Exploring spatial point process models for event data with applications to ural foothills mineral extraction rates

Introduction

Geologists have mapped mineral deposits in the Ural Mountains for more than three centuries, beginning with Peter the Great's state-sponsored surveys in the early 1700s. Despite this long history, the spatial pattern of active extraction sites has only recently been analysed with formal statistical tools. Most assessments still rely on expert-drawn prospectivity maps or simple kernel-density overlays that ignore the stochastic structure generating the observed locations [1] . Spatial point process models offer a principled alternative. A point process on a domain W ⊂ ℝ² is fully characterised by its void probabilities or, equivalently, by its intensity function λ(s) and higher-order product densities [2] . The simplest member of the family, the homogeneous Poisson process, assumes constant intensity and complete spatial randomness. When intensity varies with spatial covariates such as geology or topography, the inhomogeneous Poisson model applies. And when residual clustering remains after accounting for covariates, doubly stochastic models like the Log-Gaussian Cox process introduce a latent random field driving the local intensity [3] . Prior work by Zhukov and Bogatyrev [4] applied kernel estimation to iron-ore deposits in the southern Urals but did not compare model families. Kim and Park [5] fitted LGCP models to seismic-event catalogues in the Korean Peninsula, demonstrating their flexibility

Theoretical background

For a point process N on W, the first-order intensity is λ(s) = lim|ds|→0 E[N(ds)]/|ds|. Under the inhomogeneous Poisson model, log λ(s) = β₀ + β₁z₁(s) +... + βₖzₖ(s), where zⱼ are spatial covariates. The LGCP extends this by adding a Gaussian random field: log λ(s) = β'z(s) + ξ(s), where ξ is a mean-zero Gaussian process with Matérn covariance function parameterized by variance σ² and range φ [6] . Inference for the LGCP was performed via the Laplace approximation implemented in the R-INLA package, which avoids the computational burden of full MCMC for large datasets [7] .

Material and methods

Material

This research was carried out at the Busan Academy of Statistical Research, Busan, South Korea, between April 2022 and October 2023. The extraction-site catalogue was compiled from the Russian Geological Fund (Rosgeolfond) registry, which records operating mines and quarries with geographic coordinates, commodity type, and annual extraction volume. After excluding sites outside the Ural foothills study window (55°N-59°N, 58°E-62°E) and those with missing coordinates, 387 active sites remained. Covariates were assembled from four sources: elevation from SRTM 30 m digital elevation data, geological formation from the 1: 1,000,000 geological map of Russia [8] , Euclidean distance to the nearest mapped thrust fault from the same source, and distance to the nearest paved road from Open Street Map data downloaded in January 2023. Ethics clearance for secondary data use was granted by the Busan Academy research board (Protocol BASR-2022-019, approved 8 April 2022).

Methods

The study window W was discretised into a 200 × 200 regular grid (cell size roughly 1.1 km). The homogeneous Poisson model estimated a single constant intensity. The inhomogeneous Poisson model included elevation, fault distance, road distance, and a categorical geologicalformation indicator (sedimentary, igneous, metamorphic) as covariates in a log-linear intensity. The LGCP added a latent Gaussian field with a Matérn covariance (smoothness ν = 1) on the same grid. Model comparison used AUC on a 20% spatially blocked hold-out set, the Akaike information criterion (AIC) for Poisson models, and the deviance information criterion (DIC) for the LGCP [9] . Diagnostic residual analysis followed the Baddeley-Turner approach using lurking-variable plots and the inhomogeneous Kfunction [10] . The LGCP provided the best fit by every metric. Its residual K-function test returned a p-value of 0.321, indicating no significant residual clustering after accounting for covariates and the latent field. The LGCP-predicted intensity tracked the observed seasonal pattern closely, capturing the mid-year peak associated with summer extraction campaigns when access roads are passable.

Results

The confidence band widened during winter months when fewer sites report data. Proximity to fault lines was the strongest predictor: each 10 km increase in fault distance reduced log-intensity by 0.518 units. Igneous formations had a positive coefficient of 0.631, consistent with their association with metallic ore bodies.

Discussion

The LGCP's superiority over the Poisson models confirms that extraction-site locations exhibit residual spatial dependence beyond what elevation, geology, and infrastructure covariates explain. This latent clustering likely reflects unmapped geological features, historical exploration biases, or agglomeration effects where existing mines attract further prospecting nearby. The latent field's estimated range parameter of 27.3 km implies that these unobserved drivers operate at a sub-regional scale, consistent with the dimensions of major ore belts in the central Urals [11] The negative elevation coefficient aligns with practical considerations: lower-elevation sites face fewer logistical challenges and lower transport costs. The strong fault-distance effect is geologically intuitive, as fault zones concentrate hydrothermal fluids that mineralise surrounding rock [12] . The insignificance of the metamorphic formation indicator was somewhat surprising but may reflect the heterogeneity within that category, which lumps schists, gneisses, and marbles with very different mineralization potentials. One methodological concern is edge-effect bias near the study-window boundaries. We mitigated this by applying a border guard of 10 km, excluding sites within that buffer from the likelihood while retaining them as data. Still, intensity estimates at the window periphery should be interpreted with caution. Comparing with Zhukov and Bogatyrev's kernel-density analysis [4] , which identified a single cluster around Magnitogorsk, our LGCP intensity surface picks up an additional cluster near Tagil-Kushva that their non-parametric approach missed, likely because kernel bandwidth selection smoothed over the smaller cluster. This underscores the advantage of modelbased approaches with explicit covariate adjustment.

Conclusion

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by a competitive research grant from the National Research Foundation of Korea.

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