Using likelihood to test for Lévy flight search patterns and for general power-law distributions in nature

Abstract

1. Ecologists are obtaining ever-increasing amounts of data concerning animal movement. A movement strategy that has been concluded for a broad variety of animals is that of Lévy flights, which are random walks whose step lengths come from probability distributions with heavy power-law tails. 2. The exponent that parameterizes the power-law tail, denoted µ, has repeatedly been found to be within the Lévy range of 1 < µ ≤ 3. Here, we use Monte Carlo simulations to show that the methods used to infer the value of µ are inaccurate. 3. The widely used method of simply logarithmically transforming a standard histogram of movement lengths has been shown elsewhere to be problematic. Here, we further demonstrate how poor it is, and show that it actually biases estimates of µ towards the Lévy range of 1 < µ ≤ 3, and can bias estimates towards the value of µ = 2. Thus, previous reports of animals undergoing Lévy flights, or of µ being close to the reported optimal value of µ = 2, may simply be a consequence of the bias generated by this method. 4. A technique that has been recently recommended is to logarithmically bin the data and then normalize the resulting histogram. We show that this technique also produces biased results, and suffers from similar problems as those just outlined, although to a lesser extent. 5. The proposed solution is to use likelihood. We find that calculating the maximum likelihood estimate of µ gives the most accurate results (having also tested the rank/frequency method). Likelihood has the further advantages of being the easiest method to implement, and of yielding accurate confidence intervals. Results are applicable to power-law distributions in general, and so are not restricted to inference of Lévy flights. 6. We also re-analyse a data set of grey seal movements that was originally reported to demonstrate Lévy flight behaviour. Using Akaike weights, we test four models, and find no evidence for Lévy flights. Overall, our results suggest that Lévy flights might not be as common as previously thought