Optimisation with the Shuffle Complex Evolution method

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Description

Calibration function which searches a parameter set which is minimizing the value of an objective function

Usage

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sceua(OFUN, pars, lower, upper, maxn = 10000, kstop = 5, pcento = 0.01,
    ngs = 5, npg = 5, nps = 5, nspl = 5, mings = 5, iniflg = 1, iprint = 0, iround = 3, 
    peps = 0.0001, plog = rep(FALSE,length(pars)), implicit = NULL, timeout = NULL, ...)

Arguments

OFUN

A function to be minimized, with first argument the vector of parameters over which minimization is to take place. It should return a scalar result as an indicator of the error for a certain parameter set

pars

a vector with the initial guess the parameters

lower

the lower boundary for the parameters

upper

the upper boundary for the parameters

maxn

the maximum number of function evaluations

kstop

number of shuffling loops in which the criterion value must change by the given percentage before optimization is terminated

pcento

percentage by which the criterion value must change in given number (kstop) of shuffling loops to continue optimization

ngs

number of complexes in the initial population

npg

number of points in each complex

nps

number of points in a sub-complex

nspl

number of evolution steps allowed for each complex before complex shuffling

mings

minimum number of complexes required, if the number of complexes is allowed to reduce as the optimization proceeds

iniflg

flag on whether to include the initial point in population = 0, not included = 1, included

iprint

flag for controlling print-out after each shuffling loop = 0, print information on the best point of the population = 1, print information on every point of the population

iround

number of significant digits in print-out

peps

convergence level for parameter set (lower number means smaller difference between parameters of the population required for stop)

plog

whether optimization should be done in log10-domain. Either a single TRUE value for all parameters, or a vector with TRUE/FALSE for the different parameters

implicit

function for implicit boundaries for the parameters (e.g. sum(pars[4]+pars[5]) < 1). See below for details

timeout

if different from NULL: maximum time in seconds for execution before the optimization returns with the parameters so far.

...

arguments for the objective function, must be named

Details

sceua is an R-implementation of the Shuffle Complex Evolution - University of Arizona (Duan et al., 1992), a global optimization method which "combines the strengths of the simplex procedure of Nelder and Mead (1965) with the concepts of controlled random search (Price, 1987), competetive evolusion (Holland, 1975)" with the concept of complex shuffling, developed by Duan et al. (1992).

This implementation follows the Fortran implementation relatively close, but adds the possibility of searching in log-space for one or more of the parameters, and it uses the capability of R to pass functions as arguments, making it possible to pass implicit conditions to the parameter selection.

The objective function OFUN is a function which should give an error value for each parameter set. It should never return non-numeric values such as NA, NULL, or Inf. If some parameter combinations can give such values, the return value should rather be a large number.

The function works with fixed upper and lower boundaries for the parameters. If the possible range of a parameter might span several orders of magnitude, it might be better to search in log-space for the optimal parameter, to reduce the risk of being trapped in local optima. This can be set with the argument plog, which is either a single value (FALSE/TRUE) or a vector for all parameters. plog = c(TRUE, FALSE, FALSE, TRUE, TRUE) means that the search for parameters 1,4 and 5 should be in log10-space, whereas the search for parameters 2 and 3 are in normal space.

Implicit boundaries can be evoked by passing a function implicit to sceua. This function should give 0 when parameters are acceptable and 1 if not. If, for example, the condition is that the following sum of parameters four and five should be limited:

sum(pars[4]+pars[5]) <= 1

then the function will be implicit = function(pars) (2*pars[4] + pars[5]) > 1

Value

The function returns a list with the following elements

  • par - a vector of the best parameters combination

  • value - the value of the objective function for this parameter set

  • convergence - a list of two values

    • funConvergence - the function convergence relative to pcento

    • parConvergence - the parameter convergence relative to peps

  • counts - the number of function evaluations

  • iterations - the number of shuffling loops

  • timeout - logical; TRUE if the optimization was aborted because the timeout time was reached, FALSE otherwise

Author(s)

Jon Olav Skoien

References

Duan, Q., Sorooshian, S., and Gupta, V.K., 1992. Effective and efficient global optimization for conceptual rainfall-runoff models. Water Resour. Res. 28 (4), 1015?1031.

Holland, H.H., 1975. Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor.

Nelder, J.A. and Mead, R., 1965. A simplex method for function minimization, Comput. J., 7(4), 308-313.

Price, W.L., 1987. Global optimization algorithms for a CAD workstation, J. Optim. Theory Appl., 55(1), 133-146.

Skoien, J. O., Bloschl, G., Laaha, G., Pebesma, E., Parajka, J., Viglione, A., 2014. Rtop: An R package for interpolation of data with a variable spatial support, with an example from river networks. Computers & Geosciences, 67.

Examples

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set.seed(1)
# generate example data from a function with three parameters
# with some random noise
fun = function(x, pars) pars[2]*sin(x*pars[1])+pars[3]
x = rnorm(50, sd = 3)
y = fun(x, pars = c(5, 2, 3)) +  rnorm(length(x), sd = 0.3)
plot(x,y)
   
# Objective function, summing up squared differences
OFUN = function(pars, x, yobs) {
  yvals = fun(x, pars)  
  sum((yvals-yobs)^2)
}

sceuares = sceua(OFUN, pars = c(0.1,0.1,0.1), lower = c(-10,0,-10), 
                 upper = c(10,10,10), x = x, yobs = y)
sceuares
xx = seq(min(x), max(x), 0.1)
lines(xx, fun(xx, pars = sceuares$par))