Title:

• Part I: Diffusion models related to growth curves.
• Part II: Inference on some epidemiological statistical models.

Speaker: Giuseppina Albano (Department of Political and Social Studies/DISPS, University of Salerno).

Schedule

Abstracts:

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.

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Red Temática de Procesos Estocásticos y sus Aplicaciones.

Title:

• 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.

Speaker: Enrica Pirozzi (Dipartimento di afferenza: Dipartimento di Matematica e Fisica (DMF), Università della Campania Luigi Vanvitelli).

Schedule

Abstracts:

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. [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. [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. [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. [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. [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. [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. [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.

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Red Temática de Procesos Estocásticos y sus Aplicaciones.

Title: Velocity distributions and dynamics of spatial-temporal random fields.

Speaker: Krzysztof Podgórski (Department of Statistics, Lund University).

Schedule

Abstracts:

Part I. Velocities of moving random surfaces.

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. Dynamics of spatial temporal random fields.

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. [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. [2] Podgórski, K., Rychlik, I. (2016) Spatial size of waves. Marine Structures, 50C, pp. 55-71.
3. [3] Podgórski, K., Wegener, J. (2012) Velocities of a spatial-temporal stochastic field with embedded dynamics. Environmetrics, 23, pp. 238-252.
4. [4] Baxevani, A., Podgórski, K., Rychlik, I. (2011) Dynamically evolving Gaussian spatial fields. Extremes, 14, pp. 223-251.
5. [5] Podgórski, K., Rychlik, I. (2008) Envelope crossing distributions for Gaussian fields. Probabilistic Engineering Mechanics, 23, pp. 364-377.
6. [6] Baxevani, A., Podgórski, K., Rychlik, I. (2003) Velocities for moving random surfaces. Probabilistic Engineering Mechanics, 18, pp. 251-271.

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Red Temática de Procesos Estocásticos y sus Aplicaciones.

Title: Time–inhomogeneous Markov processes and phase–type distributions.

Speaker: Mogens Bladt (Department of Mathematical Sciences, University of Copenhagen).

Schedule: 9:30-11:00h & 11:30-13:00h.

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. [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. [2] Hansjörg Albrecher and Mogens Bladt. Inhomogeneous phase-type distributions and heavy tails. Journal of Applied Probability, 56(4):1044–1064, 2019.
3. [3] Mogens Bladt, Søren Asmussen, and Mogens Steffensen. Matrix representations of life insurance payments. European actuarial journal, 10(1):29–67, 2020.
4. [4] Mogens Bladt and Bo Friis Nielsen. Matrix-Exponential Distributions in Applied Probability. Springer Verlag, 2017.
5. [5] Jan M. Hoem. Markov chain models in life insurance. Blätter der DGVFM, 9(2):91–107, Oct 1969.
6. [6] Søren Johansen. Product integrals and Markov processes. CWI Newsletter, 12:3–13, 1986.
7. [7] Guy Latouche and Vaidyanathan Ramaswami. Introduction to matrix analytic methods in stochastic modeling, volume 5. Society for Industrial Mathematics, 1987.
8. [8] Marcel F Neuts. Matrix-geometric Solutions in Stochastic Models. An Algorithmic Approach. Courier Dover Publications, 1981.
9. [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.

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Red Temática de Procesos Estocásticos y sus Aplicaciones.

Speaker: Antonio di Crescenzo (Department of Mathematics, University of Salerno).

Schedule

Abstracts:

Seminar 1: Finite-velocity random motions and reset at the origin: Recent advances on transient and limit behaviors.

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

1. (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,
2. (ii) inclusion of the resetting mechanism to the origin for the one-dimensional process considered in (i),
3. (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.

Seminar 2: Stochastic modeling and applications based on birth-death processes.

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:

1. (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,
2. (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,
3. (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.

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Red Temática de Procesos Estocásticos y sus Aplicaciones.

Schedule

11:00 — 12:00 Francisco Torralbo (UGR)

Title: Topology of compact minimum submanifolds

Abstract: In this talk we will begin by exploring when a Riemannian manifold has compact minimal submanifolds. In the simplest affirmative cases, we will study which topological types are admissible. In particular, we will prove that any compact surface with even Euler characteristic can be minimally embedded in $$\mathbb{S}^2\times \mathbb{S}^1(r)$$ for any r > 0. If time permits, we will present how to build such embeddings.

12:30 — 13:30 Antonio Díaz Ramos (UMA)

Title: Quillen's conjecture for unitary groups

Abstract: Quillen's conjecture about the complex of p-subgroups was introduced in the 1970s and, thanks to the work of Aschbacher and Smith in the 1990s and the more recent work of Smith and Piterman, it is reduced to the case of unit groups if the prime p is odd. In this talk we will explain a geometric method that applies to soluble and p-soluble groups, several families of simple finite groups, and also to unit groups. As a consequence we will obtain that the conjecture is true for all odd primes.

Horario

11:00 — 12:00 Francisco Torralbo (UGR)

Título: Topología de subvariedades mínimas compactas.

Resumen: En esta charla comenzaremos explorando cuándo una variedad riemanniana admite subvariedades mínimas compactas. En los casos afirmativos más sencillos estudiaremos qué tipos topológicos son admisibles. En particular, probaremos que toda superficie compacta con característica de Euler par puede embeberse de forma mínima en $$\mathbb{S}^2\times \mathbb{S}^1(r)$$ para cualquier r > 0. Si el tiempo lo permite, presentaremos cómo construir dichos embebimientos.

12:30 — 13:30 Antonio Díaz Ramos (UMA)

Título: La conjetura de Quillen para los grupos unitarios

Resumen: La conjetura de Quillen sobre el complejo de p-subgrupos fue introducida en los 70 y, gracias al trabajo de los 90 de Aschbacher y Smith y al más reciente de Smith y Piterman, se reduce al caso de los grupos unitarios si el primo p es impar. En esta charla explicaremos un método geométrico que se aplica a grupos solubles y p-solubles, varias familias de grupos finitos simples, y también a los grupos unitarios. Como consecuencia obtendremos que la conjetura es cierta para todos los primos impares.

Talk 1: 12 horas día 4 de octubre

Conferenciante: Edoardo Saccenti, Associate Professor. Laboratory of
Systems and Synthetic Biology, Wageningen University & Research, the
Netherlands

Título: Multivariate experimental errors in Principal Component
Analysis: from correlations to loadings

Abstract:

Multivariate measurement errors are ubiquitous in all experimental
sciences. Depending on the experimental platform used to acquire data,
diverse types of errors are introduced, amounting to an admixture of
additive and multiplicative error components that can be uncorrelated or
correlated. In this talk I will review how experimental error affect
correlations, the building blocks of multivariate statistical methods,
and how this connect to the recovery of the subspace with Principal
Component Analysis (PCA) as obtained by using numerical simulations. I
will how show different error characteristics (variance, correlation,
correlation structure), loading structures and data distributions
influence the accuracy to estimate an error-free (true) subspace from
sampled data with PCA.

==========================================

Talk 2: 12:45 horas día 4 de octubre

Conferenciante: Age K. Smilde. Emeritus-Professor of Biosystem Data
Analysis, Swammerdam Institute for Life Sciences, University of
Amsterdam and professor of Computational Systems Biology at the
Department of Plant and Environmental Sciences at the University of
Copenhagen.

Título: Multidimensional Latent Path Models

Abstract:

In many cases in the life sciences we are confronted with measurements
performed on a biological system that can be arranged in multiple data
sets pertaining to different entities being measured, such as
metabolites, proteins, gene-expressions, microbiome compositions.
Traditionally such multiset data has been analyzed by multiblock or
multiset methods from chemometrics and psychometrics. Such an analysis
can be supervised, _e.g_., for predicting an outcome, or unsupervised,
_e.g._, exploratory analysis. Both types of analyses ignore the topology
of the individual data blocks, _i.e._, the way in which these individual
data blocks can be or must be arranged. In the social sciences methods
have emerged combining factor analysis with structural equation models
resulting in so-called latent path models, such as PLS-Path Modeling,
that deal with such topology. These latent path models usually assume
one latent variable per data block which is not sufficient for
high-dimensional life science data. In this talk, I will discuss how
this can be generalized to more than one latent variable per block. This
is not trivial and I will discuss methods, algorithms and interpretation
of such models. I will touch upon the topic of causality and will give
an example from the field of plant sciences.

SEMINARIO DE JÓVENES INVESTIGADORES

Conferenciante: Praful Gagrani

Abstract: Chemical reaction networks (CRNs) play a central role in the mathematical modeling of chemistry and biology due to their capacity to capture a wide range of nonlinear phenomena. A CRN is a hypergraph and thus exhibits a rich topological structure. In particular, we will explore a particular feature termed autocatalysis that has many consequences for dynamical organization in CRNs for theoretical and practical reasons. I will define autocatalytic networks and explain the polyhedral geometry induced by the minimal autocatalytic subnetworks (MASs). (For details, see https://arxiv.org/abs/2303.14238.) A CRN, when equipped with the appropriate kinetics, also specifies a transition matrix for the stochastic dynamics of a population state in the positive orthant of the integer lattice. Analogous to the passage of classical mechanics from quantum mechanics, one can show that, to a first order approximation, the stochastic dynamics of the system are governed by solutions of the Hamilton-Jacobi equation. I will outline our functional gradient descent algorithm to estimate these solutions. (For details, see https://doi.org/10.1103/PhysRevE.107.034305.) I will conclude with a perspective on the applications of CRNs and mention open questions in the statistical learning of biochemical systems.

SEMINARIO DE JÓVENES INVESTIGADORES

Conferenciante: Federico Nudo (University of Calabria)

Resumen: abstractmockCheb

CICLO DE CONFERENCIAS ESTADÍSTICA Y CIENCIA DE DATOS PATRICIA ROMÁN

Conferenciante: Prof. Nikolaos Limnios (Université de Technologie de Compiègne, France)

Resumen: This talk includes in a first part the presentation of Poisson processes, homogeneous and nonhomogeneous, with applications in software reliability, the compound Poisson process with applications in shock models and Cramér-Lindberg process in insurance. The second part includes the presentation of Poisson point processes and marked processes, with applications for systems undergoing shocks and cracking failures.

Reseña del autor: Nikolaos Limnios received his diploma from the AUTh, Greece in 1979; his DEA from University of Technology of Compiegne (UTC), France in 1980; and his Ph.D. in 1983 from the same university. He received his Doctorat d'Etat in 1991 from UTC. He is a Professor in Applied Mathematics at UTC since 1993. He is the co-author of the books Semi-Markov Processes and Reliability (Birkhäuser, 2001); Stochastic Systems in Merging Phase Space (Word Scientific, 2005); and Semi-Markov Chains and Hidden Semi-Markov Models toward Applications (Springer, 2008); Applied Nonparametric Statistics in Reliability (Springer, 2011); Applied Probability - From Random Sequences to Stochastic Processes (Springer, 2018).