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Benford Kanunu ve Genelleştirilmiş Benford Kanunu ile ekosistem doğallığının hesaplanması

Year 2021, Volume: 22 Issue: 2, 73 - 82, 29.06.2021
https://doi.org/10.18182/tjf.907217

Abstract

Ekosistemlerin doğallık tespitine yönelik yeni bir yöntem önerilmiştir. Bitki türleri ve örnek alanlardan oluşan (S Χ A ) üç veri seti kullanılmıştır. Bunlar Sultan Dağları Alt Bölgesi (BS) veri seti (60 Χ 96), Dedegül Dağları Alt Bölgesi (BD) veri seti (89 Χ 119) ve iki alt bölgenin birleşmesinden oluşan Beyşehir Gölü Havzası (B) veri setidir (98 Χ 215). İlk olarak BS ve BD veri setlerinden belirlenen ilk rakam olasılık değerleri (〖d_1 p〗_o ) ile Benford Kanunu teorik olasılık değerleri (d_1 p_e ) arasındaki genel uyumu belirlemek için ki kare (χ^2 ) testleri yapılmıştır. Sonuçta χ^2 (e_BS )=16,579 ve χ^2 (e_BD )=2,406 olarak bulunmuştur. İkinci olarak BS ve BD’nin gözlenen olasılık değerlerine en uyumlu teorik olasılık değerlerini belirlemek için genelleştirilmiş Benford Kanunu (GB(d;γ)) kullanılmıştır. BS ve BD için en küçük χ^2 değerleri sırasıyla γ=0,65’de ve γ=0,07’de elde edilmiştir (χ^2 (e_BD^γ )=4,992 ve χ^2 (e_BD^γ )=2,209). Beklendiği gibi genelleştirilmiş Benford Kanunu ile her iki alt bölgenin χ^2 değerleri düşmüştür. χ^2 değer düşüşü BS’de çok daha yüksek olmuştur. Alt bölgelerin örnek alan sayıları birbirlerinden farklıdır. Bu yüzden üçüncü aşamada B veri setinden her iki alt bölgenin örnek alan sayıları dikkate alınarak rastlantısal yinelemeli işlemler uygulanmış ve yineleme sayısı (K) kadar (K=10000) χ^2 değerleri elde edilmiştir. Daha sonra kalibrasyon katsayı değerlerini (kd) belirlemek için bu χ^2 değerlerinin ortalamaları ((_ ^k)(χ^2 ) ̅ (E_BS^γ )=6,747 ve (_ ^k)(χ^2 ) ̅ (E_BD^γ )=6,176) alınmıştır. Sonuçta, BS için kd=1 olduğu için BS doğallık değeri 4,992 ve BD için kd=1,093 olduğu için BD doğallık değeri 2,414 olarak bulunmuştur. Teorik olarak en doğal ekosistemler için tam doğallık değeri=0 kabul edildiğinden, elde edilen doğallık hesaplama sonuçları BD ekosistemlerinin BS ekosistemlerinden daha doğal olduğu göstermiştir.

References

  • Akkaş, M., 2007. Denetimde Benford kanunu’nun uygulaması. Gazi Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi, 9(1): 191-206.
  • Anderson, J.E., 1991. A conceptual framework for evaluating and quantifying naturalness. Conservation Biology, 5: 347−352.
  • Ausloos, M., Castellano, R., Cerqueti, R., 2016. Regularities and discrepancies of credit default swaps: A data science approach through Benford's law. Chaos, Solitons & Fractals, 90: 8-17.
  • Aybars A., Ataunal L., 2016. An application of Benford’s Law to fundamental accounting figures reported by Borsa Istanbul (BIST) companies. Journal of Economics, Finance and Accounting, 3(3): 234-243.
  • Benford, F., 1938. The law of anomalous numbers. Proceedings of The American Philosophical Society, 78: 551-572.
  • Branets, S., 2019. Detecting money laundering with Benford’s law and machine learning . Masters Thesis, University of Tartu, Faculty of Social Sciences, Estonia.
  • Cleary, R., Thibodeau, J.C., 2005. Applying digital analysis using Benford's law to detect fraud: The dangers of type I errors. Auditing: A Journal of Practice & Theory, 24(1): 77-81.
  • Clippe, P., Ausloos, M., 2012. Benford's law and theil transform of financial data. Physica A: Statistical Mechanics and Its Applications, 391: 6556- 6567.
  • Côté, S., Bélanger, L., Beauregard, R., Thiffault, É., Margni, M., 2019. A conceptual model for forest naturalness assessment and application in Quebec’s Boreal Forest. Forests, 10(4): 325. https://doi.org/10.3390/f10040325
  • Cressie, N., Read, T.R.C., 1984. Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society. Series B (Methodological), 46: 440–464.
  • Crocetti, E., Randi, G., 2016. Using the Benford's Law as a first step to assess the quality of the cancer registry data. Frontiers in Public Health, 4:225. DOI:10.3389/fpubh.2016.00225.
  • Dunlop, S., Lanfranco, S., Schembri, J.S., 2014. The role of 'Naturalness' and seral stage in the assessment and management of coastal sites, https://www.um.edu.mt/library/oar/bitstream/ 123456789/21115/1/Dunlop%2C%20Lanfranco%20%26%20Schembri%20%282014%29.%20The%20role%20of%20%27Naturalness%27%20and%20seral%20stage%20in%20the%20assessment%20and%20management%20of%20coastal%20sites.pdf, Erişim: 20.12.2020.
  • Durtschi, C., Hillison, W., Pacini, C., 2004. The effective use of Benford’s law to assist in detecting fraud in accounting data. Journal of Forensic Accounting, 5(1): 17-34.
  • Erdős, L., Bátori, Z., Penksza, K., Dénes, A., Kevey, B., Kevey, D., Magnes, M., Sengl, P., Tölgyesi, C., 2017. Can naturalness indicator values reveal habitat degradation? A test of four methodological approaches. Polish Journal Of Ecology, 65: 1-13.
  • Fanfarillo, E., Kasperski, A., Giuliani, A., Cicinelli, E., Latini, M., Abbate, G., 2018. Assessing naturalness of arable weed communities: A new index applied to a case study in central Italy. Biological Agriculture & Horticulture, 34(4): 232-244, DOI: 10.1080/01448765.2018.1434832.
  • Fattorini, L., 2003. Statistical analysis of ecological diversity (Eds., El-Shaarawi, A.H., Jureckova, J.), Environmetrics, EOLSS: Paris, France, 1, 18–29.
  • Fewster, R.M., 2009. A simple explanation of Benford’s Law. The American Statistician, 63: 26-32. http://dx.doi.org/10.1198/tast.2009.0005.
  • Fu, D., Shi, Y.Q., Su, W., 2007. A generalized Benford’s Law for JPEG coefficients and its applications in image forensics. Proceedings of SPIE, 6506, 1L1- 1L11.
  • Hill, T.P., 1995. A statistical derivation of the significant-digit law. Statistical Science, 10: 354–363.
  • Hindls, R., Hronová, S., 2015. Benford’s Law and possibilities for its use in governmental statistics. Statistika, 95(2): 54-64.
  • Hürlimann, W., 2009. Generalizing Benford’s law using power laws: Application to integer sequences. International Journal of Mathematics and Mathematical Sciences, Article ID 970284, Doi:10.1155/2009/970284.
  • Hürlimann, W., 2003. A generalized Benford law and its application. Mathematics Preprint Archive, 3(3): 217-228.
  • Hürlimann, W., 2015. On the uniform random upper bound family of first sig-nificant digit distributions. Journal of Informetrics, 9(2): 349-358.
  • Judge, G., Schechter, L., 2009. Detecting problems in survey data using Benford’s Law. Journal of Human Resources, 44(1): 1-24.
  • Leroy, B., 2016. Calculation of rarity indices for species and assemblages of species. R package version 1.3-4. Available at: https://CRAN.R-project.org/package=Rarity.
  • Machado, A., 2004. An index of naturalness. Journal for Nature Conservation, 12: 95-110.
  • Margules, C., Usher, M.B., 1981. Criteria used in assessing wildlife conservation potential: A review. Biological Conservation, 21:79–109.
  • Newcomb, S., 1881. Note on the frequency of use of the different digits in natural numbers. American Journal of Mathematics, 4(1): 39–40.
  • Nigrini, M.J., Mittermaier, L.J., 1997. The use of Benford's law as an aid in analytical procedures. Auditing, 16(2): 52.
  • Nigrini, M.J., Miller, S.J., 2007. Benford’s Law applied to hydrology data—Results and relevance to other geophysical data. Mathematical Geology, 39(5): 469–490. http://doi.org/10.1007/s11004-007-9109-5.
  • Özkan, K., 2016. Biyolojik Çeşitlilik Bileşenleri (α, β ve γ) Nasıl Ölçülür (1. Basım). Süleyman Demirel Üniversitesi, Orman Fakültesi Yayınları, Isparta.
  • Özkan, K., 2003. Beyşehir Gölü Havzasının Yetişme Ortamı Özellikleri ve Sınıflandırılması. Doktora Tezi, İÜ Araştırma Fonu Proje Numarası T-981/19022001, 189s.
  • Reif, A., Walentowski, H., 2008. The assessment of naturalness and its role for nature conservation and forestry in Europe. Waldökologie Landschaftsforschung Naturschutz, 6:63–76.
  • Shao, L., Ma, B.Q., 2010. First-digit law in nonextensive statistics, Physical Review E, 82(4): 1-4, 10.1103/PhysRevE.82.041110.
  • Smith, P.G.R., Theberge, J.B., 1986. A review of criteria for evaluating natural areas. Environmental Management, 10(6): 715-734.
  • Tans, W., 1974. Priority ranking of biotic natural areas. Michigan Botanist, 13: 31–39.
  • Tseng, H.C., Huang, W.N., Huang, D.W., 2017. Modified Benford's law for two-exponent distributions. Scientometrics, 110(3): 1403–1413.

Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law

Year 2021, Volume: 22 Issue: 2, 73 - 82, 29.06.2021
https://doi.org/10.18182/tjf.907217

Abstract

A new method was proposed to estimate ecosystem naturalness. Three species-plot (S Χ A) datasets were used. Those data sets belong to Sultan mountain sub-district (BS) (60 Χ 96) Dedegül mountain sub-district (BD) (89 Χ 119) and, Beyşehir Watershed (B) (98 Χ 215) consisting of both of the sub-districts. Firstly, chi square test (χ^2 ) was applied to define the statistical goodness of fit between the first digit observed probabilities (〖d_1 p〗_o ) and the theoretical probabilities of Benford’s Law (d_1 p_e ). It was found that χ^2 (e_BS )=16.579 and χ^2 (e_BD )=2.406.
Secondly, to find the fittest theoretical probabilities for BS and BD, generalized Benford’s Law (GB(d;γ)) was applied. Minimal χ^2 values were obtained at γ=0.65 and γ=0.07 for BS and BD respectively (χ^2 (e_BD^γ )=4.992, χ^2 (e_BD^γ )=2.209). As expected, χ^2 values of the sub-districts decreased by generalized Benford’s Law. The most dramatic χ^2 decrease occurred in BS. The number of sample plots of the sub-districts are different. Two random iterative processes happened 10000 times were therefore performed considering the number of sample plots of the sub-districts in B dataset. As a result 10000 χ^2 values were obtained for each sub-district. Average values of those χ^2 values were then used ((_ ^k)(χ^2 ) ̅ (E_BS^γ )=6.747 and (_ ^k)(χ^2 ) ̅ (E_BD^γ )=6.176) to calculate calibration coefficients of each sub-district. Naturalness values of BS and BD were found to be 4.992 and 2.414 respectively due to calibration coefficients of BS= ((_ ^k)(χ^2 ) ̅ (E_max^γ ))⁄((_ ^k)(χ^2 ) ̅ (E_BS^γ ) )=1 and BD=((_ ^k)(χ^2 ) ̅ (E_max^γ ))⁄((_ ^k)(χ^2 ) ̅ (E_BD^γ ) )=1.093. Since the perfect naturalness value is theoretically equal to 0, the obtained results indicate that BD ecosystems are more natural than BS ecosystems.

References

  • Akkaş, M., 2007. Denetimde Benford kanunu’nun uygulaması. Gazi Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi, 9(1): 191-206.
  • Anderson, J.E., 1991. A conceptual framework for evaluating and quantifying naturalness. Conservation Biology, 5: 347−352.
  • Ausloos, M., Castellano, R., Cerqueti, R., 2016. Regularities and discrepancies of credit default swaps: A data science approach through Benford's law. Chaos, Solitons & Fractals, 90: 8-17.
  • Aybars A., Ataunal L., 2016. An application of Benford’s Law to fundamental accounting figures reported by Borsa Istanbul (BIST) companies. Journal of Economics, Finance and Accounting, 3(3): 234-243.
  • Benford, F., 1938. The law of anomalous numbers. Proceedings of The American Philosophical Society, 78: 551-572.
  • Branets, S., 2019. Detecting money laundering with Benford’s law and machine learning . Masters Thesis, University of Tartu, Faculty of Social Sciences, Estonia.
  • Cleary, R., Thibodeau, J.C., 2005. Applying digital analysis using Benford's law to detect fraud: The dangers of type I errors. Auditing: A Journal of Practice & Theory, 24(1): 77-81.
  • Clippe, P., Ausloos, M., 2012. Benford's law and theil transform of financial data. Physica A: Statistical Mechanics and Its Applications, 391: 6556- 6567.
  • Côté, S., Bélanger, L., Beauregard, R., Thiffault, É., Margni, M., 2019. A conceptual model for forest naturalness assessment and application in Quebec’s Boreal Forest. Forests, 10(4): 325. https://doi.org/10.3390/f10040325
  • Cressie, N., Read, T.R.C., 1984. Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society. Series B (Methodological), 46: 440–464.
  • Crocetti, E., Randi, G., 2016. Using the Benford's Law as a first step to assess the quality of the cancer registry data. Frontiers in Public Health, 4:225. DOI:10.3389/fpubh.2016.00225.
  • Dunlop, S., Lanfranco, S., Schembri, J.S., 2014. The role of 'Naturalness' and seral stage in the assessment and management of coastal sites, https://www.um.edu.mt/library/oar/bitstream/ 123456789/21115/1/Dunlop%2C%20Lanfranco%20%26%20Schembri%20%282014%29.%20The%20role%20of%20%27Naturalness%27%20and%20seral%20stage%20in%20the%20assessment%20and%20management%20of%20coastal%20sites.pdf, Erişim: 20.12.2020.
  • Durtschi, C., Hillison, W., Pacini, C., 2004. The effective use of Benford’s law to assist in detecting fraud in accounting data. Journal of Forensic Accounting, 5(1): 17-34.
  • Erdős, L., Bátori, Z., Penksza, K., Dénes, A., Kevey, B., Kevey, D., Magnes, M., Sengl, P., Tölgyesi, C., 2017. Can naturalness indicator values reveal habitat degradation? A test of four methodological approaches. Polish Journal Of Ecology, 65: 1-13.
  • Fanfarillo, E., Kasperski, A., Giuliani, A., Cicinelli, E., Latini, M., Abbate, G., 2018. Assessing naturalness of arable weed communities: A new index applied to a case study in central Italy. Biological Agriculture & Horticulture, 34(4): 232-244, DOI: 10.1080/01448765.2018.1434832.
  • Fattorini, L., 2003. Statistical analysis of ecological diversity (Eds., El-Shaarawi, A.H., Jureckova, J.), Environmetrics, EOLSS: Paris, France, 1, 18–29.
  • Fewster, R.M., 2009. A simple explanation of Benford’s Law. The American Statistician, 63: 26-32. http://dx.doi.org/10.1198/tast.2009.0005.
  • Fu, D., Shi, Y.Q., Su, W., 2007. A generalized Benford’s Law for JPEG coefficients and its applications in image forensics. Proceedings of SPIE, 6506, 1L1- 1L11.
  • Hill, T.P., 1995. A statistical derivation of the significant-digit law. Statistical Science, 10: 354–363.
  • Hindls, R., Hronová, S., 2015. Benford’s Law and possibilities for its use in governmental statistics. Statistika, 95(2): 54-64.
  • Hürlimann, W., 2009. Generalizing Benford’s law using power laws: Application to integer sequences. International Journal of Mathematics and Mathematical Sciences, Article ID 970284, Doi:10.1155/2009/970284.
  • Hürlimann, W., 2003. A generalized Benford law and its application. Mathematics Preprint Archive, 3(3): 217-228.
  • Hürlimann, W., 2015. On the uniform random upper bound family of first sig-nificant digit distributions. Journal of Informetrics, 9(2): 349-358.
  • Judge, G., Schechter, L., 2009. Detecting problems in survey data using Benford’s Law. Journal of Human Resources, 44(1): 1-24.
  • Leroy, B., 2016. Calculation of rarity indices for species and assemblages of species. R package version 1.3-4. Available at: https://CRAN.R-project.org/package=Rarity.
  • Machado, A., 2004. An index of naturalness. Journal for Nature Conservation, 12: 95-110.
  • Margules, C., Usher, M.B., 1981. Criteria used in assessing wildlife conservation potential: A review. Biological Conservation, 21:79–109.
  • Newcomb, S., 1881. Note on the frequency of use of the different digits in natural numbers. American Journal of Mathematics, 4(1): 39–40.
  • Nigrini, M.J., Mittermaier, L.J., 1997. The use of Benford's law as an aid in analytical procedures. Auditing, 16(2): 52.
  • Nigrini, M.J., Miller, S.J., 2007. Benford’s Law applied to hydrology data—Results and relevance to other geophysical data. Mathematical Geology, 39(5): 469–490. http://doi.org/10.1007/s11004-007-9109-5.
  • Özkan, K., 2016. Biyolojik Çeşitlilik Bileşenleri (α, β ve γ) Nasıl Ölçülür (1. Basım). Süleyman Demirel Üniversitesi, Orman Fakültesi Yayınları, Isparta.
  • Özkan, K., 2003. Beyşehir Gölü Havzasının Yetişme Ortamı Özellikleri ve Sınıflandırılması. Doktora Tezi, İÜ Araştırma Fonu Proje Numarası T-981/19022001, 189s.
  • Reif, A., Walentowski, H., 2008. The assessment of naturalness and its role for nature conservation and forestry in Europe. Waldökologie Landschaftsforschung Naturschutz, 6:63–76.
  • Shao, L., Ma, B.Q., 2010. First-digit law in nonextensive statistics, Physical Review E, 82(4): 1-4, 10.1103/PhysRevE.82.041110.
  • Smith, P.G.R., Theberge, J.B., 1986. A review of criteria for evaluating natural areas. Environmental Management, 10(6): 715-734.
  • Tans, W., 1974. Priority ranking of biotic natural areas. Michigan Botanist, 13: 31–39.
  • Tseng, H.C., Huang, W.N., Huang, D.W., 2017. Modified Benford's law for two-exponent distributions. Scientometrics, 110(3): 1403–1413.
There are 37 citations in total.

Details

Primary Language English
Journal Section Orijinal Araştırma Makalesi
Authors

Kürşad Özkan 0000-0002-8526-7243

Publication Date June 29, 2021
Acceptance Date May 17, 2021
Published in Issue Year 2021 Volume: 22 Issue: 2

Cite

APA Özkan, K. (2021). Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law. Turkish Journal of Forestry, 22(2), 73-82. https://doi.org/10.18182/tjf.907217
AMA Özkan K. Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law. Turkish Journal of Forestry. June 2021;22(2):73-82. doi:10.18182/tjf.907217
Chicago Özkan, Kürşad. “Estimating Ecosystem Naturalness Using Benford’s Law and Generalized Benford’s Law”. Turkish Journal of Forestry 22, no. 2 (June 2021): 73-82. https://doi.org/10.18182/tjf.907217.
EndNote Özkan K (June 1, 2021) Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law. Turkish Journal of Forestry 22 2 73–82.
IEEE K. Özkan, “Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law”, Turkish Journal of Forestry, vol. 22, no. 2, pp. 73–82, 2021, doi: 10.18182/tjf.907217.
ISNAD Özkan, Kürşad. “Estimating Ecosystem Naturalness Using Benford’s Law and Generalized Benford’s Law”. Turkish Journal of Forestry 22/2 (June 2021), 73-82. https://doi.org/10.18182/tjf.907217.
JAMA Özkan K. Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law. Turkish Journal of Forestry. 2021;22:73–82.
MLA Özkan, Kürşad. “Estimating Ecosystem Naturalness Using Benford’s Law and Generalized Benford’s Law”. Turkish Journal of Forestry, vol. 22, no. 2, 2021, pp. 73-82, doi:10.18182/tjf.907217.
Vancouver Özkan K. Estimating ecosystem naturalness using Benford’s Law and Generalized Benford’s Law. Turkish Journal of Forestry. 2021;22(2):73-82.