least.squares.capa.ident {kappalab} | R Documentation |
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.
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)
n |
Object of class |
k |
Object of class |
C |
Object of class |
g |
Object of class |
Integral |
Object of class |
A.Shapley.preorder |
Object of class |
A.Shapley.interval |
Object of class |
A.interaction.preorder |
Object of class |
A.interaction.interval |
Object of class |
A.inter.additive.partition |
Object of class |
sigf |
Precision (default: 7 significant figures). Parameter to
be passed to the |
maxiter |
Maximum number of iterations. Parameter to
be passed to the |
epsilon |
Object of class |
The quadratic program is solved using the ipop
function of
the kernlab package.
The function returns a list structured as follows:
solution |
Object of class
|
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. |
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.
Mobius.capacity-class
,
heuristic.ls.capa.ident
,
lin.prog.capa.ident
,
mini.var.capa.ident
,
mini.dist.capa.ident
,
ls.sorting.capa.ident
,
entropy.capa.ident
.
## 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,]) ## Not run: ## 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$residuals ## End(Not run) ## a similar example based on the Sipos integral ## a randomly generated 5-criteria matrix C <- matrix(rnorm(5*n.d,0,2),n.d,5) ## the corresponding global scores g <- numeric(n.d) mu <- capacity(c(0:29,29,29)/29) for (i in 1:n.d) g[i] <- Sipos.integral(mu,C[i,]) ## Not run: ## the full solution lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos") a <- lsc$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$residuals ## End(Not run) ## additional constraints ## a Shapley preorder constraint matrix ## Sh(1) - Sh(2) >= -delta.S ## Sh(2) - Sh(1) >= -delta.S ## Sh(3) - Sh(4) >= -delta.S ## Sh(4) - Sh(3) >= -delta.S ## i.e. criteria 1,2 and criteria 3,4 ## should have the same global importances delta.S <- 0.01 Asp <- rbind(c(1,2,-delta.S), c(2,1,-delta.S), c(3,4,-delta.S), c(4,3,-delta.S) ) ## a Shapley interval constraint matrix ## 0.3 <= Sh(1) <= 0.9 Asi <- rbind(c(1,0.3,0.9)) ## an interaction preorder constraint matrix ## such that I(12) = I(45) delta.I <- 0.01 Aip <- rbind(c(1,2,4,5,-delta.I), c(4,5,1,2,-delta.I)) ## an interaction interval constraint matrix ## i.e. -0.20 <= I(12) <= -0.15 delta.I <- 0.01 Aii <- rbind(c(1,2,-0.2,-0.15)) ## an inter-additive partition constraint ## criteria 1,2,3 and criteria 4,5 are independent Aiap <- c(1,1,1,2,2) ## a more constrained solution lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos", A.Shapley.preorder = Asp, A.Shapley.interval = Asi, A.interaction.preorder = Aip, A.interaction.interval = Aii, A.inter.additive.partition = Aiap, sigf = 5) a <- lsc$solution mu.sol <- zeta(a) mu@data - mu.sol@data lsc$residuals summary(a)