least.squares.capa.ident: Least squares capacity identification

Description

Creates an object of class Mobius.capacity by means of an approach grounded on least squares optimization. More precisely, given a set of data under the form: datum=(score on criterion 1, ..., score on criterion n, overall score), and possibly additional linear constraints expressing preferences, importance of criteria, etc., this function determines, if it exists, a capacity minimizing the sum of squared errors between overall scores as given by the data and the output of the Choquet integral for those data, and compatible with the additional linear constraints. The existence is ensured if no additional constraint is given. The problem is solved using quadratic programming.

Usage

least.squares.capa.ident(n, k, C, g, Integral="Choquet",
A.Shapley.preorder = NULL, A.Shapley.interval = NULL,
A.interaction.preorder = NULL, A.interaction.interval = NULL,
A.inter.additive.partition = NULL,  sigf = 7, maxiter = 40,
epsilon = 1e-6)

Value

The function returns a list structured as follows:

solution: Object of class Mobius.capacity containing the Möbius transform of the k-additive solution, if any.
dual: The dual solution of the problem.
how: Character string describing the type of convergence.
residuals: Differences between the provided global evaluations and those returned by the obtained model.

Arguments

n: Object of class numeric containing the number of elements of the set on which the object of class Mobius.capacity is to be defined.
k: Object of class numeric imposing that the solution is at most a k-additive capacity (the Möbius transform of subsets whose cardinal is superior to k vanishes).
C: Object of class matrix containing the n-column criteria matrix. Each line of this matrix corresponds to a vector of the form: (score on criterion 1, ..., score on criterion n).
g: Object of class numeric containg the global scores associated with the vectors given in the criteria matrix.
Integral: Object of class character indicating whether the model is based on the asymmetric Choquet integral (Integral = "Choquet") or the symmetric Choquet integral (Integral = "Sipos").
A.Shapley.preorder: Object of class 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 i is greater than the Shapley importance index of criterion j with preference threshold delta.S". A line is structured as follows: the first element encodes i, the second j, and the third element contains the preference threshold delta.S.
A.Shapley.interval: Object of class 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 i lies in the interval [a,b]". The interval [a,b] has to be included in [0,1]. A line of the matrix is structured as follows: the first element encodes i, the second a, and the third b.
A.interaction.preorder: Object of class 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 ij of criteria is greater than the Shapley interaction index of the pair kl of criteria with preference threshold delta.I". A line is structured as follows: the first two elements encode ij, the second two kl, and the fifth element contains the preference threshold delta.I.
A.interaction.interval: Object of class 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 ij of criteria lies in the interval [a,b]". The interval [a,b] has to be included in [-1,1]. A line is structured as follows: the first two elements encode ij, the third element encodes a, and the fourth element encodes b.
A.inter.additive.partition: Object of class 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 {1,...,n} of length n such that two criteria belonging to the same class are "marked" by the same integer. For instance, the partition {{1,3},{2,4},{5}} can be encoded as c(1,2,1,2,3). See Fujimoto and Murofushi (2000) for more details on the concept of mu-inter-additive partition.
sigf: Precision (default: 7 significant figures). Parameter to be passed to the ipop function (quadratic programming) of the kernlab package.
maxiter: Maximum number of iterations. Parameter to be passed to the ipop function (quadratic programming) of the kernlab package.
epsilon: Object of class 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 epsilon.

Details

The quadratic program is solved using the ipop function of the kernlab package.

References

K. Fujimoto and T. Murofushi (2000) 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.

M. Grabisch, H.T. Nguyen and E.A. Walker (1995), Fundamentals of uncertainty calculi with applications to fuzzy inference, Kluwer Academic, Dordrecht.

M. Grabisch and M. Roubens (2000), Application of the Choquet Integral in Multicriteria Decision Making, in: Fuzzy Measures and Integrals: Theory and Applications, M. Grabisch, T. Murofushi, and M. Sugeno Eds, Physica Verlag, pages 415-434.

P. Miranda and M. Grabisch (1999), Optimization issues for fuzzy measures, International Journal of Fuzziness and Knowledge-based Systems 7:6, pages 545-560.

Examples

Run this code


## the number of data
n.d <- 20

## a randomly generated 5-criteria matrix
C <- matrix(rnorm(5*n.d,10,2),n.d,5)


## the corresponding global scores
g <- numeric(n.d)
mu <- capacity(c(0:29,29,29)/29)
for (i in 1:n.d)
  g[i] <- Choquet.integral(mu,C[i,])

if (FALSE) {
## the full solution 
lsc <- least.squares.capa.ident(5,5,C,g)
a <- lsc$solution
a
mu.sol <- zeta(a)

## the difference between mu and mu.sol
mu@data - mu.sol@data

## the residuals
lsc$residuals

## the mean square error
mean(lsc$residuals^2)

## a 3-additive solution 
lsc <- least.squares.capa.ident(5,3,C,g)
a <- lsc$solution
mu.sol <- zeta(a)
mu@data - mu.sol@data
lsc$residuals
}



## a similar example based on the Sipos integral

## a randomly generated 5-criteria matrix
C <- matrix(rnorm(5*n.d,0,2),n.d,5)

## the corresponding global scores
g <- numeric(n.d)
mu <- capacity(c(0:29,29,29)/29)
for (i in 1:n.d)
  g[i] <- Sipos.integral(mu,C[i,])

if (FALSE) {
## the full solution 
lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos")
a <- lsc$solution
mu.sol <- zeta(a)
mu@data - mu.sol@data
lsc$residuals

## a 3-additive solution 
lsc <- least.squares.capa.ident(5,3,C,g,Integral = "Sipos")
a <- lsc$solution
mu.sol <- zeta(a)
mu@data - mu.sol@data
lsc$residuals
}



## additional constraints

## a Shapley preorder constraint matrix
## Sh(1) - Sh(2) >= -delta.S
## Sh(2) - Sh(1) >= -delta.S
## Sh(3) - Sh(4) >= -delta.S
## Sh(4) - Sh(3) >= -delta.S
## i.e. criteria 1,2 and criteria 3,4
## should have the same global importances
delta.S <- 0.01    
Asp <- rbind(c(1,2,-delta.S),
             c(2,1,-delta.S),
             c(3,4,-delta.S),
             c(4,3,-delta.S)
            )

## a Shapley interval constraint matrix
## 0.3 <= Sh(1) <= 0.9 
Asi <- rbind(c(1,0.3,0.9))


## an interaction preorder constraint matrix
## such that I(12) = I(45)
delta.I <- 0.01
Aip <- rbind(c(1,2,4,5,-delta.I),
             c(4,5,1,2,-delta.I))

## an interaction interval constraint matrix
## i.e. -0.20 <= I(12) <= -0.15 
delta.I <- 0.01
Aii <- rbind(c(1,2,-0.2,-0.15))

## an inter-additive partition constraint
## criteria 1,2,3 and criteria 4,5 are independent 
Aiap <- c(1,1,1,2,2)





## a more constrained solution

lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos",
                                 A.Shapley.preorder = Asp,
                                 A.Shapley.interval = Asi,
                                 A.interaction.preorder = Aip,
                                 A.interaction.interval = Aii,
                                 A.inter.additive.partition = Aiap,
                                 sigf = 5)

a <- lsc$solution
mu.sol <- zeta(a)
mu@data - mu.sol@data
lsc$residuals
summary(a)

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