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Tromer RM, Barbosa MB, Bartumeus F, Catalan J, da Luz MGE, Raposo EP, Viswanathan GM. Inferring Lévy walks from curved trajectories: A rescaling method. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 92:022147. [PMID: 26382383 DOI: 10.1103/physreve.92.022147] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2015] [Indexed: 06/05/2023]
Abstract
An important problem in the study of anomalous diffusion and transport concerns the proper analysis of trajectory data. The analysis and inference of Lévy walk patterns from empirical or simulated trajectories of particles in two and three-dimensional spaces (2D and 3D) is much more difficult than in 1D because path curvature is nonexistent in 1D but quite common in higher dimensions. Recently, a new method for detecting Lévy walks, which considers 1D projections of 2D or 3D trajectory data, has been proposed by Humphries et al. The key new idea is to exploit the fact that the 1D projection of a high-dimensional Lévy walk is itself a Lévy walk. Here, we ask whether or not this projection method is powerful enough to cleanly distinguish 2D Lévy walk with added curvature from a simple Markovian correlated random walk. We study the especially challenging case in which both 2D walks have exactly identical probability density functions (pdf) of step sizes as well as of turning angles between successive steps. Our approach extends the original projection method by introducing a rescaling of the projected data. Upon projection and coarse-graining, the renormalized pdf for the travel distances between successive turnings is seen to possess a fat tail when there is an underlying Lévy process. We exploit this effect to infer a Lévy walk process in the original high-dimensional curved trajectory. In contrast, no fat tail appears when a (Markovian) correlated random walk is analyzed in this way. We show that this procedure works extremely well in clearly identifying a Lévy walk even when there is noise from curvature. The present protocol may be useful in realistic contexts involving ongoing debates on the presence (or not) of Lévy walks related to animal movement on land (2D) and in air and oceans (3D).
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Affiliation(s)
- R M Tromer
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal-RN, 59078-970, Brazil
| | - M B Barbosa
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal-RN, 59078-970, Brazil
- Centre d'Estudis Avançats de Blanes (CEAB), CSIC, Blanes, 17300, Spain
| | - F Bartumeus
- Centre d'Estudis Avançats de Blanes (CEAB), CSIC, Blanes, 17300, Spain
- CREAF, Campus UAB, Cerdanyola, 08193, Spain
| | - J Catalan
- Centre d'Estudis Avançats de Blanes (CEAB), CSIC, Blanes, 17300, Spain
- CREAF, Campus UAB, Cerdanyola, 08193, Spain
| | - M G E da Luz
- Departamento de Física, Universidade Federal do Paraná, Curitiba-PR, 81531-980, Brazil
| | - E P Raposo
- Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, Recife-PE, 50670-901, Brazil
| | - G M Viswanathan
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal-RN, 59078-970, Brazil
- National Institute of Science and Technology of Complex Systems, Universidade Federal do Rio Grande do Norte, 59078-970 Natal-RN, Brazil
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Oliveira HA, Passos JD, Barbosa MB, Do Amorim EA. [Persistent primitive trigeminal artery: critical review of the clinical significance]. Arq Neuropsiquiatr 1991; 49:315-20. [PMID: 1807233 DOI: 10.1590/s0004-282x1991000300014] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/28/2022]
Abstract
The authors report three cases of persistence of primitive trigeminal artery with no evidence of a relationship between its presence and any specific clinical syndrome. They discuss embryological aspects and establish the correlation between the presence of this malformation and other intracranial pathologies, such as "tic douloureux" and morphologic alteration of the circle of Willis. A critical review of the literature is also presented.
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Affiliation(s)
- H A Oliveira
- Departamento de Medicina Interna e Patologia, CCBS, Hospital Universitário, Universidade Federal de Sergipe, Aracaju, Brasil
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