Sigmoidal Functions: Some Computational and Modelling  Aspects

Kyurkchiev, Nikolay; Markov, Svetoslav

doi:10.11145/j.bmc.2015.03.081

Cited by 50 publications

(41 citation statements)

References 8 publications

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“…Examples of smooth sigmoidal functions, satisfying all assumptions required by the above theory, are provided by the well‐known logistic function ,

σ_{ℓ} false(x false) : = {(1 + e^{- x})}^{- 1}

x \in R

, , , , , , , and by the hyperbolic tangent function,

σ_{h} false(x false) : = false(prefixtanh x + 1 false) / 2

x \in R

, –. In particular,

σ_{ℓ}

and

σ_{h}

satisfy condition (Σ3) for all

α > 0

, in view of their exponential decay to zero, as

x \to - \infty

.…”

Section: Special Cases and Examples Of Sigmoidal Activation Functionsmentioning

confidence: 99%

Convergence for a family of neural network operators in Orlicz spaces

Costarelli

Vinti

2016

Mathematische Nachrichten

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In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators provide a constructive approximation process, in which the coefficients, the weights, and the thresholds of the networks needed in order to approximate a given function f, are known. At the end of the paper, several examples of Orlicz spaces, and of sigmoidal activation functions for which the present theory can be applied, are studied in details.

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“…Examples of smooth sigmoidal functions, satisfying all assumptions required by the above theory, are provided by the well‐known logistic function ,

σ_{ℓ} false(x false) : = {(1 + e^{- x})}^{- 1}

x \in R

, , , , , , , and by the hyperbolic tangent function,

σ_{h} false(x false) : = false(prefixtanh x + 1 false) / 2

x \in R

, –. In particular,

σ_{ℓ}

and

σ_{h}

satisfy condition (Σ3) for all

α > 0

, in view of their exponential decay to zero, as

x \to - \infty

.…”

Section: Special Cases and Examples Of Sigmoidal Activation Functionsmentioning

confidence: 99%

Convergence for a family of neural network operators in Orlicz spaces

Costarelli

Vinti

2016

Mathematische Nachrichten

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“…For a situation when κ is small it seems not natural to consider the point γ − δ as a definition of the lag time. That is why in a series of papers we propose another definition of lag time, namely γ − δ wherein δ is the Hausdorff distance between the sigmoidal function and the induced step function [17]- [25].…”

Section: Lag Timementioning

confidence: 99%

“…There exists a vast literature on sigmoidal functions. The field is characterized by a huge number of studies on real world growth phenomena and attempts to explain the intrinsic mechanisms of these phenomena using various mathematical methods [7], [8], [21], [33].…”

Section: Introductionmentioning

confidence: 99%

Building reaction kinetic models for amiloid fibril growth

Markov

2016

BIOMATH

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Abstract-In this work we discuss some methodological aspects of the creation and formulation of mathematical models describing the growth of species from the point of view of reaction kinetics. Our discussion is based on familiar examples of growth models such as logistic growth and enzyme kinetics. We propose several reaction network models for the amiloid fibrillation processes in the citoplasm. The solutions of the models are sigmoidal functions graphically visualized using the computer algebra system Mathematica.

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“…As shown in [28], equation (22) can be interpreted as y = ksy, wherein s = s(t) is the nutrient substrate used for the growth of the population; one see that s is a decay exponential function in the Gompertz model (a similar interpretation can be found in [21]), [40]). For other interpretations see [6]), [8], [20].…”

Section: Approximation Of the Step Functionmentioning

confidence: 99%

“…logistic functions in Hausdorff metric. The Hausdorff approximation of the Heaviside step function by sigmoid functions is discussed from various computational and modelling aspects in [28], [29], [30].…”

Section: Propositionmentioning

confidence: 99%

On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

Илиев

Kyurkchiev²,

Markov³

2015

BIOMATH

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Abstract-We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued step function is discussed using Hausdorff metric. Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical examples are presented using CAS MATHEMATICA.

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Sigmoidal Functions: Some Computational and Modelling Aspects

Cited by 50 publications

References 8 publications

Convergence for a family of neural network operators in Orlicz spaces

Convergence for a family of neural network operators in Orlicz spaces

Building reaction kinetic models for amiloid fibril growth

On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

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