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).
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.