caRamel optimizer

Description

Automatic Calibration by Evolutionary Multi Objective Algorithm

Details

caRamel is a package for multi-objective optimization of complex environmental models.

The algorithm is a hybrid of the MEAS algorithm (Efstratiadis and Koutsoyiannis, 2005) by using the directional search method based on the simplexes of the objective space and the epsilon-NGSA-II algorithm with the method of classification of the parameter vectors archiving management by epsilon-dominance (Reed and Devireddy, 2004).

The main function of the package is caRamel().

This function uses all the other functions of the package.

An example of an hydrological optimization is available on the following presentation: useR! 2019

Author(s)

Fabrice Zaoui, Nicolas Le Moine, Celine Monteil (EDF R&D - LNHE)

References

Efstratiadis, A. and Koutsoyiannis, D. (2005) The multi-objective evolutionary annealing-simplex method and its application in calibration hydrological models, in EGU General Assembly 2005, Geophysical Research Abstracts, Vol. 7, Vienna, 04593, European Geophysical Union. doi:10.13140/ RG.2.2.32963.81446.

Le Moine, N. (2009) Description d’un algorithme génétique multi-objectif pour la calibration d’un modèle pluie-débit (in French). Post-Doctoral Status Rep. 2, UPMC/EDF, 13 pp.

Reed, P. and Devireddy, D. (2004) Groundwater monitoring design: a case study combining epsilon-dominance archiving and automatic parameterization for the NSGA-II, in Coello-Coello C, editor. Applications of multi-objective evolutionary algorithms, Advances in natural computation series, vol. 1, pp. 79-100, World Scientific, New York. doi:10.1142/9789812567796_0004.

Extrapolation along orthogonal directions to the Pareto front in the space of the objectives

Description

gives n new candidates by extrapolation along orthogonal directions to the Pareto front in the space of the objectives

Usage

Cextrap(param, crit, directions, longu, n)

Arguments

Value

xnew : matrix [ n , NPar ] of new vectors

pcrit : matrix [ n , NObj ] estimated positions of new sets in the goal space

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
param <- matrix(rexp(100), 100, 1)
crit <- matrix(rexp(200), 100, 2)
directions <- matrix(c(1,3,2,7,13,40), nrow = 3, ncol = 2)
longu <- runif(3)
n <- 5
# Call the function
res <- Cextrap(param, crit, directions, longu, n)

Interpolation in simplexes of the objective space

Description

proposes n new candidates by interpolation in simplexes of the objective space

Usage

Cinterp(param, crit, simplices, volume, n)

Arguments

Value

xnew : matrix [ n , NPar ] of new vectors

pcrit : matrix [ n , NObj ] estimated positions of new sets in the goal space

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
param <- matrix(rexp(100), 100, 1)
crit <- matrix(rexp(200), 100, 2)
simplices <- matrix(c(15,2,1,15,22,1,18,15,2,17,13,14), nrow = 4, ncol = 3)
volume <- runif(4)
n <- 5
# Call the function
res <- Cinterp(param, crit, simplices, volume, n)

Recombination of the sets of parameters

Description

performs a recombination of the sets of parameters

Usage

Crecombination(param, blocks, n)

Arguments

Value

xnew : matrix [ n , NPar ] of new vectors

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
param <- matrix(rexp(15), 15, 1)
blocks <- NULL
n <- 5
# Call the function
res <- Crecombination(param, blocks, n)

New parameter vectors generation respecting a covariance structure

Description

proposes new parameter vectors respecting a covariance structure

Usage

Cusecovar(xref, amplif, n)

Arguments

Value

xnew : matrix [ n , NPar ] of new vectors

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
xref <- matrix(rexp(35), 35, 1)
amplif <- 2.
n <- 5
# Call the function
res <- Cusecovar(xref, amplif, n)

Determination of directions for improvement

Description

determines directions for improvement

Usage

Dimprove(o_splx, f_splx)

Arguments

Value

list of elements "oriedge": oriented edges and "ledge": length

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
o_splx <- matrix(rexp(6), 3, 2)
f_splx <- c(1,1,1)
# Call the function
res <- Dimprove(o_splx, f_splx)

Box numbering for each points individual of the population

Description

This function returns a box number for each points individual of the population

Usage

boxes(points, prec)

Arguments

Value

vector of numbers for the boxes. boxes[i] gives the number of the box containing points[i].

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
points <- matrix(rexp(200), 100, 2)
prec <- c(1.e-3, 1.e-3)
# Call the function
res <- boxes(points, prec)

MAIN FUNCTION: multi-objective optimizer

Description

Multi-objective optimizer. It requires to define a multi-objective function (func) to calibrate the model and bounds on the parameters to optimize.

Usage

caRamel(
  nobj,
  nvar,
  minmax,
  bounds,
  func,
  popsize,
  archsize,
  maxrun,
  prec,
  repart_gene = c(5, 5, 5, 5),
  gpp = NULL,
  blocks = NULL,
  pop = NULL,
  funcinit = NULL,
  objnames = NULL,
  listsave = NULL,
  write_gen = FALSE,
  carallel = 1,
  numcores = NULL,
  graph = TRUE,
  sensitivity = FALSE,
  verbose = TRUE,
  worklist = NULL
)

Arguments

Details

The optimizer was originally written for Scilab by Nicolas Le Moine. The algorithm is a hybrid of the MEAS algorithm (Efstratiadis and Koutsoyiannis (2005) <doi:10.13140/RG.2.2.32963.81446>) by using the directional search method based on the simplexes of the objective space and the epsilon-NGSA-II algorithm with the method of classification of the parameter vectors archiving management by epsilon-dominance (Reed and Devireddy <doi:10.1142/9789812567796_0004>). Reference : "Multi-objective calibration by combination of stochastic and gradient-like parameter generation rules – the caRamel algorithm" Celine Monteil (EDF), Fabrice Zaoui (EDF), Nicolas Le Moine (UPMC) and Frederic Hendrickx (EDF) June 2020 Hydrology and Earth System Sciences 24(6):3189-3209 DOI: 10.5194/hess-24-3189-2020 Documentation : "Principe de l'optimiseur CaRaMEL et illustration au travers d'exemples de parametres dans le cadre de la modelisation hydrologique conceptuelle" Frederic Hendrickx (EDF) and Nicolas Le Moine (UPMC) Report EDF H-P73-2014-09038-FR

Value

List of seven elements:

success

return value (logical, length = 1) : TRUE if successfull

parameters

Pareto front (matrix, nrow = archsize, ncol = nvar)

objectives

objectives of the Pareto front (matrix, nrow = archsize, ncol = nobj+nadditional)

derivatives

list of the Jacobian matrices of the Pareto front if the sensitivity parameter is TRUE or NA otherwise

save_crit

evolution of the optimal objectives

total_pop

total population (matrix, nrow = popsize+archsize, ncol = nvar+nobj+nadditional)

gpp

the calling period for the third generation rule (independent sampling with a priori parameters variance)

Author(s)

Fabrice Zaoui - Celine Monteil

Examples

# Definition of the test function
viennet <- function(i) {
  val1 <- 0.5*(x[i,1]*x[i,1]+x[i,2]*x[i,2])+sin(x[i,1]*x[i,1]+x[i,2]*x[i,2])
  val2 <- 15+(x[i,1]-x[i,2]+1)*(x[i,1]-x[i,2]+1)/27+(3*x[i,1]-2*x[i,2]+4)*(3*x[i,1]-2*x[i,2]+4)/8
  val3 <- 1/(x[i,1]*x[i,1]+x[i,2]*x[i,2]+1) -1.1*exp(-(x[i,1]*x[i,1]+x[i,2]*x[i,2]))
  return(c(val1,val2,val3))
}
# Number of objectives
nobj <- 3
# Number of variables
nvar <- 2
# All the objectives are to be minimized
minmax <- c(FALSE, FALSE, FALSE)
# Define the bound constraints
bounds <- matrix(data = 1, nrow = nvar, ncol = 2)
bounds[, 1] <- -3 * bounds[, 1]
bounds[, 2] <- 3 * bounds[, 2]

# Caramel optimization
results <-
  caRamel(nobj = nobj,
          nvar = nvar,
          minmax =  minmax,
          bounds = bounds,
          func = viennet,
          popsize = 100,
          archsize = 100,
          maxrun = 500,
          prec = matrix(1.e-3, nrow = 1, ncol = nobj),
          carallel = 0)

Decreasing of the population of parameters sets

Description

decreases the population of parameters sets

Usage

decrease_pop(matobj, minmax, prec, archsize, popsize)

Arguments

Value

A list containing two elements:

ind_arch

indices of individuals in the updated Pareto front

ind_pop

indices of individuals in the updated population

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
matobj <- matrix(rexp(200), 100, 2)
prec <- c(1.e-3, 1.e-3)
archsize <- 100
minmax <- c(FALSE, FALSE)
popsize <- 100
# Call the function
res <- decrease_pop(matobj, minmax, prec, archsize, popsize)

Successive Pareto fronts of a population

Description

calculates the successive Pareto fronts of a population (classification "onion peel"), when objectives need to be maximized.

Usage

dominate(matobj)

Arguments

Value

f : vector of dimension NInd of dominances

Author(s)

Alban de Lavenne, Fabrice Zaoui

Examples

# Definition of the parameters
matobj <- matrix(runif(200), 100, 2)
# Call the function
pareto_rank <- dominate(matobj)

Rows domination of a matrix by a vector

Description

indicates which rows of the matrix Y are dominated by the vector (row) x

Usage

dominated(x, Y)

Arguments

Value

D : vector of booleans

Author(s)

Alban de Lavenne, Fabrice Zaoui

Examples

# Definition of the parameters
Y <- matrix(rexp(200), 100, 2)
x <- Y[1,]
# Call the function
res <- dominated(x, Y)

Downsizing of a population to only one individual per box up to a given accuracy

Description

reduces the number of individuals in a population to only one individual per box up to a given accuracy

Usage

downsize(points, Fo, prec)

Arguments

Value

vector indices

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
points <- matrix(rexp(200), 100, 2)
prec <- c(1.e-3, 1.e-3)
Fo <- sample(1:100, 100)
# Call the function
res <- downsize(points, Fo, prec)

Calculation of the variances-covariances matrix on the reference population

Description

calculates the variances-covariances matrix on the reference population

Usage

matvcov(x, g)

Arguments

Value

rr : variances-covariances matrix on the reference population

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
x <- matrix(rexp(30), 30, 1)
g <- mean(x)
# Call the function
res <- matvcov(x, g)

Generation of a new population of parameter sets following the five rules of caRamel

Description

generates a new population of parameter sets following the five rules of caRamel

Usage

newXval(param, crit, isperf, sp, bounds, repart_gene, blocks, fireworks)

Arguments

Value

xnew : matrix of new vectors [ sum(Repart_Gene) + eventually (nobj+1)*nvar if fireworks , NPar ]

project_crit: assumed position of the new vectors in the criteria space: [ sum(Repart_Gene)+ eventually (nobj+1)*nvar if fireworks , NObj ];

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
param <- matrix(rexp(100), 100, 1)
crit <- matrix(rexp(200), 100, 2)
isperf <- c(FALSE, FALSE)
bounds <- matrix(data = 1, nrow = 1, ncol = 2)
bounds[, 1] <- -5 * bounds[, 1]
bounds[, 2] <- 10 * bounds[, 2]
sp <- (bounds[, 2] - bounds[, 1]) / (2 * sqrt(3))
repart_gene <- c(5, 5, 5, 5)
fireworks <- TRUE
blocks <- NULL
# Call the function
res <- newXval(param, crit, isperf, sp, bounds, repart_gene, blocks, fireworks)

Indicates which rows are Pareto

Description

indicates which rows of the X criterion matrix are Pareto, when objectives need to be maximized

Usage

pareto(X)

Arguments

Value

Ft : vector [NInd], TRUE when the set is on the Pareto front.

Author(s)

Alban de Lavenne, Fabrice Zaoui

Examples

# Definition of the parameters
X <- matrix(runif(200), 100, 2)
# Call the function
is_pareto <- pareto(X)

Plotting of caRamel results

Description

Plot graphs of the Pareto front and a graph of optimization evolution

Usage

plot_caramel(caramel_results, nobj = NULL, objnames = NULL)

Arguments

Examples

# Definition of the test function
viennet <- function(i) {
  val1 <- 0.5*(x[i,1]*x[i,1]+x[i,2]*x[i,2])+sin(x[i,1]*x[i,1]+x[i,2]*x[i,2])
  val2 <- 15+(x[i,1]-x[i,2]+1)*(x[i,1]-x[i,2]+1)/27+(3*x[i,1]-2*x[i,2]+4)*(3*x[i,1]-2*x[i,2]+4)/8
  val3 <- 1/(x[i,1]*x[i,1]+x[i,2]*x[i,2]+1) -1.1*exp(-(x[i,1]*x[i,1]+x[i,2]*x[i,2]))
  return(c(val1,val2,val3))
}
nobj <- 3 # Number of objectives
nvar <- 2 # Number of variables
minmax <- c(FALSE, FALSE, FALSE) # All the objectives are to be minimized
bounds <- matrix(data = 1, nrow = nvar, ncol = 2) # Define the bound constraints
bounds[, 1] <- -3 * bounds[, 1]
bounds[, 2] <- 3 * bounds[, 2]

# Caramel optimization
results <- caRamel(nobj, nvar, minmax, bounds, viennet, popsize = 100, archsize = 100,
          maxrun = 500, prec = matrix(1.e-3, nrow = 1, ncol = nobj), carallel = FALSE)

# Plot of results
plot_caramel(results)

Plotting of a population of objectives and Pareto front

Description

Plots graphs the population regarding each couple of objectives and emphasizes the Pareto front

Usage

plot_pareto(MatObj, nobj = NULL, objnames = NULL, maximized = NULL)

Arguments

Author(s)

Celine Monteil

Examples

# Definition of the population
Pop <- matrix(runif(300), 100, 3)

# Definition of objectives to maximize (Obj1, Obj2) and to minimize (Obj3)
maximized <- c(TRUE, TRUE, FALSE)

# Call the function
plot_pareto(MatObj = Pop, maximized = maximized)

Plotting of a population of objectives

Description

Plot graphs the population regarding each couple of objectives

Usage

plot_population(
  MatObj,
  nobj,
  ngen = NULL,
  nrun = NULL,
  objnames = NULL,
  MatEvol = NULL,
  popsize = 0
)

Arguments

Author(s)

Celine Monteil

Examples

# Definition of the population
Pop <- matrix(runif(300), 100, 3)
# Call the function
plot_population(MatObj = Pop, nobj = 3, objnames = c("Obj1", "Obj2", "Obj3"))

Selection of n points

Description

performs a selection of n points in facp

Usage

rselect(n, facp)

Arguments

Value

ix : ranks of selected points (vector of dimension n)

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
n <- 5
facp <- runif(30)
# Call the function
res <- rselect(n, facp)

Converting the values of a vector into their rank

Description

converts the values of a vector into their rank

Usage

val2rank(X, opt)

Arguments

Value

R : rank vector

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
X <- matrix(rexp(100), 100, 1)
opt <- 3
# Call the function
res <- val2rank(X, opt)

Volume of a simplex

Description

calculates the volume of a simplex

Usage

vol_splx(S)

Arguments

Value

V : simplex volume

Author(s)

Fabrice Zaoui

Examples

# Definition of the parameters
S <- matrix(rexp(6), 3, 2)
# Call the function
res <- vol_splx(S)