ls.sorting.treatment: Least squares capacity identification in the framework of a...
In kappalab: Non-Additive Measure and Integral Manipulation Functions

ls.sorting.treatment

R Documentation

Least squares capacity identification in the framework of a sorting procedure: evaluation of the determined capacity

Description

This function assigns alternatives to classes and optionally compares the obtained classification to a given one. The classes are described by a set of prototypes (well-known alternatives for the decision maker) and a capacity (which can, for instance, be determined by ls.sorting.capa.ident). This function (in combination with ls.sorting.capa.ident) is an implementation of the TOMASO method; see Meyer and Roubens (2005).

Usage

ls.sorting.treatment(P, cl.proto, a, A, cl.orig.A = NULL)

Arguments

`P`	Object of class `matrix` containing the `n`-column criteria matrix. Each line of this matrix corresponds to a prototype. A prototype is an alternative for which the class is known beforehand.
`cl.proto`	Object of class `numeric` containing the indexes of the classes the prototypes `P` are belonging to (the greater the class index, the better the prototype is considered by the decision maker).
`a`	Object of class `Mobius.capacity` used to model the classes determined by `P` by a Choquet integral.
`A`	Object of class `matrix` containing the `n`-column criteria matrix representing the alternatives which have to be classified.
`cl.orig.A`	Object of class `numeric` containing the "true" classes of the alternatives of `A`. This can be used for the evaluation of the quality of the model.

Value

The function returns a list structured as follows:

`correct.A`	Object of class `matrix` which contains the different types of assignments of the elements of `A`. In columns the alternatives. First line: correct assignment to a single class (assignment of degree 0). Second line: correct ambiguous assignment to two classes (assignment of degree 1), etc. Last line: bad assignment. In case no `orig.class.A` is given, `correct.A` is `NULL`.
`class.A`	Object of class `matrix` which contains the assignments of the elements of `A`. In columns, the alternatives. First line, lower class. Second line, upper class.
`eval.correct`	Object of class `numeric` which contains the ratio assignments over number of elements of `A`(for each type of assignment; the first element is the ratio of correct assignments). In case no `orig.class.A` is given, `eval.correct` is `NULL`.
`minmax.P`	Object of class `matrix` which contains the min and the max of each class, according to the prototypes. In columns, the classes, first line: the minimum, second line: the maximum.
`Choquet.A`	Object of class `numeric` which contains the Choquet integral of the evaluations of the alternatives of `A`.

References

P. Meyer, M. Roubens (2005), 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.

Examples


## generate a random problem with 10 prototypes and 4 criteria
n.proto <- 10 ## prototypes
n <- 4  ## criteria
P <- matrix(runif(n.proto*n,0,1),n.proto,n)

## the corresponding global scores, based on a randomly generated
## capacity a
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,P[i,])

## based on these global scores, let us create a classification (3 classes)
cl.proto<-numeric(n.proto)
cl.proto[glob.eval <= 0.33] <- 1
cl.proto[glob.eval > 0.33 & glob.eval<=0.66] <-2
cl.proto[glob.eval > 0.66] <- 3

## search for a capacity which satisfies the constraints
lsc <- ls.sorting.capa.ident(n ,4, P, cl.proto, 0.1)

## output of the QP
lsc$how

## analyse the quality of the model (classify the prototypes by the
## model and compare both assignments)
lst <- ls.sorting.treatment(P,cl.proto,lsc$solution,P,cl.proto)

## assignments of the prototypes
lst$class.A
## assignment types
lst$correct.A
## evaluation
lst$eval.correct

## generate a second set of random alternatives (A)
## their "correct" class is determined as beforehand with the
## randomly generated capacity a
## the goal is to see if we can reproduce this classification
## by the capacity learnt from the prototypes

## a randomly generated criteria matrix of 10 alternatives
A <- matrix(runif(10*n,0,1),10,n)
cl.orig.A <-numeric(10)
	
## the corresponding global scores
glob.eval.A <- numeric(10)
for (i in 1:10)
  glob.eval.A[i] <- Choquet.integral(a,A[i,])

## based on these global scores, let us determine a classification
cl.orig.A[glob.eval.A <= 0.33] <- 1
cl.orig.A[glob.eval.A>0.33 & glob.eval.A<=0.66] <-2
cl.orig.A[glob.eval.A > 0.66] <- 3

## let us now classify the alternatives of A according to the model
## built on P

lst <- ls.sorting.treatment(P,cl.proto,lsc$solution,A,cl.orig.A)

## assignment of the alternatives of A
lst$class.A
## type of assignments
lst$correct.A
## evaluation
lst$eval.correct

## show the learnt capacity
## x11()
## barplot(Shapley.value(lsc$solution), main="Learnt capacity", sub="Shapley")
## summary of the learnt capacity
lsc$solution
summary(lsc$solution)

kappalab documentation built on Nov. 8, 2023, 1:07 a.m.