A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.

Where did they not go? Considerations for generating pseudo-absences for telemetry-based habitat models

Background Habitat suitability models give insight into the ecological drivers of species distributions and are increasingly common in management and conservation planning. Telemetry data can be used in habitat models to describe where animals were present, however this requires the use of presence-only modeling approaches or the generation of ‘pseudo-absences’ to simulate locations where animals did not go. To highlight considerations for generating pseudo-absences for telemetry-based habitat models, we explored how different methods of pseudo-absence generation affect model performance across species’ movement strategies, model types, and environments. Methods We built habitat models for marine and terrestrial case studies, Northeast Pacific blue whales (Balaenoptera musculus) and African elephants (Loxodonta africana). We tested four pseudo-absence generation methods commonly used in telemetry-based habitat models: (1) background sampling; (2) sampling within a buffer zone around presence locations; (3) correlated random walks beginning at the tag release location; (4) reverse correlated random walks beginning at the last tag location. Habitat models were built using generalised linear mixed models, generalised additive mixed models, and boosted regression trees. Results We found that the separation in environmental niche space between presences and pseudo-absences was the single most important driver of model explanatory power and predictive skill. This result was consistent across marine and terrestrial habitats, two species with vastly different movement syndromes, and three different model types. The best-performing pseudo-absence method depended on which created the greatest environmental separation: background sampling for blue whales and reverse correlated random walks for elephants. However, despite the fact that models with greater environmental separation performed better according to traditional predictive skill metrics, they did not always produce biologically realistic spatial predictions relative to known distributions. Conclusions Habitat model performance may be positively biased in cases where pseudo-absences are sampled from environments that are dissimilar to presences. This emphasizes the need to carefully consider spatial extent of the sampling domain and environmental heterogeneity of pseudo-absence samples when developing habitat models, and highlights the importance of scrutinizing spatial predictions to ensure that habitat models are biologically realistic and fit for modeling objectives.

Some recent advances for limit theorems

We present some recent developments for limit theorems in probability theory, illustrating the variety of this field of activity. The recent results we discuss range from Stein’s method, as well as for infinitely divisible distributions as applications of this method in stochastic geometry, to asymptotics for some discrete models. They deal with rates of convergence, functional convergences for correlated random walks and shape theorems for growth models.