First passage time moments of asymmetric Lévy flights

Padash, Amin; Chechkin, Aleksei V.; Dybiec, Bartłomiej; Magdziarz, Marcin; Shokri, Babak; Metzler, Ralf

doi:10.1088/1751-8121/ab9030

Cited by 23 publications

(13 citation statements)

References 150 publications

(325 reference statements)

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“…The models proposed here demonstrate normal diffusive movements; however, some animals exhibit super-diffusive movements such as Lévy flights or walks. In future research, the emergence of Lévy walks using a memory-based model could be investigated since the first-passage times from the stochastic processes of Lévy walks were also well-studied [3,4] and [43]. Further research could also focus on the investigation of the moments of the first-return time [3].…”

Section: Discussionmentioning

confidence: 99%

“…The first-passage and first-return times of particles or walkers provide understanding of the spatial properties of a system [1][2][3]. The former indicates the time required for particles or walkers to move across a target site for the first time, whereas the latter indicates the time required for them to return to the target site for the first time.…”

Section: Introductionmentioning

confidence: 99%

“…For example, animals conduct random searches for various purposes such as finding food resources or relocating towards their goal when they become lost [14]. Therefore, first-passage and first-return times are important properties of random walkers, and they were extensively studied using random walk algorithms [1][2][3][4][5][6][7][8]15].…”

Section: Introductionmentioning

confidence: 99%

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A vague memory can affect first-return time

Sakiyama

2020

J. Phys. Commun.

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First-return time is an important property for the return of particles or walkers to a start point. Recursive walks, which may be related to first-return time, are found in both random walk models and memory-based walk models. Achieving a balance between recursive walks and diffusive movements is a crucial but difficult modeling problem. Here, starting with a simple Brownian-walk model, I investigated how vague memorized information influences the first-return times of a walker. In the proposed model, the walker memorizes recently visited positions and recalls the direction in which it previously moved when returning to those positions. Using the recalled information, the walker then moves in the opposite direction to that previously traveled. In addition, the walker considers its recent experience and modifies its directional rules, i.e., memorized information, when the rule disturbs the recent flow of its movement. Thus, the proposed model effectively produces recursive walks in which a walker returns to a start point while demonstrating diffusive movements.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A vague memory can affect first-return time

Sakiyama

2020

J. Phys. Commun.

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“…Mathematically, the Lévy Flights model is a generalized continuous-time random walk process, a sequential sampling model (Voss et al, 2019) that uses an α-stable jump length distribution (or Lévy distribution) (Gnedenko and Kolmogorov, 1954) with a long-tailed, power-law asymptote λ(x) ∼ |x| −1−α (0 < α ≤ 2) to describe noise in the accumulation process (Padash et al, 2019(Padash et al, , 2020. Therefore, information accumulation in LF model can be formulated as follows:…”

Section: Lévy Flights Modelmentioning

confidence: 99%

Are there jumps in evidence accumulation, and what, if anything, do they reflect psychologically? An analysis of Lévy-Flights models of decision-making

Rasanan¹,

Ja²,

Dk³

2021

Preprint

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According to existing theories of simple decision-making, decisions are initiated by continuously sampling and accumulating perceptual evidence until a threshold value has been reached. Many models, such as the diffusion decision model, assume a noisy accumulation process, described mathematically as a stochastic Wiener process with Gaussian distributed noise. Recently, an alternative account of decision-making has been proposed in the Lévy Flights (LF) model, in which accumulation noise is characterized by a heavy-tailed power-law distribution, controlled by a parameter, α. The LF model produces sudden large “jumps” in evidence ac- cumulation that are not produced by the standard Wiener diffusion model, which some have argued provide better fits to data. It remains unclear, however, whether jumps in evidence accumulation have any real psychological meaning. Here, we investigate the conjecture by Voss et al. (2019) that jumps might reflect sudden shifts in the source of evidence people rely on to make decisions. We reason that if jumps are psychologically real, we should observe systematic reductions in jumps as people become more practiced with a task (i.e., as people converge on a stable decision strategy with experience). We fitted four versions of the LF model to behavioral data from a study by Evans and Brown (2017), using a five-layer deep inference neural network for parameter estimation. The analysis revealed systematic reductions in jumps as a function of practice, such that the LF model more closely approximated the standard Wiener model over time. This trend could not be attributed to other sources of parameter variability, speaking against the possibility of trade-offs with other model parameters. Our analysis suggests that jumps in the LF model might be capturing strategy instability exhibited by relatively inexpe- rienced observers early on in task performance. We conclude that further investigation of a potential psychological interpretation of jumps in evidence accumulation is warranted.

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“…The evolution in time of Lévy flights emerges to be governed by a fractional diffusion equation [66,46,47,71]. A number of properties of Lévy flights has been studied, e.g., [10,11,13,57,55,54]. However, in the probabilistic derivation of the fractional diffusion equation, the distinctive singularity of the fractional Laplacian is obtained, with a constant time-step, when the distribution of jumps is bi-modal and equal to zero in zero [68], see also Appendix B.…”

Section: Introductionmentioning

confidence: 99%

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

Pagnini,

Vitali

2021

Preprint

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We study Markovian continuous-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.

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First passage time moments of asymmetric Lévy flights

Cited by 23 publications

References 150 publications

A vague memory can affect first-return time

A vague memory can affect first-return time

Are there jumps in evidence accumulation, and what, if anything, do they reflect psychologically? An analysis of Lévy-Flights models of decision-making

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

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