Minimizing errors in identifying Lévy flight behaviour of organisms

  1. Figure 1

    Figure 1
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    Figure 1

    (a) Histogram plot of 1000 random numbers generated from a power-law distribution with Lévy exponent µ = −2·00 (see eqn 1 in text for explanation). (b) Histogram of the same data after log-10 transformation of both axes (LT), as for eqn 2 in text. (c) Histogram of the data in (a) after placing frequency data into bins of width 2k (termed logarithmic binning) (LB), where k is an integer, then log-10 transforming the results (eqn 3). (d) Histogram of the data in (a) after dividing the frequency of observations in the untransformed variable in each logarithmic bin by the bin width and N (total number of steps) before log-10 transformation (LBN). (e) A cumulative distribution (CD) of the same data given in (a).

  2. Figure 2

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    Figure 2

    Estimated values of the Lévy exponent µ recovered from known Lévy distributions using different plotting methods. The actual µ-values of power laws subjected to the different plotting methods are joined by the solid grey line. Open square symbols denote values after log-10 transformation (LT), circles represent values from log-10 transformation after dividing through the 2k binned data by each bin width and total N (LBN). Triangle symbols illustrate µ-values derived from a cumulative distribution function (CD). Note that CD values for µ < 2 are not shown because biasing due to the imposition of a maximum step length make these values unreliable. Estimates for each µ-value were derived from 10 000 sets of 1000 random numbers. Estimates for the method log-10 transformation after 2k binning (LB) is not shown because the corrected slope was the same as the results for LBN. LB and CD methods yield slopes with 1 −µ, so correction is required to estimate µ.

  3. Figure 3

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    Figure 3

    (a) Summary of the mean percentage error (± 1 SD; n = 7) of each plotting method across the seven Lévy distribution datasets (µ: 1·5, 1·75, 2·0, 2·25, 2·5, 2·75, 3·0). Plotting method: LT, log-10 transformation of frequency vs. step (move) length; LB, log-10 transformation after 2k (logarithmic) binning using the geometric midpoint of each bin; LBN, log-10 transformation after dividing through the 2k binned data by each bin width and total N (normalization); CD, cumulative distribution function. The effect of sample size on (b) the magnitude of error variation about recovered µ-values following 2k binning and normalization, and on (c) kurtosis of the frequency distribution of recovered µ-values.

  4. Figure 4

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    Figure 4

    Reanalysis of empirical data from published studies. (a) Frequency histogram of movement lengths of a grey seal from Austin et al. (2004), together with (b) the log-transformed data indicate the absence of a power-law distribution of movement lengths (cf. Figure 1). Austin et al. calculated a Lévy flight exponent of 1·26; however, (c) reanalysis of the seal data using 2k logarithmic binning and normalization prior to log-10 transformation provides no evidence for the presence of Lévy flights in the data (e.g. µ < 1) and confirms the absence of a power-law distribution. (d) Frequency histogram of distances between feeding events of wandering albatrosses (Weimerskirch et al. 2005), and (e) the calculated µ-value of 1·26 using the log-10 transformation method. (f) Reanalysis confirms presence of a Lévy distribution in prey encounter by albatrosses, but the more accurate plotting method shows the exponent is higher and closer to the theorized optimal search pattern of µ≈ 2.