Geographically weighted regression (GWR) facilitates spatial modeling by providing location-specific coefficients to capture spatial non-stationarity. GWR incorporates a distance decay effect, assigning greater weights to proximal observations under the assumption they exert more influence on the regression parameters. However, distant observations may share significant context similarities, such as socioeconomic or environmental factors, which can influence the regression model. This study introduces an attention-based architecture to address context similarity between samples. A deep learning model termed Context-Attention Geographically Weighted Regression (CatGWR) is proposed to integrate context similarity with distance-based proximity to enhance the estimation of spatial non-stationarity in spatial regression models. Such an integration results in contextualized spatial weights for CatGWR to identify the varying patterns of nonstationary relationships across different spatial locations and context conditions. Validation through simulation experiments and an empirical study on housing prices in Shenzhen, China, shows the superior predictive accuracy and robustness of CatGWR in modeling complex spatial interactions, especially under contextual influences, in which CatGWR improves the R2 of fit and prediction results by at least 6% compared to existing models. Future work will focus on optimizing bandwidth selection and exploring additional attention mechanisms to enhance model performance.

The estimation of spatial heterogeneity within real estate markets holds significant importance in house price modelling. However, employing a single or straightforward distance to measure spatial proximity is probably insufficient in complex urban areas, thereby resulting in an inadequate modelling of spatial heterogeneity. To address this issue, this paper incorporates multiple distance measures within a neural network framework to achieve an optimized measure of spatial proximity (OSP). Consequently, a geographically neural network weighted regression model with optimized measure of spatial proximity (osp-GNNWR) is devised for the purpose of spatially heterogeneous modeling. Trained as a unified model, osp-GNNWR obviates the need for separate pretraining of OSP. This enables OSP to delineate the modeled spatial process through a post hoc calculated value. Through simulation experiments and a real-world case study on house prices, the proposed model reaches more accurate descriptions of diverse spatial processes and exhibits better overall performance. The interpretable results of the case study in Wuhan demonstrate the efficacy of the osp-GNNWR model in addressing spatial heterogeneity within real estate markets, suggesting its potential for modelling and predicting complex geographical phenomena.

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.