PASTA is a joint research team between Inria - Nancy Grand Est, CNRS and University of Lorraine, located at Institut Élie Cartan of Lorraine.

PASTA aims to construct and develop new methods and techniques by promoting and interweaving stochastic modeling and statistical tools to integrate, analyze and enhance real data.

The specificity and the identity of PASTA are:

The leading direction of our research is to develop the topic of data enriched spatio-temporal stochastic models, through a mathematical perspective. Specifically, we jointly leverage major tools of probability and statistics: data analysis and the analytical study of stochastic processes. We aim at exploring the three different aspects, namely: shape, time and environment, of the same phenomenon. These mathematical methodologies will be intended for solving real-life problems through inter-disciplinary and industrial partnerships.

Our research program develops three interwoven axes:

In particular, we are interested in the evolution of stochastic dynamical systems evolving in intricate configuration spaces. These configuration spaces could be spatial positions, graphs, physical spaces with singularities, space of measures, space of chemical compounds, and so on.

While facing a new modeling question, we have to construct the appropriate class of models among
what we call the meta-models.
Meta-models and then models are selected according to the properties to be simulated or inferred
as well as the objectives to be reached.
Among other examples of such meta-models which we regularly use, let us mention
Markov processes (diffusion, jump, branching processes), Gibbs measures, and random graphs.
On these topics, the team has an intensive research experience from different perspectives.

Finding the balance between usability, interpretability and realism is our first guide. This is the keystone in modeling, and the main difference with black-box approaches in machine learning. Our second guide is to study the related mathematical issues in modeling, simulation and inference. Models are sources of interesting open mathematical questions. We are eager to expand the “capacity” of the models by exploring their mathematical properties, providing simulation algorithms or proposing more efficient ones, as well as new inference procedures with statistical guarantees.

To study and apply the class of stochastic models we have to handle the following questions:

Our main application domains are: insurance, geophysics, geology, medicine, astronomy and finance.

We aim at providing new tools regarding the modeling, simulation and inference of spatio-temporal stochastic processes and other dynamical random systems living in large state-spaces. As such, there are many application domains which we consider.

In particular, we have partnerships with practitioners in: cosmology, geophysics, healthcare systems, insurance, and telecom networks.

We detail below our actions in the most representative application domains.

Geophysics is a domain which requires the application of a broad range of mathematical tools related to probability and statistics while more and more data are collected. There are several directions in which we develop our methodology in relation with practitioners in the field.

On such topics, we hold long standing interdisciplinary collaborations with INRAE Grenoble, the RING Team (GeoRessources, University of Lorraine), IMAR (Institute of Mathematics of the Romanian Academy) in Bucharest. We also have the support of the interdisciplinary LUE Deepsurf project (University of Lorraine).

We have longstanding and continuous cooperation with astronomers and cosmologists in France, Spain and Estonia. In particular, we are interested in using spatial statistics tools to detect galaxies and other star patterns such as filaments detection. Such developments require to design specific point processes giving appropriate morpho-statistical distributions, as well as specific inference algorithms which are based on Monte Carlo simulations and able to handle the large volume of data.

Graphs are essential to model complex systems such as the relations between agents, the spatial distribution of points that are connected such as stars, the connections in telecommunication networks, and so on. We develop various directions of the study of random graphs that are motivated by a large class of applications:

We have longstanding collaborations on these topics with Agence de Biomédecine (ABM), Le Foyer (insurance company, Luxembourg), INRAE (Avignon), Dyogene (Inria Paris), Lip 6, UTC, LORIA (computer science laboratory, Nancy), University of Buenos Aires, Northwestern University and LAAS (CNRS, Toulouse).

We have a strong interest in the fragmentation equation for understanding snow or rock avalanches. Our point of view is to explore the probabilistic representations of transport equations in this framework as well as the possibilities they offer. We developed a stochastic process that represents the typical evolution of the mass of a rock or of a snow aggregate subject to successive random breakages.

In particular, we have studied from analytic viewpoint the size of a population with a fixed rate of fragmentation, as well as models with a diffusion in space and a fragmentation in size 35.

We have also set-up a preliminary program toward the use of neural networks to recover the fragmentation rate and the fragmentation kernel from data using deep recurrent neural networks.

Furthermore, we obtained new results with Oana Lupaşcu-Stamate (Institute of Mathematical Statistics and Applied Mathematics, Bucharest) for a binary coagulation-fragmentation equation which describes the avalanches phenomena. We use tools from self-organized critical systems and construct an adapted stochastic process for this phenomenon. We obtain in particular the asymptotic behaviour of the stochastic process and develop a numerical method in order to approximate the solution 28.

Hawkes processes represent a common class of self-excited stochastic processes.

We have studied the use of Hawkes processes in the context of insurance. In particular, we built a recommendation system for insurance products based on individual probabilities of life events. This system was successfully tested on the database from the insurance company Le Foyer (Luxembourg) 24 and it is now used by the company. Other applications of Hawkes processes in insurance have also been studied by introducing and developing the study of multi-variate Hawkes processes 17.

The numerical approximation of stochastic differential equations (SDEs) and in particular new methodologies to approximate hitting times of SDEs is a challenging problem which is important for a large class of practical issues such as: geophysics, finance, insurance, biology, etc.

With Samuel Herrmann (University of Burgundy) we made important progress on this topic by developing new methods. Recently we introduced methods for the strong convergence and pathwise approximation of one-dimensional SDEs.

In particular, we develop new techniques for the path approximation of Bessel processes of arbitrary dimension, as such a process represents the norm of a multi-dimensional Brownian motion 13. These are part of the family of the so-called ε-strong approximations. Our approach constructs jointly the sequences of exit times and corresponding exit positions of some well-chosen domains, the construction of these domains being an important step. We construct the algorithm for any dimension and treat separately the integer dimension case and the non integer framework, each situation requiring appropriate techniques. We prove the convergence of the scheme and provide the control of the efficiency with respect to the parameter ε. We expand the theoretical part by a series of numerical developments.

Together with Samuel Herrmann (University of Burgundy) and Cristina Zucca (University of Torino) we pursue our work on the exact simulation of the hitting times of multi-dimensional diffusions.

With Denis Villemonais (IECL, University of Lorraine), we constructed an estimator of the stickyness parameter of the sticky Brownian motion and other general diffusion processes from high frequency observations. This work is based on the construction of suitable estimators for the local time and the occupation time. Besides, this work provides a construction of sticky stochastic differential equations 22, 26, 11.

With Renaud Marty (IECL, University of Lorraine), we are studying an invariant embedding problem, which consists in solving a differential equation whose initial and terminal conditions are linked by a linear relation. Using tools from rough paths an global analysis theories, we consider Rough Differential Equations which extend ordinary differential equations driven by rough signals. In particular, we use our development in the context of equations driven by a Brownian motion while avoiding all the difficulties related to the use of anticipative stochastic calculus 29.

We are studying an expansion of the maximum likelihood estimator using formal series expansions. The aim of this work is to understand the lack of Gaussianity in the non-asymptotic regime 30. We apply this expansion to the estimator of the skewness parameter of a skew Brownian motion, whose asymptotic mixed normality is also proved with a rate of convergence of order

In collaboration with Paolo Pigato (University Tor Vergata, Roma) we previously studied parameter estimation for the linear drift of the self Vasicek model which follows a several-regime Ornstein-Uhlenbeck dynamic. The model fits well the behavior in financial markets related to crisis periods. In addition, in the case of two regimes, we provided a test for detecting the change in the regime. These results, improved with respect to the ones of last year (given only for two regimes) are contained in

32.

After considering high frequency observations, we study new estimators for low frequency observations and the presence of several regimes for different threshold models which show mean-reversions features. This work is in progress.

We have made the following advances regarding stochastic matching models:

In 31 we have constructed an original perfect sampling algorithm for CTMC's under a given control condition. This condition is reminiscent of the so-called “Dominated coupling from the past” (DCFTP) of Kendall et al. With this algorithm we are able to perfectly simulate the stationary state of stochastic matching models with reneging, allowing to estimate, by Monte Carlo techniques, the loss probability at equilibrium under various matching policies, a result that have crucial applications in practice, for instance for organ transplants models, which are subject to heavy time constraints.

In 18, with Jean Mairesse (CNRS/Lip6), we build a research program on the general topic of stochastic matching models with reneging, by proposing a new approach of this class of models, generating additional, fictional, classes in the compatibility graph, to account for the balking of items.

In the thesis of Jocelyn Begeot 23, we build on a previous article 36 to obtain an explicit characterization of the stability regions of a wide class of stochastic matching models and skill-based queueing systems.

In 19, with Ohad Perry (Northwestern University) we draw a research program on the essential aspect of correlations of patience times and service times in service systems. To model this, we propose a measure-valued Markov modeling to address services systems with reneging, to account for the correlation
between the patience times of the tasks and their service times. This is a theoretically tedious, but essential, task for practical purposes, as most models proposed in the literature address independent patience and service times.

In the article 21 we propose a first modeling of the Trustworthiness in the IOT by game-theoretical approaches, in the presence of intelligent and malicious intrusions. Our construction is based in a dynamic phase-based modeling of the behavior of malicious nodes.

Following this line of thought, in the submitted papers 34, 19, we propose a framework for extensive simulations of concrete networks, showing the efficiency of our trust allocation management algorithm, based on game theory.

In 12 with Mohamed Habib Diallo Aoudi and Vincent Robin (University of Technology of Compiègne), we obtain a large graph estimate of the matching coverage of local matching algorithms (greedy and minimal residual) on large bipartite random graphs constructed by the configuration model. We proceed by deriving the hydrodynamic limit, in the large-graph asymptotic, of a measure-valued Markov process representing the exploration of the graph together with its sequential construction. We are currently completing a second paper including the proof of convergence (in spaces of measure-valued processes) to the hydrodynamic limit, for general, non-necessarily bipartite, graphs, and for a large class of matching algorithms.

In an ongoing collaboration with Matthieu Jonckheere and Nahuel Soprano-Loto (CNRs/LAAS), we address the question of maximal marriages on the stochastic block model (SBM), in the large graph asymptotic. Here again, we proceed by a sequential construction of the graph, coupled with its exploration, and propose, under certain explicit feasibility condition on the connectivity matrix of the SBM, a matching algorithm that is able to achieve, infinitely often with probability 1, a perfect marriage. A draft is in preparation and will be submitted shortly for publication.

Detecting the number and composition of multiple sources in groundwaters from hydrochemical data has remained highly challenging. In this work we develop a new interaction point process that integrates geological knowledge for the purpose of automatic sources detection of multiple sources in groundwaters from hydrochemical data. The key assumption of this approach is to consider the unknown sources to be the realization of a point process. The probability density describing the sources distribution is built in order to take into account the multidimensional nature of the data and specific physical rules. The method was first calibrated on synthetic data and then tested on real data from geothermal and ore-forming hydrothermal systems 33, 25.

This work is a collaboration with Antonin Richard (University of Lorraine, GeoRessources).

Galaxy peculiar velocities are excellent cosmological probes provided that biases inherent to their measurements are contained before any study. We propose a new algorithm based on an object point process model whose probability density is built to minimize the effects of Malmquist biases and uncertainties due to lognormal errors in radial peculiar velocity catalogs. The resulting configurations are bias-minimized catalogs. Tests are conducted on synthetic catalogs mimicking the second and third distance modulus catalogs of the Cosmicflows project, and then on observational catalogs. The large scale structure reconstructed with the Wiener filter technique applied to the bias-minimized observational catalogs matches with great success the local Universe cosmic web as depicted by redshift surveys of local galaxies. These new bias-minimized versions of Cosmicflows catalogs can now be used as a starting point for several studies including the production of simulations constrained to reproduce the local Universe.

This work is a collaboration with Jenny Sorce (CNRS Cristal - Lille) and Elmo Tempel (Tartu Observatory - University of Tartu).

Economic forests represent today an important part of the wood production on a global scale. Measuring and understanding the impact of diseases that occur in this type of forest with a very low variety of species is therefore an economic matter today. Our interest is focused on a parasitic fungus impacting coniferous forests, Armillaria Ostoyae.

The approach consists in considering a Markovian model with multiple labels on a lattice of the same size as the data, a Potts-type model: each tree will have a life span and a spatial neighborhood. We use statistical inference techniques, both classical (pseudo-likelihood) and recent (ABC methods: ABC Shadow), which we apply to a 20-year dataset.

Seismic fault interpretation is an important input to subsurface models. Since in seismic images
the dominant features are reflection events corresponding to horizons, fault interpretation can
be achieved by computing a fault probability image.
We aim at quantifying the uncertainties related to the number and connectivity of faults honoring a
given probability image, as all the possible fault networks can yield different outcomes in terms of
subsurface behavior (e.g., reservoir flow).
Fault networks are seen as realization of marked-point processes whose density are defined by an energy function.
We use a simulated annealing framework based
on Metropolis-Hastings algorithms, which makes it possible to find the global maximum of the
probability density, built in the form of a Gibbs density. We apply the proposed approach to a 2-D
seismic cross-section extracted from the Volve seismic cube (provided by Equinor).

This work is a collaboration with Fabrice Taty-Moukati, François Bonneau, Guillaume Caumon (University of Lorraine, GeoRessources) and Xinming Wu (University of Science and Technology of China, Hefei).

Fracture networks (FN) are systems of complex mechanical discontinuities in rocks. They dramatically impact fluid flow acting as a drain or a barrier. With François Bonneau (University of Lorraine, Georessources), we propose to use marked-point processes to build a stochastic mathematical model for fracture characterization and to approximate FN with a collection of marked-points standing for straight-line segment with a record of the length and the strike azimuth of the horizon. Geologists usually use a characterization workflow integrating density or mark distributions, which are first-order metrics, to describe the number and the geometry of fractures. Recently, second-order characteristics have been used to characterize FN's inner correlation and spatial variability. We then investigate stochastic models whose realizations reproduce the first and second order characteristics of the observations. This work may open the path to a thinner classification of FN and to predictive stochastic simulations.

We are extending our respective results on high frequency approximation of the local time of oscillating-skew-sticky diffusion processes. The purpose is to estimate the parameters of stickiness and skewness separately and to model some critical behavior in financial markets related to crisis periods. We managed to extend the results obtained during the PhD thesis of Alexis Anagnostakis in the context of sticky BM to more general estimators of local time and to oscillating-skew-sticky Brownian motion. Our goal is to reach rates of convergence obtained in 37. We are also dealing with the different problem non-uniqueness of solutions in this context.

With Lucian Beznea (IMAR, Bucharest) and Oana Lupaşcu-Stamate (Institute of Mathematical Statistics and Applied Mathematics, Bucharest) we are developing a stochastic approach for the two-dimensional Navier-Stokes equation in a bounded domain. More precisely we consider the vorticity equation and construct a specific non-local branching process. This approach is new and can conduct to important advances as it will give also a new numerical algorithm if successful.

In particular, we obtained several results concerning the construction of a duality - time reversal process and also in the development of a numerical algorithm with a non-local branching process involving creation and disapearance of particles that mimic the physics of the vorticity in the boundary layer.

Madalina Deaconu is Deputy Head of Science at Inria Nancy - Grand Est since January 2022. She is also, at the national level, member of the Evaluation Commission of Inria.

Since January 2018, Madalina Deaconu is the Head of the Fédération Charles Hermite, a federation of research within CNRS and University of Lorraine,
gathering three research laboratories: CRAN (control theory), IECL (mathematics) and LORIA
(computer science) with the goal of creating interdisciplinary projects and partnerships with industry.

She is also member of the IECL Laboratory Council, and of the Conseil de Pôle AM2I, University of Lorraine (2018-2022).

At the national level she is member of the Scientific Committee of the CNRS GdR MathGéoPhys in mathematics
in interaction with geophysics.

Antoine Lejay is Head of the GdR TRAG
(INSMI-CNRS).
He is also member
of the board the AMIES
and of the executive committee of Impact LUE Digitrust project (University of Lorraine).
He is also representative of IECL for program submission on Interactions: Humans and Sytems in a Digital World for the LUE (Lorraine Université d'Excellence) call for 2023-2028.

He was also the Head of
the Probability and Statistics team of IECL up to October 2022.

Pascal Moyal is Head of the Probability and Statistics team at IECL (2022-). As such, he is member of the Laboratory Council of IECL.

He is also the Head of the Master 2 Ingénierie Mathématique et Sciences des Données at University of Lorraine.

Sara Mazzonetto is assistant professor, Pascal Moyal and Radu Stoica are professors. They have full teaching duties with lectures at all the levels of the university. We mention here only lectures at Master 1 and Master 2 levels as well as responsabilities.

The following PhD have been defended in 2022:

Regarding the PhD in progress: