sceua function

Optimisation with the Shuffle Complex Evolution method

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

sceua(OFUN, pars, lower, upper, maxn = 10000, kstop = 5, pcento = 0.01,
    ngs = 5, npg = 2 * length(pars) + 1, nps = length(pars) + 1, 
    nspl = 2 * length(pars) + 1, mings = ngs, 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. iniflg = 0, not included. iniflg= 1, included
• iprint: flag for controlling print-out after each shuffling loop. iprint < 0: no output. iprint = 1: print information on the best point of the population. iprint > 0: 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

Returns

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

There are also two elements returned as attributes:

• parset: - the entire set of parameters from the last evolution step
• xf: - the values of the objective function from the last evolution step

The last two can be accessed as attr(sceuares, "parset") and attr(sceuares, "xf"), if the result is stored as sceuares.

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

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.

Author(s)

Jon Olav Skoien

Examples

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))