Benford Kanunu ve Genelleştirilmiş Benford Kanunu ile ekosistem doğallığının hesaplanması

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.

Anahtar Kelimeler:

Hemerobi, Doğallık, Orman ekosistemleri, İlk sayı kanunu, Genelleştirme, Rastgele seçim

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

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.

Keywords:

Hemeroby, Naturalness, Forest ecosystems, First digit rule, Generalization, Randomization,

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