emma: Evolutionary Model-based Multiresponse Approach

Description Usage Arguments Details Value Author(s) References Examples

View source: R/emma.R

Description

EMMA designs the experiments using a procedure based on the Particle Swarm Optimization (PSO) algorithm. Firstly, EMMA selects a set of initial experimental points (see emmat0) that define the initial position of the particles; subsequently, for a given number of iterations, the particles are moved and new experimental points are selected (see emmatn).

Usage

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emma(in.name, nlev, lower, upper, out.name, opt="mn", nd=10, na=5, 
	weight, C = 20, w1 = 0.7, w2 = 0.4, c1i = 2.5, c1f = 0.5, 
	c2i = 0.5, c2f = 2.5, b = 5, pr.mut, graph, fn1 = NULL, 
	fn2 = NULL, fn3 = NULL, fn4 = NULL, nresp) 

Arguments

in.name

A vector containing the names of the input variables (factors).

nlev

A numeric vector of the same length as in.name, containing the number of factor levels.

lower

A numeric vector of the same length as in.name, containing the lower values of the factors.

upper

A numeric vector of the same length as in.name, containing the upper values of the factors.

out.name

A vector containing the name(s) of the output/response variable(s).

opt

A character vector of the same length as the number of responses, indicating for each response function, if the response must be minimized ('mn') or maximized ('mx').

nd

Number of experimental points to be selected when t = 0.

na

A numeric value indicating the number of experimental points to be selected when t > 0.

weight

A numerical vector of the same length as the number of responses, reflecting the relevance of each response. Use weight = 1 if only one response is investigated; if multiple responses are investigated, the sum of the values in weight must be 1.

C

A numeric value indicating the maximum number of iterations.

w1

The first numeric value used to calculate the inertia weight parameter of the time variant PSO algorithm; the default is w1 = 0.7.

w2

The second numeric value used to calculate the inertia weight parameter of the time variant PSO algorithm; The default is w2 = 0.4.

c1i

The first numeric value used to calculate the acceleration coefficient c1 of the time variant PSO algorithm; the default is c1i = 2.5.

c1f

The second numeric value used to calculate the acceleration coefficient c1 of the time variant PSO algorithm; the default is c1f = 0.5.

c2i

The first numeric value used to calculate the acceleration coefficient c2 of the time variant PSO algorithm; the default is c2i = 0.5.

c2f

The second numeric value used to calculate the acceleration coefficient c2 of the time variant PSO algorithm; the default is c2f = 2.5.

b

A numeric value, used in the mutation operator, that determines the degree of dependence of the mutation on the iteration number; the default is b = 5.

pr.mut

A numeric vector of the same length as the number of iterations C containing the probability of mutation for each time instant.

graph

Logical; if 'yes', a plot of the MARS model is produced. A plot is produced only if the model contains more than one explanatory variable.

fn1

The first function to be optimised. Use fn1 = NULL if the function is unknown (e.g. when designing experiments in applied problems).

fn2

The second function to be optimised. Use fn2 = NULL if the function is unknown (e.g. when designing experiments in applied problems).

fn3

The third function to be optimised. Use fn3 = NULL if the function is unknown (e.g. when designing experiments in applied problems).

fn4

The forth function to be optimised. Use fn4 = NULL if the function is unknown (e.g. when designing experiments in applied problems).

nresp

The response to be plotted. Use nresp = 1 to plot the first response...

Details

To select the new experimental points to be investigated, the following steps are iterated. A MARS model is fitted to the collected data so that an approximated function is obtained for each response; these approximated functions are used to predict the response values at the non-investigated experimental points. Each point in the experimental region E (xspace) is now associated with a vector of response values that has been either measured or estimated. The best (measured or estimated) value of each response is selected and used to identify the target. Subsequently, for each experimental point in E, the scalar distance between the response values and the target is computed and the solution that is closest to the target is selected. If such solution has not been tested yet (see emmacheck), the experiment needs to be performed and its response values are measured. The target is then updated, as well as the scalar distances of all the experimental points from the target. The scalar distances are used to identify the good performing experimental points. The experimental point whose response values are closest to the target is referred to as the global best. Similarly, a personal best is identified for each particle by considering the experimental points visited by that particle and selecting that point featuring the response values that are closest to the target. Finally, the particles velocity and position are updated and a new set of experimental points is identified.

The parameters w1 and w2 are used to calculate the inertia weight w of the PSO algorithm, namely the parameter that controls the influence of the previous particle velocity on the present velocity. High values of w favour a global search, whereas lower values of w encourage a local search. In EMMA the inertia weight is allowed to decrease linearly with iteration from w1 to w2 thus favouring the exploration initially and the exploitation subsequently. The parameters c1i and c1f are used to calculate the cognitive acceleration coefficient c1 of the PSO algorithm, whereas the parameters c2i and c2f are used to calculate the social acceleration coefficient c2 of the PSO algorithm.Higher values of c1 ensure larger deviation of the particle in the search space (exploration), while higher values of c2 signify the convergence to the current global best (exploitation). In EMMA c1 is allowed to decrease from c1i to c1f and c2 is allowed to increase from c2i to c2f. See Tripathi et al. (2007) for more details.

Value

An object of class emma with the components listed below:

xpop

Experimental points investigated.

ypop

Response values observed at the experimental points investigated.

xspace

Experimental region. It is given by all the possible combinations of the factors' levels and contains xpop. The rownames uniquely identify the experimental points and are reported also in xpop.

yspace

Response values that have been either observed or predicted. Observed response values are stored also in ypop. Predicted response values are obtained using a MARS model fitted to the available data.

opt

Indicates if each single function is either minimized ('mn') or maximized ('mx').

nd

Number of experimental points selected initially (t=0).

na

Number of experimental points selected in subsequent iterations (t>0).

tested

IDs of the tested experimental points.

time

Current time instant of the EMMA procedure.

weight

Relative importance of each response. If only one response is investigated, then weight = 1; if multiple responses are investigated, the sum of the values in weight must be 1.

Gb

ID of the best experimental point investigated (global best). Use xspace[Gb,] to visualise the global best and use yspace[Gb,] to visualise its measured response value(s). Gb identifies the experimental point whose response values are closest to the target; the target is a set of desirable response values which are automatically selected on the basis of the measured and predicted response values.

Pb

ID of the best experimental point investigated by each particle (personal best). Use xspace[Pb,] to identify the personal bests and use yspace[Pb,] to visualise their measured response values. Among the experimental points associated to one particle, the Pb identifies the experimental point that is whose response values are closest to the target.

Gb.arch

Archive of the global bests identified. Because the global best changes as new experimental points are investigated, an archive is maintained.

Pb.arch

Archive of the personal bests identified. Because the personal bests change as new experimental points are investigated, an archive is maintained.

v

Velocities used to update the particles position. The position of a particle is uniquely determined by the predictors' values; it also defines the experiment to be performed. At each step of EMMA, the position of a particle is updated by adding a numerical value (velocity) to the current value of each single predictor.

sam.x

IDs of the experiments that have been selected in the current iteration of the procedure. Use xspace[sam.x,] to visualise the experiments to be performed.

add

Logical. If '0' indicates that an additional experimental point needs to be investigated; if '1' indicates that an additional experimental point is not required.

Author(s)

Laura Villanova, Kate Smith-Miles and Rob J Hyndman

References

Villanova L., Falcaro P., Carta D., Poli I., Hyndman R., Smith-Miles K. (2010) 'Functionalization of Microarray Devices: Process Optimization Using a Multiobjective PSO and Multiresponse MARS Modelling', IEEE CEC 2010, DOI: 10.1109/CEC.2010.5586165

Carta D., Villanova L., Costacurta S., Patelli A., Poli I., Vezzu' S., Scopece P., Lisi F., Smith-Miles K., Hyndman R. J., Hill A. J., Falcaro P. (2011) 'Method for Optimizing Coating Properties Based on an Evolutionary Algorithm Approach', Analytical Chemistry 83 (16), 6373-6380.

Friedman J. H. (1991) 'Multivariate adaptive regression splines' (with discussion), The Annals of Statistics 19, 1:141.

Tripathi P. K., Bandyopadhyay S., Pal S. K. (2007) 'Multi-objective particle swarm optimization with time variant inertia and acceleration coefficients' Information Sciences, 177, 5033:5049.

Examples

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#########################
## 1 response variable ##
#########################
in.name <- c("x1","x2")
nlev <- c(20, 20)
lower <- c(-2.048, -2.048)
upper <- c(2.048, 2.048)
out.name <- "y"
weight <- 1
C <- 10
pr.mut <- c(0.1, 0.07, 0.04, rep(0.01, C-3))

emma(in.name, nlev, lower, upper, out.name, opt = "mn", nd = 10, na = 5, 
	weight, C , w1 = 0.7, w2 = 0.4, c1i = 2.5, c1f = 0.5, c2i = 0.5, 
	c2f = 2.5, b = 5, pr.mut, graph = "yes", fn1 = ackley) 

##########################
## 2 response variables ##
##########################
in.name <- c("x1", "x2")
nlev <- c(20, 20)
lower <- c(-3, -3)
upper <- c(3, 3)
out.name <- c("y1", "y2")
weight <- c(0.2, 0.8)
C <- 10
pr.mut <- c(0.1, 0.07, 0.04, rep(0.01, C-3))

emma(in.name, nlev, lower, upper, out.name, opt = c("mn", "mx"), nd = 10, 
	na = 5, weight, C , w1 = 0.7, w2 = 0.4, c1i = 2.5, c1f = 0.5, 
	c2i = 0.5, c2f = 2.5, b = 5, pr.mut, graph = "yes", fn1 = ackley, 
	fn2 = peaks, nresp = 2)

emma documentation built on May 30, 2017, 5:51 a.m.