Spatial clustering has been widely used for spatial data mining and knowledge discovery. An ideal multivariate spatial clustering should consider both spatial contiguity and aspatial attributes. Existing spatial clustering approaches may face challenges for discovering repeated geographic patterns with spatial contiguity maintained. In this paper, we propose a Spatial Toeplitz Inverse Covariance-Based Clustering (STICC) method that considers both attributes and spatial relationships of geographic objects for multivariate spatial clustering. A subregion is created for each geographic object serving as the basic unit when performing clustering. A Markov random field is then constructed to characterize the attribute dependencies of subregions. Using a spatial consistency strategy, nearby objects are encouraged to belong to the same cluster. To test the performance of the proposed STICC algorithm, we apply it in two use cases. The comparison results with several baseline methods show that the STICC outperforms others significantly in terms of adjusted rand index and macro-F1 score. Join count statistics is also calculated and shows that the spatial contiguity is well preserved by STICC. Such a spatial clustering method may benefit various applications in the fields of geography, remote sensing, transportation, and urban planning, etc.

Geographically explicit, taxonomically resolved fossil occurrences are necessary for reconstructing macroevolutionary patterns and for testing a wide range of hypotheses in the Earth and life sciences. Heterogeneity in the spatial and temporal distribution of fossil occurrences in the Paleobiology Database (PBDB) is attributable to several different factors, including turnover among biological communities, socioeconomic disparities in the intensity of paleontological research, and geological controls on the distribution and fossil yield of sedimentary deposits. Here we use the intersection of global geological map data from Macrostrat and fossil collections in the PBDB to assess the extent to which the potentially fossil-bearing, surface-expressed sedimentary record has yielded fossil occurrences. We find a significant and moderately strong positive correlation between geological map area and the number of fossil occurrences. This correlation is consistent regardless of map unit age and binning protocol, except at period level; the Neogene and Quaternary have non-marine map units covering large areas and yielding fewer occurrences than expected. The sedimentary record of North America and Europe yields significantly more fossil occurrences per sedimentary area than similarly aged deposits in most of the rest of the world. However, geographic differences in area and age of sedimentary deposits lead to regionally different expectations for fossil occurrences. Using the sampling of surface-expressed sedimentary units in North America and Europe as a predictor for what might be recoverable from the surface-expressed sedimentary deposits of other regions, we find that the rest of the globe is approximately 45% as well sampled in the PBDB. Using age and area of bedrock and sampling in North America and Europe as a basis for prediction, we estimate that more than 639,000 occurrences from outside these regions would need to be added to the PBDB to achieve global geological parity in sampling. In general, new terrestrial fossil occurrences are expected to have the greatest impact on our understanding of macroevolutionary patterns.