Desirée Romero

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

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 > or = 1, allows for modeling both length and weight for some animal species in both isometric (b = 3) and allometric (b not = 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.

FORECASTING FOR THE UNIVARIATE LOGNORMAL DIFFUSION PROCESS WITH EXOGENOUS FACTORS

The forecasting problem for the univariate lognormal diffusion process with exogenous factors is studied. For point predictions we propose the use of the mode function together with the mean function and, for some particular cases, the conditional version of these functions. For the interval predictions we obtain confidence bands for the mentioned functions; taking into consideration the percentile functions, intervals containing the variable of the process, for each time, with a specific probability are also obtained.

Estimating the parameters of a Gompertz-type diffusion process by means of Simulated Annealing

Abstract This paper explores the application of the Simulated Annealing algorithm for the maximum likelihood estimation of the parameters of a Gompertz-type process. Firstly, the solution space is bounded using relevant information about the process provided by the sample data. Secondly, a proposal for improvement is made, namely the application of a second cycle of the algorithm, including a refinement factor. Finally, both the specifications for the application of the algorithm and the proposed improvement are validated through their application to simulated and real data.

SOME TIME RANDOM VARIABLES RELATED TO A GOMPERTZ-TYPE DIFFUSION PROCESS

In the context of a Gompertz-type diffusion process (associated with a particular expression of the Gompertz curve for which its limit value depends on the initial value), we consider the study of two time-random variables: the inflection time of the model and the time at which it achieves a certain percentage of the total growth. Once we have shown the difficulties in the approach of the former, we deal with it as a particular case of the second. Furthermore, because of the peculiar characteristics of the considered process, we prove that these two problems can be formulated as a first-passage-time problem through a constant boundary. Finally, we conclude with an application to the growth of rabbits, in which we obtain the density functions of the inflection time and the time at which a rabbit achieves the half of its total growth.

APPROXIMATING THE NONHOMOGENEOUS LOGNORMAL DIFFUSION PROCESS VIA POLYNOMIAL EXOGENOUS FACTORS

ABSTRACT In this article we propose a methodology for building a lognormal diffusion process with polynomial exogenous factors in order to fit data that present an exponential trend and show deviations with respect to an exponential curve in the observed time interval. We show that such a process approaches a nonhomogeneous lognormal diffusion and proves that it is specially useful in the case when external information (exogenous factors) about the process is not available even though the existence of these influences is clear. An application to the global man-made emissions of methane is provided.

Comparing Optimization Methods, in Continuous Space, for Modelling with a Diffusion Process

Many probabilistic models are frequently used for natural growth-patterns modelling and their forecasting such as the diffusion processes. The maximum likelihood estimation of the parameters of a diffusion process requires a system of equations that, for some cases, has no explicit solution to be solved. Facing that situation, we can approximate the solution using an optimization method. In this paper we compare five optimization methods: an Iterative Method, an algorithm based on Newton-Raphson solver, a Variable Neighbourhood Search method, a Simulated Annealing algorithm and an Evolutionary Algorithm. We generate four data sets following a Gompertz-lognormal diffusion process using different noise level. The methods are applied with these data sets for estimating the parameters which are present into the diffusion process. Results show that bio-inspired methods gain suitable solutions for the problem every time, even when the noise level increase. On the other hand, some analytical methods as Newton-Raphson or the Iterative Method do not always solve the problem whether their scores depend on the starting point for initial solution or the noise level hinders the resolution of the problem. In these cases, the bio-inspired algorithms remain as a suitable and reliable approach.

Modellation and Forecast of Traffic Series by a Stochastic Process

Traffic in a road point is counted, by a device, for more than 1 year, giving us time series. The obtained data have a trend with a clear seasonality. This allows us to split taking each season (1 week or 1 day, depending on the series) as a separated path. We see the set of paths we have like a random sample of paths for a stochastic process. In this case seasonality is a common behaviour for every path that allow us to estimate the parameters of the model. We use a Gompertz-lognormal diffusion process to model the paths. With this model we use a parametric function for short-time forecasts that improves some classical methodologies.

Comparing Some Estimate Methods in a Gompertz-Lognormal Diffusion Process

This paper compares four different methods to obtain maximum likelihood estimates of the parameters of a Gompertz-lognormal diffusion process, where no analytical solution for the likelihood equations exists. A recursive method, a Newton-Raphson algorithm, a Simulated Annealing algorithm and an Evolutionary Algorithm to obtain estimates are proposed. The four methods are compared using a simulated data set. The results are compared with simulated paths of the process in terms of several error measurements.

A Gompertz‐type diffusion process for the study of social phenomena

We propose a growth stochastic model associated to the Gompertz curve (bounded and S-shaped growth curve) that improves, in certain aspects, other models in the literature. Growth is relevant to many applications. For this reason, many mathematical models have been proposed. For instance, Capocelli and Ricciardi [1] proposed a diffusion process associated with the Gompertz curve, including a noise term in the differential equation associated to the deterministic model. On the other hand, Tan [2] defined the stochastic Gompertz birth-death process as that whose mean function is a Gompertz curve. The curves considered in these two works have different expressions; the essential difference between them is its limiting value. For the first one, this is independent of the initial value x0, but not for the second one. There are many practical situations in the Social Sciences in which phenomena may be modeled by dynamic variables, with continuous state space, following a Gompertz pattern growing whose limit value depends on x0. Following the methodology in [1], we present here a new diffusion process whose mean function is a Gompertz curve, a property which is not fulfilled by the above diffusion process. This has the mentioned feature, and it may well be used for forecasting aims. The resulting process is a lognormal diffusion process with an exogenous factor of the kind h(t) = e−βt, for which the known inferential results [3 and references herein] are not valid. Our work also involves an inferential study of the process as well as an application to real data concerning the salaries along several years in the different Spanish communities. This application allows one to find groups with similar behavior patterns.