The walking pattern of an organism is typically designated as either a Lévy walk or a Brownian walk based on whether the frequency distribution of its linear step lengths follows a power law distribution or an exponential distribution. However, there are many cases where actual data cannot be classified into either of these categories. In this paper, we propose a general distribution that includes the power law and exponential distributions as special cases. This distribution has two parameters: one parameter represents the exponent, similar to the power law and exponential distributions, and the other is a shape parameter representing the shape of the distribution. By introducing this distribution, an intermediate distribution model can be interpolated between the power law and exponential distributions. In this study, the proposed distribution was fitted to the frequency distribution of the step length calculated from the walking data of pill bugs. The autocorrelation coefficients were also calculated from the time-series data of the step length, and the relationship between the shape parameter and time dependency was investigated. The results indicate that individuals whose step length frequency distributions are closer to the power law distribution have stronger time dependence.

C++ program for parameter estimation of generalized distributions and source code for statistical analysis using R.

The programs for the parameter estimation and autocorrelation coefficient calculation described above were developed using C++. The compiler was MinGW 8.1.0 64-bit for C++. The Qt library (Qt version Qt 5.15.2 MinGW 64-bit) was also used for development.

The Wilcoxon's rank sum test was used to test the difference in the means. For all analyses, the statistical significance was set at p < 0.01. The following analyses were performed using the R 3.6.1 statistical software unless otherwise specified. We used the R packages of exactRankTests version 0.8.31 for the Wilcoxon rank sum test. The operating system used was Windows 10.

The data shown in 'sample_length_time_series_data.csv' is the walking data of a pill bug.
The values represent time series data of step length. For more information about step length, please refer to the literature (https://www.biorxiv.org/content/10.1101/2021.11.29.470497v1).