# Benford’s Law for Telemetry Data of Wildlife

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{10}is decadic logarithm):

## 2. Materials and Methods

#### 2.1. Data

^{2}at altitudes between 880 and 3000 m above sea level in the valley of Kaprun (province of Salzburg, Austria). It was adjacent to the core area of Upper Tauern National Park.

_{i}, y

_{i}, z

_{i}), the time (t

_{i}), and (starting with row 2) the distance (2) to the previous position. The leading nonzero digits of the successive distances were also computed in the spreadsheet. If the time between two consecutive measurements exceeded one day, this distance was discarded. Distance 0 (one observation) was ignored.

#### 2.2. Statistics

^{2}, defined by the equation:

^{2}and for each simulated BL-distributed logbook, we computed the p-value from the chi-square distribution with eight degrees of freedom and used it to check if BL was refuted (i.e., p-value below 0.05). To this end, we counted how many of the 1132 simulated logbooks refuted BL. We repeated this simulation 500 times. We thereby obtained a sample of 500 counts. We compared this sample with the observed count of refutations of BL for our logbook data. To this end, we estimated a Poisson distribution for the sample of 500 counts, verified its goodness of fit using probability plots [45] and statistical tests [46] implemented in a Mathematica [37] function (DistributionFitTest), and used this distribution to compute the p-value for the hypothesis of a higher number of refutations than observed for the animal logbooks. For p < 0.05, we concluded that the observed data were unlikely to fulfil BL.

## 3. Results

^{−30}for a higher χ

^{2}than observed), whereas in terms of MAD = 0.0048, the deviation of the data from BL was barely discernible. However, in a simulation of 500 BL-distributed samples of size 73,393 each, much smaller values of MAD were observed; 95% of the simulated MAD-values were below 0.00131. Fitting a lognormal distribution to the sample of simulated MAD-values, MAD > 0.0048 was unlikely (p-value 4.6 × 10

^{−10}).

^{−7}). Thus, although the chi-square test supported BL for most weekly samples, the number of observed refutations was too large to be explained by spurious tests alone.

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Newcomb, S. Note on the frequency of use of different digits in natural numbers. Am. J. Math.
**1881**, 4, 39–40. [Google Scholar] [CrossRef][Green Version] - Benford, F.A. The law of anomalous numbers. Proc. Am. Philos. Soc.
**1938**, 78, 551–572. [Google Scholar] - Hill, T.P.; Berger, A. Benford Online Bibliography. Available online: www.benfordonline.net/ (accessed on 25 September 2021).
- Nigrini, M.J. Forensic Analytics: Methods and Techniques for Forensic Accounting Investigations; Wile: New York, NY, USA, 2020. [Google Scholar]
- Varian, H.R. Benford’s Law. Am. Stat.
**1970**, 26, 65–66. [Google Scholar] - Carslaw, C.A.P.N. Anomalies in Income Numbers: Evidence of Goal Oriented Behavior. Account. Rev.
**1988**, 63, 321–327. [Google Scholar] - Nigrini, M.J. Taxpayer compliance application of Benford’s law. J. Am. Tax. Assoc.
**1996**, 18, 72–92. [Google Scholar] - Akinadewo, I.S.; Akinkoye, E.Y. Tax Evasion Detection in Nigeria: Analysis of the Specific Forensic Accounting Techniques Used. Bus. Manag. Rev.
**2020**, 11, 131–139. [Google Scholar] - Beber, B.; Scacco, A. What the Numbers Say: A Digit-Based Test for Election Fraud. Political Anal.
**2012**, 20, 211–234. [Google Scholar] [CrossRef][Green Version] - Buyse, M.; George, S.L.; Evans, S.; Geller, N.L.; Edler, L.; Hutton, J. The Role of Biostatistics in the Prevention, Detection and Treatment of Fraud in Clinical Trials. Stat. Med.
**1999**, 18, 3435–3451. [Google Scholar] [CrossRef] - Hindls, R.; Hronová, S. Benford’s Law and Possibilities for Its Use in Governmental Statistics. Statistika
**2015**, 95, 54–64. [Google Scholar] - Shao, L.; Ma, B.Q. Empirical mantissa distributions of pulsars. Astropart. Phys.
**2010**, 33, 255–262. [Google Scholar] [CrossRef][Green Version] - Pain, J.C. Regularities and symmetries in atomic structure and spectra. High Energy Density Phys.
**2013**, 9, 392–401. [Google Scholar] [CrossRef][Green Version] - Arita, M. Scale-freeness and biological networks. J. Biochem.
**2005**, 138, 1–4. [Google Scholar] [CrossRef] - Campanario, J.M.; Coslado, M.A. Benford’s law and citations, articles and impact factor of scientific journals. Scientometrics
**2011**, 88, 421–423. [Google Scholar] [CrossRef] - Margellou, A.G.; Pomonis, P.J. Benford’s law, Zipf’s law and the pore properties in solids. Microporous Mesoporous Mater.
**2020**, 292, 109735. [Google Scholar] [CrossRef] - Salsburg, D. Digit Preference in the Bible. Chance
**1997**, 10, 46–48. [Google Scholar] [CrossRef] - Gómez-Camponovo, M.; Moreno, J.; Idrovo, A.J.; Páez, M.; Achkar, M. Monitoring the Paraguayan epidemiological dengue surveillance system using Benford’s law. Biomédica
**2016**, 36, 583–592. [Google Scholar] [CrossRef][Green Version] - Sambridge, M.; Jackson, A. National COVID numbers—Benford’s law looks for errors. Nature
**2020**, 581, 384. [Google Scholar] [CrossRef] - Hill, T.P. The significant-digit phenomenon. Am. Math. Mon.
**1995**, 102, 322–327. [Google Scholar] [CrossRef][Green Version] - Hill, T.P. A statistical derivation of the significant-digit law. Stat. Sci. A Rev. J. Inst. Math. Stat.
**1995**, 10, 354–363. [Google Scholar] [CrossRef] - Berger, A.; Hill, T.P. An Introduction to Benford’s Law; Princeton University Press: Princeton, NJ, USA, 2015. [Google Scholar]
- Engel, H.A.; Leuenberger, C. Benford’s law for exponential random variables. Stat. Probab. Lett.
**2003**, 63, 361–365. [Google Scholar] [CrossRef] - Lemons, D.; Lemons, N.; Peter, W. First Digit Oscillations. Stats
**2021**, 4, 595–601. [Google Scholar] [CrossRef] - Pietronero, L.; Tosatti, E.; Tosatti, V.; Vespignanic, A. Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf. Phys. A
**2001**, 293, 297–304. [Google Scholar] [CrossRef][Green Version] - Goodman, W. The promises and pitfalls of Benford’s law. Significance
**2016**, 13, 38–41. [Google Scholar] [CrossRef] - Viswanathan, G.M.; Buldyrev, S.V.; Havlin, S.; da Luz, M.G.E.; Raposo, E.P.; Stanley, H.E. Optimizing the success of random searches. Nature
**1999**, 401, 911–914. [Google Scholar] [CrossRef] - Mårell, A.; Ball, J.; Hofgaard, A. Foraging and movement paths of female reindeer: Insights from fractal analysis, correlated random walks, and Lévy flights. Can. J. Zool.
**2002**, 80, 854–865. [Google Scholar] [CrossRef] - Schürger, K. Lévy Processes and Benford’s Law. In Benford’s Law: Theory and Applications; Miller, S.J., Ed.; Princeton University Press: Princeton, NJ, USA, 2015; pp. 135–173. [Google Scholar]
- Edwards, A.M.; Phillips, R.A.; Watkins, N.W.; Freeman, M.P.; Murphy, E.J.; Afanasyev, V.; Buldyrev, S.V.; da Luz, M.G.E.; Raposo, E.P.; Stanley, H.E.; et al. Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature
**2007**, 449, 1044–1048. [Google Scholar] [CrossRef][Green Version] - Berger, A.; Bunimovich, L.A.; Hill, T.P. One-dimensional dynamical systems and Benford’s law. Trans. Am. Math. Soc.
**2004**, 357, 197–219. [Google Scholar] [CrossRef][Green Version] - Brähler, G.; Bensmann, M.; Jakobi, H.R. Das Benfordsche Gesetz und Seine Anwendbarkeit bei der Digitalen Prüfung von Fahrtenbüchern; Ilmenauer Schriften zur Betriebswirtschaftslehre; Technische Universität: Ilmenau, Germany, 2011. [Google Scholar]
- Sambridge, M.; Tkalčić, H.; Jackson, A. Benford’s Law in the natural sciences. Geophys. Res. Lett.
**2010**, 37, L22301–L22306. [Google Scholar] [CrossRef] - Griesberger, P.; Hackländer, K. Integrales Rotwildmanagement: Strategievernetzung Zwischen Forst-, Land-, Jagd- und Tourismuswirtschaft; FFG Project Number 848464; Final Project Report; BOKU: Vienna, Austria, 2018. [Google Scholar]
- Pröger, L. Anwendbarkeit des Benford-Gesetzes auf Bewegungsdaten von Wildtieren. Master’s Thesis, Institute of Mathematics, Department of Integrative Biology and Biodiversity Research, University of Natural Resources and Life Sciences, Vienna, Austria, 2021. [Google Scholar]
- Department of Defense. Its definition and relationships with local geodetic systems. In World Geodetic System 1984; Technical Report; DoD: Rockville, MD, USA, 1991. [Google Scholar]
- Wolfram Research Inc. Mathematica; Version 12.3; Wolfram Research Inc.: Champaign, IL, USA, 2021. [Google Scholar]
- Lindley, D.V. A Statistical Paradox. Biometrika
**1957**, 44, 187–192. [Google Scholar] [CrossRef] - Kossovsky, A.E. On the Mistaken Use of the Chi-Square Test in Benford’s Law. Stats
**2021**, 4, 419–453. [Google Scholar] [CrossRef] - Nigrini, M.J. Audit Sampling Using Benford’s Law: A Review of the Literature with Some New Perspectives. J. Emerg. Technol. Account.
**2017**, 14, 29–46. [Google Scholar] [CrossRef] - Druică, E.; Oancea, B.; Vâlsan, C. Benford’s law and the limits of digit analysis. Int. J. Account. Inf. Syst.
**2018**, 31, 75–82. [Google Scholar] [CrossRef] - Cerqueti, R.; Lupi, C. Some New Tests of Conformity with Benford’s Law. Stats
**2021**, 4, 745–761. [Google Scholar] [CrossRef] - Bickel, P.J.; Götze, F.; van Zwet, W.R. Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses. Stat. Sin.
**1997**, 7, 1–31. [Google Scholar] - Yates, F. Contingency table involving small numbers and the χ
^{2}test. Suppl. J. R. Stat. Soc.**1934**, 1, 217–235. [Google Scholar] [CrossRef] - Chambers, J.; Cleveland, W.; Kleiner, B.; Tukey, P. Graphical Methods for Data Analysis; Chapman & Hall: Boca Raton, FL, USA, 2017. [Google Scholar]
- D’Agostino, R.B.; Stephens, M.A. Goodness-of-Fit.-Techniques; Taylor & Francis (Informa): London, UK, 1986. [Google Scholar]
- Clopper, C.; Pearson, E.S. The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika
**1934**, 26, 404–413. [Google Scholar] [CrossRef] - Joenssen, D.W. Testing for Benford’s Law: A Monte Carlo Comparison of Methods; Working Paper; No. 2545243; SSRN: Rochester, NY, USA, 2015. [Google Scholar]

**Figure 1.**Probability plots comparing the simulated counts of refutations for BL-distributed logbooks with a Poisson distribution for (

**a**) refutations using χ

^{2}; (

**b**) refutations using YCC; plots using Mathematica [30].

Digit: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Observed: | 21,473 | 13,461 | 9516 | 7669 | 5941 | 4691 | 3976 | 3576 | 3090 |

Theoretical: | 22,093.5 | 12,923.8 | 9169.6 | 7112.5 | 5811.4 | 4913.4 | 4256.2 | 3754.2 | 3358.3 |

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**MDPI and ACS Style**

Pröger, L.; Griesberger, P.; Hackländer, K.; Brunner, N.; Kühleitner, M.
Benford’s Law for Telemetry Data of Wildlife. *Stats* **2021**, *4*, 943-949.
https://doi.org/10.3390/stats4040055

**AMA Style**

Pröger L, Griesberger P, Hackländer K, Brunner N, Kühleitner M.
Benford’s Law for Telemetry Data of Wildlife. *Stats*. 2021; 4(4):943-949.
https://doi.org/10.3390/stats4040055

**Chicago/Turabian Style**

Pröger, Lasse, Paul Griesberger, Klaus Hackländer, Norbert Brunner, and Manfred Kühleitner.
2021. "Benford’s Law for Telemetry Data of Wildlife" *Stats* 4, no. 4: 943-949.
https://doi.org/10.3390/stats4040055