1D analysis of 2D isotropic random walks

TL;DR

This paper introduces a 1D projection method for analyzing 2D isotropic random walks, preserving essential information and revealing temporal properties through pattern counting, offering an alternative to traditional 2D analysis techniques.

Contribution

The authors propose a rotationally invariant 1D reduction of 2D trajectories that retains key stochastic properties and enables new pattern-based analysis methods.

Findings

No essential information is lost in the 1D reduction.

Pattern counting reveals temporal properties not evident from autocorrelation.

Method applied successfully to a restricted turning angle model.

Abstract

Many stochastic systems in physics and biology are investigated by recording the two-dimensional (2D) positions of a moving test particle in regular time intervals. The resulting sample trajectories are then used to induce the properties of the underlying stochastic process. Often, it can be assumed a priori that the underlying discrete-time random walk model is independent from absolute position (homogeneity), direction (isotropy) and time (stationarity), as well as ergodic. In this article we first review some common statistical methods for analyzing 2D trajectories, based on quantities with built-in rotational invariance. We then discuss an alternative approach in which the two-dimensional trajectories are reduced to one dimension by projection onto an arbitrary axis and rotational averaging. Each step of the resulting 1D trajectory is further factorized into sign and magnitude. The statistical properties of the signs and magnitudes are mathematically related to those of the step lengths and turning angles of the original 2D trajectories, demonstrating that no essential information is lost by this data reduction. The resulting binary sequence of signs lends itself for a pattern counting analysis, revealing temporal properties of the random process that are not easily deduced from conventional measures such as the velocity autocorrelation function. In order to highlight this simplified 1D description, we apply it to a 2D random walk with restricted turning angles (RTA model), defined by a finite-variance distribution $p(L)$ of step length and a narrow turning angle distribution $p(\phi)$, assuming that the lengths and directions of the steps are independent.