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In studies that produce data with spatial structure, it is common that covariates of interest vary spatially in addition to the error. Because of this, the error and covariate are often correlated. When this occurs, it is difficult to distinguish the covariate effect from residual spatial variation. In an i.i.d. normal error setting, it is well known that this type of correlation produces biased coefficient estimates, but predictions remain unbiased. In a spatial setting, recent studies have shown that coefficient estimates remain biased, but spatial prediction has not been addressed. The purpose of this paper is to provide a more detailed study of coefficient estimation from spatial models when covariate and error are correlated and then begin a formal study regarding spatial prediction. This is carried out by investigating properties of the generalized least squares estimator and the best linear unbiased predictor when a spatial random effect and a covariate are jointly modelled. Under this setup, we demonstrate that the mean squared prediction error is possibly reduced when covariate and error are correlated.

PM10 prediction has attracted special legislative and scientific attention due to its harmful effects on human health. Statistical techniques have the potential for high-accuracy PM10 prediction and accordingly, previous studies on statistical methods for temporal, spatial and spatio-temporal prediction of PM10 are reviewed and discussed in this paper. A review of previous studies demonstrates that Support Vector Machines, Artificial Neural Networks and hybrid techniques show promise for suitable temporal PM10 prediction. A review of the spatial predictions of PM10 shows that the LUR (Land Use Regression) approach has been successfully utilized for spatial prediction of PM10 in urban areas. Of the six introduced approaches for spatio-temporal prediction of PM10, only one approach is suitable for high-resolved prediction (Spatial resolution < 100 m; Temporal resolution ≤ 24 h). In this approach, based upon the LUR modeling method, short-term dynamic input variables are employed as explanatory variables alongside typical non-dynamic input variables in a non-linear modeling procedure.

Haemonchus contortus is a gastrointestinal nematode parasite of small ruminants, which feeds on blood and causes significant disease and production loss in sheep and goats, especially in warmer parts of the world. The life cycle includes free-living immature stages, which are subject to climatic influences on development, survival and availability, and this species therefore exhibits spatio-temporal heterogeneity in its infection pressure based on the prevailing climate. Models that better explain this heterogeneity could predict future epidemiological changes. The basic reproduction quotient (Q0) was used as a simple process-based model to predict climate-driven changes in the potential transmission of H. contortus across widely different geo-climatic zones, and showed good agreement with the observed frequency of this species in the gastrointestinal nematode fauna of sheep (r = 0.81, P <0.01). Averaged monthly Q0 output was further used within a geographical information system (GIS) to produce preliminary haemonchosis risk maps for the United Kingdom (UK) over a four-year historical span and under future climate change scenarios. Prolonged transmission seasons throughout the UK are predicted, especially in the south although with restricted transmission in peak summer due to rainfall limitation. Additional predictive ability might be achieved if information such as host density and distribution, grazing pattern and edaphic conditions were included as risk layers in the GIS-based risk map. However, validation of such risk maps presents a significant challenge, with georeferenced observed data of sufficient spatial and temporal resolution rarely available and difficult to obtain.

There is no standard methodology which allows the incorporation of geological information into digital soil mapping (DSM) despite the great potential of geology as environmental covariate in DSM. To All this gap, in this study, a geochemical parent material classification scheme was tested on the watershed area of Lake Balaton, for which soil maps at a finer scale have not yet been created. A parent material map was prepared on the basis of a 1:100 000 surface geology map in order to make the incorporation of soil modelling and mapping possible. Legacy data of 12400 soil sample points was used in order to examine the role of geology in the quantitative distribution of some soil properties and element content (liquid limit, soil organic carbon, pH(KCL), CaCO3, Mg, Cu, Zn, Mn). Results confirm that the SiO2 content of the parent material influences the properties of the derived soils. In the second part of the study Random Forest models were developed for three major soil properties (liquid limit, soil organic carbon, pH) with the use of additional environmental covariates: elevation, slope, aspect, curvature, topographic position index (TPI), annual average temperature, annual average precipitation, remote sensing based normalized difference vegetation index (NDVI) and land cover information. The performance and accuracy of the models were evaluated on the basis of the coefficient of determination (R2) and root mean square error (RMSE), calculated on a randomly selected validation dataset (20% of the database). The models performed with R2 values of 0.72, 0.6 and 0.68 for liquid limit, soil organic carbon and pH respectively. The importance of variables was also examined in the RF models, and this demonstrated that while geology is among the best-performing predictors, in neither case is it the most important variable. Ninety metre resolution maps of the three major soil properties were compiled by making spatial predictions with the RF models developed here. For validation of the maps, an independent soil database was used, which showed that the prediction performed well on the cultivated area where the concordance correlation coefficients (CCC) were 0.73, 0.73 and 0.69 for liquid limit, pH and soil organic carbon respectively.

Spatial predictions of soil macro and micro-nutrient content across Sub-Saharan Africa at 250 m spatial resolution and for 0–30 cm depth interval are presented. Predictions were produced for 15 target nutrients: organic carbon (C) and total (organic) nitrogen (N), total phosphorus (P), and extractable—phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg), sulfur (S), sodium (Na), iron (Fe), manganese (Mn), zinc (Zn), copper (Cu), aluminum (Al) and boron (B). Model training was performed using soil samples from ca. 59,000 locations (a compilation of soil samples from the AfSIS, EthioSIS, One Acre Fund, VitalSigns and legacy soil data) and an extensive stack of remote sensing covariates in addition to landform, lithologic and land cover maps. An ensemble model was then created for each nutrient from two machine learning algorithms—random forest and gradient boosting, as implemented in R packages ranger and xgboost—and then used to generate predictions in a fully-optimized computing system. Cross-validation revealed that apart from S, P and B, significant models can be produced for most targeted nutrients (R-square between 40–85%). Further comparison with OFRA field trial database shows that soil nutrients are indeed critical for agricultural development, with Mn, Zn, Al, B and Na, appearing as the most important nutrients for predicting crop yield. A limiting factor for mapping nutrients using the existing point data in Africa appears to be (1) the high spatial clustering of sampling locations, and (2) missing more detailed parent material/geological maps. Logical steps towards improving prediction accuracies include: further collection of input (training) point samples, further harmonization of measurement methods, addition of more detailed covariates specific to Africa, and implementation of a full spatio-temporal statistical modeling framework.

As the university campus is a place for learning, conducting scientific research, and communication, campus street spatial quality has an impact on its users. Therefore, refinement evaluations of campus spatial quality are essential for constructing high-quality campuses. In this study, machine learning was used to conduct semantic segmentation and spatial perception prediction on street view images. The physical features and perception quality of the surrounding areas of the Chongshan campus of Liaoning University were obtained. The study found that the visual beautiful quality (VBQ) of the student living area was the highest, and the VBQ of the teacher living area was the lowest when compared to the research and study area, student living area, sports area, and surrounding area. Greenness and openness had positive influences on VBQ, while enclosure had a negative influence. This study analyzed the influence mechanism operating between spatial physical features and VBQ. The results provide theoretical and technical support for campus space spatial quality construction and improvement.

Spatial statistics for very large spatial data sets is challenging. The size of the data set, n, causes problems in computing optimal spatial predictors such as kriging, since its computational cost is of order [graphic removed] . In addition, a large data set is often defined on a large spatial domain, so the spatial process of interest typically exhibits non-stationary behaviour over that domain. A flexible family of non-stationary covariance functions is defined by using a set of basis functions that is fixed in number, which leads to a spatial prediction method that we call fixed rank kriging. Specifically, fixed rank kriging is kriging within this class of non-stationary covariance functions. It relies on computational simplifications when n is very large, for obtaining the spatial best linear unbiased predictor and its mean-squared prediction error for a hidden spatial process. A method based on minimizing a weighted Frobenius norm yields best estimators of the covariance function parameters, which are then substituted into the fixed rank kriging equations. The new methodology is applied to a very large data set of total column ozone data, observed over the entire globe, where n is of the order of hundreds of thousands.

Depth to bedrock serves as the lower boundary of land surface models, which controls hydrologic and biogeochemical processes. This paper presents a framework for global estimation of depth to bedrock (DTB). Observations were extracted from a global compilation of soil profile data (ca. 1,30,000 locations) and borehole data (ca. 1.6 million locations). Additional pseudo‐observations generated by expert knowledge were added to fill in large sampling gaps. The model training points were then overlaid on a stack of 155 covariates including DEM‐based hydrological and morphological derivatives, lithologic units, MODIS surface reflectance bands and vegetation indices derived from the MODIS land products. Global spatial prediction models were developed using random forest and Gradient Boosting Tree algorithms. The final predictions were generated at the spatial resolution of 250 m as an ensemble prediction of the two independently fitted models. The 10–fold cross‐validation shows that the models explain 59% for absolute DTB and 34% for censored DTB (depths deep than 200 cm are predicted as 200 cm). The model for occurrence of R horizon (bedrock) within 200 cm does a good job. Visual comparisons of predictions in the study areas where more detailed maps of depth to bedrock exist show that there is a general match with spatial patterns from similar local studies. Limitation of the data set and extrapolation in data spare areas should not be ignored in applications. To improve accuracy of spatial prediction, more borehole drilling logs will need to be added to supplement the existing training points in under‐represented areas. Key Points Observations from soil and geological surveys are combined for developing global spatial prediction models of depth to bedrock Machine learning explains 59% of variation in spatial distribution of depth to bedrock for interpolation but much less for extrapolation The framework proposed can be used to gradually improve accuracy by adding more ground observations

In spatial statistics, a common objective is to predict values of a spatial process at unobserved locations by exploiting spatial dependence. Kriging provides the best linear unbiased predictor using covariance functions, and is often associated with Gaussian processes. However, for nonlinear predictions for nonGaussian and categorical data, the Kriging prediction is no longer optimal, and the associated variance is often overly optimistic. Although deep neural networks (DNNs) are widely used for general classification and prediction, they have not been studied thoroughly for data with spatial dependence. In this work, we propose a novel DNN structure for spatial prediction, where we capture the spatial dependence by adding an embedding layer of spatial coordinates with basis functions. We show in theory and simulation studies that the proposed DeepKriging method has a direct link to Kriging in the Gaussian case, and has multiple advantages over Kriging for nonGaussian and nonstationary data. That is, it provides nonlinear predictions, and thus has smaller approximation errors. Furthermore, it does not require operations on covariance matrices, and thus is scalable for large data sets. With sufficiently many hidden neurons, the proposed method provides an optimal prediction in terms of model capacity. In addition, we quantify prediction uncertainties based on density prediction, without assuming a data distribution. Finally, we apply the method to PM2.5 concentrations across the continental United States.

Gathering information about forest variables is an expensive and arduous activity. Therefore, directly collecting the data required to produce high-resolution maps over large spatial domains is infeasible. Next-generation collection initiatives for remotely sensed light detection and ranging (LiDAR) data are specifically aimed at producing complete-coverage maps over large spatial domains. Given that Li-DAR data and forest characteristics are often strongly correlated, it is possible to use the former to model, predict, and map forest variables over regions of interest. This entails dealing with high-dimensional (∼10²) spatially dependent LiDAR outcomes over a large number of locations (∼10⁵ ∼10⁶). With this in mind, we develop the spatial factor nearest neighbor Gaussian process (SF-NNGP) model, which we embed in a two-stage approach that connects the spatial structure found in LiDAR signals with forest variables. We provide a simulation experiment that demonstrates the inferential and predictive performance of the SF-NNGP, and use the two-stage modeling strategy to generate complete-coverage maps of the forest variables, with associated uncertainty, over a large region of boreal forests in interior Alaska.