Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on Estimation of Multi-Sigmoidal Gompertz Functions with Random Noise

Patricia Román-Román, Juan José Serrano-Pérez, Francisco Torres-Ruiz

Abstract

The behaviour of many dynamic real phenomena shows different phases, with each one following a sigmoidal type pattern. This requires studying sigmoidal curves with more than one inflection point. In this work, a diffusion process is introduced whose mean function is a curve of this type, concretely a transformation of the well-known Gompertz model after introducing in its expression a polynomial term. The maximum likelihood estimation of the parameters of the model is studied, and various criteria are provided for the selection of the degree of the polynomial when real situations are addressed. Finally, some simulated examples are presented.

A Note on Estimation of Multi-Sigmoidal Gompertz Functions with Random Noise

Abstract

The behaviour of many dynamic real phenomena shows different phases, with each one following a sigmoidal type pattern. This requires studying sigmoidal curves with more than one inflection point. In this work, a diffusion process is introduced whose mean function is a curve of this type, concretely a transformation of the well-known Gompertz model after introducing in its expression a polynomial term. The maximum likelihood estimation of the parameters of the model is studied, and various criteria are provided for the selection of the degree of the polynomial when real situations are addressed. Finally, some simulated examples are presented.
Paper Structure (8 sections, 42 equations, 9 figures, 7 tables)

This paper contains 8 sections, 42 equations, 9 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Multi-sigmoidal Gompertz curve
  3. Multi-sigmoidal Gompertz diffusion process
  4. Maximum likelihood estimation
  5. System of equations and numerical computations
  6. About $t_0$
  7. Simulations
  8. Conclusions

Figures (9)

  • Figure S1: Several examples of multi-sigmoidal Gompertz curves.
  • Figure S2: Example 1. Simulated sample-paths. Black lines represent the sample mean.
  • Figure S3: Example 1. Theoretical, sample and estimated mean functions.
  • Figure S4: Example 1. Resistor-average distances between the theoretical and estimated models (up) and between the sample and estimated models (down).
  • Figure S5: Example 1. Theoretical, sample, and estimated inflection time instants.
  • ...and 4 more figures