Nothing
library(kappalab)
## the number of alternatives
n.a <- 300
## a randomly generated 5-criteria matrix
C <- matrix(rnorm(5*n.a,10,2),n.a,5)
## the corresponding global scores
g <- numeric(n.a)
mu <- capacity(c(0:29,29,29)/29)
for (i in 1:n.a)
g[i] <- Choquet.integral(mu,C[i,])
## 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$residuals
## a similar example based on the Sipos integral
n.a <- 300
## a randomly generated 5-criteria matrix
C <- matrix(rnorm(5*n.a,0,2),n.a,5)
## the corresponding global scores
g <- numeric(n.a)
mu <- capacity(c(0:29,29,29)/29)
for (i in 1:n.a)
g[i] <- Sipos.integral(mu,C[i,])
## the full solution
lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos")
a <- lsc$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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