Spatio-temporal modeling of the topology of swarm behavior with persistence landscapes

We propose a method for modeling the topology of swarm behavior in a manner which facilitates the application of machine learning techniques such as clustering. This is achieved by modeling the persistence of topological features, such as connected components and holes, of the swarm with respect to time using zig-zag persistent homology. The output of this model is subsequently transformed into a representation known as a persistence landscape. This representation forms a vector space and therefore facilitates the application of machine learning techniques. The proposed model is validated using a real data set corresponding to a swarm of 300 fish. We demonstrate that it may be used to perform clustering of swarm behavior with respect to topological features

Modelling topological features of swarm behaviour in space and time with persistence landscapes

This paper presents a model of swarm behaviour that encodes the spatial-temporal characteristics of topological features such as holes and connected components. Specifically, the persistence of topological features with respect to time are computed using zig-zag persistent homology. This information is in turn modelled as a persistence landscape which forms a normed vector space and facilitates the application of statistical and data mining techniques. Validation of the proposed model is performed using a real data set corresponding to a swarm of fish. It is demonstrated that the proposed model may be used to perform retrieval and clustering of swarm behaviour in terms of topological features. In fact, it is discovered that clustering returns clusters corresponding to the swarm behaviours of flock, torus and disordered. These are the most frequently occurring types of behaviour exhibited by swarms in general

The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Stability and statistical inferences in the space of topological spatial relationships

Modelling topological properties of the spatial relationship between objects, known as the extit{topological relationship}, represents a fundamental research problem in many domains including Artificial Intelligence (AI) and Geographical Information Science (GIS). Real world data is generally finite and exhibits uncertainty. Therefore, when attempting to model topological relationships from such data it is useful to do so in a manner which is both extit{stable} and facilitates extit{statistical inferences}. Current models of the topological relationships do not exhibit either of these properties. We propose a novel model of topological relationships between objects in the Euclidean plane which encodes topological information regarding connected components and holes. Specifically, a representation of the persistent homology, known as a persistence scale space, is used. This representation forms a Banach space that is stable and, as a consequence of the fact that it obeys the strong law of large numbers and the central limit theorem, facilitates statistical inferences. The utility of this model is demonstrated through a number of experiments

Robust tracking of objects with dynamic topology

In many instances of the object tracking problem the topological properties of objects can change over time. Such changes include the splitting of an object into multiple objects or merging of multiple objects into a single object. We propose a novel tracking model which is robust to such changes. This model is formulated terms of homology theory whereby 0-dimensional homology classes, which correspond to path-connected components, are tracked. A generalisation of this model for tracking spatially close objects lying in an ambient metric space is also proposed. This generalisation is particularly suitable for tracking spatial-temporal phenomena such as weather phenomena. The utility of the proposed model is demonstrated with respect to tracking rain clouds in radar imagery

Towards the prediction of critical transitions in spatially extended populations with cubical homology

The prediction of critical transitions, such as extinction events, is vitally important to preserving vulnerable populations in the face of a rapidly changing climate and continuously increasing human resource usage. Predicting such events in spatially distributed populations is challenging because of the high dimensionality of the system and the complexity of the system dynamics. Here, we reduce the dimensionality of the problem by quantifying spatial patterns via Betti numbers ($\beta_0$ and $\beta_1$), which count particular topological features in a topological space. Spatial patterns representing regions occupied by the population are analyzed in a coupled patch population model with Ricker map growth and nearest-neighbors dispersal on a two-dimensional lattice. We illustrate how Betti numbers can be used to characterize spatial patterns by type, which in turn may be used to track spatiotemporal changes via Betti number time series and characterize asymptotic dynamics of the model parameter space. En route to a global extinction event, we find that the Betti number time series of a population exhibits characteristic changes. We hope these preliminary results will be used to aide in the prediction of critical transitions in spatially extended systems. Additional applications of this technique include analysis of spatial data (e.g., GIS) and model validation.Comment: Published in Contemporary Mathematics: Dynamical Systems and Random Processes, Volume 736, 201

Spatio-temporal analysis of vegetation dynamics of selected successional stages of dry acidic grasslands : experimental studies and model simulations

Austenfeld M. Spatio-temporal analysis of vegetation dynamics of selected successional stages of dry acidic grasslands : experimental studies and model simulations. Bielefeld (Germany): Bielefeld University; 2009.Eine allgemeine Entwicklungsumgebung wurde für die Analyse und Simulation von räumlich-zeitlichen Phänomenen in ökologischen Systemen entwickelt. Die gesamte Plattform basiert auf einer "Rich-Client-Platform" (RCP), die neue Konzepte der Modularisierung und allgemeinen Programmarchitektur mitbringt. Damit bietet sie die Grundlage für eine nachhaltige Weiterentwicklung und ist somit eine solide Basis für eine integrierte Entwicklungsumgebung für ökologische Modelle. Die Integration verschiedener statistischer Werkzeuge, Methoden der Bildverarbeitung und spezielle Visualisierungen qualifizieren diese Umgebung besonders für die Analyse der oben genannten räumlich-zeitlichen Prozesse. Aufgrund ihrer vergleichsweise geringen Komplexität wurden Sandlebensräume wiederholt für Studien von Vegetationsmustern und ihrer zugrunde liegenden biotischen Interaktionen genutzt. Für einen integrativen Überblick und weitere integrative Ansätze mit Hilfe von Simulationsmodellen wurde die oben genannte Plattform genutzt, um eine individuenbasierte Modellstruktur für die Analyse von Langzeiteffekten aufgrund von Umweltveränderungen auf die Stabilität von Sandlebensräumen zu entwickeln, die typischerweise von zwei Pionierarten, Corynephorus canescens und Polytrichum piliferum, dominiert werden. Das Modell wurde mit experimentellen Daten verifiziert, und die vom Modell erzeugten räumlich-zeitlichen Muster zeigten eine hohe Übereinstimmung mit natürlich gemessenen Mustern. Das Modell wurde dann genutzt, um Langzeiteffekte von Veränderungen der Temperatur, Nährstoffversorgung und Störungsraten in diesem System zu untersuchen. Die Ergebnisse zeigten eine generell hohe Stabilität des Systems unter veränderten Temperatur- und Nährstoffbedingungen, wobei temporal wiederkehrende, kleinräumige Störungen als Grundlage notwendig waren. Schließlich wurde noch eine Untersuchung über die Auswirkungen von Herbivorie und Konkurrenz auf Corynephorus canescens durchgeführt. In einem kontrollierten Freilandexperiment wurden die Auswirkungen von entfernter Biomasse von Blättern sowie die An- oder Abwesenheit eines intraspezifischen und interspezifischen Konkurrenten (Hieracium pilosella) auf die überirdische und unterirdische Allokation von Biomasse in der folgenden Regenerationsphase analysiert. Die Ergebnisse zeigten, dass Corynephorus canescens die Fähigkeit besitzt, leichte bis mittlere Blattverluste (die typische natürliche Herbivorie von Kaninchen und Paarhufern simulieren sollten) zu kompensieren, ohne dabei an Konkurrenzstärke zu verlieren. Unterirdisch konnten keine Auswirkungen der simulierten Herbivorie bzw. Konkurrenz festgestellt werden. Aufgrund dieser zu vernachlässigenden Effekte wurde Herbivorie nicht in dem Modell berücksichtigt.A generic modeling environment for the analysis and simulation of spatio-temporal phenomena in ecosystems was developed. This framework was built upon a Rich Client Platform (RCP) which uses new concepts of extensibility and software architecture for sustainable development and provides a solid basis for an Integrated Development Environment (IDE) for ecological models. The integration of various statistical tools, imaging routines and several specialized drawing panels makes this environment particularly suitable for the analysis of the above mentioned spatio-temporal ecological processes. Because of their comparatively low complexity, dry acidic grassland ecosystems have been repeatedly used for studying vegetation pattern formation and the underlying biotic interactions. In order to obtain an integrative view of the existing knowledge as well as to provide a possibility for further integrative analysis with the help of model simulations, the above described platform was used to develop an individual based Model structure for the investigation of long term effects of environmental changes on the stability of early successional stages of such dry acidic grasslands which are typically dominated by the two pioneer species Corynephorus canescens and Polytrichum piliferum. The model was validated with experimental data and the spatio-temporal patterns created by the model were in good accordance with the measured natural patterns. The model was then used to analyze the effect of changes in temperature, nutrient supply and disturbance rate on the long term behavior of this ecosystem. The results showed an overall high stability of this system under different temperature and nutrient scenarios as long as an intermediate disturbance frequency is assured. Finally, an experimental study on the effect of herbivory and competition on the Corynephorus canescens was conducted. In a controlled field experiment, the effects of the removal of various amounts of aboveground biomass on the above and belowground biomass allocation during the following regeneration phase was analyzed in the presence or absence of an intraspecific and interspecific competitor (Hieracium pilosella). The results show a rather high ability of C. canescens to compensate low to medium amounts of foliage loss (reflecting the typical natural herbivory induced by grasshoppers and rabbits) without significant changes in its competitive ability. Belowground, no biomass effects of foliage removal and/or competition could be detected. Because of these negligible effects, herbivory was not implemented in the above described model

Computational aspects of cellular intelligence and their role in artificial intelligence.

The work presented in this thesis is concerned with an exploration of the computational aspects of the primitive intelligence associated with single-celled organisms. The main aim is to explore this Cellular Intelligence and its role within Artificial Intelligence. The findings of an extensive literature search into the biological characteristics, properties and mechanisms associated with Cellular Intelligence, its underlying machinery - Cell Signalling Networks and the existing computational methods used to capture it are reported. The results of this search are then used to fashion the development of a versatile new connectionist representation, termed the Artificial Reaction Network (ARN). The ARN belongs to the branch of Artificial Life known as Artificial Chemistry and has properties in common with both Artificial Intelligence and Systems Biology techniques, including: Artificial Neural Networks, Artificial Biochemical Networks, Gene Regulatory Networks, Random Boolean Networks, Petri Nets, and S-Systems. The thesis outlines the following original work: The ARN is used to model the chemotaxis pathway of Escherichia coli and is shown to capture emergent characteristics associated with this organism and Cellular Intelligence more generally. The computational properties of the ARN and its applications in robotic control are explored by combining functional motifs found in biochemical network to create temporal changing waveforms which control the gaits of limbed robots. This system is then extended into a complete control system by combining pattern recognition with limb control in a single ARN. The results show that the ARN can offer increased flexibility over existing methods. Multiple distributed cell-like ARN based agents termed Cytobots are created. These are first used to simulate aggregating cells based on the slime mould Dictyostelium discoideum. The Cytobots are shown to capture emergent behaviour arising from multiple stigmergic interactions. Applications of Cytobots within swarm robotics are investigated by applying them to benchmark search problems and to the task of cleaning up a simulated oil spill. The results are compared to those of established optimization algorithms using similar cell inspired strategies, and to other robotic agent strategies. Consideration is given to the advantages and disadvantages of the technique and suggestions are made for future work in the area. The report concludes that the Artificial Reaction Network is a versatile and powerful technique which has application in both simulation of chemical systems, and in robotic control, where it can offer a higher degree of flexibility and computational efficiency than benchmark alternatives. Furthermore, it provides a tool which may possibly throw further light on the origins and limitations of the primitive intelligence associated with cells

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized Criticality and Turbulence" (2012, 2013, Bern, Switzerland