1- PoMoS & GloMo
  2- Algorithmes
  3- Tests & validations
  4- Autres codes en développement :
  5- References
 

PoMoS & GloMo sont deux modules complémentaires développés au CESBIO pour la modélisation déterministe de systèmes mal connus et partiellement observés. Ces modules prennent à la fois appui sur une approche système complexe et sur la théorie des systèmes dynamiques. Mangiarotti S., Coudret R., Drapeau L.

 

  • PoMoS ( Polynomial Model Search) vise à construire un réseau de liens déterministes entre variables observées. Il est basé sur un algorithme de recherche évolutionnaire combiné à une approche par les moindres carrés.
  • GloMo ( Global Modelling) vise à construire un réseau de liens déterministes entre variables observées. Il est basé sur un algorithme de recherche évolutionnaire combiné à une approche par les moindres carrés.
 

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Figure : Reconstruction dans l’espace des phases du signal de blé pluvial, en zone semi-aride : signal observé par télédétection spatiale (noir), signal modélisé avec PoMoS & GloMo par approche globale (rouge).

Indépendants dans leur fonctionnement, ces deux modules ont été construits afin d’être utilisable en synergie. D’autres modules sont en cours de développement dans l’objet de constituer progressivement une plateforme d’analyse, de modélisation et de prévision spatialisée et multicritère.

PoMoS vient d'être validé par le Comprehensive R Archive Network (CRAN), il est également téléchargeable sur le site du CRAN. GloMo est en cours de validation.

Référence: G. Gouesbet & C. Letellier, Global vector field reconstruction by using a multivariate polynomial L2-approximation on nets, Physical Review E, 49(6), 4955-4972, 1994.

Si vous êtes intéréssé par le code source de ce modèle, vous pouvez vous le procurer auprès de Sylvain Mangiarotti du CESBIO, en remplissant ce formulaire, Merci de fournir en quelques mots le futur domaine d'application de cet outil.

Dans le supplément "Sciences et Médecine" du journal "Le Monde" de ce 20 janvier 2014, un article sur l' application de la théorie du chaos aux prévisions de rendement agricoles. Ce travail se base sur les travaux de Sylvain Mangiarotti chercheur au
LMI-TREMA/CESBIO/UCAM

Découvrir l'article

 

 

Algorithmes...

PoMoS (Polynomial Model Search) and GloMo (Global Modelling) are two algorithms developed as packages under R language for the modelling of dynamical systems observed from a restricted number of time series.

This package is dedicated to multiple time series datasets but also applies to single time series. It is based on an algorithm aiming at identifying the models’ structures of Ordinary Differential Equations (ODE) in a polynomial formulation (1) :

 
  where

The heuristic underlying the algorithm is an evolutionary algorithm combined to a least square technique. The quality of the models is estimated based on a criterion accounting for the fitting and the parsimony of the models. A convivial interface is provided allowing for an efficient parameterization of the heuristic and an interactive search of models’ structures.

This package is dedicated to global modelling from single time series. Only polynomial Ordinary Differential Equations are considered in the present version of the package. Global modelling approach was developed in the early 1990s by Gouesbet & Letellier 1994 (see also Letellier et al. 2009) and consists in getting a system of equations equivalent to the original system (Eq. 1) in the following canonical form (2) :

 
where , and a function of  one of the variables of the system.

When applying the approach to a real dataset,can advantageously be approximated by a polynomial. The GloMo package can be used separately or in interaction with the PoMoS package.

 


 

 

 

 

 

 

 

 

 

 

 

Tests & validations

The two packages were tested on three different systems for validation (Mangiarotti et al. 2012). An illustration of the result is provided for each system in Figure 1.

(1) On a theoretic model : the three variables of the Rössler system (Rössler 1976) were considered. Models of small size (which is generally a display of robustness) were obtained for each variable, even in the quite difficult case of variable z (see Lainscsek et al. 2003).

(2) On an experimental system : the electrodissolution of copper in phosphoric acid. The dataset obtained in John Hudson’s group (Basset & Hudson 1989) made available on the atomosyd website was used for this purpose. A model of small size was obtained also.

(3) On a real environmental system : the cycle of rainfed wheat in North Morocco as observed from satellite data. Data from the Global Inventory Modeling and Mapping Studies were used for this purpose (Tucker et al. 2005). A chaotic model of parsimonious size was also obtained for this latter system.

image  

 

Original phase portraits (in black) and models' phase portraits obtained by global modelling (in red).

Three cases are presented :

the z variable of Rössler system (left panels),

the current intensity measured in an experiment of copper electrodissolution in phosphoric acid (middle panels)

and the vegetation index measured from satellite corresponding to the cycle of rainfed wheat in North Morocco (left panels).

 

 

 

 

 

 

 

 

Autres codes en developpement

LyapFD computes the Lyapunov exponents of an ODE model of polynomial formulation

Poinca provides the Poincaré sections and the first return maps from an embedded flow

AMoPo packages dedicated to the data assimilation in a polynomial ODE model

SpatioGloMo for mutli-series / spatialized Global Modelling

SpatioPoMo for mutli-series / spatialized Polynomial Modelling

Réferences

 

Bassett M. R. & Hudson J. L., 1989. Quasi-Periodicity and chaos during an electrochemical reaction, The Journal of Physical Chemistry, 93, 2731–2737.

Gouesbet G. & Letellier C., 1994. Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets, Physical Review E, 49 (6), 4955–4972.

Lainscsek C., Letellier C. & Gorodnitsky I., 2003. Global modeling of the Rössler system from the z-variable, Physics Letters A, 314 (5-6), 409–427.

Letellier C., Aguirre L.A. & Freitas U.S., 2009. Frequently asked questions about global modeling. Chaos 19, doi:10.1063/1.3125705.

www.atomosyd.net

Mangiarotti S., Coudret R., Drapeau L. & Jarlan L., 2012. Polynomial Model Search & Global Modelling, two algorithms for modeling Chaos. Physical review E 86(4), 046205. (abstract)

Rössler O.E., 1976. An Equation for Continuous Chaos, Physics Letters, 57A(5), 397–398.

Tucker C.J., Pinzon J.E., Brown M.E., Slayback D.A., Pak E.W., Mahoney R., Vermote E.F. & Saleous N.E., 2005. An extended AVHRR 8-km NDVI dataset compatible with MODIS and SPOT vegetation NDVI data. International Journal of Remote Sensing, 26:20, 4485–4498