The exponential growth of geographic data reveals limitations in traditional spatial indices. Spatial learned indices that incorporate machine learning models have been proposed to enhance index performance. However, due to the considerable overhead of fine-grained data partitioning and the complexity of hierarchical model structures, existing spatial learned indices still exhibit bottlenecks in index construction and query processing. To address the aforementioned issues, we propose TBSI, an in-memory Transformer-based spatial learned index with an end-to-end structure. TBSI employs an enhanced quadtree to optimize data partitioning and utilizes a Transformer-based position prediction model to manage each data partition, preserving a simple yet effective index structure. TBSI exhibits superior performance in both index construction and query processing. We also design spatial query algorithms based on a filtering-refinement mechanism and data update algorithms based on buffers and flag arrays to support efficient query processing and index maintenance. Extensive experiments on real-world and synthetic datasets demonstrated that, compared to baselines, TBSI achieved up to 23.4 times speedup in build time, up to 24.3 times reduction in index size, up to 5.9 times improvement in range queries, and up to 4.5 times improvement in kNN queries. Also, TBSI exhibited robust adaptability to dynamic data updates.

Spatial databases are the main means to manage geo-big data, and learned spatial indices are a novel approach to improve the spatial retrieval performance of spatial databases by modeling the data distribution. However, the complex hierarchical structures in current learning models pose significant limitations, including prolonged construction times, slow data updates, and suboptimal dynamic query performance. Consequently, improving the efficiency of both index construction and updates is essential. We addressed these challenges by introducing a new method, the Spatial Uniform Partition Learned Index (SUPLI). SUPLI utilizes an iterative uniform partitioning algorithm that simplifies data distribution by uniformly segmenting space and applies a linear regression function—instead of a neural network model—to enable efficient index construction. Additionally, SUPLI incorporates query load optimization and historical query learning strategies, which dynamically adjust the spatial query algorithm to enhance query efficiency. Furthermore, a buffer structure is employed to store change information, facilitating efficient updates. Comparative evaluations conducted on three synthetic datasets and two real-world datasets show that SUPLI outperforms the classic R-tree by an order of magnitude in construction, query, and update performance, and demonstrates additional advantages over similar spatial learned indices, such as SPRIG and LISA.

The cross-fertilization of the fast-developing AI technology and spatial indexing has given rise to spatial learned indexes. However, these indexes rely on historical data distributions to build models, which limits their ability to anticipate data that has not yet arrived. To address this, we propose a novel Spatio-Temporal Update Method (STUM) that enhances conventional spatial learned indexes by introducing a Spatial Delta Area (SDA) for updates without altering their hierarchical structure. STUM learns spatio-temporal auto-correlation from historical data and integrates predicted future distributions. We apply STUM to the Spatial Learned Block Range INdex (SLBRIN), resulting in the development of the Spatio-Temporal Updatable learned Block Range INdex (STUBRIN), which adopts Revmap to integrate spatio-temporal sequence predictions with the spatial block range. STUBRIN optimizes the retraining process by learning the temporal continuity from spatial distribution and fusing it into the error threshold control mechanism and historical delta learning mechanism. Our results show that STUBRIN achieves 1.9-2.4×, 1.8-13.3×, 3.4-6.7× better build, query and update performance compared to state-ofthe-art methods. Additionally, STUBRIN offers superior query and update stability. For concurrent learned indexes, we have also designed parallel scheduling for STUBRIN, which improves build, query and update performance by 6.2-6.8×, 0.3-4.2×, 2.6-5.5×, without increasing the index size.

Analyzing and accurately predicting the spatiotemporal dynamics of PM2.5 remain challenging. The existing spatiotemporal prediction approaches are associated with high model complexity and limited interpretability. Conventional methods combining Koopman theory and deep learning often neglect spatial correlations in spatiotemporal data. This study used the hourly PM2.5 dataset of the Beijing-Tianjin-Hebei region to reveal its spatiotemporal hierarchy using Koopman mode decomposition to identify the key dynamic modes. Furthermore, a Spatial Physics Constrained Learning (SPCL) model utilizing a graph representation learning method was proposed to combine the graph topological information of the PM2.5 spatial features with the Koopman feature function. The results showed that PM2.5 has growth, decay, and oscillation modes as well as daily, weekly, monthly, and yearly periods. SPCL achieved mean absolute error, root mean square error (RMSE), correlation r, and index of agreement values of 9.678, 13.922, 0.864, and 0.921, respectively. The average RMSE at 12 h improved by 16.1%, 12.7%, 0.9%, and 3.5% compared with using Long short-term Memory, Graph Convolutional Networks and Long Short-Term Memory Networks, Spatio-Temporal Graph Convolutional Networks, and Dynamic Spatiotemporal Graph Convolution Network, respectively. By discretizing the neural network hidden layers, the explanatory key of PM2.5 modes was elucidated, which demonstrated enhanced stability.

Geographically weighted regression (GWR) is a classical method of modeling spatially non-stationary relationships. The geographically neural network weighted regression (GNNWR) model solves the problem of the inaccurate construction of spatial weight kernels using a spatially weighted neural network. However, when the spatial distribution of observations is uneven, the spatial proximity expression in the input of GWR and GNNWR models does not fully represent the impact of the whole research space on the estimating point. Therefore, we established a global spatial proximity grid (GSPG) to express the spatial proximity of each estimating point and proposed a spatially weighted convolutional neural network (SWCNN) to extract the relationship between the GSPG and spatial weights. Finally, we proposed a geographically convolutional neural network weighted regression (GCNNWR) model combining SWCNN and ordinary linear regression (OLR) model to estimate spatial non-stationarity. We used two case studies of simulated data and real environment data to demonstrate the advancements of the GCNNWR model. The GCNNWR model achieved higher estimation accuracy and greater predictive power than the OLR, GWR, multi-scale GWR (MGWR), and GNNWR models. Moreover, the GCNNWR model maintained its better stability and accuracy in estimating spatially non-stationary relationships when the distribution of observations was uneven.

Geographically weighted regression (GWR) and geographically and temporally weighted regression (GTWR) are classic methods for estimating non-stationary relationships. Although these methods have been widely used in geographical modeling and spatiotemporal analysis, they face challenges in adequately expressing space-time proximity and constructing a kernel with optimal weights. This probably results in an insufficient estimation of spatiotemporal non-stationarity. To address complex non-linear interactions between time and space, a spatiotemporal proximity neural network (STPNN) is proposed in this paper to accurately generate space-time distance. A geographically and temporally neural network weighted regression (GTNNWR) model that extends geographically neural network weighted regression (GNNWR) with the proposed STPNN is then developed to effectively model spatiotemporal non-stationary relationships. To examine its performance, we conducted two case studies of simulated datasets and environmental modeling in coastal areas of Zhejiang, China. The GTNNWR model was fully evaluated by comparing with ordinary linear regression (OLR), GWR, GNNWR, and GTWR models. The results demonstrated that GTNNWR not only achieved the best fitting and prediction performance but also exactly quantified spatiotemporal non-stationary relationships. Further, GTNNWR has the potential to handle complex spatiotemporal non-stationarity in various geographical processes and environmental phenomena.

Geographically weighted regression (GWR) is a classic and widely used approach to model spatial non-stationarity. However, the approach makes no precise expressions of its weighting kernels and is insufficient to estimate complex geographical processes. To resolve these problems, we proposed a geographically neural network weighted regression (GNNWR) model that combines ordinary least squares (OLS) and neural networks to estimate spatial non-stationarity based on a concept similar to GWR. Specifically, we designed a spatially weighted neural network (SWNN) to represent the nonstationary weight matrix in GNNWR and developed two case studies to examine the effectiveness of GNNWR. The first case used simulated datasets, and the second case, environmental observations from the coastal areas of Zhejiang. The results showed that GNNWR achieved better fitting accuracy and more adequate prediction than OLS and GWR. In addition, GNNWR is applicable to addressing spatial non-stationarity in various domains with complex geographical processes.

Spatiotemporal kriging (STK) is recognized as a fundamental space-time prediction method in geo-statistics. Spatiotemporal regression kriging (STRK), which combines space-time regression with STK of the regression residuals, is widely used in various fields, due to its ability to take into account both the external covariate information and spatiotemporal autocorrelation in the sample data. To handle the spatiotemporal non-stationary relationship in the trend component of STRK, this paper extends conventional STRK to incorporate it with an improved geographically and temporally weighted regression (I-GTWR) model. A new geo-statistical model, named geographically and temporally weighted regression spatiotemporal kriging (GTWR-STK), is proposed based on the decomposition of deterministic trend and stochastic residual components. To assess the efficacy of our method, a case study of chlorophyll-a (Chl-a) prediction in the coastal areas of Zhejiang, China, for the years 2002 to 2015 was carried out. The results show that the presented method generated reliable results that outperform the GTWR, geographically and temporally weighted regression kriging (GTWR-K) and spatiotemporal ordinary kriging (STOK) models. In addition, employing the optimal spatiotemporal distance obtained by I-GTWR calibration to fit the spatiotemporal variograms of residual mapping is confirmed to be feasible, and it considerably simplifies the residual estimation of STK interpolation.