APPLICATIONS OF SOME NEW TRANSMUTED CUMULATIVE DISTRIBUTION FUNCTIONS IN POPULATION DYNAMICS

1. Introduction

In literature, several transformations exists to obtain a new cumulative distribution function (cdf) using other(s) well-known 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 R², 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 R² is ||A-B||= 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 H-distance 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 H-distance 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 H-distance 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 H-distance  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 H-distance  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 H-distance  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 H-distance 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 t₀;

- calculation of the H-distance 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.