Genelleştirilmiş Lojistik Büyüme Eğrisinin Birinci Türevinin Fourier Dönüşümü

Öz

Genelleştirilmiş lojistik büyüme eğrisi simetrisi olmayan sigmoid eğrileri için tipik bir örnektir ve genellikle lineer olmayan regresyon için kullanılır. Bir sigmoid eğrisinin “kritik noktası” kısaca, türevlerinin mutlak ekstremum noktalarının (eğer varsa) limiti olarak tanımlanır. Bir sigmoid eğrisinin kritik noktasının varlığı ve konumu Fourier dönüşümü ile ifade edilebilir. Bu çalışmada, genelleştirilmiş lojistik büyüme eğrisinin birinci türevinin Gama fonksiyonları cinsinden Fourier dönüşümü elde edilmiş ve bazı özel durumlar tartışılmıştır. 

Anahtar Kelimeler

• Lojistik büyüme,Fourier dönüşümü,Sigmoid eğrisi

The Fourier Transform of the First Derivative of the Generalized Logistic Growth Curve

Öz

The “generalized logistic growth curve” or the “5-point sigmoid” is a typical example for sigmoidal curves without symmetry and it is commonly used for non-linear regression. The “critical point” of a sigmoidal curve is defined as the limit, if it exists, of the points where its derivatives reach their absolute extreme values. The existence and the location of the critical point of a sigmoidal curve is expressed in terms of its Fourier transform. In this work, we obtain the Fourier transform of the first derivative of the generalized logistic growth curve in terms of Gamma functions and we discuss special cases.

Anahtar Kelimeler

• Logistic growth,Fourier transform,Sigmoidal curve

Kaynakça

1. Abramowitz, M., Stegun, I. A. (1972). Handbook of Mathematical Functions, Dover, New York, USA.
2. Beukers, F. (2007). Gauss’ Hypergeometric Function. Progress in Mathematics. 260, 23–42.
3. Bilge, A.H., Pekcan, O., Gurol, M.V. (2012). Application of epidemic models to phase transitions. Phase Transitions. 85(11), 1009–1017.
4. Bilge, A.H., Pekcan, O. (2013). A Mathematical Description of the Critical Point in Phase Transitions. Int. J. Mod. Phys. C. 24.
5. Bilge, A.H., Pekcan, O. (2015). A mathematical characterization of the gel point in sol-gel transition, Edited by: Vagenas, EC; Vlachos, DS; Bastos, C; et al., 3rd International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE 2014) August 28-31, 2014, Madrid, SPAIN, Journal of Physics Conference Series. 574.
6. Bilge, A.H., Ozdemir, Y. (2016). Determining the Critical Point of a Sigmoidal Curve via its Fourier Transform, Edited by Vagenas, E.C. and Vlachos, D.S., 5th International Conference on Mathematical Modeling in Physical Sciences(IC-MSQUARE 2016) May 23-26, 2016, Athens, GREECE, Journal of Physics Conference Series. 738.
7. Bilge, A.H., Pekcan, O., Kara, S., Ogrenci, S. (2017). Epidemic models for phase transitions: Application to a physical gel, 4th Polish-Lithuanian-Ukrainian Meeting on Ferroelectrics Physics Location: Palanga, LITHUANIA, 05-09 September 2016, Phase Transitions. 90(9), 905–913.
8. Gradshteyn, I.S., Ryzhik I.M. (2007). Table of Integrals, Series, and Products. A. Jeffrey, D. Zwillinger (ed.), Elsevier Inc., USA.

9. Papoulis, A. (1962). The Fourier Integral and its Applications. McGraw-Hill Co., New York, USA.
10. Pearson J. (2009). Computation of Hypergeometric Functions. MSc Thesis, Oxford University, UK.