APPLICATIONS OF SOME NEW TRANSMUTED CUMULATIVE DISTRIBUTION FUNCTIONS IN POPULATION DYNAMICS
APPLICATIONS OF SOME NEW TRANSMUTED CUMULATIVE DISTRIBUTION FUNCTIONS IN POPULATION DYNAMICS
Funding
This work has been supported by the project FP17FMI008 of Department for Scientific Research, Paisii Hilendarski University of Plovdiv.
Conflict of Interest
None declared.
Vesselin Kyurkchiev^{1}, Anton Iliev^{1,2*}, Nikolay Kyurkchiev^{1}
^{1}Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski,
24, Tzar Asen Str., 4000 Plovdiv, Bulgaria
^{2}Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,
Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
*To whom correspondence should be addressed.
Associate editor: Giancarlo Castellano
Received on 02 April 2017, revised on 10 April 2017, accepted on 25 April 2017.
Abstract
Motivation: In literature, several transformations exists to obtain a new cumulative distribution function (cdf) using other(s) wellknown cdf(s).
Results: In this note we find applications of some new cumulative distribution function transformations to construct a family of sigmoidal functions based on the Verhulst logistic function.
We prove estimates for the Hausdorff approximation of the shifted Heaviside step function by means of this family. Numerical examples, illustrating our results are given.
Keywords: Cumulative Distribution Function, Logistic Function, Shifted Heaviside Step Function, Hausdorff Distance, Upper and Lower Bounds.
Contact: aii@uniplovdiv.b
1. Introduction
In literature, several transformations exists to obtain a new cumulative distribution function (cdf) using other(s) wellknown cdf(s) (Aryal & Tsokos, 2009; Aryal, 2013; Gupta, R. G., Gupta, P. L. & Gupta, R. D., 1998; Khan & King, 2013; Kumar, Singh, & Singh, 2015a; Kumar, Singh & Singh, 2015b; Kumar, Singh & Singh 2017).
Definition 1 Another popular transformation by using a (cdf) F(t) is (Kumar, Singh, & Singh, 2015a):
The transformation (1) has great applications in data analysis.
Definition 2 Define the logistic (Verhulst) function f on R as
The logistic function belongs to the important class of smooth sigmoidal functions arising from population and cell growth models.
Since then the logistic function finds applications in many scientific fields, including biology, population dynamics, chemistry, demography, economics, geoscience, mathematical psychology, probability, financial mathematics, statistics, insurance mathematics to name a few (Anguelov & Markov, 2016; Lente, 2015; Kyurkchiev & Markov, 2016a; Kyurkchiev, 2016a; Costarelli & Spigler, 2013; Kyurkchiev & Markov, 2014; Kyurkchiev & Markov, 2015; Kyurkchiev & Markov, 2016b).
Definition 3 The (interval) step function is:
usually known as shifted Heaviside step function.
Definition 4 (Hausdorff, 2005; Sendov, 1990) The Hausdorff distance (the H–distance) ρ(f,g) between two interval functions f,g on Ω⊆R, is the distance between their completed graphs F(f) and F(g) considered as closed subsets of Ω×R. More precisely,
wherein . is any norm in R^{2}, e. g. the maximum norm (t,x)=max{t,x}; hence the distance between the points A=(t_{A},x_{A}), B=(t_{B},x_{B}) in R^{2} is AB= max (t_{A}t_{B},x_{A}x_{B}).
In this paper we discuss several computational, modelling and approximation issues related to two familiar classes of sigmoidal functions–these are the families of transmuted cumulative distribution functions.
2. Methods
1. Let us consider the following sigmoid
based on (1) with the Verhulst logistic function f(t).
The one–sided Hdistance d=ρ(h_{t0},G) between the shifted Heaviside step function h_{t0} and the sigmoidal function G satisfies the relation:
The following theorem gives upper and lower bounds for d=d(k).
Theorem 2.1 The one–sided Hdistance d(k) between the function h_{t0 }and the function G can be expressed in terms of the rate parameter k for any real k≥2 as follows:
Proof. We define the functions
From Taylor expansion
we see that the function G1(d) approximates F1(d) with d→0 as O(d2) (see Figure 1).
In addition G'1d>0 and for k≥2.
G1(dl)<0; G1(dr)>0.
This completes the proof of the inequalities (7).
Fig. 1  The functions and for .
The generated sigmoidal function G(t) for k=20 is visualized on Figure 2.
Fig. 2  The Hdistance d(k) between the functions and G for k=20 is d=0.105561; ; .
Some computational examples using relations (7) are presented in Table 1. The third column of Table 1 contains the value of d for prescribed values of k computed by solving the nonlinear equation (6).
Table 1. Bounds for d(k) computed by (6) and (7) for various rates k
k 
dl 
d computed by (6) 
dr 
30 
0.0462014 
0.0747728 
0.142182 
40 
0.0357291 
0.0627923 
0.119042 
50 
0.0291032 
0.0541761 
0.102935 
100 
0.015101 
0.0296749 
0.0633183 
500 
0.00311425 
0.00617859 
0.0179747 
1000 
0.00156321 
0.0048109 
0.0100999 
Definition 5 Another popular transformation by using a (cdf) F(t) is (Kumar, Singh & Singh, 2017):
2. Let us consider the following sigmoid
with
based on (10) with the Verhulst logistic function f(t).
The one–sided Hdistance between the shifted Heaviside step function and the sigmoidal function G1 satisfies the relation:
The following theorem gives upper and lower bounds for
Theorem 2.2 The one–sided Hdistance between the function and the function can be expressed in terms of the rate parameter for any real as follows:
The proof follows the ideas given in this paper and will be omitted.
Fig. 3  The Hdistance between the functions and for is .
Fig. 4  Comparison between G (red) and G1 (green) for k=20.
3. Results
To achieve our goal, we obtain new estimates for the one–sided Hdistance between a shifted Heaviside step function and its best approximating family of transmuted cumulative distribution functions–these are the families of functions G(t) and G1(t) based on the Verhulst logistic function.
Numerical examples, illustrating our results are given.
In some cases the approximation of shifted Heaviside function by G1(t) is better in comparison to its approximation by G(t) (see Figure 4).
For other results, see (Iliev, Kyurkchiev & Markov, 2017a; Kyurkchiev, 2015; Kyurkchiev & Iliev, 2016; Kyurkchiev, V. & Kyurkchiev, N., 2015; Kyurkchiev & Markov, 2016c; Iliev, Kyurkchiev & Markov, 2017b; Kyurkchiev, V. & Kyurkchiev N., 2017; Kyurkchiev, 2016b).
Fig. 5  Software tools in CAS Mathematica.
We propose a software module within the programming environment CAS Mathematica for the analysis of the considered families of transmuted cumulative distribution functions.
The module offers the following possibilities:
 generation of the functions G(t) and G1(t) under user defined values of the reaction rate k and t_{0};
 calculation of the Hdistance between the Heaviside function h_{t0} and the sigmoidal functions G(t) and G1(t);
 software tools for animation and visualization.
4. Appendix
Focusing on the shifted logistic function
and the shifted function
Fig. 6  Interpolation of the experimental data by model (16).
We examine the following experimental data (biomass) for Xantobacter autotrophycum by the model .
Table 2. The experimental data (biomass) for Xantobacter autotrophycum and approximation by for and
t 
Biomass 

0 
0.104 
0.0736774 
0.065 
0.233 
0.229003 
0.099 
0.39 
0.372755 
0.125 
0.507 
0.50254 
0.145 
0.618 
0.602883 
0.188 
0.766 
0.784021 
0.233 
0.88 
0.899886 
From Figure 6 it can be seen that the results are satisfactory. We point out that in similar ”exponential” data type the results are near to Gompertz growth model.
Acknowledgment. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the paper quality.
References
Anguelov, R. & Markov, S. (2016). Hausdorff Continuous Interval Functions and Approximations, In: M. Nehmeier et al. (Eds), Scientific Computing, Computer Arithmetic, and Validated Numerics, 16th International Symposium, SCAN 2014, LNCS 9553, 313, Springer. doi:10.1007/9783319317694
Aryal, G. R. (2013). Transmuted loglogistic distribution. Journal of Statistics Applications and Probability, 2 (1), 1120.
Aryal, G. R. & Tsokos, C. P. (2009). On the transmuted extreme value distribution with application, Nonlinear Analysis: Theory, Methods and Applications, 71 (12), 14011407.
Costarelli, D. & Spigler, R. (2013). Constructive Approximation by Superposition of Sigmoidal Functions. Analysis in Theory and Applications, 29 (2), 169196. doi:10.4208/ata.2013.v29.n2.8
Gupta, R. G., Gupta, P. L. & Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics, Theory and Methods, 27, 887904.
Hausdorff, F. (2005). Set Theory (2 ed.), Chelsea Publishing, New York, Republished by AMSChelsea.
Iliev, A., Kyurkchiev, N. & Markov, S. (2017a). On the Approximation of the step function by some sigmoid functions. Mathematics and Computers in Simulation, 133, 223234. doi:10.1016/j.matcom.2015.11.005
Iliev, A., Kyurkchiev, N. & Markov, S. (2017b). A family of recurrence generated parametric activation functions with applications to neural networks. International Journal on Research Innovations in Engineering Science and Technology (IJRIEST), 2 (1), 6068.
Khan, M. & King, R. (2013). Transmuted modified Weibull distribution: A generalization of the modified Weibull probability distribution. European Journal of Pure and Applied Mathematics, 6 (1), 6688.
Kumar, D., Singh, U. & Singh, S. (2015a). A method of proposing new distribution and its application to Bladder cancer patients data. Journal of Statistics Applications & Probability Letters, 2 (3), 235245.
Kumar, D., Singh, U. & Singh, S. (2015b). A new distribution using sine function and its application to Bladder cancer patients data. Journal of Statistics Applications & Probability, 4 (3), 417427.
Kumar, D., Singh, U. & Singh, S. (2017). Lifetime distribution: derived from some minimum guarantee distribution. Sohag Journal of Mathematics, 4 (1), 711.
Kyurkchiev, N. (2015). On the Approximation of the step function by some cumulative distribution functions. Comptes rendus de l’Académie bulgare des Sciences, 68 (12), 14751482.
Kyurkchiev, N. (2016a). Mathematical Concepts in Insurance and Reinsurance. Some Moduli in Programming Environment MATHEMATICA. LAP LAMBERT Academic Publishing, Saarbrucken, 136 pp.
Kyurkchiev, N. (2016b). A family of recurrence generated sigmoidal functions based on the Verhulst logistic function. Some approximation and modelling aspects. Biomath Communications, 3 (2), 18 pp. doi:10.11145/bmc.2016.12.171
Kyurkchiev, N. & Iliev, A. (2016). On the Hausdorff distance between the shifted Heaviside function and some generic growth functions. International Journal of Engineering Works, 3 (10), 7377.
Kyurkchiev, N. & Markov S. (2016b). On the Hausdorff distance between the Heaviside step function and Verhulst logistic function. Journal of Mathematical Chemistry, 54 (1), 109119. doi:10.1007/S1091001505520
Kyurkchiev, N. & Markov, S. (2014). Sigmoidal functions: some computational and modelling aspects. Biomath Communications, 1 (2), 3048. doi:10.11145/j.bmc.2015.03.081
Kyurkchiev, N. & Markov, S. (2015). Sigmoid Functions: Some Approximation and Modelling Aspects. Some Moduli in Programming Environment Mathematica. LAP LAMBERT Academic Publishing, Saarbrucken.
Kyurkchiev, N. & Markov, S. (2016a). On the numerical solution of the general kinetic Kangle reaction system. Journal of Mathematical Chemistry, 54 (3), 792805. doi:10.1007/s1091001605920
Kyurkchiev, N. & Markov, S. (2016c). Hausdorff approximation of the sign function by a class of parametric activation functions. Biomath Communications, 3 (2), 11 pp.
Kyurkchiev, V. & Kyurkchiev N. (2017). A family of recurrence generated functions based on the ”halfhyperbolic tangent activation function”. Biomedical Statistics and Informatics, 2 (3). doi:10.11648/j.bsi.20170203.12
Kyurkchiev, V. & Kyurkchiev, N. (2015). On the Approximation of the Step function by RaisedCosine and Laplace Cumulative Distribution Functions. European International Journal of Science and Technology, 4 (9), 75–84.
Lente, G. (2015). Deterministic Kinetics in Chemistry and Systems Biology. Springer, New York.
Sendov, B. (1990). Hausdorff Approximations. Kluwer, Boston. doi:10.1007/9789400906730