A new Gompertz-type diffusion process with application to random growth

Jáimez, Ramón Gutiérrez; Román, Pablo; Romero, Desirée; Serrano, Juan José Pretel; Torres, F.

doi:10.1016/j.mbs.2006.09.020

Cited by 49 publications

(31 citation statements)

References 17 publications

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“…In the practical application of this type of models, the noise term has a moderate magnitude (see Gutiérrez et al [9] in the context of Gompertz-type growth). This feature makes the sample paths of the resulting processes show von Bertalanffy-like behavior patterns, except for the presence of small random fluctuations.…”

Section: The Von Bertalanffy Growth Curve Ismentioning

confidence: 99%

“…To this end, we follow the methodology considered in Gutiérrez et al [9] in which several methods are presented in order to introduce a diffusion process in the context of gompertzian growth. Specifically, we look for a process in which the solution of the Fokker-Planck equation, without noise, is such a curve.…”

Section: The Generalized Von Bertalanffy Diffusion Processmentioning

confidence: 99%

“…Firstly, the introduction of a stochastic model (specifically a diffusion process) to model behavior patterns associated with the generalized von Bertalanffy curve (2) that allows the consideration of both length and weight for some animal species. To this end, the model is obtained by applying the methodology developed by Gutiérrez et al [9] in the context of gompertzian growth: from the deterministic equation, whose solution is the curve of interest, a stochastic component is introduced as well as the condition that the mean function of the resulting diffusion process be a curve of the type (2) (that fits well the sample data and can be used for forecasting purposes) is imposed. The building of the model as well as the study of some of its characteristics (probability distribution of the process, mean, mode and quantile functions) is presented in sections 2 and 3.…”

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confidence: 99%

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A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

Román-Román

Romero

Torres-Ruiz

2010

Journal of Theoretical Biology

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The von Bertalanffy growth curve has been commonly used for modeling animal growth (particularly fish). Both deterministic and stochastic models exist in association with this curve, the latter allowing for the inclusion of fluctuations or disturbances that might exist in the system under consideration which are not always quantifiable or may even be unknown. This curve is mainly used for modeling the length variable whereas a generalized version, including a new parameter b ≥ 1, allows for modeling both length and weight for some animal species in both isometric (b = 3) and allometric (b = 3) situations.In this paper a stochastic model related to the generalized von Bertalanffy growth curve is proposed. This model allows to investigate the time evolution of growth variables associated both with individual behaviors and mean population behavior. Also, with the purpose of fitting the above mentioned model to real data and so be able to forecast and analyze particular characteristics, we study the maximum likelihood estimation of the parameters of the model. In addition, and regarding the numerical problems posed by solving the likelihood equations, a strategy is developed for obtaining initial solutions for the usual numerical procedures. Such strategy is validated by means of simulated examples. Finally, an application to real data of mean weight of swordfish is presented.

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Section: The Von Bertalanffy Growth Curve Ismentioning

confidence: 99%

Section: The Generalized Von Bertalanffy Diffusion Processmentioning

confidence: 99%

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confidence: 99%

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A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

Román-Román

Romero

Torres-Ruiz

2010

Journal of Theoretical Biology

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“…Later, Gutiérrez et al [6] introduced a non-homogeneous Gompertztype diffusion process in order to include some exogenous factors. The main difference between the two models is the limit value of the process: in the former this value is independent on the initial distribution, whereas in the second one a dependence on this distribution exists.…”

Section: Introductionmentioning

confidence: 99%

“…This is the case, for example, of the dynamics of growth of individuals in the same species. It must be pointed out that the methodology used in Gutiérrez et al [6] can be extended to other contexts in order to model several modes of population growth (see, for instance, Roman et al [14]). …”

Section: Introductionmentioning

confidence: 99%

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

Albano¹,

Giorno²,

Román-Román³

et al. 2011

Journal of Theoretical Biology

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The present work deals with a Gompertz-type diffusion process, which includes in the drift term a time-dependent function C(t) representing the effect of a therapy able to modify the dynamics of the underlying process. However, in experimental studies is not immediate to deduce the functional form of C(t) from a treatment protocol. So a statistical approach is proposed in order to estimate this function, when a control group and one or more treated groups are observed. In order to validate the proposed strategy a simulation study for several interesting functional forms of C(t) has been carried out. Finally, an application to infer the net effect of cisplatine and doxorubicin+ cyclophosphamide in actual murine models is presented.

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A bivariate stochastic Gompertz diffusion model: statistical aspects and application to the joint modeling of the Gross Domestic Product and CO₂ emissions in Spain

2008

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SUMMARYIn this paper we propose a bivariate stochastic Gompertz diffusion model as the solution for a system of two Itô stochastic differential equations (SDE) that are similar as regards the drift and diffusion coefficients to those considered in the univariate Gompertz diffusion model, which has been the object of much study in recent years. We establish the probabilistic characteristics of this model, such as the bivariate transition density, the bidimensional moment functions, the conditioned trend functions and in particular, the correlation function between each of the components of the model. We then go on to study the maximum likelihood estimation of the bidimensional drift and the diffusion matrix of the diffusion in question, proposing a computational statistical methodology for this purpose based on discrete observations over time, for both components of the model. By these means we are able to achieve the maximum likelihood estimation of the trend and correlation functions and thus establish a method for trend analysis, which we apply to the real case of two dependent variables, Gross Domestic Product (GDP) and CO 2 emission in Spain, the joint dynamic evolution of which is modeled by the proposed Gompertz bidimensional model. This implementation is carried out on the basis of annual observations of the variables over the period [1986][1987][1988][1989][1990][1991][1992][1993][1994][1995][1996][1997][1998][1999][2000][2001][2002][2003]. The application is a new methodology in environmental and climate change studies, and provides an alternative to other approaches of a more econometric nature, or those corresponding to the methodology of secular trends in Time Series.

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A new Gompertz-type diffusion process with application to random growth

Cited by 49 publications

References 17 publications

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

A bivariate stochastic Gompertz diffusion model: statistical aspects and application to the joint modeling of the Gross Domestic Product and CO₂ emissions in Spain

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A new Gompertz-type diffusion process with application to random growth

Cited by 49 publications

References 17 publications

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

A bivariate stochastic Gompertz diffusion model: statistical aspects and application to the joint modeling of the Gross Domestic Product and CO2 emissions in Spain

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A bivariate stochastic Gompertz diffusion model: statistical aspects and application to the joint modeling of the Gross Domestic Product and CO₂ emissions in Spain