Papers by Francisco Torres Ruiz
Predicción en el proceso de difusión lognormal con factores exógenos
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Inference on a non-homogeneous Gompertz process with jumps as model of tumor dynamics
We propose an inhomogeneous stochastic process with jumps based on the Gompertz diffusion to desc... more We propose an inhomogeneous stochastic process with jumps based on the Gompertz diffusion to describe the evolution of a solid tumor subject to an intermittent therapeutic program. Each jumps represents an application of a therapy that shifts the cancer mass to a return state. The intermittent treatment leads to a reduction in tumor size, but produces also an increase in the growth rate represented inthe model by the inclusion of time functions depending on the cycle of applicationof the therapy. To find a compromise between these two aspects a strategy to select the inter-jump intervals is proposed and an estimation procedure of the model is provided, supported by several simulations illustrating the validity of the proposed procedure.
Mathematics, 2021
A proposal is made to employ stochastic models, based on diffusion processes, to represent the ev... more A proposal is made to employ stochastic models, based on diffusion processes, to represent the evolution of the SARS-CoV-2 virus pandemic. Specifically, two diffusion processes are proposed whose mean functions obey multi-sigmoidal Gompertz and Weibull-type patterns. Both are constructed by introducing polynomial functions in the ordinary differential equations that originate the classical Gompertz and Weibull curves. The estimation of the parameters is approached by maximum likelihood. Various associated problems are analyzed, such as the determination of initial solutions for the necessary numerical methods in practical cases, as well as Bayesian methods to determine the degree of the polynomial. Additionally, strategies are suggested to determine the best model to fit specific data. A practical case is developed from data originating from several Spanish regions during the first two waves of the COVID-19 pandemic. The determination of the inflection time instants, which correspon...
Mathematics, 2020
Stochastic models based on deterministic ones play an important role in the description of growth... more Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is ad...
Journal of Sensors, 2019
Low back pain affects around 50% of pregnant women and presents significant morbidity and persist... more Low back pain affects around 50% of pregnant women and presents significant morbidity and persistence for years in 20% of postpartum women who report that pain. Numerous studies have documented gait alterations during pregnancy and postpartum. Therefore, an analysis of the relationship between certain gait parameters and low back pain was attempted using low-cost validated instrumented insoles. This work presents a longitudinal cohort study carried out during routine gynecological follow-up visits in the first, second, and third trimesters of pregnancy at an Obstetrics and Gynecology Service. Sample size was 62. Plantar pressure data were collected with specially designed instrumented insoles containing four force sensors to measure peak pressure, center of pressure, and stance phase time in each foot and in each pregnancy trimester. The analysis was carried out on a two-dimensional level, simultaneously considering the data from both feet using Hotelling’s T2 test. This longitudina...
Inference on some parametric functions in the univeriate lognormal diffusion process with exogenous factors
Test, 2001
In this paper we consider, for the univariate ]ognorma] diffusion process with exogenous factors,... more In this paper we consider, for the univariate ]ognorma] diffusion process with exogenous factors, the inference for some parametric functions that include as par-ticular cases the trend and the covariance function of the process. Concretely, we obtain the UMVU estimators of ...
Statistical Papers
We consider a lognormal diffusion process having a multisigmoidal logistic mean, useful to model ... more We consider a lognormal diffusion process having a multisigmoidal logistic mean, useful to model the evolution of a population which reaches the maximum level of the growth after many stages. Referring to the problem of statistical inference, two procedures to find the maximum likelihood estimates of the unknown parameters are described. One is based on the resolution of the system of the critical points of the likelihood function, and the other is on the maximization of the likelihood function with the simulated annealing algorithm. A simulation study to validate the described strategies for finding the estimates is also presented, with a real application to epidemiological data. Special attention is also devoted to the first-passage-time problem of the considered diffusion process through a fixed boundary.
Some Time Random Variables Related to a Gompertz-Type Diffusion Process
Cybernetics and Systems
In the context of a Gompertz-type diffusion process (associated with a particular expression of t... more 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.
A Gompertz-type diffusion process for the study of social phenomena
More general problems on first-passage times for diffusion processes: A new version of the fptdApprox R package
ABSTRACT Version 1.2 of the fptdApprox R package implemented a general heuristic strategy for the... more ABSTRACT Version 1.2 of the fptdApprox R package implemented a general heuristic strategy for the efficient application of numerical schemes aimed at solving Volterra integral equations whose solutions are first-passage-time (f.p.t.) density functions for a diffusion process. This paper presents a new version of this package, developed with the goal of addressing interesting f.p.t. problems that cannot be solved with the previous version. More specifically, it addresses f.p.t. problems for diffusion processes defined by means of not-closed-form time expressions (which appear in application fields such as tumor growth or the evolution of economic variables), and unconditioned f.p.t. problems for diffusion processes with non-degenerate initial distributions (of great interest in the study of population growth in Biology and Ecology). Here, the functionality of this new version is described, with a focus on its numerical and computational aspects. Moreover, by means of several examples, we show how the package can be used to solve problems such as those mentioned above.
Mathematical biosciences, 2012
We consider a diffusion model based on a generalized Gompertz deterministic growth in which carry... more We consider a diffusion model based on a generalized Gompertz deterministic growth in which carrying capacity depends on the initial size of the population. The drift of the resulting process is then modified by introducing a time-dependent function, called "therapy", in order to model the effect of an exogenous factor. The transition probability density function and the related moments for the proposed process are obtained. A study of the influence of the therapy on several characteristics of the model is performed. The first-passage-time problem through time-dependent boundaries is also analyzed. Finally, an application to real data concerning a rabbit population subject to particular therapies is presented.
Estimating and determining the effect of a therapy on tumor dynamics by means of a modified Gompertz diffusion process
Journal of theoretical biology, Jan 7, 2015
A modified Gompertz diffusion process is considered to model tumor dynamics. The infinitesimal me... more A modified Gompertz diffusion process is considered to model tumor dynamics. The infinitesimal mean of this process includes non-homogeneous terms describing the effect of therapy treatments able to modify the natural growth rate of the process. Specifically, therapies with an effect on cell growth and/or cell death are assumed to modify the birth and death parameters of the process. This paper proposes a methodology to estimate the time-dependent functions representing the effect of a therapy when one of the functions is known or can be previously estimated. This is the case of therapies that are jointly applied, when experimental data are available from either an untreated control group or from groups treated with single and combined therapies. Moreover, this procedure allows us to establish the nature (or, at least, the prevalent effect) of a single therapy in vivo. To accomplish this, we suggest a criterion based on the Kullback-Leibler divergence (or relative entropy). Some sim...
A stochastic model of cancer growth subject to an intermittent treatment with combined effects: reduction in tumor size and rise in growth rate
Bulletin of mathematical biology, 2014
A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analy... more A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy that shifts the cancer mass to a return state and it produces an increase in the growth rate of the cancer cells. For the resulting process, consisting in a combination of different Gompertz processes characterized by different growth parameters, the first passage time problem is considered. A strategy to select the inter-jump intervals is given so that the first passage time of the process through a constant boundary is as large as possible and the cancer size remains under this control threshold during the treatment. A computational analysis is performed for different choices of involved parameters. Finally, an estimation of parameters based on the maximum likelihood method is provided and some simulations are performed to illus...
Scandinavian Journal of Statistics, 2006
An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on in... more An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on intrinsic model posterior probabilities is considered. It is proven that these posterior probabilities are simple functions of the classical F-statistic, thus the evaluation of the procedure can be carried out analytically through the frequentist analysis of the posterior probability of the null. An asymptotic analysis proves that, under mild conditions on the design matrix, the procedure is consistent. For any testing hypothesis it is also seen that there is a one-to-one mapping -which we call calibration curve -between the posterior probability of the null hypothesis and the classical p-value. This curve adds substantial knowledge about the possible discrepancies between the Bayesian and the p-value measures of evidence for testing hypothesis. It permits a better understanding of the serious difficulties that are encountered in linear models for interpreting the p-values. A specific illustration of the variable selection problem is given.
<I>TOURCOMP</I>: A program for estimating end-member proportions in tourmalines
Mineralogical Magazine, 2008
ABSTRACT A Visual Basic program (TOURCOMP) has been written to recast the tourmaline composition ... more ABSTRACT A Visual Basic program (TOURCOMP) has been written to recast the tourmaline composition into end-member components from electron microprobe data or more complete tourmaline analyses. TOURCOMP is a program based on a linear algebraic model that directly calculates the end-member proportions of tourmalines from their structural formulae. The program is developed for IBM-compatible personal computers running under the Windows™ operating system. The source code has been also translated and compiled in order to run on an Apple computer. Analytical problems, uncertainties concerning site occupancies, and the normalization procedure to determine the structural formula are the main error sources. However, the method of recalculating tourmaline end-members presented in this paper is considered to provide reasonably good results, bearing in mind the chemical complexity of tourmaline.
On the effect of a therapy able to modify both the growth rates in a Gompertz stochastic model
Mathematical Biosciences, 2013
A Gompertz-type diffusion process characterized by the presence of exogenous factors in the drift... more A Gompertz-type diffusion process characterized by the presence of exogenous factors in the drift term is considered. Such a process is able to describe the dynamics of populations in which both the intrinsic rates are modified by means of time-dependent terms. In order to quantify the effect of such terms the evaluation of the relative entropy is made. The first passage time problem through suitable boundaries is studied. Moreover, some simulation results are shown in order to capture the dependence of the involved functions on the parameters. Finally, an application to tumor growth is presented and simulation results are shown.
Journal of Theoretical Biology, 2012
Modeling the effect of therapies in cancer animal models remains a challenge. This point may be a... more Modeling the effect of therapies in cancer animal models remains a challenge. This point may be addressed by considering a diffusion process that models the tumor growth and a modified process that includes, in its infinitesimal mean, a time function modeling the effect of the therapy. In the case of a Gompertz diffusion process, where a control group and one or more treated groups are examined, a methodology to estimate this function has been proposed by . This method has been applied to infer the effect of cisplatin and doxorubicin þ cyclophosphamide on breast cancer xenografts. Although this methodology can be extended to other diffusion processes, it has an important restriction: it is necessary that a known diffusion process adequately fits the control group. Here, we propose the use of a stochastic process for a hypothetical control group, in such a way that both the control and the treated groups can be modeled by modified processes of the former. Thus, the comparison between models would allow estimating the real effect of the therapy. The new methodology has been validated by inferring the effects in breast cancer models, and we have checked the robustness of the procedure against the choice of stochastic model for the hypothetical control group. Finally, we have also applied the methodology to infer the effect of a therapeutic peptide and ovariectomy on the growth of a breast cancer xenograft, and its efficiency in modeling the effect of different treatments in the absence of control group data is shown.
First-Passage-Time Densities for Time-Non-Homogeneous Diffusion Processes
Journal of Applied Probability, 1997
In this paper we study a Volterra integral equation of the second kind, including two arbitrary c... more In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].
Cybernetics and Systems, 2005
We consider an extension of the Gompertz homogeneous diffusion process by introducing time functi... more We consider an extension of the Gompertz homogeneous diffusion process by introducing time functions (exogenous factors) that affect its trend. After obtaining its transition probability density function, the inference on the parameters of the process is obtained by considering discrete sampling of the sample paths. Finally, we apply this stochastic process to model housing price in Spain.
Approximate and generalized confidence bands for the mean and mode functions of the lognormal diffusion process
Computational Statistics & Data Analysis, 2007
Approximate and generalized confidence bands for the mean and mode functions of the univariate lo... more Approximate and generalized confidence bands for the mean and mode functions of the univariate lognormal diffusion process are obtained. To this end, the already existing methods for building confidence intervals for the mean of the lognormal distribution have been suitably adapted. Moreover, a new method has been proposed. The bands obtained from the above procedures are compared through a simulation
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Papers by Francisco Torres Ruiz