The kriging model can accommodate various spatial supports and has been extensively applied in hydrology, meteorology, soil science, and other domains. With the expansion of applications, it is essential to extend the kriging model for new spatial support of high-dimensional data. Geographical flows can depict the movements of geographical objects and imply the underlying mobility patterns in geographical phenomena. However, due to the bias, sparsity, and uneven quality of flow data in the real world, research about flows remains hindered by the lack of complete flow data and effective flow interpolation methods. In this study, we design a kriging interpolation model for flows based on several flow-related concepts and the autocorrelation of flows. We also analyze the second-order stationarity and anisotropy in the flow spatial random field. To illustrate the effectiveness and applicability of our method, we conduct two case studies. The former case study compares several experiments of flow density interpolation using Beijing mobile signaling data and illustrates the conditions of applicable areas. The latter case study extends our model to other flow attributes, such as travel time uncertainty, using Beijing taxi origin-destination flow data. The results of these cases demonstrate the effectiveness and high accuracy of our model.

Geographical flows reflect the movements, spatial interactions or connections among locations and are generally abstracted as origin-destination (OD) flows. In this context, clustering is a spatial pattern describing a group of flows with adjacent O and D points. For data composed of two types of flows (bivariate-flow data), a bivariate-flow cluster is a cluster comprising two types of flows, at least one of which exhibits a clustering pattern. In a bivariate-flow cluster, varying flow density combinations imply different meanings. For instance, a cluster with high-density travel flows on both weekdays (type A) and weekends (type B) may be associated with entertainment, whereas high-density flows on weekdays and sparse flows on weekends may reveal work-related travel. However, identifying bivariate-flow clusters with different flow density combinations is still an unsolved problem. To this end, we extend a bivariate-point clustering method and propose a density-based clustering method for bivariate flows. The simulation experiments verify model robustness. In a case study, we apply this method to extract clusters of bivariate-flow data comprising Beijing taxi OD flows of different periods, and identify clusters of work-related, entertainment, tourism, or egress and return travels. These results demonstrate the capability of our method in detecting bivariate-flow clusters.

Geographical flow (hereafter flow) can be modeled as an orderly connected point pair composed of an origin (O) and a destination (D). Aggregation is the most common form of spatial heterogeneity of flows, which we define as their deviation from complete spatial randomness (CSR), and the aggregation scale is an important indicator for its perception. Nevertheless, quantifying the aggregation scale of flows is still an unsolved problem. In this paper, we propose the L-function for flows as a solution, derive theoretical null models of the K-function and L-function in a flow space. We conduct simulation experiments to validate the L-function and its capability to detect aggregation scales. Finally, we apply the solution in a case study with taxi data in Beijing and identify nine aggregation scales of taxi OD flows, ranging from 170 m to 22.1 km. These scales correspond to three classes: less than 300 m, from 600 m to 700 m and more than 1500 m. The classes are related to the sizes of the urban facilities where the dominant flow clusters occur, indicating that the L-function in flow space can detect the aggregation scale of flows at the building scale, the block scale and the district scale.