Actividades del semestre 4

SEMESTRE 4 . Procesos y campos aleatorios de recuento y difusión. Algoritmos para la toma de decisiones y aplicaciones en fiabilidad, supervivencia y medioambiente 

Fechas 06/03/2025-28/03/2025

 (Nodos UGr-Prob y UGr-IO)

Sede IMAG (Instituto de Matemáticas de la Universidad de Granada) 

https://wpd.ugr.es/~imag/es/

INAUGURACIÓN SEMESTRE 4    06/03/2025   SALA DE CONFERENCIAS DE IMAG

Seminario 1 SPA Series. 06/03/2025. 9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Part I. Finite-velocity random motions and reset at the origin: recent advances on transient and limit behaviors 

Part II. Stochastic modeling and applications based on birth-death processes

Abstract

Part I

Stochastic processes for the description of finite-velocity random motions have been largely studied during the last decades. Usually, they refer to the motion of a particle moving with finite speed on the real line, or on more general domains, with alternations between various possible velocities or directions at random times. The basic model is concerning the so-called (integrated) telegraph process, in which the changes of directions of the two possible velocities are governed by the Poisson process.

In this talk we aim to present some recent results in this research area, involving

(i) one-dimensional and two-dimensional finite-velocity random motions, such that the random intertimes between consecutive changes of directions are governed by geometric counting processes,

(ii) inclusion of the resetting mechanism to the origin for the one-dimensional process considered in (i),

(iii) one-dimensional finite-velocity random motions with instantaneous reset to the origin regulated by Bernoulli trials.

We focus on various features of the considered stochastic processes, including the analysis of the probability laws, the behavior under limit conditions, the mean-square distance between processes, connections with the classical telegraph process.

Part II

Birth-death processes are a flexible tool for stochastic modeling, being a continuous-time analog of random walks. They are largely employed in applications, such as in evolutionary dynamics, neuronal modeling, queueing, reliability and risk theory. The developments in pre-existing techniques for the analysis of birth-death processes have allowed us to achieve new results and new models in this field. Along this direction, the talk will focus on the illustration of recent studies aimed at:

(i) discussing growth-evolution models characterized by time-dependent growth rates and their stochastic counterpart described by birth-death processes and diffusive approximations, with special attention to a modified Richards growth model involving a time-dependent perturbation in the growth rate,

(ii) the analysis and the application to real data of a two-dimensional time inhomogeneous birth-death process to model the time-evolution of fake news in a population,

(iii) the study of the behavior of a multispecies birth-death-immigration process and of a continuous-time multi-type Ehrenfest model, whose diffusion approximations lead to stochastic processes belonging to the class of Pearson diffusions on the spider.

Seminario 2 SPA Series. 19/03/2025.  9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Time–inhomogeneous Markov processes and phase–type distributions

Abstract

1. Introduction to inhomogeneous Markov jump processes and product integration.

2. Inhomogeneous phase-type distributions, IPH.

3. The distribution of rewards.

4. Application to life insurance (survival analysis).

5. Heavy-tailed IPH distributions and insurance risk.

6. Estimation of IPH using the EM algorithm.

7. Stochastic interest rates and IPH.

8. Fitting stochastic interest rates from observed bond prices (IPH fitting)

9. Outlook towards stochastic mortality rates.

We briefly introduce the theory of time–inhomogeneous Markov jump processes using product integration. The principal application of these processes has traditionally been to multi–state life–insurance modelling (introduced in [5]), where, by nature, transition rates are time–inhomogeous (mortality rates e.g. vary according to age). The product integral [6] of a matrix function is the solution to a linear system of differential equations with varying coefficients, and transition probabilities of Markov processes can obtained this way by the well–known Kolmogorov forward and backward equations.

Though time–inhomogeneous phase–type distributions (IPH) [2] are nearly always present as part of multi–state life–insurance models (since the state of death is absorbing), they were never considered as such or treated in their own right. However, they provide a nice addition to standard tools in Markov processes for computing probabilities of first entrance times to different states.

In life insurance, payments are made in the different states of an insured: continuous rates during sojourns or lump sums at transitions. All payments are deterministic and known (by contract). In other fields of Applied Probability, such payments would be known as rewards. The total expected payments (in present value) of a contract is called the reserve. Computing the reserve is hence the same as computing the total expected reward in a time–inhomogeneous Markov process, where the reward structure is as described. One can derive a Laplace transform for this quantity, from which formulas for the expectation (and reserve) and higher order moments can be obtained, [3].

Leaving behind the life–insurance applications, we consider the IPH distributions in more detail. They generalise the well–established time–homogeneous phase–type distributions,[8, 4, 7], and thereby inherit the denseness in the class of distributions on the positive reals, i.e. we may approximate any distribution with positive support arbitrarily well by a IPH distribution. As opposed to ordinary phase–type distributions that all have exponential tails, the tail behaviour of IPH distributions can be almost any. This extension comes with a price: some useful results involving renewal theory and ladder heights are no longer valid for general IPH. However, they can be estimated by an EM algorithm similar to the one applied to ordinary phase-type distributions when only the absorption times are observed (and nothing else). We will show how this algorithm  can be applied to estimating heavy-tailed data (insurance risk), light-tailed data (mortality), or fitting stochastic interest rates.

Both in finance and insurance, the calibration of stochastic interest rates plays an important role. The models applied are usually based on solutions to stochastic differential equations (SDE). Instead, we consider a dense class of stochastic interest rate models (so-called Markovian interest rate models) in which an underlying Markov jump process dictates the type of a (deterministic or constant) spot rate at different times. This model goes back at least to [9]. It turns out that this model is intimately connected to phase–type distributions: the bond price in a Markovian interest model equals the survival function of a phase–type distribution. This connection, which was not noticed until recently, [1], allows us to provide a dense class of stochastic interest rate models that can be calibrated to data (observed bond prices), and which also fits perfectly into the life insurance set-up.

In the seminar, we will demonstrate how to perform the fitting using R (with the package ”ma-trixdist”). The R programs will be made available.

References

[1] Jamaal Ahmad and Mogens Bladt. Phase-type representations of stochastic interest rates with applications to life insurance. European Actuarial Journal, pages 1–36, 2023.

[2] Hansj¨org Albrecher and Mogens Bladt. Inhomogeneous phase-type distributions and heavy tails. Journal of Applied Probability, 56(4):1044–1064, 2019.

[3] Mogens Bladt, Søren Asmussen, and Mogens Steffensen. Matrix representations of life insurance payments. European actuarial journal, 10(1):29–67, 2020.

[4] Mogens Bladt and Bo Friis Nielsen. Matrix-Exponential Distributions in Applied Probability. Springer Verlag, 2017.

[5] Jan M. Hoem. Markov chain models in life insurance. Bl¨atter der DGVFM, 9(2):91–107, Oct 1969.

[6] Søren Johansen. Product integrals and Markov processes. CWI Newsletter, 12:3–13, 1986.

[7] Guy Latouche and Vaidyanathan Ramaswami. Introduction to matrix analytic methods in stochastic modeling, volume 5. Society for Industrial Mathematics, 1987.

[8] Marcel F Neuts. Matrix-geometric Solutions in Stochastic Models. An Algorithmic Approach. Courier Dover Publications, 1981.

[9] Ragnar Norberg. A time-continuous Markov chain interest model with applications to insur-ance. Applied Stochastic Models and Data Analysis, 11(3):245–256, 1995. 

Seminario 3  SPA Series. 20/03/2025.  9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Velocity distributions and dynamics of spatial-temporal random fields

Part I. Velocities of moving random surfaces

Part II. Dynamics of spatial temporal random fields

Abstract 

Part I

For a stationary two-dimensional random field evolving in time, one can derive statistical distributions of appropriately defined velocities utilizing a generalization of the Rice formula. The theory can be applied to practical problems where evolving random fields are considered to be adequate models. Examples include changes of atmospheric pressure, variation of air pollution, or dynamical models of the sea surface elevation. In particular, statistical properties of velocities can be obtained both for the sea surface and for the envelope field based on this surface. Additional extensions can be obtained by studying three-dimensional geometry of spatial waves. Their statistical distributions can be presented in explicit integral forms for the deep water seas modeled as Gaussian fields. The proposed approach allows for investigation of the effect that shape and directionality of the sea spectrum have on the joint distributions of the size characteristics. 

Part II

General non-stationary spatial-temporal surfaces can have dynamics governed by velocity fields. By applying an extension of the standard moving average construction models which are stationary in time are obtained. The resulting surface changes with time  is dynamically inactive since its velocities, when sampled across the field, have distributions centered at zero. A dynamical evolution can be added to such a field by composing it with a dynamical flow governed by a given velocity field. The models are extensions of the earlier discretized autoregressive models which account for a local velocity of traveling surface. For such a surface, its dynamics is a combination of dynamics introduced by the flow and the dynamics resulting from the covariance structure of the underlying stochastic field. An important connection between the resulting stochastic field and underlying deterministic dynamics is obtained by demonstrating that in the case of isotropic spatial dependencies, the observed random velocities are centered at the velocities of the underlying physical flow. Strategies for simulation of such fields are provided and foundation for statistical fitting and prediction procedures are obtained.

References

[1] Lindgren, G., Podgórski, K., Rychlik, I. (2022) Effective persistency evaluation via exact excursion distributions for random processes and fields. Journal of Physics Communications, 6, 035007.

[2] Podgórski, K., Rychlik, I. (2016) Spatial size of waves. Marine Structures, 50C, pp. 55-71.

[3] Podgórski, K., Wegener, J. (2012) Velocities of a spatial-temporal stochastic field with embedded dynamics. Environmetrics, 23, pp. 238-252.

[4] Baxevani, A., Podgórski, K., Rychlik, I. (2011) Dynamically evolving Gaussian spatial fields. Extremes, 14, pp. 223-251.

[5] Podgórski, K., Rychlik, I. (2008) Envelope crossing distributions for Gaussian fields. Probabilistic Engineering Mechanics, 23, pp. 364-377.

[6] Baxevani, A., Podgórski, K., Rychlik, I. (2003) Velocities for moving random surfaces. Probabilistic Engineering Mechanics, 18, pp. 251-271.

Seminario 4 SPA Series. 24/03/2025.  9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Part I. Fractional stochastic processes for modeling some biological dynamics: theoretical setting, modeling approaches, numerical comparisons and simulations

Part II. Time-changed stochastic models and fractionally integrated processes to model the actin-myosin  interaction and  dwell times

Abstract 

Part I

Motivated by the need to model some neurophysiological evidences not included in classical neuronal models, we construct  stochastic models based on coupled fractional stochastic differential equations, with different fractional orders. Indeed, one of the main motivations is that to model  neuronal dynamics on different time-scales. We give explicit expressions of the process representing the voltage variation in the neuronal membrane.  Numerical evaluations of the average behaviors of involved processes are presented in order to put in evidence the features of the proposed models.

In order to refine the theoretical setting, we focus on Mittag–Leffler (ML) fractional integrals involved in the solution processes of a more general system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed.

References

[1] Abundo, M.; Pirozzi, E. Fractionally integrated Gauss-Markov processes and applications. In Communications in Nonlinear Science and Numerical Simulation; Elsevier: Amsterdam, The Netherlands, 2021; Volume 101, p. 105862. ISSN 1007-5704.

[2] Pirozzi, E. Some Fractional Stochastic Models for Neuronal Activity with Different Time-Scales and Correlated Inputs. Fractal Fract. 2024, 8, 57. https://doi.org/10.3390/fractalfract8010057.

[3] Anh, P.T.; Doan, T.S.; Huong, P.T. A variation of constant formula for Caputo fractional stochastic differential equations. Stat. Probab. Lett. 2019, 145, 351–358.

[4] Pirozzi, E. Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications. Mathematics 2024, 12, 3094. https://doi.org/10.3390/math12193094.

Part II

We propose two stochastic models for the interaction between the myosin head and the actin filament, the physio-chemical mechanism triggering the muscle contraction and at the moment not completely understood. We make use of the fractional calculus approach with the purpose to construct non-Markov processes for models with memory. A time-changed process and a fractionally integrated process are proposed for the two models. Each of these include memory effects in different way. We describe such features from a theoretical point of view and also with simulations of sample paths. Mean functions and covariances are provided taking into account constant and time-dependent tilting forces by which effects of external loads are included. The investigation of the dwell time of such phenomenon is carried out by means of density estimations of the first exit time (FET) of the processes from a strip: this  mimics the times of the steps of the myosin head during the sliding movement outside a potential well due to the interaction with the actin. For the case of the time changed diffusion process we specialize an equation for the probability density function of the FET from a strip. The schemes of two simulation algorithms are provided and performed. Some numerical and simulation results are given and discussed.

References

[1] Leonenko N., Pirozzi .E., The time-changed stochastic approach and fractionally integrated processes to model the actin-myosin  interaction and  dwell times. Mathematical Biology and Engineering. (Submitted).

[2] Leonenko, N.,  Pirozzi, E. (2021). First passage times for some classes of fractional time-changed diffusions. Stochastic Analysis and Applications, 40(4), 735–763. https://doi.org/10.1080/07362994.2021.1953386.

[3] Kobayashi, K. Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations. J Theor Probab 24, 789–820 (2011). https://doi.org/10.1007/s10959-010-0320-9.

Seminario 5  SPA Series. 26/03/2025.  9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Part I. Diffusion models related to growth curves

Part II. Inference  on some epidemiological statistical  models 

Abstract 

Part I

The seminar focuses on a general growth curve able to unify the classical cases of Malthusian, Richards, Gompertz, Logistic and several of their generalizations. Here two stochastic generalizations of the deterministic growth are obtained by introducing a multiplicative and an additive noise, respectively, to the deterministic equation. We show that the resulting processes are lognormal and gaussian respectively, and their mean is the deterministic trend of the curve. Since the distributions of the two processes are known, the problem of estimating the parameters of the model is analyzed by means of the maximum likelihood method. Further, due to the parametric structure of the processes, the resulting systems of equations are quite complex, and numerical solutions must be searched. Estimating procedures for the involved parameters make use of specific metaheuristic algorithms.  They try to find the solution of an optimization problem by iterative procedures in which the current solution must be better than the previous one. Most of such algorithms are based on the behavior of search agents moving randomly across the parametric space. Following certain rules of movement, life or death, the algorithm evolves until reach a stable state. The values of the parameters at that time become the final solution. An extensive simulation study is then made to evaluate the goodness of the proposed procedure, i.e. the convergence of the algorithms and the properties of the estimates. Another interesting focus of the talk is the first passage of passage of processes across suitable boundaries. In particular, by suitable choice of parameters, we show that the threshold can be a percentage of the average size of the population, and this is often of interest in applications in population growth. For such cases we provide explicit solutions for the probability density function of the random variable First Passage Time. 

Part II

In this seminar the focus is on epidemiological stochastic models. We firstly consider a time-inhomogeneous diffusion process able to describe the dynamics of infected people in a susceptible-infectious epidemic model. Such a model describes epidemics evolution in micro-parasites or the evolution of the size of individuals who have contracted a disease that has made them immune. Here we focus on the inference for such a process, by providing an estimation procedure for the involved parameters. Such procedure is based on the transformation of the original process to a non-homogeneous Wiener process and on the subsequent Generalized Method of Moments to find suitable estimate for the infinitesimal drift and variance of the transformed process.

In the second part, a Susceptible-Infected- Removed stochastic model is discussed, emphasizing the role of the contribution of each subpopulation. Such a model is able to include the natural growth of the Susceptible population, and it is shown that it captures multi-peaks dynamics in the epidemic spreads. The inference is addressed by means of a quasi-maximum likelihood method. Numerical procedures for determining the local maxima of the likelihood function are discussed. Emphasis is placed on the problems associated with the use of such techniques and some insights are provided.

Seminario 6 SPA Series. 27/03/2025.  9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Stochastic processes in software reliability

Dr. Tadashi Dohi has served as a Full Professor at Hiroshima University, Japan, since 2002. He is currently appointed as Dean of School of Informatics and  Data   Science and Associate Dean of Graduate School of Advanced Science and Engineering, Hiroshima University.      He     received a Doctor       of     Engineering degree from Hiroshima University in 1995.  His research interests include Software Reliability, Dependable Computing, Performance  Evaluation, Operations Research. To date,  his  research   has led to 280 journal papers, 340 peer-reviewed conference papers, 25 book editions, and 47 book chapters in the above    research fields. Dr. Dohi is a Regular        Member of IEICE, IPSJ, REAJ, a Fellow Member of ORSJ, and a Senior Member of IEEE (Computer Society and Reliability Society). He was acting President of REAJ in 2018 and 2019. He has served as the General Chair of 15 international conferences, including ISSRE 2011, ATC 2012, DASC 2019, and ICECCS 2022. Of note, he was a founding member    of  the Internationa Symposium on Advanced Reliability and Maintenance Modeling (APARM) and International Workshop on Software Aging and    Rejuvenation  (WoSAR).  He has  been a  steering committee member in AIWARM/APARM, ISSRE, DASC, DSA. He has also worked as a program committee member in several  premier      international conferences such as DSN, ISSRE, COMPSAC, SRDS, QRS, EDCC, PRDC, HASE, SAC, ICPE, among numerous others. He is an Associate Editor/Editorial Board Member of over 20 international journals, including IEEE Transactions on Reliability.

Abstract

Software reliability engineering plays a central role to quantify software product reliability in the verification/validation phase before the release. During the last five decades, in fact, over two hundred software reliability models have been proposed in the literature. First, we quickly overview the software reliability modeling based on the software fault count data observed in the testing phase, where homogeneous Markov processes (HMPs) and non-homogeneous Poisson processes (NHPPs) have gained much popularity to assess quantitative software reliability. Next, we provide generalization frameworks on software reliability models, by introducing non-homogeneous Markov processes (NHMPs). Finally, we give some examples on non-Markovian process modeling, including geometric processes, trend renewal processes (TRPs) and Hawkes processes. We show how these generalized software reliability models can work better than the classical ones in terms of goodness-of-fit and predictive performances, through several empirical studies with actual software fault count data.

Seminario 7 SPA Series. 28/03/2025.  9:30-11:00; 11:30-13:00. Sala de Conferencias de IMAG

Fractional hyperbolic diffusions on sphere with random data

Part I.  Introduction to statistical analysis of Gaussian isotropic spherical random fields 

Part II. Fractional hyperbolic diffusions on sphere with random data

Abstract

Part I

Spherical random fields are very useful for modelling some phenomena in areas such as earth sciences (like, for example, in geophysics and climatology and cosmology). In fact, the application of statistical methods in cosmology has become increasingly important due to the many experimental data obtained in recent years, and spherical random fields are of particular interest regarding the analysis of Cosmic Microwave Background (CMB) radiation discovered by the astronomers Arno Penzias and Robert Wilson in 1964.

As well-known, the CMB is a spatially isotropic radiation field spread throughout the visible universe, originated around 14 billion years ago, and it is the main source of information we have about the evolution of the universe. The CMB radiation can be mathematically modelled as an isotropic spherical random field for which there is a spectral representation by means of spherical harmonics.

We consider statistical analysis of Gaussian isotropic spherical random fields, when only one observation of the field is available. The characteristic function of these empirical covariances is given in terms of cumulants. It turns out that our estimator follows a Rosenblatt type distribution [1,2,3]. Our goal is to estimate the covariance function of the random field, given a single observation at each point of the discretized sphere. We present a methodology for handling the problem of cosmic variance in this framework. 

This is joint results with M.Taqqu (Boston University, USA) and G.Terdik (Debrecen University, Hungary). Some results used in the talk are obtained jointly with M.D.Ruiz-Medina (Granada University, Spain).

Part II

Our next objective is to study the fundamental solutions to fractional hyperbolic diffusion equation in the time variable using the Caputo derivative, and its properties. The exact solutions of the fractional hyperbolic diffusion equation with random data are derived  in terms of series expansions of isotropic in space  random fields on the unit sphere. The results are illustrated through  motivating numerical examples of interest in several applied fields.

This is joint results with A. Olenko (La Trope University, Melbourne, Australia) and J.Vaz  (UNICAMP, Brazil) [4,5]

References

[1] Leonenko, N.N., Taqqu, M.S. and Terdik, Gy (2018) Estimation of the covariance function of Gaussian isotropic random fields, related Rosenblatt-type distributions and the cosmic variance problem, Electronic Journal of Statistics, Vol. 12 (2018) 3114–3146 

[2] Leonenko, N.N.,  Ruiz-Medina,M.D. and  Taqqu, M.S. (2017). Rosenblatt distribution subordinated to gaussian random fields with long-range dependence. Stoch. Anal. Appl., 35(1):144–177, 2017

[3] Leonenko, N., Ruiz-Medina, M.D. and Taqqu,M.S.  (2017) Non-central limit theorems for random fields subordinated to gamma-correlated random fields, Bernoulli, 2017, Vol. 23, No. 4B, 3469-3507

[4] Leonenko, N. and Vaz, J. (2020) Spectral analysis of fractional hyperbolic diffusion equations with random data, Journal of Statistical Physics, 179,155-175

[5] Leonenko, N., Olenko, A. and Vaz, J. (2024) On fractional spherically restricted hyperbolic diffusion random field, Communications in Nonlinear Science and Numerical Simulation, 131,107866