mini.var.capa.ident: Minimum variance capacity identification

Description

Creates an object of class Mobius.capacity using a maximum like quadratic entropy principle, which is equivalent to the minimization of the variance. More precisely, this function determines, if it exists, the minimum variance capacity compatible with a set of linear constraints. The problem is solved using strictly convex quadratic programming.

Usage

mini.var.capa.ident(n, k, A.Choquet.preorder = NULL,
A.Shapley.preorder = NULL, A.Shapley.interval = NULL,
A.interaction.preorder = NULL, A.interaction.interval = NULL,
A.inter.additive.partition = NULL, 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.
value: Value of the objective function.
iterations: Information returned by solve.QP.
iact: Information returned by solve.QP.

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).
A.Choquet.preorder: Object of class matrix containing the constraints relative to the preorder of the alternatives. Each line of the matrix corresponds to one constraint of the type "alternative a is preferred to alternative b with preference threshold delta.C". A line is structured as follows: the first n elements encode alternative a, the next n elements encode alternative b, and the last element contains the preference threshold delta.C.
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.
epsilon: Object of class numeric containing the thresold 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 solve.QP function of the quadprog 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.

I. Kojadinovic (2005), Minimum variance capacity identification, European Journal of Operational Research, in press.

Examples

Run this code

## some alternatives
a <- c(18,11,18,11,11)
b <- c(18,18,11,11,11)
c <- c(11,11,18,18,11)
d <- c(18,11,11,11,18)
e <- c(11,11,18,11,18)
    
## preference threshold relative
## to the preorder of the alternatives
delta.C <- 1

## corresponding Choquet preorder constraint matrix 
Acp <- rbind(c(d,a,delta.C),
             c(a,e,delta.C),
             c(e,b,delta.C),
             c(b,c,delta.C)
            )

## 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(34)
delta.I <- 0.01
Aip <- rbind(c(1,2,3,4,-delta.I),
             c(3,4,1,2,-delta.I))

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


## a minimum variance 2-additive solution
min.var <- mini.var.capa.ident(5,2,A.Choquet.preorder = Acp)              
m <- min.var$solution
m

## the resulting global evaluations
rbind(c(a,mean(a),Choquet.integral(m,a)),
      c(b,mean(b),Choquet.integral(m,b)),
      c(c,mean(c),Choquet.integral(m,c)),
      c(d,mean(d),Choquet.integral(m,d)),
      c(e,mean(e),Choquet.integral(m,e)))

## the Shapley value
Shapley.value(m)

## a minimum variance 3-additive more constrained solution
min.var2 <- mini.var.capa.ident(5,3,
                                   A.Choquet.preorder = Acp,
                                   A.Shapley.preorder = Asp)
m <- min.var2$solution
m
rbind(c(a,mean(a),Choquet.integral(m,a)),
      c(b,mean(b),Choquet.integral(m,b)),
      c(c,mean(c),Choquet.integral(m,c)),
      c(d,mean(d),Choquet.integral(m,d)),
      c(e,mean(e),Choquet.integral(m,e)))
Shapley.value(m)

## a minimum variance 5-additive more constrained solution
min.var3 <- mini.var.capa.ident(5,5,
                                   A.Choquet.preorder = Acp,
                                   A.Shapley.preorder = Asp,
                                   A.Shapley.interval = Asi,
                                   A.interaction.preorder = Aip,
                                   A.interaction.interval = Aii)

m <- min.var3$solution
m
rbind(c(a,mean(a),Choquet.integral(m,a)),
      c(b,mean(b),Choquet.integral(m,b)),
      c(c,mean(c),Choquet.integral(m,c)),
      c(d,mean(d),Choquet.integral(m,d)),
      c(e,mean(e),Choquet.integral(m,e)))
summary(m)

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