Statistical Methods in Topological Data Analysis for Complex, High-Dimensional Data

The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data spaces. This paper provides an introductory overview of the mathematical underpinnings of Topological Data Analysis, the workflow to convert samples of data to topological summary statistics, and some of the statistical methods developed for performing inference on these topological summary statistics. The intention of this non-technical overview is to motivate statisticians who are interested in learning more about the subject.Comment: 15 pages, 7 Figures, 27th Annual Conference on Applied Statistics in Agricultur

Los modelos de evolución del paisaje y la supervivencia de paleoformas

[Abstract] Qne ofthe factots rnitigating against the recognition and acceptance ofvery old palaeosurfaces was, and in sorne rneasure rernains, the tacit acceptance ofsorne ofthe better known rnodels oflandscape evolution. Thus both the steady state and peneplanation rnodels irnply virtual conternporaneity of surface, and though scarp retreat allows for a greater age, the rnaxirnurn is deterrnined by the duration of a cycle, probably of the order of 33 Ma; rnuch younger than rnany fitrnly dated epigene surfaces. Basically landforrn and landscape persistence involves the stability, or only slow rate ofchange, ofsurfaces (divides) ofbounding scarps, or both. Such slow rates ofchange are induced by such factors as resistant bedrock: hence the preservation of rnany palaeoforrns on quartzites, etc. In addition, several rnechanisrns, such as uplift, through drainage, and consequent local «aridity>,; incision and unequal activity; and reinforcernent or positive feedback rnechanisrns, enhance the persistence of surfaces

Persistence Flamelets: multiscale Persistent Homology for kernel density exploration

In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, aimed at estimating topological invariants of a fixed object; when we allow this object to change, however, little has been done to investigate the evolution in its topology. In this work we define the Persistence Flamelets, a multiscale version of one of the most popular tool in TDA, the Persistence Landscape. We examine its theoretical properties and we show how it could be used to gain insights on KDEs bandwidth parameter

Spatial memory shapes density dependence in population dynamics

Most population dynamics studies assume that individuals use space uniformly, and thus mix well spatially. In numerous species, however, individuals do not move randomly, but use spatial memory to visit renewable resource patches repeatedly. To understand the extent to which memorybased foraging movement may affect density-dependent population dynamics through its impact on competition, we developed a spatially explicit, individual-based movement model where reproduction and death are functions of foraging efficiency. We compared the dynamics of populations of with- and without-memory individuals. We showed that memory-based movement leads to a higher population size at equilibrium, to a higher depletion of the environment, to a marked discrepancy between the global (i.e. measured at the population level) and local (i.e. measured at the individual level) intensities of competition, and to a nonlinear density dependence. These results call for a deeper investigation of the impact of individual movement strategies and cognitive abilities on population dynamics

A Landscape Persistence Assessment of Częstochowa Upland : a Case Study of Ogrodzieniec, Poland

Landscape permanence is understood as the temporal extent of the dominance of a given type of landscape, expressed by the temporal continuity of its use. This issue, already being the subject of much research, is important in proper landscape protection and management. In this paper, spatial landscape persistence and persistence of particular landscape types are presented for the Ogrodzieniec municipality, Cz˛estochowa Upland, Poland. In addition, a background of landscape types and their changes in the Cz˛estochowa Upland has been presented. Based on current and historical topographic maps, landscape types (forest, agriculture, settlement, fortified and industrial) were identified for the following studied periods: 1831, 1944, 1965, 2007, 2014 and 2020. After overlapping the maps, the persistence index was calculated, and isochrones of landscape persistence were determined. The term ‘landscape isochrones’ introduced in this paper is defined as theoretical lines of equal landscape time duration (iso-persistence line). The results show that the landscape of Ogrodzieniec can be considered to be persistent. The largest area of the municipality is occupied by the most permanent landscapes dating from before 1831. The most persistent is the fortified landscape. The method applied is important for planning sustainable development of the region, which is currently under intense tourist and economic pressure

Driving forces of landscape change - current and new directions

The concept of driving forces is gaining increasing attention in landscape-change research. We summarize the state of the art of this field and present new conceptual and methodological directions for the study of driving forces of landscape changes. These new directions address four major challenges faced by landscape-change studies, i.e., studying processes and not merely spatial patterns, extrapolating results in space and time, linking data of different qualities, and considering culture as a driver of landscape change. The proposed research directions include: studying landscape change across borders and transects, focusing on persistence as well as change, investigating rates of change, considering attractors of landscape change, targeting correlation and causality, and searching for precursors of landscape change. Based on established knowledge and the new approaches we outline a standard procedure to study driving forces of landscape change. We anticipate that our analytical and systematic approach increases the relevance of studies of landscape change for science as well as for the solution of real world problem

Lithology and the evolution of bedrock rivers in post-orogenic settings: Constraints from the high elevation passive continental margin of SE Australia

Understanding the role of lithological variation in the evolution of topography remains a fundamental issue, especially in the neglected post-orogenic terrains. Such settings represent the major part of the Earth's surface and recent modelling suggests that a range of interactions can account for the presence of residual topography for hundreds of millions of years, thereby explaining the great antiquity of landscapes in such settings. Field data from the inland flank of the SE Australian high-elevation continental margin suggest that resistant lithologies act to retard or even preclude the headward transmission of base-level fall driven by the isostatic response to regional denudation. Rejuvenation, be it episodic or continuous, is ‘caught up’ on these resistant lithologies, meaning in effect that the bedrock channels and hillslopes upstream of these ‘stalled’ knickpoints have become detached from the base-level changes downstream of the knickpoints. Until these knickpoints are breached, therefore, catchment relief must increase over time, a landscape evolution scenario that has been most notably suggested by Crickmay and Twidale. The role of resistant lithologies indicates that detachment-limited conditions are a key to the longevity of some post-orogenic landscapes, whereas the general importance of transport-limited conditions in the evolution of post-orogenic landscapes remains to be evaluated in field settings. Non-steady-state landscapes may lie at the heart of widespread, slowly evolving post-orogenic settings, such as high-elevation passive continental margins, meaning that non-steady-state landscapes, with increasing relief through time, are the ‘rule’ rather than the exception