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Spatial point process models are widely applied to point pattern data from various fields in the social and environmental sciences. However, a serious hurdle in fitting point process models is the presence of duplicated points, wherein multiple observations share identical spatial coordinates. This often occurs because of decisions made in the geo-coding process, such as assigning representative locations (e.g., aggregate-level centroids) to observations when data producers lack exact location information. Because spatial point process models like the Log-Gaussian Cox Process (LGCP) assume unique locations, researchers often employ {\it ad hoc} solutions (e.g., jittering) to address duplicated data before analysis. As an alternative, this study proposes a Modified Minimum Contrast (MMC) method that adapts the inference procedure to account for the effect of duplicates without needing to alter the data. The proposed MMC method is applied to LGCP models, with simulation results demonstrating the gains of our method relative to existing approaches in terms of parameter estimation. Interestingly, simulation results also show the effect of the geo-coding process on parameter estimates, which can be utilized in the implementation of the MMC method. The MMC approach is then used to infer the spatial clustering characteristics of conflict events in Afghanistan (2008-2009).

The ideal Galton board and Lorentz gas billiard models have been studied numerically and analytically primarily in settings where friction and rotational velocity are neglected. We eliminate these simplifying assumptions and study the resulting dynamics of a more general model using no-slip collisions, in which particles rotate and may exchange linear and angular momentum at collisions while adhering to certain conservation laws. Using numerical experiments and phase portrait analysis we show that (in contrast to specular dispersing billiards) regularity persists when a small force is introduced while (consistent with specular billiards) under a stronger force new structure including invariant regions may arise. We also show analytically that with the introduction of an external force periodicity proliferates, with new types of periodic orbits not present in the no-force case., Comment: 20 pages, 16 figures

Compared to widely used likelihood-based approaches, the minimum contrast (MC) method offers a computationally efficient method for estimation and inference of spatial point processes. These relative gains in computing time become more pronounced when analyzing complicated multivariate point process models. Despite this, there has been little exploration of the MC method for multivariate spatial point processes. Therefore, this article introduces a new MC method for parametric multivariate spatial point processes. A contrast function is computed based on the trace of the power of the difference between the conjectured $K$-function matrix and its nonparametric unbiased edge-corrected estimator. Under standard assumptions, we derive the asymptotic normality of our MC estimator. The performance of the proposed method is demonstrated through simulation studies of bivariate log-Gaussian Cox processes and five-variate product-shot-noise Cox processes.

We develop flexible multivariate spatio-temporal Hawkes process models to analyze patterns of terrorism. Previous applications of point process methods to political violence data mainly utilize temporal Hawkes process models, neglecting spatial variation in these attack patterns. This limits what can be learned from these models as any effective counter-terrorism strategy requires knowledge on both when and where attacks are likely to occur. Even the existing work on spatio-temporal Hawkes processes imposes restrictions on the triggering function that are not well-suited for terrorism data. Therefore, we generalize the structure of the spatio-temporal triggering function considerably, allowing for nonseprability, nonstatitionarity, and cross-triggering (i.e., across the groups). To demonstrate the utility of our models, we analyze two samples of real-world terrorism data: Afghanistan (2002-2013) for univariate analysis and Nigeria (2009-2017) for bivariate analysis. Jointly, these two studies demonstrate that our models outperform standard Hawkes process models, besting widely-used alternatives in overall model fit and revealing spatio-temporal patterns that are, by construction, masked in these models (e.g., increasing dispersion in cross-triggering over time).

Event data are increasingly common in applied political science research. While these data are inherently locational, political scientists predominately analyze aggregate summaries of these events as areal data. In so doing, they lose much of the information inherent to these point pattern data, and much of the flexibility that comes analyzing events using point process models. Recognizing these advantages, applied statisticians have increasingly utilized point process models in the analysis of political events. Yet, this work often neglects inherent limitations of political event data (e.g, geolocation accuracy), which can complicate the direct application of point process models. In this paper, we attempt to bridge this divide: introducing the benefits of point process modeling for political science research, and highlighting the unique challenges political science data pose for these approaches. To ground our discussion, we focus the Global Terrorism Database, using a univariate and bivariate log-Gaussian Cox process model (LGCP) to analyze terror attacks in Nigeria during 2014.

The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar configurations. A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down. The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart. Our main results are as follows: for circular cylinders in dimension $3$, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration. Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two---a very common occurrence in planar no-slip billiards---the motion in the longitudinal direction, under no forces, is generically not bounded. This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions. While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a no-slip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics., Comment: 34 pages, 13 figures

The giant impact hypothesis is the dominant theory explaining the formation of our Moon. However, its inability to produce an isotopically similar Earth-Moon system with correct angular momentum has cast a shadow on its validity. Computer-generated impacts have been successful in producing virtual systems that possess many of the physical properties we observe. Yet, addressing the isotopic similarities between the Earth and Moon coupled with correct angular momentum has proven to be challenging. Equilibration and evection resonance have been put forth as a means of reconciling the models. However, both were rejected in a meeting at The Royal Society in London. The main concern was that models were multi-staged and too complex. Here, we present initial impact conditions that produce an Earth-Moon system whose angular momentum and isotopic properties are correct. The model is straightforward and the results are a natural consequence of the impact., Comment: 7 pages, 2 figures, 2 tables