A new class of activation functions. Some related problems and applications

The cumulative distribution function (cdf) of the discrete two--parameter bathtub hazard distribution has important role in the fields of population dynamics, reliability analysis and life testing experiments. Also of interest to the specialists is the task of approximating the Heaviside function by new (cdf) in Hausdorff sense. We define new activation function and family of new recurrence generated functions and study the ''saturation'' by these families. In this paper we analyze some intrinsic properties of the new Topp-Leone-G-Family with baseline ''deterministic-type'' (cdf) - (NTLG-DT). Some numerical examples with real data from Biostatistics, Population dynamics and Signal theory, illustrating our results are given. It is shown that the study of the two characteristics - "confidential curves" and ''super saturation'' is a must when choosing the right model. Some related problems are discussed, as an example to the Approximation Theory.

Definition 3. [3] The Hausdorff distance (the Hdistance) 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 Definition 4. We define the following activation function: Definition 5. Define the following family of new recurrence generated functions based on the function A(t; β).

II. MAIN RESULTS
When studying the intrinsic properties of the family M β (t), it is also appropriate to study the "saturation" to the horizontal asymptote.
In this Section we give upper and lower estimates for the one-sided Hausdorff approximation of the Heaviside step-function h t0 (t) by means of family (1), where t 0 is the level of the "median".
Let t 0 is the unique positive root of the nonlinear equation M 1 (t 0 ) − 1 2 = 0. The one-sided Hausdorff distance d between h t0 (t) and the function (1) satisfies the relation The following theorem gives upper and lower bounds for d Theorem 1. Let β = 1, q < 2 e 2 ( e 1.05 Then, for the one-sided Hausdorff distance d between h t0 (t) and the (cdf) -(1) the following inequalities hold: Proof. In order to express d in terms of q, let us examine the function Consider then the function Fig. 1).
We look for two reals d l and d r such that g(d l ) < 0 and g(d r ) > 0 (leading to g(d l ) < d < g(d r )).
For given β = 1 the one-sided Hausdorff distance d satisfies the relation The reader may formulate the corresponding approximation problem following the ideas given in Theorem 1, and will be omitted.
It is well known that in many cases the existing modifications to the classical logistic and Gompertz models do not give very reliable results in approximating "specific data".
We examine the following "specific datasets": Example 1. We analyze the following data [4] data The cdf M β (t) for β = 0.484411 and q = 0.82547 is visualized on Fig. 3.

The new activation function.
We define the following activation function: In antenna-feeder technique most often occurred signals are of types shown on Fig. 6 -Fig. 7.
For β even, the corresponding approximation using model (7) is shown in Fig. 6.
For β odd, the corresponding approximation using new activation function A(t; β) is shown in Fig. 7.
A family of recurrence generated functions based on the A(t; β).
Let us consider the following family of recurrence generated functions based on the function A(t; β).
We study the Hausdorff approximation of the Heaviside step function h t0 (t) where t 0 is the "median" by families of the new Topp-Leone-G-Family with baseline "deterministic-type" (cdf) -(NTLG-DT).
The obtained two-sides estimations (see Proposition 1. [1] ) in particular case with usage of the baseline "deterministic-type" (cdf) for α = 0.9; β = 0.3; q = 0.1 are given in Fig. 9 a. Let t 0 is the value for which Q(t 0 ) = 1 2 . The Hausdorff distance d between the function h t0 (t) and Q(t) satisfies the relation For fixed α = 0.9; β = 0.3; q = 0.1 we find t 0 = 0.0852097 and from the nonlinear equation (15) we have d = 0.116811 (see, Fig. 9 b). From Fig. 9 it can be seen that these estimations can be used as "confidence bounds", which are extremely useful for the specialists in the choice of model for cumulative data approximating in areas of Biostatistics, Population dynamics, Growth theory, Debugging and Test theory, Computer viruses propagation, Financial and Insurance mathematics.
The results obtained in this article can be successfully continued.
Exploring both features -"confidential curves" and "super saturation" is a must when choosing the right model.
The general solution of the differential equation (18) is of the following form: where Ei(.) is the traditional exponential integral. The new "growth" function y(t) and the "input function" s(t) = q e t −1 are visualized on Fig. 12-Fig. 13. Example 4. We will analyze a sample of experimental data obtained by the biologist T. Carlson in 1913 about the development of Saccharomyces culture in nutrient medium (see, for example [58], [57]).
After that using the model M * (t) for ω = 0.305247, k = 3.01914 and q = 0.83 we obtain the fitted model (see, Fig. 15).
The general solution y(t) has been applied widely in life testing experiments and debugging theory.

ACKNOWLEDGMENT
The author would like to thank the anonymous referees for their valuable comments. This paper is supported by the National Scientific Program "Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)", financed by the Ministry of Education and Science.