On the Hausdorff distance between the shifted Heaviside step function and the transmuted Stannard growth function

—In this paper we study the one-sided Hausdorff distance between the shifted Heaviside step–function and the transmuted Stannard growth function. Precise upper and lower bounds for the Hausdorff distance have been obtained. We present a software module (intellectual property) within the programming environment CAS Mathematica for the analysis of the growth curves. Numerical examples, illustrating our results are given, too.


I. INTRODUCTION AND PRELIMINARIES
The Stannard function finds numerous applications in many scientific fields, including population dynamics, bacterial growth, population ecology, plant biology, chemistry, agriculture, demography, financial mathematics, statistics and fuzzy set theory [1]- [5]. Definition 1. For γ ∈ R define the shifted Heaviside step function as [12]: Definition 2. Define the shifted Stannard growth function S(t) as [1]- [5]: where β, k and m ∈ R are the growth parameters. We note that the slope of (2) at t = γ is equal to: Definition 3. A random variable T is said to have a transmuted distribution if its cumulative distribution function (cdf) is given by [6], [7]: "λ -transmuting" of (cdf) is a familiar technique from the field of probability distributions with application to insurance mathematics.
Definition 4. The Hausdorff 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 [8], [9], [12]. More precisely, we have wherein ||.|| is any norm in R 2 , e. g. the maximum norm ||(t, x)|| = max{|t|, |x|}; hence the distance between the points Sigmoidal growth curves typically have three parts (phases, time intervals): lag, log and stationary parts. It is a challenging question to characterize mathematically these phases. The lag time (interval) is practically important in many medical and biotechnological applications as this time is responsible for the acceleration or inhibition of the process and the possibility of controlling the lag time depends on the understanding of the hidden mechanisms of the corresponding process [10], [11].
Usually the lag time is defined by means of the uniform distance between the sigmoidal function and the induced cut function. We propose a new definition for the lag time by means of the Hausdorff distance between the sigmoidal function and the induced step function.
In this work we prove estimates for the onesided Hausdorff approximation of the shifted Heaviside step-function by transmuted Stannard growth function.
Let us point out that the Hausdorff distance is a natural measuring criteria for the approximation of bounded discontinuous functions [12], [13].  Function S * (t) from (5) satisfies: We study the Hausdorff approximation d of the Heaviside step function h γ (t) by the transmuted Stannard function (5)-(7) and look for an expression for the error of the best one-sided approximation. Let The following Theorem gives upper and lower bounds for d.
Proof. We need to express d in terms of k, β and m. The Hausdorff distance d satisfies the relation Consider the function By means of Taylor expansion we obtain . Fig. 3). Further, for |λ| ≤ 1 and B > 4 we have This completes the proof of the theorem.
Some computational examples using relations (9) are presented in Table 1. The last column of Table 1 contains the values of d computed by solving the nonlinear equation (10). III. CONCLUSION REMARKS New estimates for the Hausdorff distance between an interval Heviside step function and its best approximating Stannard function are obtained.
On Fig. 1 and Fig. 2 appropriate illustrations of some approximations of the shifted Heaviside step function by transmuted Stannard growth function are given.
We propose a software module within the programming environment CAS Mathematica for the analysis of the considered growth curves (see Fig.  4). The module offers the following possibilities: i) generation of the shifted Stannard curve under user-defined values for k, m, β; ii) automatic check of the condition |λ| ≤ 1 that guarantees the existence of sigmoidality of the transmuted Stannard curve; iii) software tools for animation and visualization.
The Hausdorff approximation of the interval step function by the logistic and other sigmoidal functions is discussed from various approximation, computational and modelling aspects in [14]- [27].