The measurement of “sinuosity” in correlated random walks

Abstract

Many animals appear to move according to a correlated random walk, that is to say, on a sufficiently fine time scale the direction of successive steps will be in approximately the same but on a sufficiently coarse time scale the direction of successive steps will be randomly distributed around a circle. Bovet & Benhamou (1988, J. theor. Biol. 131, 419–433) have examined the technical problems associated with analysing correlated random walks and present a useful technique for measuring the sinuosity of the path by sampling the path at successive points each the same Cartesian distance from the next. Bovet and Benhamou use simulations to obtain an empirical expression relating the true sinuosity to the value obtained using their sampling technique. In this paper we derive an alternative technique based on sampling the path at successive points equally spaced in time. We are able to derive rigorous expressions for the sinuosity and the variance in the estimate of the sinuosity.