GPoM (version 1.0)

gPoMo: gPoMo: Generalized Polynomial Modeling

Description

Algorithm for a Generalized Polynomial formulation of multivariate Global Modeling. Global modeling aims to obtain multidimensional models from single time series [1-2]. In the generalized (polynomial) formulation provided in this function, it can also be applied to multivariate time series [3-4].

Example: Note that nS provides the number of dimensions used from each variable

case I For nS=c(2,3) means that 2 dimensions are reconstructed from variable 1: the original variable X1 and its first derivative X2), and 3 dimensions are reconstructed from variable 2: the original variable X3 and its first and second derivatives X4 and X5. The generalized model will thus be such as: \(dX1/dt = X2\) \(dX2/dt = P1(X1,X2,X3,X4,X5)\) \(dX3/dt = X4\) \(dX4/dt = X5\) \(dX5/dt = P2(X1,X2,X3,X4,X5).\)

case II For nS=c(1,1,1,1) means that only the original variables X1, X2, X3 and X4 will be used. The generalized model will thus be such as: \(dX1/dt = P1(X1,X2,X3,X4)\) \(dX2/dt = P2(X1,X2,X3,X4)\) \(dX3/dt = P3(X1,X2,X3,X4)\) \(dX4/dt = P4(X1,X2,X3,X4).\)

Usage

gPoMo(data, tin = NULL, dtFixe = NULL, dMax = 2, nS = c(3), winL = 9,
  weight = NULL, show = 1, verbose = 1, underSamp = NULL, EqS = NULL,
  IstepMin = 2, IstepMax = 2000, nPmin = 1, nPmax = 14,
  method = "lsoda")

Arguments

data

Input Time series: Each column corresponds to one input variable.

tin

Input date vector which length should correspond to the variables of the input data (same number of lines).

dtFixe

Time step used for the analysis. In principle, it should correspond to the time step of the input data. Modification of this time step may be used to stabilize the numerical computation. Such modification should be performed in full consciouness that it will lead to change the time scale accordingly.

dMax

Maximum degree of the polynomial functions allowed in the model (see poLabs).

nS

A vector providing the number of dimensions that will be used for each input variables (see Examples 1 and 2). The dimension of the resulting model will be nVar = sum(nS).

winL

Total number of points used for computing the derivatives of the input time series (data). This parameter will be used as an input in function drvSucc.

weight

Weighting function of input data series. By default uniform weight is applied. This weighting function can also be used to apply the analysis piecewise using zeros and ones.

show

Indicates (2) or not (0-1) the algorithm progress

verbose

Gives information (if set to 1) about the algorithm progress and keeps silent if set to 0

underSamp

Points used for undersampling the data. For undersamp = 1 the complete time series is used. For undersamp = 2, only one data out of two is kept, etc.

EqS

Model template including all allowed regressors. Each column corresponds to one equation. Each line corresponds to one regressor following the convention given by poLabs(nVar,dMax).

IstepMin

Minimum step of integration at the beginning of the analysis (by default IstepMin=10).

IstepMax

Maximum step of integration before stopping the analysis, with the interface this value can be changed during the analysis

nPmin

Corresponds to the minimum number of parameters (and thus of regressor) of the model

nPmax

Corresponds to the maximum number of parameters (and thus of regressor) of the model

method

The integration technique used for the numerical integration. Default is 'lsoda'. Others such as 'rk4' or 'ode45' may also be used. See package deSolve for details.

Value

A list containing:

$tin The time vector of the input time series

$inputdata The input time series

$tfiltdata The time vector of the filtered time series (boudary removed)

$filtdata A matrix of the filtered time series with its derivatives

$okMod A vector classifying the models: diverging models (0), periodic models of period-1 (-1), unclassified models (1).

$coeff A matrix with the coefficients of one selected model

$models A list of all the models to be tested $mToTest1, $mToTest2, etc. and all selected models $model1, $model2, etc.

$tout The time vector of the output time series (vector length corresponding to the longest numerical integration duration)

$stockoutreg A list of matrices with the integrated trajectories (variable X1 in column 1, X2 in 2, etc.) of all the models $model1, $model2, etc.

References

[1] 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. [2] Mangiarotti S., Coudret R., Drapeau L. & Jarlan L., Polynomial search and Global modelling: two algorithms for modeling chaos. Physical Review E, 86(4), 046205. [3] Mangiarotti S., Le Jean F., Huc M. & Letellier C., Global Modeling of aggregated and associated chaotic dynamics. Chaos, Solitons and Fractals, 83, 82-96. [4] S. Mangiarotti, M. Peyre & M. Huc, 2016. A chaotic model for the epidemic of Ebola virus disease in West Africa (2013-2016). Chaos, 26, 113112.

See Also

gloMoId, autoGPoMoSearch, autoGPoMoTest

Examples

# NOT RUN {
#Example 1
data("Ross76")
tin <- Ross76[,1]
data <- Ross76[,3]
dev.new()
out1 <- gPoMo(data, tin=tin, dMax = 2, nS=c(3), show = 1,
              IstepMax = 1000, nPmin = 9, nPmax = 11)
visuEq(3, 2, out1$models$model1, approx = 4)

# }
# NOT RUN {
#Example 2
data("Ross76")
tin <- Ross76[,1]
data <- Ross76[,2:4]
dev.new()
out2 <- gPoMo(data, tin=tin, dMax = 2, nS=c(1,1,1), show = 1,
              IstepMin = 10, IstepMax = 3000, nPmin = 7, nPmax = 8)
# the simplest model able to reproduce the observed dynamics is model #5
visuEq(3, 2, out2$models$model5, approx = 4) # the original Rossler system is thus retrieved
# }
# NOT RUN {
# }
# NOT RUN {
#Example 3
data("Ross76")
tin <- Ross76[,1]
data <- Ross76[,2:3]
# model template:
EqS <- matrix(1, ncol = 3, nrow = 10)
EqS[,1] <- c(0,0,0,1,0,0,0,0,0,0)
EqS[,2] <- c(1,1,0,1,0,1,1,1,1,1)
EqS[,3] <- c(0,1,0,0,0,0,1,1,0,0)
visuEq(3, 2, EqS, substit = c('X','Y','Z'))
dev.new()
out3 <- gPoMo(data, tin=tin, dMax = 2, nS=c(2,1), show = 1,
      EqS = EqS, IstepMin = 10, IstepMax = 2000,
      nPmin = 9, nPmax = 11)
# }
# NOT RUN {
# }
# NOT RUN {
#Example 4
data(sprottK)
data(rossler)
data <- cbind(rossler,sprottK)[1:400,]
dev.new()
out4 <- gPoMo(data, dt=1/20, dMax = 2, nS=c(1,1,1,1,1,1),
             show = 1, method = 'rk4',
             IstepMin = 2, IstepMax = 3, nPmin = 13, nPmax = 13)
# the original Rossler (variables x, y and z) and Sprott (variables X, Y and Z)
# systems are retrieved:
visuEq(6, 2, out4$models$model347, approx = 4,
           substit = c('x', 'y', 'z', 'X', 'Y', 'Z'))
   # to check the robustness of the model, the integration duration
   # should be chosen longer (at least IstepMax = 4000)
# }
# NOT RUN {
# }