70 research outputs found
Asymptotic distribution of global errors in the numerical computations of dynamical systems
International audienceWe propose an analysis of the effects introduced by finite accuracy and round off arithmetic on numerical computations of discrete dynamical systems. Our method, that uses the statistical tool of the decay of fidelity, computes the error comparing directly the numerical orbit with the exact one (or, more precisely, with another numerical orbit computed with a much higher accuracy). Furthermore, as a model of the effects of round off arithmetic on the map, we also consider a random perturbation of the exact orbit with an additive noise, for which exact results can be obtained for some prototype maps. We investigate the decay laws of fidelity and their relationship with the error probability distribution for regular and chaotic maps, both for additive and numerical noise. In particular, for regular maps we find an exponential decay for additive noise, and a power law decay for numerical noise. For chaotic maps numerical noise is equivalent to additive noise, and our method is suitable to identify a threshold for the reliability of numerical results, i.e. a number of iterations below which global errors can be ignored. This threshold grows linearly with the number of bits used to represent real numbers
Intrinsic group behaviour: dependence of pedestrian dyad dynamics on principal social and personal features
Being determined by human social behaviour, pedestrian group dynamics depends
on "intrinsic properties" of the group such as the purpose of the pedestrians,
their personal relation, their gender, age, and body size. In this work we
quantitatively study the dynamical properties of pedestrian dyads by analysing
a large data set of automatically tracked pedestrian trajectories in an
unconstrained "ecological" setting (a shopping mall), whose relational group
properties have been coded by three different human observers. We observed that
females walk slower and closer than males, that workers walk faster, at a
larger distance and more abreast than leisure oriented people, and that inter
group relation has a strong effect on group structure, with couples walking
very close and abreast, colleagues walking at a larger distance, and friends
walking more abreast than family members. Pedestrian height (obtained
automatically through our tracking system) influences velocity and abreast
distance, both growing functions of the average group height. Results regarding
pedestrian age show as expected that elderly people walk slowly, while active
age adults walk at the maximum velocity. Groups with children have a strong
tendency to walk in a non abreast formation, with a large distance (despite a
low abreast distance). A cross-analysis of the interplay between these
intrinsic features, taking in account also the effect of extrinsic crowd
density, confirms these major effects but reveals also a richer structure. An
interesting and unexpected result, for example, is that the velocity of groups
with children {\it increases} with density, at least in the low-medium density
range found under normal conditions in shopping malls. Children also appear to
behave differently according to the gender of the parent
Social group behaviour of triads. Dependence on purpose and gender
We analysed a set of uninstructed pedestrian trajectories automatically tracked in a public area, and we asked a human coder to assess their group relationships. For those pedestrians who belong to the groups, we asked the coder to identify their apparent purpose of visit to the tracking area and apparent gender. We studied the quantitative dependence of the group dynamics on such properties in the case of triads (three people groups) and compared them to the two pedestrian group case (dyads), studied in a previous work. We found that the group velocity strongly depends on relation and gender for both triads and dyads, while the influence of these properties on spatial structure of groups is less clear in the triadic case. We discussed the relevance of these results to the modelling of pedestrian and crowd dynamics, and examined the possibility of the future works on this subject
Thermodynamics of a gas of pedestrians: theory and experiment
In this paper, we perform an experiment on the interaction of pedestrians in a chaotic environment and investigate the possibility to study its results using a thermodynamic model. In contrast to simple single-file unidirectional scenarios, where only distance and time are relevant to adjust walking speed, bidirectional cases are much more complex since pedestrians can perform evading manoeuvres to avoid collisions. To better understand collision avoidance in a bidimensional environment we designed a set of experiments where people need to move chaotically for the whole time. Trajectories of moving pedestrians were obtained by tracking their head position, but a method to obtain body orientation failed, thus limiting reliable information on average quantities, i.e. average density and speed. By analysing those data, we showed that equations for thermodynamic processes can be used to describe pedestrian dynamics from medium densities or a state where interaction distances are very small. To allow combining low density cognitive aspects of collision avoidance with semi-random motion at medium densities we also developed a microscopic simulation model inspired by physics. Results show that, after calibrations, the simulation model allows to reproduce the fundamental diagram of different studies despite the very simple rules implemented. This shows that describing the statistical nature of specific crowds requires a relatively small set of rules and research should focus on cognitive/psychological aspects which are essential for understanding crowds of people
Identification of social relation within pedestrian dyads
This study focuses on social pedestrian groups in public spaces and makes an effort to identify the type of social relation between the group members. As a first step for this identification problem, we focus on dyads (i.e. 2 people groups). Moreover, as a mutually exclusive categorization of social relations, we consider the domain-based approach of Bugental, which precisely corresponds to social relations of colleagues, couples, friends and families, and identify each dyad with one of those relations. For this purpose, we use anonymized trajectory data and derive a set of observables thereof, namely, inter-personal distance, group velocity, velocity difference and height difference. Subsequently, we use the probability density functions (pdf) of these observables as a tool to understand the nature of the relation between pedestrians. To that end, we propose different ways of using the pdfs. Namely, we introduce a probabilistic Bayesian approach and contrast it to a functional metric one and evaluate the performance of both methods with appropriate assessment measures. This study stands out as the first attempt to automatically recognize social relation between pedestrian groups. Additionally, in doing that it uses completely anonymous data and proves that social relation is still possible to recognize with a good accuracy without invading privacy. In particular, our findings indicate that significant recognition rates can be attained for certain categories and with certain methods. Specifically, we show that a very good recognition rate is achieved in distinguishing colleagues from leisure-oriented dyads (families, couples and friends), whereas the distinction between the leisure-oriented dyads results to be inherently harder, but still possible at reasonable rates, in particular if families are restricted to parent-child groups. In general, we establish that the Bayesian method outperforms the functional metric one due, probably, to the difficulty of the latter to learn observable pdfs from individual trajectories
Social aspects of collision avoidance: A detailed analysis of two-person groups and individual pedestrians
Pedestrian groups are commonly found in crowds but research on their social
aspects is comparatively lacking. To fill that void in literature, we study the
dynamics of collision avoidance between pedestrian groups (in particular dyads)
and individual pedestrians in an ecological environment, focusing in particular
on (i) how such avoidance depends on the group's social relation (e.g.
colleagues, couples, friends or families) and (ii) its intensity of social
interaction (indicated by conversation, gaze exchange, gestures etc). By
analyzing relative collision avoidance in the ``center of mass'' frame, we were
able to quantify how much groups and individuals avoid each other with respect
to the aforementioned properties of the group. A mathematical representation
using a potential energy function is proposed to model avoidance and it is
shown to provide a fair approximation to the empirical observations. We also
studied the probability that the individuals disrupt the group by ``passing
through it'' (termed as intrusion). We analyzed the dependence of the
parameters of the avoidance model and of the probability of intrusion on
groups' social relation and intensity of interaction. We confirmed that the
stronger social bonding or interaction intensity is, the more prominent
collision avoidance turns out. We also confirmed that the probability of
intrusion is a decreasing function of interaction intensity and strength of
social bonding. Our results suggest that such variability should be accounted
for in models and crowd management in general. Namely, public spaces with
strongly bonded groups (e.g. a family-oriented amusement park) may require a
different approach compared to public spaces with loosely bonded groups (e.g. a
business-oriented trade fair).Comment: 25 pages, 15 figures, 3 table
Estimating social relation from trajectories
This study focuses on social pedestrian groups in public spaces and makes an effort to identify the social relation between the group members. We particularly consider dyads having coalitional or mating relation. We derive several observables from individual and group trajectories, which are suggested to be distinctive for these two sorts of relations and propose a recognition algorithm taking these observables as features and yielding an estimation of social relation in a probabilistic manner at every sampling step. On the average, we detect coalitional relation with 87% and mating relation with 81% accuracy. To the best of our knowledge, this is the first study to infer social relation from joint (loco)motion patterns and we consider the detection rates to be a satisfactory considering the inherent challenge of the problem
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