The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets

Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.     

Towards spatial geochemical modelling: Use of geographically weighted regression for mapping soil organic carbon contents in Ireland

It is challenging to perform spatial geochemical modelling due to the spatial heterogeneity features of geochemical variables. Meanwhile, high quality geochemical maps are needed for better environmental management. Soil organic C (SOC) distribution maps are required for improvements in soil management and for the estimation of C stocks at regional scales. This study investigates the use of a geographically weighted regression (GWR) method for the spatial modelling of SOC in Ireland. A total of 1310 samples of SOC data were extracted from the National Soil Database of Ireland. Environmental factors of rainfall, land cover and soil type were investigated and included as the independent variables to establish the GWR model. The GWR provided comparable and reasonable results with the other chosen methods of ordinary kriging (OK), inverse distance weighted (IDW) and multiple linear regression (MLR). The SOC map produced using the GWR model showed clear spatial patterns influenced by environmental factors and the smoothing effect of spatial interpolation was reduced. This study has demonstrated that GWR provides a promising method for spatial geochemical modelling of SOC and potentially other geochemical parameters.     

Accounting for exhaustive secondary data into the mapping of water table elevation

This paper presents the incorporation of a digital elevation model into the spatial prediction of water table elevation in Mazandaran province (Iran) using a range of interpolation techniques. The multivariate methods used are: linear regression (LR), cokriging (COK), kriging with an external drift (KED) and regression kriging (RK). The analysis is performed on 3 years (1987, 1997 and 2007) of water table elevation data from about 260 monitoring wells. Prediction performances of the different algorithms are compared with two univariate techniques, i.e. inverse distance weighting and ordinary kriging (OK), through cross validation and examination of the consistency of the generated maps with the natural phenomena. Significantly smaller prediction errors are obtained for four multivariate algorithms but, in particular, KED and RK outperform LR and COK for 3 years. The results show the potential for using elevation for a more precise mapping of water table elevation.     

基于地质统计学的NDVI图像估值技术

将疏采样后的NDVI图像作为未受云层影响的已知数据,分别用普通克里格、泛克里格、指示克里格和序贯指示模拟对NDVI图像进行恢复并比较其效果。研究发现,各种克里格法对NDVI图像的估值效果由高到低依次为泛克里格、普通克里格、指示克里格,通常计算方便的普通克里格法就能够满足图像恢复所要求的精度;普通克里格方差和泛克里格方差只能反映数据的构型,不能很好地衡量估值图像的不确定性,指示克里格的条件方差的分布和实际误差的分布基本一致,能够较好地衡量估值图像的不确定性,并且其大小与NDVI影像数据的不确定性大小的分布一致。序贯指示模拟得到的多个等概率实现表现出很大的空间变异性,多个实现的均值图像光滑效应明显,估值精度不高,但是多个实现的方差分布可以很好地表征空间数据的不确定性分布。     

Comparison of cokriging and adaptive neuro-fuzzy inference system models for suspended sediment load forecasting

The present research was carried out by using artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS), cokriging (CK) and ordinary kriging (OK) using the rainfall and streamflow data for suspended sediment load forecasting. For this reason, the time series of daily rainfall (mm), streamflow (m³/s), and suspended sediment load (tons/day) data were used from the Kojor forest watershed near the Caspian Sea between 28 October 2007 and 21 September 2010 (776 days). Root mean square error, efficiency coefficient, mean absolute error, and mean relative error statistics are used for evaluating the accuracy of the ANN, ANFIS, CK, and OK models. In the first part of the study, various combinations of current daily rainfall, streamflow and past daily rainfall, streamflow data are used as inputs to the neural network and neuro-fuzzy computing technique so as to estimate current suspended sediment. Also, the accuracy of the ANN and ANFIS models are compared together in suspended sediment load forecasting. Comparison results reveal that the ANFIS model provided better estimation than the ANN model. In the second part of the study, the ANN and ANFIS models are compared with OK and CK. The comparison results reveal that CK was a better estimation than the OK. The ANFIS and ANN models also provided better estimation than the OK and CK models.     

Trend removal in spatially correlated datasets

Typically, datasets originated from mining exploration sites, industrially polluted and hazardous waste sites are correlated spatially over the region under investigation. Ordinary kriging (OK) is a well-established geostatistical tool used for predicting variables, such as precious metal contents, biomass, species counts, and environmental pollutants at unsampled spatial locations based on data collected from the neighboring sampled locations at these sites. One of the assumptions required to perform OK is that the mean of the characteristic of concern is constant for the entire region under consideration (e.g., there is no spatial trend present in the contaminant distribution across the site). This assumption may be violated by dalasets obtained from environmental applications. The occurrence of spatial trend in a dataset collected from a polluted site is an indication of the presence of two or more statistical populations (strata) with significantly different mean concentrations. Use of OK in these situations can result in inaccurate kriging estimates with higher SDs which, in turn, can lead to incorrect decisions regarding all subsequent environmental monitoring and remediation activities. A univariate and a multivariate approach have been described to identify spatial trend that may be present at the site. The trend then is removed by subtracting the respective means from the corresponding populations. The results of OK before and after trend removal are being compared. Using a real dataset, it is shown that standard deviations (SDs) of the kriging estimates obtained after trend removal are uniformly smaller than the corresponding SDs of the estimates obtained without the trend removal.     

Predicting Threshold Exceedance by Local Block Means in Soil Pollution Surveys

Soil contamination by heavy metals and organic pollutants around industrial premises is a problem in many countries around the world. Delineating zones where pollutants exceed tolerable levels is a necessity for successfully mitigating related health risks. Predictions of pollutants are usually required for blocks because remediation or regulatory decisions are imposed for entire parcels. Parcel areas typically exceed the observation support, but are smaller than the survey domain. Mapping soil pollution therefore involves a local change of support. The goal of this work is to find a simple, robust, and precise method for predicting block means (linear predictions) and threshold exceedance by block means (nonlinear predictions) from data observed at points that show a spatial trend. By simulations, we compared the performance of universal block kriging (UK), Gaussian conditional simulations (CS), constrained (CK), and covariance-matching constrained kriging (CMCK), for linear and nonlinear local change of support prediction problems. We considered Gaussian and positively skewed spatial processes with a nonstationary mean function and various scenarios for the autocorrelated error. The linear predictions were assessed by bias and mean square prediction error and the nonlinear predictions by bias and Peirce skill scores.     

不同采样点数量下土壤有机质含量空间预测方法对比

运用普通克里格、泛克里格、协同克里格和回归克里格4种方法,结合由DEM获取的高程因子以及土壤全氮和阳离子交换量（CEC）,预测了黑龙江省海伦市耕地有机质含量的空间分布。不同样点数量下海伦市土壤有机质含量的空间变异结构分析表明,样点数量多并不一定能够识别土壤有机质含量的结构性连续组分,最优化的布置采样点位置可能比单纯增加...     

Mapping soil electric conductivity using Bayesian kriging: A case study from Qasur,Pakistan

Quality of soil data is vital to formulate agricultural policies at different scales. Current agricultural applications in Pakistan depend however, on average values of soil estimates over larger areas. In this work, model-based ordinary kriging (OK) and Bayesian kriging (BK) to interpolate soil data is used. The aim is to compare the two different methods for the accuracy of soil data prediction. For this soils were sampled for Electrical Conductivity (EC, dS m ^–1) at 759 different locations in the rural agricultural areas of Qasur Tehsil, Pakistan. Cross validation was used to compare the performance of OK and BK. Our results show strong skewness and spatial dependency of soil EC values in heterogeneous regions. Box-Cox transformation successfully reduced the level of skewness in the soil EC data (from 14.1 to 0.11). Contrary to OK, under-estimation of soil EC values was not evident in the BK interpolation. Mean square prediction error for BK (1.45) was significantly reduced as compared to that for OK (6.1). Considering these findings, BK is a better model to explain the sub-regional soil EC variability and estimating strategies for sustainable agricultural planning in Pakistan.     

Examining the rainfall–topography relationship using non-stationary modelling technique in semi-arid Aseer region,Saudi Arabia

The mountainous region of Aseer, corresponding to the Afromontane phytogeographic region, is an eco-sensitive zone and has complex relationship between topography and rainfall. The region is located inland of the red sea escarpment edge in the west. Therefore, rainfall can occur during any month of the year in the mountain of the high Aseer region when moist air forces up the escarpment from the red sea. Monitoring the rainfall data and its topographical elevation variable in Aseer region is an essential requirement for feasible and accurate rainfall-based data for different applications, such as hydrological and ecological resource management in rugged terrain and remote areas. The relationship of elevation and rainfall are spatially non-stationary, non-linear, scale dependent, and often modelled by conventional regression models. Therefore, a local modelling technique, geographically weighted regression (GWR), was applied to deal with non-stationary, non-linear, scale-dependent problems. The GWR using topoclimatic data (elevation and rainfall) to analyse the cumulative rainfall data for rainy months (March to June) of the 4 years estimated from CHIRPS (Climate Hazards Group InfraRed Precipitation with Stations) product for Aseer region. The bandwidth (scale-size) of the Aseer region rainfall–elevation relationship has stabilised at round off 12 km. By selecting the suitable bandwidth, the spatial pattern of the rainfall–elevation relationship was significantly enhanced by using the GWR than the traditional ordinary least squares (OLS) regression model. GWR local modelling techniques estimated well in terms of accuracy, predictive power and decreased residual autocorrelation. Additionally, GWR assesses the significance of local statistic at each location and identified the location of spatial clusters with local regression coefficients significantly improved as compared with global OLS model, thereby highlighting local variations. Therefore, the GWR, local modelling approach managed to produce more accurate estimates by taking into account local characteristics.     

Indicator Kriging without Order Relation Violations

Indicator kriging (IK) is a spatial interpolation technique aimed at estimating the conditional cumulative distribution function (ccdf) of a variable at an unsampled location. Obtained results form a discrete approximation to this ccdf, and its corresponding discrete probability density function (cpdf) should be a vector, where each component gives the probability of an occurrence of a class. Therefore, this vector must have positive components summing up to one, like in a composition in the simplex. This suggests a simplicial approach to IK, based on the algebraic-geometric structure of this sample space: simplicial IK actually works with log-odds. Interpolated log-odds can afterwards be easily re-expressed as the desired cpdf or ccdf. An alternative but equivalent approach may also be based on log-likelihoods. Both versions of the method avoid by construction all conventional IK standard drawbacks: estimates are always within the (0,1) interval and present no order-relation problems (either with kriging or co-kriging). Even the modeling of indicator structural functions is clarified.     

Monotone Regression and Correction for Order Relation Deviations in Indicator Kriging

The indicator kriging (IK) is one of the most efficient nonparametric methods in geo-statistics. The order relation problem in the conditional cumulative distribution values obtained by IK is the most severe drawback of it. The correction of order relation deviations is an essential and important part of IK approach. A monotone regression was proposed as a new correction method which could minimize the deviation from original quintiles value, although, ensuring all order relations.     

Geostatistical approach for identification of transmissivity structure at Dulliu area in Taiwan

 A thorough understanding of the characteristics of transmissivity makes groundwater deterministic models more accurate. These transmissivity data characteristics occasionally possess a complicated spatial variation over an investigated site. This study presents both geostatistical estimation and conditional simulation methods to generate spatial transmissivity maps. The measured transmissivity data from the Dulliu area in Yun-Lin county, Taiwan, is used as the case study. The spatial transmissivity maps are simulated by using sequential Gaussian simulation (SGS), and estimated by using natural log ordinary kriging and ordinary kriging. Estimation and simulation results indicate that SGS can reproduce the spatial structure of the investigated data. Furthermore, displaying a low spatial variability does not allow the ordinary kriging and natural log kriging estimates to fit the spatial structure and small-scale variation for the investigated data. The maps of kriging estimates are smoother than those of other simulations. A SGS with multiple realizations has significant advantages over ordinary kriging and even natural log kriging techniques at a site with a high variation in investigated data. These results are displayed in geographic information systems (GIS) as basic information for further groundwater study. Received: 27 August 1999 · Accepted: 22 February 2000     

Simplicial Indicator Kriging

Indicator kriging (IK) is a spatial interpolation technique devised for estimating a conditional cumulative distribution function at an unsampled location. The result is a discrete approximation, and its corresponding estimated probability density function can be viewed as a composition in the simplex. This fact suggested a compositional approach to IK which, by construction, avoids all its standard drawbacks (negative predictions, not-ordered or larger than one). Here, a simple algorithm to develop the procedure is presented.     

Probabilistic assessment of the water quality of bathing beaches on the Taiwanese coast using indicator-based geostatistics with one-dimensional cyclic coordinates

Because Taiwan is a subtropical island, many pleasure beaches are situated on its coast. However, according to long-term monitoring data, fecal contamination at Taiwanese coastal beaches frequently exceeds the U.S. Environmental Protection Agency (EPA) guidelines. To avoid public health hazards, mapping the spatial extent of this contamination is crucial. This study applied indicator kriging (IK) to probabilistically assess the water quality of bathing beaches on the Taiwanese coast. Moreover, because the discontinuity of the traditional Cartesian coordinate established on an island coastline is difficult for geostatistical estimates, this study proposed a novel kriging estimation approach to deal with this problem. First, a one-dimensional (1-D) cyclic coordinate system of the Taiwanese coast was established using primary and secondary coordinates at each beach site. Escherichia coli (E. coli) and enterococci concentrations at coastal beaches were converted into indicator variables according to the U.S. EPA guidelines. IK was then used to spatially model the occurrence probabilities that exceeded the U.S. EPA guidelines for E. coli and enterococci. Finally, the water quality of bathing beaches on the Taiwanese coast was classified on the basis of the estimated probabilities. The study results indicated that bathing on the central western, northeastern, and southeastern Taiwanese coasts poses a potential threat to human health caused by high levels of fecal contamination. Moreover, primary and secondary coordinates established at beach sites were capable of analyzing the spatial variability and kriging estimates of the 1-D cyclic coordinates along the coastline.     

Quadratic covariance estimation and equivalence of predictions

This article illustrates the use of linear and nonlinear regression models to obtain quadratic estimates of covariance parameters. These models lead to new insights into the motivation behind estimation methods, the relationships between different methods, and the relationship of covariance estimation to prediction. In particular, we derive the standard estimating equations for minimum norm quadratic unbiased translation invariant estimates (MINQUEs) from an appropriate linear model. Connections between the linear model, minimum variance quadratic unbiased translation invariant estimates (MIVQUEs), and MINQUEs are examined and we provide a minimum norm justification for the use of one-step normal theory maximum likelihood estimates. A nonlinear regression model is used to define MINQUEs for nonlinear covariance structures and obtain REML estimates. Finally, the equivalence of predictions under various models is examined when covariance parameters are estimated. In particular, we establish that when using MINQUE, iterative MINQUE, or restricted maximum likelihood (REML) estimates, the choice between a stationary covariance function and an intrinsically stationary semivariogram is irrelevant to predictions and estimated prediction variances.     

Restricted kriging: A link between sample value and sample configuration

Restricted kriging provides a simple and quick remedy for the problem known as the weight independence of data in ordinary kriging. A major consequence of this problem is the effect of over-smearing in estimates, which, in turn, adds one uncertain factor to subsequent mine decisions. A detailed count is reported here on a restricted kriging system that incorporates two restrictions—one for high-grade samples and the other for low-grade samples. The restriction of high grade samples is because of their low priori likelihoods, whereas the main reason to restrict low grade samples is their nature as being waste and low analysis precisions. The two constraints reinforce each other in terms of enhancing the variables of estimates. A detailed case study on an epithermal gold deposit is carried out in terms of both cross validation and block modeling, showing that restricted kriging is superior over OK in mimicking the variables of original data.     

Addressing issues of spatial spillover effects and non-stationarity in analysis of residential burglary crime

The aim of this paper is to address two critical but largely neglected issues in the spatial analysis of urban crime which are spatial spillover effects of crime penetrating neighborhood boundaries and non-stationarity regarding the relationships between contextual factors and neighborhood crime. We use a GIS-based spatial approach to normalize the estimate of burglary crime at block group level and use the geographically weighted regression (GWR) to investigate the correlates of neighborhood crime. Results suggest that the use of normalized measure of neighborhood crime helps better reveal the spatial patterns of burglary crime and the use of GWR accounts for the spatial variations of relationships between contextual factors and crime. In particular, the normalized measure of crime has implications for improving the measurement accuracy of the risk of crime across urban neighborhoods and can be applied to the spatial analysis of other socioeconomic issues such as housing foreclosures and environmental hazards which are also plagued by the spatial spillover issue when geographically contiguous data are analyzed.     

Investigation of general regression neural network architecture for grade estimation of an Indian iron ore deposit

The mineral resource estimation requires accurate prediction of the grade at location from limited borehole information. It plays the dominant role in the decision-making process for investment and development of various mining projects and hence become an important and crucial stage. This paper evaluvates the use of two distinct artificial neural network (ANN)-based models, general regression neural network (GRNN) and multilayer perceptron neural network (MLP NN), to improve the grade estimation from Koira iron ore region in Sundargarh district, Odisha. ANN-based models capture the inherent complex structure of mineral deposits and provide a reliable generalization of the iron grade. The ANN-based approach does not require any preliminary geological study and is free from any statistical assumption on the raw data before its application. The GRNN is a one-pass learning algorithm and does not require any iterative procedure for training less complex structure and requires only one learning parameter for optimization. In this investigation, the spatial coordinates and multiple lithological units were taken as input variables and the iron grade was taken as the output variable. The comparative analysis of these models has been carried out and the results obtained were validated with traditional geostatistical method ordinary kriging (OK). The GRNN model outperforms the other methods, i.e. MLP and OK, with respect to generalization and predictability of the grades at an un-sampled location.