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.

For bivariate origin-destination (OD) movement data composed of two types of individual OD movements, a bivariate cluster can be defined as a group of two types of OD movements, at least one of which has a high density. The identification of such bivariate clusters can provide new insights into the spatial interactions between different movement patterns. Because of spatial heterogeneity, the effective detection of inhomogeneous and irregularly shaped bivariate clusters from bivariate OD movement data remains a challenge. To fill this gap, we propose a network-constrained method for clustering two types of individual OD movements on road networks. To adaptively estimate the densities of inhomogeneous OD movements, we first define a new network-constrained density based on the concept of the shared nearest neighbor. A fast Monte Carlo simulation method is then developed to statistically estimate the density threshold for each type of OD movements. Finally, bivariate clusters are constructed using the density-connectivity mechanism. Experiments on simulated datasets demonstrate that the proposed method outperformed three state-of-the-art methods in identifying inhomogeneous and irregularly shaped bivariate clusters. The proposed method was applied to taxi and ride-hailing service datasets in Xiamen. The identified bivariate clusters successfully reveal competition patterns between taxi and ride-hailing services.

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.

Detecting regional co-location patterns on urban road networks is challenging because it is computationally prohibitive to search all potential co-location patterns and their localities, and effective statistical methods for evaluating the prevalence of regional co-location patterns are lacking. To overcome these challenges, this study developed an adaptive method for detecting network-constrained regional co-location patterns. Specifically, an alternate prevalence measure of regional co-location patterns was defined based on the likelihood ratio statistic. A network-constrained k-nearest neighbor method was used to construct instances of candidate co-location patterns, and a heuristic two-phase expansion method was proposed to identify candidate localities of regional co-location patterns. The statistical significance of regional co-location patterns was evaluated using a Monte Carlo simulation. Experiments using extensive simulated datasets showed that our method was superior to three state-of-the-art methods. The proposed method was also applied to a Beijing points of interest (POI) dataset. The identified regional POI co-location patterns could support a better understanding of the spatial organization of urban functions and may be useful for facilitating urban planning.

A bivariate flow cluster is a group of two types of spatial flows, where both types of flows have high (or low) values, or one type of flow has a high value while the other has a low value. Identifying bivariate flow clusters aids in understanding the complex interactions between different flow patterns. Detecting bivariate flow clusters remains challenging because statistics for quantitatively assessing bivariate flow clusters are lacking and the shapes and sizes of clusters vary. This study proposes a novel bivariate flow clustering method (BiFlowAMOEBA) by improving a multidirectional optimum ecotope-based algorithm (AMOEBA) which embeds local Getis-Ord statistic in an iterative procedure to detect irregular-shaped clusters. We define a bivariate local Getis-Ord statistic for quantitatively assessing bivariate flow clusters, use a hierarchical clustering strategy to construct clusters, and evaluate the statistical significance of clusters using a Monte Carlo simulation. Experimental results of simulated datasets show that BiFlowAMOEBA can identify bivariate flow clusters of different shapes more accurately and completely, compared with two state-of-the-art methods. Two case studies show that BiFlowAMOEBA helps not only unveil the interactions between public transport and taxi services but also identifies competition patterns between taxis and ride-hailing services.

Identifying clusters from individual origin–destination (OD) flows is vital for investigating spatial interactions and flow mapping. However, detecting arbitrarily-shaped and non-uniform flow clusters from network-constrained OD flows continues to be a challenge. This study proposes a shared nearest-neighbor-based clustering method (SNN_flow) for inhomogeneous OD flows constrained by a road network. To reveal clusters of varying shapes and densities, a normalized density for each OD flow is defined based on the concept of shared nearest-neighbor, and flow clusters are constructed using the density-connectivity mechanism. To handle large amounts of disaggregated OD flows, an efficient method for searching the network-constrained k-nearest flows is developed based on a local road node distance matrix. The parameters of SNN_flow are statistically determined: the density threshold is modeled as a significance level of a significance test, and the number of nearest neighbors is estimated based on the variance of the kth nearest distance. SNN_flow is compared with three state-of-the-art methods using taxicab trip data in Beijing. The results show that SNN_flow outperforms existing methods in identifying flow clusters with irregular shapes and inhomogeneous distributions. The clusters identified by SNN_flow can reveal human mobility patterns in Beijing.

Multilevel co-location patterns embedded in spatial datasets are difficult to discern due to the complexity of neighboring relationships among spatial features. The neighboring relationships are used to determine whether instances of different spatial features are located in close geographic proximity. When spatial features are distributed unevenly, the neighboring relationships among spatial features cannot be constructed appropriately. Correspondingly, the instances of co-location patterns cannot be generated correctly, and the prevalence of multilevel co-location patterns cannot be measured accurately. To overcome this challenge, this study develops a method to adaptively detect multilevel co-location patterns based on natural neighborhoods. First, locally adaptive neighboring relationships for instances of different spatial features, called ‘natural neighborhoods’, are defined by considering the formation mechanism of co-location patterns and the local-distribution characteristics of spatial features. Using the natural neighborhoods, we propose a multilevel refining method to identify all global and local co-location patterns algorithmically. We compare the proposed method against three state-of-the-art methods using both simulated and real-life datasets. The comparison shows that the proposed method can discover multilevel co-location patterns from unevenly distributed spatial features more completely and accurately with less a priori knowledge for the construction of the natural neighborhoods.

Urban black holes and volcanoes are typical traffic anomalies that are useful for optimizing urban planning and maintaining public safety. It is still challenging to detect arbitrarily shaped urban black holes and volcanoes considering the network constraints with less prior knowledge. This study models urban black holes and volcanoes as bivariate spatial clusters and develops a network-constrained bivariate clustering method for detecting statistically significant urban black holes and volcanoes with irregular shapes. First, an edge-expansion strategy is proposed to construct the network-constrained neighborhoods without the time-consuming calculation of the network distance between each pair of objects. Then, a network-constrained spatial scan statistic is constructed to detect urban black holes and volcanoes, and a multidirectional optimization method is developed to identify arbitrarily shaped urban black holes and volcanoes. Finally, the statistical significance of multiscale urban black holes and volcanoes is evaluated using Monte Carlo simulation. The proposed method is compared with three state-of-the-art methods using both simulated data and Beijing taxicab spatial trajectory data. The comparison shows that the proposed method can detect urban black holes and volcanoes more accurately and completely and is useful for detecting spatiotemporal variations of traffic anomalies.

Spatiotemporal co-occurrence patterns (STCOPs) are subsets of Boolean features whose instances frequently co-occur in both space and time. The detection of STCOPs is crucial to the investigation of the spatiotemporal interactions among different features. However, prevalent STCOPs reported by available methods do not necessarily indicate the statistically significant dependence among different features, which is likely to result in highly erroneous assessments in practice. To improve the reliability of results, this paper develops a statistical method to detect STCOPs and discern their statistical significance. The proposed method detects STCOPs against the null hypothesis that the spatiotemporal distributions of different features are independent of each other. To construct the null hypothesis, suitable spatiotemporal point-process models considering spatiotemporal autocorrelation are employed to model the distributions of different features. The performance of the proposed statistical method is assessed by synthetic experiments and a case study aimed at identifying crime patterns among multiple crime types in Portland City. The experimental results demonstrate that the proposed method is more effective for detecting meaningful STCOPs than the available alternative methods.

The discovery of spatial clusters formed by proximal spatial units with similar non-spatial attribute values plays an important role in spatial data analysis. Although several spatial contiguity-constrained clustering methods are currently available, almost all of them discover clusters in a geographical dataset, even though the dataset has no natural clustering structure. Statistically evaluating the significance of the degree of homogeneity within a single spatial cluster is difficult. To overcome this limitation, this study develops a permutation test approach Specifically, the homogeneity of a spatial cluster is measured based on the local variance and cluster member permutation, and two-stage permutation tests are developed to determine the significance of the degree of homogeneity within each spatial cluster. The proposed permutation tests can be integrated into the existing spatial clustering algorithms to detect homogeneous spatial clusters. The proposed tests are compared with four existing tests (i.e., Park’s test, the contiguity-constrained nonparametric analysis of variance (COCOPAN) method, spatial scan statistic, and q-statistic) using two simulated and two meteorological datasets. The comparison shows that the proposed two-stage permutation tests are more effective to identify homogeneous spatial clusters and to determine homogeneous clustering structures in practical applications.

Regional co-location patterns represent subsets of feature types that are frequently located together in sub-regions in a study area. These sub-regions are unknown a priori, and instances of these co-location patterns are usually unevenly distributed across a study area. Regional co-location patterns remain challenging to discover. This study developed a multi-level method to identify regional co-location patterns in two steps. First, global co-location patterns were detected, and other non-prevalent co-location patterns were identified as candidates for regional co-location patterns. Second, an adaptive spatial clustering method was applied to detect the sub-regions where regional co-location patterns are prevalent. To improve computational efficiency, an overlap method was developed to deduce the sub-regions of (k + 1)-size co-location patterns from the sub-regions of k-size co-location patterns. Experiments based on both synthetic and ecological data sets showed that the proposed method is effective in the detection of regional co-location patterns.