Approximation of the Cut Function by Some Generic Logistic Functions and Applications
Nikolay Kyurkchiev, Svetoslav Markov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
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To cite this article:
Nikolay Kyurkchiev, Svetoslav Markov. Approximation of the Cut Function by Some Generic Logistic Functions and Applications. Advances in Applied Sciences. Vol. 1, No. 2, 2016, pp. 24-29. doi: 10.11648/j.aas.20160102.11
Received: August 17, 2016; Accepted: August 27, 2016; Published: September 12, 2016
Abstract: In this paper we study the uniform approximation of the cut function by smooth sigmoid functions such as Nelder and Turner–Blumenstein–Sebaugh growth functions. To illustrate the use of one of the models we have fitted the model to the "classical Verhulst data". Several numerical examples are presented throughout the paper using the contemporary computer algebra system MATHEMATICA.
Keywords: Sigmoid Functions, Cut Function, Step Function, Nelder Growth Function, Turner–Blumenstein–Sebaugh Generic Function, Uniform Approximation
1. Introduction
We study the uniform approximation of the cut function by Nelder and Turner–Blumenstein–Sebaugh growth functions. We find an expression for the error of the best uniform approximation.
The estimates obtained give more insight on the lag phase, growth phase and plateau phase in the growth process [1–4].
2. Preliminaries
2.1. Sigmoid Functions
In this work we consider sigmoidal functions of a single variable defined on the real line, that is functions of the form . Sigmoid functions can be defined as bounded monotone non-decreasing functions on . One usually makes use of normalized sigmoidal functions defined as monotone non-decreasing functions , such that and (in some applications the left asymptote is assumed to be : ).
2.2. The Cut and the Nelder and Turner–Blumenstein–Sebaugh Growth Functions
The cut (ramp) function is the simplest piece-wise linear sigmoidal function. Let be an interval on the real line with centre and radius . A cut function is defined as follows:
Definition. The cut function is defined for by
(1)
Note that the slope of function on the interval is (the slope is constant in the whole interval ).
Two special cases are of interest for present discussion in the sequel.
Special case 1. For we obtain the special cut function on the interval :
Special case 2. For we obtain the special cut function on the interval :
In 1961 Nelder [6] consider the differential equation
with the solution
(2)
When the ordinary logistic equation is obtained. An attractive choice for is given by Turner–Blumenstein–Sebaugh in [5]:
The generalization (the generic logistic equation) of the Verhulst logistic equation has the form [5]:
(3)
where are growth parameters. If and either or then (3) reduces to the ordinary logistic equation.
Definition. Define the following shifted modification of (3) with jump at point as:
(4)
for which .
Figure 1. The cut and the function with , , , , , , , uniform distance .
Figure 2. The cut and the function with , , , , , , , uniform distance .
3. Approximation of the Cut Function by Shifted Turner–Blumenstein–Sebaugh Function (4)
We next focus on the approximation of the cut function (1) by shifted Turner–Blumenstein–Sebaugh (STBS) growth function defined by (4).
Note that the slope of at is
and the slope of at is
Let choose . The function defined by (4) has an inflection at point (see Fig. 1 and Fig. 2).
Consider functions (1) and (4) with same centres , that is functions and .
In addition chose and to have same slopes at their coinciding centres.
Then, noticing that the largest uniform distance between the cut and (STBS) functions is achieved at the endpoints of the underlying interval we have:
The above can be summarized in the following
Theorem 1. The function defined by (4): i) is the (STBS) function of best uniform one-sided approximation to function in the interval (as well as in the interval ); ii) approximates the cut function in uniform metric with an error
(5)
4. Approximation of the Cut Function by Nelder Function [6]
Definition. Define the special shifted Nelder growth function with jump at point as:
(6)
Then .
We next focus on the approximation of the cut function (1) by shifted Nelder growth function defined by (6).
Note that the slope of at is and the slope of at is
Let choose . The function defined by (5) has an inflection at point (see Fig. 3 and Fig. 4).
Consider functions (1) and (6) with same centres , that is functions and .
In addition chose and to have same slopes at their coinciding centres.
Then, noticing that the largest uniform distance between the cut and Nelder functions is achieved at the endpoints of the underlying interval :
The above can be summarized in the following
Theorem 2. The function defined by (6): i) is the Nelder function of best uniform one-sided approximation to function in the interval (as well as in the interval ); ii) approximates the cut function in uniform metric with an error
(7)
Figure 3. The cut and the function (6) with , , , , , uniform distance .
Figure 4. The cut and the function (6) with , , , , , uniform distance .
Remark. Let us point out that estimate (7) does not make use of the parameter .
5. Computational Issues and Fitting the Model (3) (with Jump at γ) Against Verhulst Data
To illustrate the use of the model (3) we have fitted the model to the Verhulst data by use of software module in programming environment CAS Mathematica.
In his famous work [7] Verhulst applies the logistic model to fit census data for the population in France. The given data in column 3 (Fig. 5) will be briefly called Verhulst data.
The appropriate fitting of Verhulst data by the function (3) with , , , , , , is visualized on Fig. 6.
The following scales are appropriate: on the abscissa 1000 units correspond to one division; on the ordinate 1000000 units correspond to one division.
We may expect somewhat more accuracy in predicting the population of metropolitan France for 1841, 1851 and 2010 (see Table 1).
Year | Population by TBS model | Population — National Inst. of Statistics |
1841 | 34559900 | 34912000 |
1851 | 36489500 | 36472000 |
2010 | 62850000 | 62765000 |
Let us note that the results are unexpectedly reliable, especially in relation to extrapolation for the year 2010. In this case the relative error is on the order of 0.135%.
The experts from the National Institute of Statistics - France, when studying in detail the population of metropolitan France pay special attention to existing two centuries of population growth: First World War 1914–1918 and Second World War 1939–1945, (see Fig. 7; https://en.wikipedia.org/wiki/Demographics of France).
It can be concluded from Fig. 7 that knowledge of the lag-time is very substantial for choosing the correct empirical growth model.
We hope that the results on the approximation of the cut function by smooth sigmoidal functions can be useful for choosing the correct population model.
For some approximation, computational and modelling aspects, see [8–25].
Acknowledgments
The authors would like to thank the anonymous reviewers for their helpful comments that contributed to improving the final version of the presented paper.
References