# 1 Introduction

## 1.1 Scope

The Michel’s calibration strategy (Calibration_Michel() function) is the calibration algorithm proposed in airGR. However, other optimization methods can be used in combination with airGR. We show here how to use different R packages to perform parameter estimation.

In this article, we use the GR4J model to illustrate the different optimization strategies. In particular, we assume that the R global environment contains input climate data, observed discharge and functions from the Get Started article, as shown below. Please note that the calibration period is defined in the CreateRunOptions() function .

example("Calibration_Michel")

In order for the RunModel_*() functions to run faster during the parameter estimation process, it is recommended that the outputs contain only the simulated flows (see the Outputs_Sim argument in the CreateRunOptions() help page).

RunOptions <- airGR::CreateRunOptions(FUN_MOD = RunModel_GR4J, InputsModel = InputsModel,
IndPeriod_Run = Ind_Run,
Outputs_Sim = "Qsim")

Regarding the different optimization strategies presented here, we refer to each package for in-depth information about the description of the methods used.

## 1.2 Definition of the necessary function

Parameter estimation can be performed by defining a function that takes a parameter set as input and returns the value of the performance criterion. There are two important steps: the transformation of parameters to real space and the computation of the value of the performance criterion. Here we choose to minimize the root mean square error.

The change of the repository from the “real” parameter space to a “transformed” space ensures homogeneity of displacement in the different dimensions of the parameter space during the step-by-step procedure of the calibration algorithm of the model.

OptimGR4J <- function(ParamOptim) {
## Transformation of the parameter set to real space
RawParamOptim <- airGR::TransfoParam_GR4J(ParamIn = ParamOptim,
Direction = "TR")
## Simulation given a parameter set
OutputsModel <- airGR::RunModel_GR4J(InputsModel = InputsModel,
RunOptions = RunOptions,
Param = RawParamOptim)
## Computation of the value of the performance criteria
OutputsCrit <- airGR::ErrorCrit_RMSE(InputsCrit = InputsCrit,
OutputsModel = OutputsModel,
verbose = FALSE)
return(OutputsCrit$CritValue) } In addition, we need to define the lower and upper bounds of the four GR4J parameters in the transformed parameter space: lowerGR4J <- rep(-9.99, times = 4) upperGR4J <- rep(+9.99, times = 4) # 2 Local optimization We start with a local optimization strategy by using the PORT routines (using the nlminb() of the stats package) and by setting a starting point in the transformed parameter space: startGR4J <- c(4.1, 3.9, -0.9, -8.7) optPORT <- stats::nlminb(start = startGR4J, objective = OptimGR4J, lower = lowerGR4J, upper = upperGR4J, control = list(trace = 1)) The RMSE value reaches a local minimum value after 35 iterations. We can also try a multi-start approach to test the consistency of the local optimization. Here we use the same grid used for the filtering step of the Michel’s calibration strategy (Calibration_Michel() function). For each starting point, a local optimization is performed. startGR4JDistrib <- TransfoParam_GR4J(ParamIn = CalibOptions$StartParamDistrib,
Direction = "RT")
startGR4J <- expand.grid(data.frame(startGR4JDistrib))
optPORT_ <- function(x) {
opt <- stats::nlminb(start = x,
objective = OptimGR4J,
lower = lowerGR4J, upper = upperGR4J,
control = list(trace = 1))
}
listOptPORT <- apply(startGR4J, MARGIN = 1, FUN = optPORT_)

We can then extract the best parameter sets and the value of the performance criteria:

parPORT <- t(sapply(listOptPORT, function(x) x$par)) objPORT <- sapply(listOptPORT, function(x) x$objective)
resPORT <- data.frame(parPORT, RMSE = objPORT)

As can be seen below, the optimum performance criterion values (column objective) can differ from the global optimum value in many cases, resulting in various parameter sets.

summary(resPORT)
##        X1              X2               X3                  X4                 RMSE
##  Min.   :5.548   Min.   :0.1240   Min.   :-0.038062   Min.   :-8.242667   Min.   :0.7864
##  1st Qu.:5.548   1st Qu.:0.1243   1st Qu.:-0.003741   1st Qu.:-8.242667   1st Qu.:0.7864
##  Median :5.548   Median :0.8866   Median : 4.478766   Median :-8.242667   Median :0.7864
##  Mean   :5.682   Mean   :0.6325   Mean   : 2.977265   Mean   :-5.820890   Mean   :0.9220
##  3rd Qu.:5.946   3rd Qu.:0.8866   3rd Qu.: 4.478766   3rd Qu.:-1.965413   3rd Qu.:1.1630
##  Max.   :5.953   Max.   :0.8866   Max.   : 4.478767   Max.   : 0.009845   Max.   :1.2235

The existence of several local minima illustrates the importance of defining an appropriate starting point or of using a multi-start strategy or a global optimization strategy.

# 3 Global optimization

Global optimization is most often used when facing a complex response surface, with multiple local mimina. Here we use the following R implementation of some popular strategies:

## 3.1 Differential Evolution

optDE <- DEoptim::DEoptim(fn = OptimGR4J,
lower = lowerGR4J, upper = upperGR4J,
control = DEoptim::DEoptim.control(NP = 40, trace = 10))

## 3.2 Particle Swarm

# to install the package temporary removed from CRAN
# Rtools needed (windows : https://cran.r-project.org/bin/windows/Rtools/)
# install.packages("https://cran.r-project.org/src/contrib/Archive/hydroPSO/hydroPSO_0.5-1.tar.gz",
#                  repos = NULL, type = "source", dependencies = TRUE)
optPSO <- hydroPSO::hydroPSO(fn = OptimGR4J,
lower = lowerGR4J, upper = upperGR4J,
control = list(write2disk = FALSE, verbose = FALSE))

## 3.3 MA-LS-Chains

optMALS <- Rmalschains::malschains(fn = OptimGR4J,
lower = lowerGR4J, upper = upperGR4J,
maxEvals = 2000)

# 4 Results

As it can be seen in the table below, the four additional optimization strategies tested lead to very close optima.

##    Algo      X1    X2     X3    X4
## 1 airGR 257.238 1.012 88.235 2.208
## 2  PORT 256.840 1.007 88.126 2.205
## 3    DE 256.840 1.007 88.126 2.205
## 4   PSO 256.799 1.007 88.147 2.205
## 5 MA-LS 256.820 1.007 88.116 2.205

# 5 Multiobjective optimization

Multiobjective optimization is used to explore possible trade-offs between different performances criteria. Here we use the following R implementation of an efficient strategy:

Motivated by using the rainfall-runoff model for low flow simulation, we explore the trade-offs between the KGE values obtained without any data transformation and with the inverse transformation.

First, the OptimGR4J() function previously used is modified to return two values.

InputsCrit_inv <- InputsCrit
InputsCrit_inv$transfo <- "inv" MOptimGR4J <- function(i) { if (algo == "caRamel") { ParamOptim <- x[i, ] } ## Transformation of the parameter set to real space RawParamOptim <- airGR::TransfoParam_GR4J(ParamIn = ParamOptim, Direction = "TR") ## Simulation given a parameter set OutputsModel <- airGR::RunModel_GR4J(InputsModel = InputsModel, RunOptions = RunOptions, Param = RawParamOptim) ## Computation of the value of the performance criteria OutputsCrit1 <- airGR::ErrorCrit_KGE(InputsCrit = InputsCrit, OutputsModel = OutputsModel, verbose = FALSE) ## Computation of the value of the performance criteria OutputsCrit2 <- airGR::ErrorCrit_KGE(InputsCrit = InputsCrit_inv, OutputsModel = OutputsModel, verbose = FALSE) return(c(OutputsCrit1$CritValue, OutputsCrit2$CritValue)) } ## 5.1 caRamel caRamel is a multiobjective evolutionary algorithm combining the MEAS algorithm and the NGSA-II algorithm. algo <- "caRamel" optMO <- caRamel::caRamel(nobj = 2, nvar = 4, minmax = rep(TRUE, 2), bounds = matrix(c(lowerGR4J, upperGR4J), ncol = 2), func = MOptimGR4J, popsize = 100, archsize = 100, maxrun = 15000, prec = rep(1.e-3, 2), carallel = FALSE, graph = FALSE) The algorithm returns parameter sets that describe the pareto front, illustrating the trade-off between overall good performance and good performance for low flow. ggplot() + geom_point(aes(optMO$objectives[, 1], optMO$objectives[, 2])) + coord_equal(xlim = c(0.4, 0.9), ylim = c(0.4, 0.9)) + xlab("KGE") + ylab("KGE [1/Q]") + theme_bw() The parameter sets can be viewed in the parameter space, illustrating different populations. param_optMO <- apply(optMO$parameters, MARGIN = 1, FUN = function(x) {
airGR::TransfoParam_GR4J(x, Direction = "TR")
})
GGally::ggpairs(data.frame(t(param_optMO)), diag = NULL) + theme_bw()

RunOptions$Outputs_Sim <- "Qsim" run_optMO <- apply(optMO$parameters, MARGIN = 1, FUN = function(x) {
airGR::RunModel_GR4J(InputsModel = InputsModel,
RunOptions = RunOptions,
Param = x)
}$Qsim) run_optMO <- data.frame(run_optMO) ggplot() + geom_line(aes(x = as.POSIXct(InputsModel$DatesR[Ind_Run]),
y = run_optMO$X1)) + geom_line(aes(x = as.POSIXct(InputsModel$DatesR[Ind_Run]),
y = run_optMO\$X54),
colour = "darkred") +
scale_x_datetime(limits = c(as.POSIXct("1998-01-01"), NA)) +
ylab("Discharge [mm/d]") + xlab("Date") +
scale_y_sqrt() +
theme_bw()