PoMoS & GloMo

 

 

 

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.

PoMoS (Polynomial Model Search)

 

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.

 

 

GloMo (Global Modelling)

 

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 and Validation

 

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).

 


 

How to get the packages ?

 

If you are interested by the packages of these two routines, you could obtain them by writing to Sylvain Mangiarotti in filling the form available at this address. Please, provide few words at the end of the form about the fields in which the source code will be used.

 

Other packages under development:

 

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

 

References :

 

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.

 


 

Retour au site web du CESBIO