Race Lévy flights: A mathematically tractable framework for studying heavy-tailed accumulation noise

Abstract

Levy flights is a particular exemplar of the generalised random walk processes, in which the jump lengths are drawn from a power-law asymptote ($\alpha$-stable) distribution. While employing this heavy-tailed distribution for accumulation noise within the diffusion decision model framework provides improved fitting performance over the standard Gaussian accumulation noise, the Levy flight model contains two key limitations. Specifically, the use of the diffusion framework limits the model to only being applicable to two alternative decision-making tasks, and the lack of an exact likelihood function can make fitting behavioral data a challenging and computationally costly task. This paper aims to provide a mathematically tractable framework for modeling $n$-alternative ($n\geq1$) decision-making tasks with an $\alpha$-stable distribution for the accumulation noise. We propose a race Levy flight model, with a racing architecture, making the model directly applicable to decision tasks with any number of alternatives. Moreover, with the corresponding space-fractional diffusion-advection equation for the accumulation process, a numerical scheme based on approximating of the joint probability distribution for each accumulator is provided. In addition, we fit our proposed model to two perceptual decision-making data sets, and show its improved performance compared to the standard racing diffusion model.

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

“…This method was first used for the DDM with constant drift and boundary [55] and a finite difference method was applied for approximating the solution. More recently, this method has been applied to more general diffusion models with time-dependent drift rate or threshold [70,71,72] and race Lévy flight model [73]. However, the numerical methods that are used for solving the PDE is based on spatial discretizing (e.g., finite difference method [70,71,73] or finite element method [72]).…”

“…More recently, this method has been applied to more general diffusion models with time-dependent drift rate or threshold [70,71,72] and race Lévy flight model [73]. However, the numerical methods that are used for solving the PDE is based on spatial discretizing (e.g., finite difference method [70,71,73] or finite element method [72]). Therefore, an additional step for interpolating the spatial dimension to obtain a continuous solution is required in these methods, which reduces the time efficiency of these methods.…”