R/mini.var.capa.ident.R

In kappalab: Non-Additive Measure and Integral Manipulation Functions

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#
# Copyright © 2005 Michel Grabisch and Ivan Kojadinovic    
#
# Ivan.Kojadinovic@polytech.univ-nantes.fr
#
# This software is a package for the statistical system GNU R:
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## Minimum variance capacity identification

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## Constructs a Mobius.capacity object by means of a quadratic program 

mini.var.capa.ident <- function(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) {

    ## check n and k
    if (!(as.integer(n) == n && k %in% 1:n))
        stop("wrong arguments")

    ## check A.Choquet.preorder
    if (!((is.matrix(A.Choquet.preorder)
           && dim(A.Choquet.preorder)[2] == 2*n+1) 
          || is.null(A.Choquet.preorder)))
        stop("wrong Choquet preorder constraint matrix")
    
    ## check A.Shapley.preorder
    if (!((is.matrix(A.Shapley.preorder) && dim(A.Shapley.preorder)[2] == 3) 
          || is.null(A.Shapley.preorder)))
        stop("wrong Shapley preorder constraint matrix")

    ## check A.Shapley.interval
    if (!((is.matrix(A.Shapley.interval) && dim(A.Shapley.interval)[2] == 3) 
          || is.null(A.Shapley.interval)))
        stop("wrong Shapley interval constraint matrix")

    ## check A.interaction.preorder
    if (!((is.matrix(A.interaction.preorder)
           && dim(A.interaction.preorder)[2] == 5)
          || is.null(A.interaction.preorder)))
        stop("wrong interaction preorder constraint matrix")

    ## check A.interaction.interval
    if (!((is.matrix(A.interaction.interval)
           && dim(A.interaction.interval)[2] == 4) 
          || is.null(A.interaction.interval)))
        stop("wrong interaction interval constraint matrix")

    ## check A.inter.additive.partition
    if (!((is.numeric(A.inter.additive.partition)
           && sum(levels(factor(A.inter.additive.partition))
                  == 1:max(A.inter.additive.partition))
           == max(A.inter.additive.partition))
          || is.null(A.inter.additive.partition)))
        stop("wrong inter-additive partition")

    ## check epsilon
    if (!(is.positive(epsilon) && epsilon <= 1e-3))
        stop("wrong epsilon value")

    ## number of variables
    n.var <- binom.sum(n,k) - 1

    ## number of monotonicity constraints
    n.con <- n*2^(n-1)

    ## k power set in natural order
    subsets <-  .C("k_power_set", 
                   as.integer(n),
                   as.integer(k),
                   subsets = integer(n.var+1),
                   PACKAGE="kappalab")$subsets

    ## monotonicity constraints
    A <- .C("monotonicity_constraints", 
            as.integer(n), 
            as.integer(k),
            as.integer(subsets),
            A = integer(n.var * n.con),
            PACKAGE="kappalab")$A


    A <- matrix(A,n.con,n.var,byrow=TRUE)

    ## quadratic form
    D.Shapley <- .C("objective_function_Choquet_coefficients", 
                    as.integer(n), 
                    D = double(n.con),
                    PACKAGE="kappalab")$D		

    Dmat <- t(A) %*% diag(D.Shapley) %*% A

    ## add the normalization constraint sum a(T) = 1
    A <- rbind(rep(1,n.var),A)
    bvec <- c(1,rep(epsilon,n.con))
    meq <- 1
    
    ## add the constraints relative to the preorder of the alternatives
    if (!is.null(A.Choquet.preorder)) {
        
        for (i in 1:dim(A.Choquet.preorder)[1]) {
            
            cpc <- Choquet.preorder.constraint(n,k,subsets,
                                               A.Choquet.preorder[i,][1:n],
                                               A.Choquet.preorder[i,][(n+1):(2*n)],
                                               A.Choquet.preorder[i,2*n+1])

            A <- rbind(A,cpc$A)
            bvec <- c(bvec,cpc$b)	
        }	
    }
    
    ## add the constraints relative to the preorder of the criteria
    if (!is.null(A.Shapley.preorder)) {
        
        for (i in 1:dim(A.Shapley.preorder)[1]) {
            
            spc <- Shapley.preorder.constraint(n,k,subsets,
                                             A.Shapley.preorder[i,1],
                                             A.Shapley.preorder[i,2],
                                             A.Shapley.preorder[i,3])
            
            A <- rbind(A,spc$A)
            bvec <- c(bvec,spc$b)	
        }	
    }

    ## add the constraints relative to the importance of the criteria
    if (!is.null(A.Shapley.interval)) {
        
        for (i in 1:dim(A.Shapley.interval)[1]) {
            
            sic <- Shapley.interval.constraint(n,k,subsets,
                                               A.Shapley.interval[i,1],
                                               A.Shapley.interval[i,2],
                                               A.Shapley.interval[i,3])

            ## Sh(i) >= a
            A <- rbind(A,sic$A)
            bvec <- c(bvec,sic$b)
            ## - Sh(i) >= -b
            A <- rbind(A,-sic$A)
            bvec <- c(bvec,-(sic$b + sic$r))
            
        }	
    }

    ## add the constraints relative to the preorder of the interactions
    if (!is.null(A.interaction.preorder)) {
        
        for (i in 1:dim(A.interaction.preorder)[1]) {
            
            ipc <- interaction.preorder.constraint(n,k,subsets,
                                             A.interaction.preorder[i,1],
                                             A.interaction.preorder[i,2],
                                             A.interaction.preorder[i,3],
                                             A.interaction.preorder[i,4],
                                             A.interaction.preorder[i,5])
            
            A <- rbind(A,ipc$A)
            bvec <- c(bvec,ipc$b)	
        }	
    }

    ## add the constraints relative to the magnitude of the interaction
    if (!is.null(A.interaction.interval)) {
        
        for (i in 1:dim(A.interaction.interval)[1]) {
            
            iic <- interaction.interval.constraint(n,k,subsets,
                                                   A.interaction.interval[i,1],
                                                   A.interaction.interval[i,2],
                                                   A.interaction.interval[i,3],
                                                   A.interaction.interval[i,4])

            ## I(ij) >= a
            A <- rbind(A,iic$A)
            bvec <- c(bvec,iic$b)
            ## I(ij) <= b
            A <- rbind(A,-iic$A)
            bvec <- c(bvec,-(iic$b+iic$r))
        }	
    }

    ## add the constraints relative to the inter-addtive partition
    if (!is.null(A.inter.additive.partition)) {
        
            
        iapc <- inter.additive.partition.constraint(n,k,subsets,
                                                 A.inter.additive.partition)

        A <- rbind(iapc$A,A)
        bvec <- c(iapc$b,bvec)
        meq <- meq + length(iapc$b)
    }
    

    ## quadprog
    qp <- solve.QP(2*Dmat,rep(0,n.var),t(A),bvec,meq = meq)

    return(list(solution = Mobius.capacity(c(0,qp$solution),n,k),
                value = qp$value, iterations = qp$iterations,
                iact = qp$iact))    
    
} 

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