help of the Voronoi tessellation, we construct a natural parametrization, that is, space as well as time resolution consistent, for data lying on irregular grid points. In the application, the statistical model combines forecasts of three other meteorological variables obtained from a numerical weather prediction model with past precipitation observations. The model is then used to predict three-hourly precipitation over 24 hours. It performs better than a separable, stationary and isotropic version, and it performs comparably to a deterministic numerical weather prediction model for precipitation and has the advantage that it quantifies prediction uncertainty.