https://github.com/cran/kappalab
Tip revision: d2fc5e4ce0f6c794342013b6dccb28fc6b39c9ba authored by Ivan Kojadinovic on 07 May 2005, 00:00:00 UTC
version 0.1-3
version 0.1-3
Tip revision: d2fc5e4
ls.sorting.capa.ident.Rd
\name{ls.sorting.capa.ident}
\alias{ls.sorting.capa.ident}
\title{Least squares capacity identification in the framework of a sorting procedure}
\description{
Sorting alternatives means assigning each alternative to a predefined
ordered class. The aim of the implemented method is to model a given classification (sorting)
of the alternatives by means of a Choquet integral. The result of the function
is an object of class \code{Mobius.capacity}. This function (in
combination with \code{ls.sorting.treatment}) is an implementation of
the TOMASO method; see Meyer and Roubens (2005). The input data are
given under the form of a set of alternatives and associated classes,
each alternative being described according to a set of criteria. These
well-known alternatives are called "prototypes". They represent
alternatives for which the decision maker has an a priori knowledge
and that he/she is able to assign to one of the
ordered classes. If the provided classification of the prototypes cannot be
described by a Choquet integral, an approximative solution, which minimizes
the "gap" between the given classification and the one derived from
the Choquet integral, is proposed. The problem is solved by quadratic
programming. This function should be used in combination with
\code{ls.sorting.treatment} which allows to evaluate the model which
has been built and to assign other alternatives to the ordered classes.
}
\usage{
ls.sorting.capa.ident(n, k, C, cl, d, A.Shapley.preorder = NULL,
A.Shapley.interval = NULL, A.interaction.preorder = NULL,
A.interaction.interval = NULL, A.inter.additive.partition = NULL,
sigf = 5, maxiter = 20, epsilon = 1e-6)
}
\arguments{
\item{n}{Object of class \code{numeric} containing the
number of elements of the set on which the object of class
\code{Mobius.capacity} is to be defined (in short, the number of criteria).}
\item{k}{Object of class \code{numeric} imposing that the solution is at
most a k-additive capacity (the \enc{Möbius}{Mobius} transform of subsets whose cardinal is
superior to \code{k} vanishes).}
\item{C}{Object of class \code{matrix} containing the
\code{n}-column criteria matrix. Each line of this matrix
corresponds to a prototype.}
\item{cl}{Object of class \code{numeric} containing the indexes of the
classes the alternatives are belonging to (the greater the class
index, the better the prototype is considered by the decision
maker). Each class index between min(cl) and max(cl) must be present.}
\item{d}{Object of class \code{numeric} containing the threshold value
for the classes, i.e. the minimal "distance" between two neighbor
classes (e.g. the difference in terms of the Choquet integral of the
worst prototype of class 3 and the best prototype of
class 2 should be at least \code{d}).}
\item{A.Shapley.preorder}{Object of class \code{matrix} containing the
constraints relative to the preorder of the criteria. Each line
of this 3-column matrix corresponds to one constraint of the type
"the Shapley importance index of criterion \code{i} is greater than
the Shapley importance index of criterion \code{j} with preference threshold
\code{delta.S}". A line is structured as follows: the first element
encodes \code{i}, the second \code{j}, and the third element contains
the preference threshold \code{delta.S}.}
\item{A.Shapley.interval}{Object of class \code{matrix} containing the
constraints relative to the quantitative importance of the
criteria. Each line of this 3-column matrix corresponds to one
constraint of the type "the Shapley importance index of criterion
\code{i} lies in the interval \code{[a,b]}". The interval
\code{[a,b]} has to be included in \code{[0,1]}. A line of the
matrix is structured as follows: the first element encodes \code{i},
the second \code{a}, and the third \code{b}.}
\item{A.interaction.preorder}{Object of class \code{matrix}
containing the constraints relative to the preorder of the pairs of
criteria in terms of the Shapley interaction index. Each line of this 5-column matrix
corresponds to one constraint of the type "the Shapley interaction
index of the pair \code{ij} of criteria is greater than the Shapley interaction
index of the pair \code{kl} of criteria with preference threshold \code{delta.I}".
A line is structured as follows: the first two elements encode
\code{ij}, the second two \code{kl}, and the fifth element contains
the preference threshold \code{delta.I}.}
\item{A.interaction.interval}{Object of class \code{matrix}
containing the constraints relative to the type and the magnitude of
the Shapley interaction index for pairs of criteria. Each line of
this 4-column matrix corresponds to one constraint of the type
"the Shapley interaction index of the pair \code{ij} of criteria
lies in the interval \code{[a,b]}". The interval \code{[a,b]} has to
be included in \code{[-1,1]}. A line is structured as follows: the first two elements encode
\code{ij}, the third element encodes \code{a}, and the fourth element
encodes \code{b}.}
\item{A.inter.additive.partition}{Object of class \code{numeric}
encoding a partition of the set of criteria imposing that there be
no interactions among criteria belonging to different classes
of the partition. The partition is to be given under the form of a
vector of integers from \code{{1,\dots,n}} of length \code{n} such
that two criteria belonging to the same class are "marked" by the
same integer. For instance, the partition \code{{{1,3},{2,4},{5}}} can
be encoded as \code{c(1,2,1,2,3)}. See Fujimoto and Murofushi (2000)
for more details on the concept of mu-inter-additive partition.}
\item{sigf}{Precision (default: 5 significant figures). Parameter to
be passed to the \code{ipop} function (quadratic programming)
of the \pkg{kernlab} package.}
\item{maxiter}{Maximum number of iterations. Parameter to
be passed to the \code{ipop} function (quadratic programming)
of the \pkg{kernlab} package.}
\item{epsilon}{Object of class \code{numeric} containing the
threshold value for the monotonicity constraints, i.e. the
difference between the "weights" of two subsets whose cardinals
differ exactly by 1 must be greater than \code{epsilon}.}
}
\details{
The quadratic program is solved using the \code{ipop} function of
the \pkg{kernlab} package.
}
\value{
The function returns a list structured as follows:
\item{solution}{Object of class
\code{Mobius.capacity} containing the \enc{Möbius}{Mobius} transform of the
\code{k}-additive solution.}
\item{glob.eval}{The global evaluations satisfying the given classification.}
\item{how}{Information returned by \code{ipop} (cf. \pkg{kernlab}) on
the convergence of the solver.}
}
\references{
K. Fujimoto and T. Murofushi (2000) \emph{Hierarchical decomposition of the
Choquet integral}, in: Fuzzy Measures and Integrals: Theory and
Applications, M. Grabisch, T. Murofushi, and M. Sugeno Eds, Physica
Verlag, pages 95-103.
P. Meyer, M. Roubens (2005), \emph{Choice, Ranking and Sorting in Fuzzy Multiple
Criteria Decision Aid}, in: J. Figueira, S. Greco, and M. Ehrgott,
Eds, Multiple Criteria Decision Analysis: State of the Art
Surveys, volume 78 of International Series in Operations Research and
Management Science, chapter 12, pages 471-506. Springer Science +
Business Media, Inc., New York.
}
\seealso{
\code{\link{Mobius.capacity-class}},
\cr \code{\link{mini.var.capa.ident}},
\cr \code{\link{least.squares.capa.ident}},
\cr \code{\link{ls.sorting.treatment}},
\cr \code{\link{entropy.capa.ident}}.
}
\examples{
## generate a random problem with 10 prototypes and 4 criteria
n.proto <- 10 ## prototypes
n <- 4 ## criteria
k <- 4
d <- 0.1
## generating random data for the prototypes
C <- matrix(runif(n.proto*n,0,1),n.proto,n)
cl <- numeric(n.proto)
## the corresponding global evaluations
glob.eval <- numeric(n.proto)
a <- capacity(c(0:(2^n-3),(2^n-3),(2^n-3))/(2^n-3))
for (i in 1:n.proto)
glob.eval[i] <- Choquet.integral(a,C[i,])
## and the classes for the prototypes
cl[glob.eval <= 0.33] <- 1
cl[glob.eval > 0.33 & glob.eval <= 0.66] <-2
cl[glob.eval > 0.66] <- 3
cl
\dontrun{
# starting the calculations
# search for a capacity which satisfies the constraints
lsc <- ls.sorting.capa.ident(n ,k, C, cl, d)
## output of the quadratic program (ipop, package kernlab)
lsc$how
## the capacity satisfying the constraints
lsc$solution
summary(lsc$solution)
## the global evaluations satisfying the constraints
lsc$glob.eval
}
## let us now add some constraints
## a Shapley preorder constraint matrix
## Sh(1) > Sh(2)
## Sh(3) > Sh(4)
delta.S <-0.01
Asp <- rbind(c(1,2,delta.S), c(3,4,delta.S))
## a Shapley interval constraint matrix
## 0.1 <= Sh(1) <= 0.2
Asi <- rbind(c(1,0.1,0.2))
## an interaction preorder constraint matrix
## such that I(12) > I(34)
delta.I <- 0.01
Aip <- rbind(c(1,2,3,4,delta.I))
## an interaction interval constraint matrix
## i.e. 0.2 <= I(12) <= 0.4
## and 0 < I(34) <= 1
Aii <- rbind(c(1,2,0.2,0.4), c(3,4,delta.I,1))
## an inter-additive partition constraint
## criteria 1,2 and criteria 3,4 are independent
Aiap <- c(1,1,2,2)
## starting the calculations
## search for a capacity which satisfies the constraints
lsc <- ls.sorting.capa.ident(n ,k, C, cl, d,
A.Shapley.preorder = Asp,
A.Shapley.interval = Asi,
A.interaction.preorder = Aip,
A.interaction.interval = Aii,
A.inter.additive.partition = Aiap)
## output of ipop
lsc$how
## the capacity satisfying the constraints
lsc$solution
summary(lsc$solution)
## the global evaluations satisfying the constraints
lsc$glob.eval
}
\keyword{math}