A bivariate community in a bilayer mobility network is defined as a group of nodes encompassing two types of movements, at least one of which exhibits a densely connected structure characterized by dense internal connections and sparse connections to other groups. Detecting such communities helps reveal relationships between different human movement patterns, offering insights for urban planning and transport optimization. However, identifying bivariate communities in the presence of noise remains challenging. To address this issue, we developed a density-based community detection method for bilayer mobility networks. We first identified high- and low-density nodes for each type of movements based on Monte Carlo simulations and the strength of connections between node pairs to reduce the interference of noise. Then, we upgraded the density-based clustering procedure to construct bivariate communities with varying combinations of interaction intensities. In the simulation experiments, quantitative evaluation using Adjusted Rand Index and Normalized Mutual Information showed that the proposed method outperformed three state-of-the-art methods in identifying bivariate communities. We further applied this method to bilayer mobility networks constructed from bus smart card data and taxi trajectory data in Beijing. The identified bivariate communities revealed the spatial correlations between different transportation modes.

Modeling nonstationary and nonlinear variations in spatial processes is challenging. To overcome this long-standing challenge in spatial interpolation, we introduced the spatial compound variable theory, which assumes that a spatial variable can be divided into global regular and local irregular components. We argue that nonstationary and nonlinear variations in these two components have distinct properties: the global regular component describes a nonlinear trend that is locally predictable, whereas the local irregular component represents local hotspots, cold spots, or outliers. Following the spatial compound variable theory, we developed a novel nonlinear and nonstationary spatial interpolation model by integrating Yang Chizhong filtering and an inductive graph convolution network (YangNet). Specifically, we used the Yang Chizhong filtering and interpolation method to estimate the global regular component with a nonlinear trend, built an inductive graph convolution network to model the nonlinear relationship between the estimated global regular component and the corresponding observation data, and used the nonlinear relationship to estimate the local irregular component. A case study of the Xiadian gold deposit in China demonstrates that YangNet outperforms four representative gold grade interpolation methods in terms of accuracy and smoothing effect mitigation. YangNet exhibits excellent adaptability and can be applied widely in geoscience.

Determining the optimal number of regions is a challenging issue in regionalization. Although cluster validity indices developed for non-spatial clustering have been used to determine the optimal number of regions, spatial contiguity constraints for regionalization are often neglected. Consequently, different regionalization results can share the same validity index value, which reduces the reliability of identifying the optimal number of regions in regionalization. To overcome this limitation, this study proposes a spatially constrained statistical approach for determining the optimal number of regions using two metrics: (i) a permutation-based variance for measuring the homogeneity within regions and (ii) a proportion index based on spatially constrained k-nearest neighbors to quantify the separation between regions. Furthermore, a distance-based method is employed to balance these two metrics to automatically determine the optimal number of regions. Experimental results on five synthetic datasets, the US presidential election and climate datasets show that the statistical approach developed in this study outperforms three widely used cluster validity indices in determining the optimal number of regions. The proposed statistical approach is straightforward to implement and can effectively reduce subjectivity in regionalization.

The prediction of river water quality is key in water resource management. Data-driven machine learning models have been widely used for predicting river water quality. However, these models seldom consider the physical mechanisms of water quality variation, which degrades the accuracy and stability of the prediction results. Hence, we develop a physics-guided spatio–temporal neural network (PGSTNN) model to predict a critical parameter for water quality assessment, i.e. dissolved oxygen. Physical information regarding spatio–temporal interactions in a hydrological network is explicitly considered to construct the architecture of PGSTNN. Two physical rules of dissolved oxygen variation (i.e. Henry’s law and power-scaling law) are established for the loss function of PGSTNN to guarantee the physical consistency of the prediction results. Experiments on the 2020–2021 water quality dataset in Atlanta, USA show that PGSTNN outperforms seven baseline neural network models in terms of prediction accuracy and stability. PGSTNN typically brings at least 10% accuracy (e.g. root mean square error and mean absolute error) improvement over the comparison methods. The proposed PGSTNN may not only improve the emergency response ability of water resource management, but also provide useful ideas for integrating scientific knowledge with machine learning.

In multivariate spatial interpolation, the accuracy of a variable of interest can be improved using ancillary variables. Although geostatistical methods are widely used for multivariate spatial interpolation, these methods usually require second-order stationary assumption of spatial processes, which is difficult to satisfy in practice. We developed a new multivariate spatial interpolation method based on Yang-Chizhong filtering (CoYangCZ) to overcome this limitation. CoYangCZ does not solve the multivariate spatial interpolation problem from a purely statistical point of view but integrates geometry and statistics-based strategies. First, we used a weighted moving average method based on binomial coefficients (i.e. Yang-Chizhong filtering) to fit the spatial autocorrelation structure of each spatial variable from a geometric perspective. We then quantified the spatial autocorrelation of each spatial variable and the correlations between different spatial variables by analyzing the variances of different spatial variables. Finally, we obtain the best linear unbiased estimators at the unsampled locations. Experiments on air pollution and meteorological datasets show that CoYangCZ has a higher interpolation accuracy than cokriging, regression kriging, gradient plus-inverse distance squared, sequential Gaussian co-simulation, and the kriging convolutional network. CoYangCZ can adapt to second-order non-stationary spatial processes; therefore, it has a wider scope of application than purely statistical methods.

The stationarity assumption of geostatistical methods is difficult to satisfy in practice. To overcome this limitation, this study proposed a geometric and statistical coupling strategy for modeling spatial dependence structures and developed a generalized Yang Chizhong filtering and interpolation (GYangCZ) method without the assumption of stationarity. In this work, we theoretically prove the effectiveness of Yang Chizhong filtering in fitting spatial dependence structures from a geometric perspective, and develop an orientation-constrained Yang Chizhong filtering to fit the local and discontinuous spatial dependence structures. To measure nonstationary spatial dependence structure, we define a local statistical indicator (i.e., fundamental variation function) by comparing the variance of the original data and the fitted geometric surfaces obtained under different filtering radii. The fundamental variation function is used as the kernel function to obtain the approximate best linear unbiased estimators at unobserved locations. We theoretically demonstrate that when only a linear drift exists in local areas, GYangCZ does not require the stationarity assumption. GYangCZ was used to estimate the gold grade of the Xiadian gold deposit in China. The results show that GYangCZ outperformed ordinary kriging, moving window kriging, and kriging convolution networks. GYangCZ is easy to implement with wide applications in geoscience.

The presence of global spatial autocorrelation usually leads to the spurious identification of spatial hotspots and hinders the identification of local hotspots. Despite the use of statistical methods to address global spatial autocorrelation in spatial hotspot detection, accurately modeling global spatial autocorrelation structure without the stationarity assumption of spatial processes is difficult. To overcome this challenge, we fitted the global spatial autocorrelation structure from a geometric perspective and identified the optimal global spatial autocorrelation structure by analyzing the variances in spatial data. Hotspots were detected from the residuals obtained by removing the global spatial autocorrelation structure from the original dataset. We upgraded a weighted moving average method based on binomial coefficients (Yang Chizhong filtering) to fit the global spatial autocorrelation structure for field-like geographic phenomena. A variance decay indicator, based on the variance in the original and filtered data, was used to identify the optimal global spatial autocorrelation structure. Yang Chizhong filtering does not require a spatial stationarity assumption and can preserve local autocorrelation structures in the residuals as much as possible. Experimental results showed that hotspot detection methods combined with Yang Chizhong filtering can effectively reduce type-I and -II errors in the results and discover implicit and valuable urban hotspots.