Spatial interpolation is a crucial task in geography. As perhaps the most widely used interpolation methods, geostatistical models -- such as Ordinary Kriging (OK) -- assume spatial stationarity, which makes it difficult to capture the nonstationary characteristics of geographic variables. A common solution is trend surface modeling (e.g., Regression Kriging, RK), which relies on external explanatory variables to model the trend and then applies geostatistical interpolation to the residuals. However, this approach requires high-quality and readily available explanatory variables, which are often lacking in many spatial interpolation scenarios -- such as estimating heavy metal concentrations underground. This study proposes a Feature-Free Regression Kriging (FFRK) method, which automatically extracts geospatial features -- including local dependence, local heterogeneity, and geosimilarity -- to construct a regression-based trend surface without requiring external explanatory variables. We conducted experiments on the spatial distribution prediction of three heavy metals in a mining area in Australia. In comparison with 17 classical interpolation methods, the results indicate that FFRK, which does not incorporate any explanatory variables and relies solely on extracted geospatial features, consistently outperforms both conventional Kriging techniques and machine learning models that depend on explanatory variables. This approach effectively addresses spatial nonstationarity while reducing the cost of acquiring explanatory variables, improving both prediction accuracy and generalization ability. This finding suggests that an accurate characterization of geospatial features based on domain knowledge can significantly enhance spatial prediction performance -- potentially yielding greater improvements than merely adopting more advanced statistical models.

Dynamic MRI reconstruction, one of inverse problems, has seen a surge by the use of deep learning techniques. Especially, the practical difficulty of obtaining ground truth data has led to the emergence of unsupervised learning approaches. A recent promising method among them is implicit neural representation (INR), which defines the data as a continuous function that maps coordinate values to the corresponding signal values. This allows for filling in missing information only with incomplete measurements and solving the inverse problem effectively. Nevertheless, previous works incorporating this method have faced drawbacks such as long optimization time and the need for extensive hyperparameter tuning. To address these issues, we propose Dynamic-Aware INR (DA-INR), an INR-based model for dynamic MRI reconstruction that captures the spatial and temporal continuity of dynamic MRI data in the image domain and explicitly incorporates the temporal redundancy of the data into the model structure. As a result, DA-INR outperforms other models in reconstruction quality even at extreme undersampling ratios while significantly reducing optimization time and requiring minimal hyperparameter tuning.

Spatial prediction is a fundamental task in geography. In recent years, with advances in geospatial artificial intelligence (GeoAI), numerous models have been developed to improve the accuracy of geographic variable predictions. Beyond achieving higher accuracy, it is equally important to obtain predictions with uncertainty measures to enhance model credibility and support responsible spatial prediction. Although geostatistic methods like Kriging offer some level of uncertainty assessment, such as Kriging variance, these measurements are not always accurate and lack general applicability to other spatial models. To address this issue, we propose a model-agnostic uncertainty assessment method called GeoConformal Prediction, which incorporates geographical weighting into conformal prediction. We applied it to two classic spatial prediction cases, spatial regression and spatial interpolation, to evaluate its reliability. First, in the spatial regression case, we used XGBoost to predict housing prices, followed by GeoConformal to calculate uncertainty. Our results show that GeoConformal achieved a coverage rate of 93.67%, while Bootstrap methods only reached a maximum coverage of 81.00% after 2000 runs. Next, we applied GeoConformal to spatial interpolation models. We found that the uncertainty obtained from GeoConformal aligned closely with the variance in Kriging. Finally, using GeoConformal, we analyzed the sources of uncertainty in spatial prediction. We found that explicitly including local features in AI models can significantly reduce prediction uncertainty, especially in areas with strong local dependence. Our findings suggest that GeoConformal holds potential not only for geographic knowledge discovery but also for guiding the design of future GeoAI models, paving the way for more reliable and interpretable spatial prediction frameworks.

Reservoir modeling is a critical process in the development of subsurface reservoirs, such as those found in oil and gas fields [1, 2]. Its primary objective is to characterize the spatial distribution of key reservoir properties, including porosity and permeability, which are essential for assessing reservoir reserves, evaluating properties, and determining overall potential [3, 4]. By integrating data from core samples, well logs, seismic surveys, and other sources, reservoir model offer a detailed representation of the spatial relationships between the essential reservoir properties. This modeling process is not only fundamental for understanding the current condition of the reservoir but also serves as the foundation for subsequent numerical simulations [5, 6] and the development of effective management strategies [7, 8]. Spatial interpolation is a widely used technique in reservoir modeling, involving the estimation of reservoir property distributions across a reservoir based on observations at discrete points [9].

Estimating spatially distributed information through the interpolation of scattered observation datasets often overlooks the critical role of domain knowledge in understanding spatial dependencies. Additionally, the features of these data sets are typically limited to the spatial coordinates of the scattered observation locations. In this paper, we propose a hybrid framework that integrates data-driven spatial dependency feature extraction with rule-assisted spatial dependency function mapping to augment domain knowledge. We demonstrate the superior performance of our framework in two comparative application scenarios, highlighting its ability to capture more localized spatial features in the reconstructed distribution fields. Furthermore, we underscore its potential to enhance nonlinear estimation capabilities through the application of transformed fuzzy rules and to quantify the inherent uncertainties associated with the observation data sets. Our framework introduces an innovative approach to spatial information estimation by synergistically combining observational data with rule-assisted domain knowledge.

Predictions in the form of probability distributions are crucial for decision-making. Quantile regression enables this within spatial interpolation settings for merging remote sensing and gauge precipitation data. However, ensemble learning of quantile regression algorithms remains unexplored in this context. Here, we address this gap by introducing nine quantile-based ensemble learners and applying them to large precipitation datasets. We employed a novel feature engineering strategy, reducing predictors to distance-weighted satellite precipitation at relevant locations, combined with location elevation. Our ensemble learners include six stacking and three simple methods (mean, median, best combiner), combining six individual algorithms: quantile regression (QR), quantile regression forests (QRF), generalized random forests (GRF), gradient boosting machines (GBM), light gradient boosting machines (LightGBM), and quantile regression neural networks (QRNN). These algorithms serve as both base learners and combiners within different stacking methods. We evaluated performance against QR using quantile scoring functions in a large dataset comprising 15 years of monthly gauge-measured and satellite precipitation in contiguous US (CONUS). Stacking with QR and QRNN yielded the best results across quantile levels of interest (0.025, 0.050, 0.075, 0.100, 0.200, 0.300, 0.400, 0.500, 0.600, 0.700, 0.800, 0.900, 0.925, 0.950, 0.975), surpassing the reference method by 3.91% to 8.95%. This demonstrates the potential of stacking to improve probabilistic predictions in spatial interpolation and beyond.

The acquisition of accurate rainfall distribution in space is an important task in hydrological analysis and natural disaster pre-warning. However, it is impossible to install rain gauges on every corner. Spatial interpolation is a common way to infer rainfall distribution based on available raingauge data. However, the existing works rely on some unrealistic pre-settings to capture spatial correlations, which limits their performance in real scenarios. To tackle this issue, we propose the SSIN, which is a novel data-driven self-supervised learning framework for rainfall spatial interpolation by mining latent spatial patterns from historical observation data. Inspired by the Cloze task and BERT, we fully consider the characteristics of spatial interpolation and design the SpaFormer model based on the Transformer architecture as the core of SSIN. Our main idea is: by constructing rich self-supervision signals via random masking, SpaFormer can learn informative embeddings for raw data and then adaptively model spatial correlations based on rainfall spatial context. Extensive experiments on two real-world raingauge datasets show that our method outperforms the state-of-the-art solutions. In addition, we take traffic spatial interpolation as another use case to further explore the performance of our method, and SpaFormer achieves the best performance on one large real-world traffic dataset, which further confirms the effectiveness and generality of our method.

Gridded satellite precipitation datasets are useful in hydrological applications as they cover large regions with high density. However, they are not accurate in the sense that they do not agree with ground-based measurements. An established means for improving their accuracy is to correct them by adopting machine learning algorithms. This correction takes the form of a regression problem, in which the ground-based measurements have the role of the dependent variable and the satellite data are the predictor variables, together with topography factors (e.g., elevation). Most studies of this kind involve a limited number of machine learning algorithms, and are conducted for a small region and for a limited time period. Thus, the results obtained through them are of local importance and do not provide more general guidance and best practices. To provide results that are generalizable and to contribute to the delivery of best practices, we here compare eight state-of-the-art machine learning algorithms in correcting satellite precipitation data for the entire contiguous United States and for a 15-year period. We use monthly data from the PERSIANN (Precipitation Estimation from Remotely Sensed Information using Artificial Neural Networks) gridded dataset, together with monthly earth-observed precipitation data from the Global Historical Climatology Network monthly database, version 2 (GHCNm). The results suggest that extreme gradient boosting (XGBoost) and random forests are the most accurate in terms of the squared error scoring function. The remaining algorithms can be ordered as follows from the best to the worst: Bayesian regularized feed-forward neural networks, multivariate adaptive polynomial splines (poly-MARS), gradient boosting machines (gbm), multivariate adaptive regression splines (MARS), feed-forward neural networks, linear regression.

The interpolation of spatial data can be of tremendous value in various applications, such as forecasting weather from only a few measurements of meteorological or remote sensing data. Existing methods for spatial interpolation, such as variants of kriging and spatial autoregressive models, tend to suffer from at least one of the following limitations: (a) the assumption of stationarity, (b) the assumption of isotropy, and (c) the trade-off between modelling local or global spatial interaction. Addressing these issues in this work, we propose the use of Markov reward processes (MRPs) as a spatial interpolation method, and we introduce three variants thereof: (i) a basic static discount MRP (SD-MRP), (ii) an accurate but mostly theoretical optimised MRP (O-MRP), and (iii) a transferable weight prediction MRP (WP-MRP). All variants of MRP interpolation operate locally, while also implicitly accounting for global spatial relationships in the entire system through recursion. Additionally, O-MRP and WP-MRP no longer assume stationarity and are robust to anisotropy. We evaluated our proposed methods by comparing the mean absolute errors of their interpolated grid cells to those of 7 common baselines, selected from models based on spatial autocorrelation, (spatial) regression, and deep learning. We performed detailed evaluations on two publicly available datasets (local GDP values, and COVID-19 patient trajectory data). The results from these experiments clearly show the competitive advantage of MRP interpolation, which achieved significantly lower errors than the existing methods in 23 out of 40 experimental conditions, or 35 out of 40 when including O-MRP.

We propose a probabilistic model for refining coarse-grained spatial data by utilizing auxiliary spatial data sets. Existing methods require that the spatial granularities of the auxiliary data sets are the same as the desired granularity of target data. The proposed model can effectively make use of auxiliary data sets with various granularities by hierarchically incorporating Gaussian processes. With the proposed model, a distribution for each auxiliary data set on the continuous space is modeled using a Gaussian process, where the representation of uncertainty considers the levels of granularity. The fine-grained target data are modeled by another Gaussian process that considers both the spatial correlation and the auxiliary data sets with their uncertainty. We integrate the Gaussian process with a spatial aggregation process that transforms the fine-grained target data into the coarse-grained target data, by which we can infer the fine-grained target Gaussian process from the coarse-grained data. Our model is designed such that the inference of model parameters based on the exact marginal likelihood is possible, in which the variables of fine-grained target and auxiliary data are analytically integrated out. Our experiments on real-world spatial data sets demonstrate the effectiveness of the proposed model.