IMDS: Multidimensional scaling of interval-valued dissimilarities.
In smds: Symbolic Multidimensional Scaling

Description Usage Arguments Details Value Author(s) References Examples

Performs MDS for given interval-valued dissimilarities.

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IMDS(IDM, p=2,eps= 1e-5 ,maxit =1000,model=c("sphere","box"),
opt.method=c("MM", "BFGS"), ini = "auto",report=100,grad.num=FALSE,
rel=0, dil=1)

`IDM`	The interval-valued dissimilarity matrix (an object of class "array": `IDM[1,,]`: the lower dissmilarity matrix; `IDM[2,,]`: the upper dissmilarity matrix).
`p`	Number of dimensions.
`eps`	Convergence criterion for the majorization minimization algorithm.
`maxit`	Maximum number of iteretions.
`model`	If "sphere", then the hypersphere model is used. If "box", then the hyperbox model is used.
`opt.method`	If "BFGS", then the BFGS method is used for optimizing the stress function. If "MM", then the majorization minimization algortihm is used.
`ini`	List which consists of an initial center coordinate matrix `ini[[1]]` and an initial radius vector `ini[[2]]` (optional). If "auto", then a configuration of the classical MDS for `(IDM[1,,]+IDM[2,,])/2` is used as the center coordinate matrix and radii are drawn from the uniform distribution U(0,1).
`report`	The frequency of reports. Defaults to every 100 iterations.
`grad.num`	If `FALSE`, then exact gradient function is used in the BFGS method. If `TRUE`, then a numerical gradient is used in the BFGS method.
`rel`	If `acc=1`, accelerate by the relaxed update. If `acc=0`, the relaxed update is not used. Not used for the BFGS method. Defaults to `0`.
`dil`	If `acc=1`, accelerate by the optimal dilation of the configration. If `acc=0`, the optimal dilation is not used. Not used for the BFGS method. Defaults to `1`.

The default optimization method is a majorization-minimization algorithm with the optimal dilation.

Method "MM" is a majorization-minimization (MM) algortihm for the specified model. If model="box", method "MM" is a MM algorithm, called I-Scal, which is proposed by Groenen et al. (2006). If model="sphere", method "MM" is a MM algorithm which can be considered as I-Scal for the hypersphere model.

Method "BFGS" is a quasi-Newton method (also known as a variable metric algorithm), specifically that published simultaneously in 1970 by Broyden, Fletcher, Goldfarb and Shanno. For more details, see Chapter 15 of Nash (1990).

IMDS returns a list with components:

`X`	The best corrdinate matrix with p columns whose rows give the coordinates of the vertexes.

If model="sphere",: r The best radius vector.
If model="box",: R The best radius matrix with p columns whose rows give the radii of objects.

`str`	The value of the stress function of IMDS corresponding to `X` is returned.
`str.vec`	If "MM", then the vector of values on each iteration is returned.
`EIDM`	If "MM", then the interval-valued dissimilarity matrix correspondint to the estimated parameters.

Yoshikazu Terada

Groenen, P. J. F., Winsberg, S., Rodriguez, O., and Diday, E. (2006). I- scal: Multidimensional scaling of interval dissimilarities. Computational Statistics & Data Analysis, 51, 360–378.

Nash, J. C. (1990) Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation. Adam Hilger.

###################
#Fats and Oil data#
###################
###############################
data(oil.idiss)
#Apply the hypersphere model via the BFGS method
set.seed(1)
res.bfgs <- IMDS(IDM=oil.idiss, p=2,model="sphere",opt.method="BFGS", ini = "auto")
plot(res.bfgs,main="Sph_bfgs")
#Apply the hypersphere model via the MM algorithm
set.seed(1)
res.mm <- IMDS(IDM=oil.idiss, p=2,model="sphere",opt.method="MM", ini = "auto")
plot(res.mm,main="Sph_MM")
#Apply the hyperbox model via the BFGS method
set.seed(1)
res.bfgs <- IMDS(IDM=oil.idiss, p=2,model="box",opt.method="BFGS", ini = "auto")
plot(res.bfgs,main="Box_bfgs")
#Apply the hyperbox model via the MM algorithm
set.seed(1)
res.mm <- IMDS(IDM=oil.idiss, p=2,model="box",opt.method="MM", ini = "auto")
plot(res.mm,main="Box_MM")
###############################

Loading required package: MASS
initial  value 66.561821 
final  value 4.497577 
converged
initial value 66.561821 
iter 100 stress = 3.816240 
iter (final) 104 stress = 3.816196 
converged
initial  value 69.269403 
final  value 2.066804 
converged
initial value 69.269403 
iter 100 stress = 2.270354 
iter 200 stress = 2.070075 
iter 300 stress = 2.037790 
iter 400 stress = 2.019342 
iter 500 stress = 2.003719 
iter 600 stress = 1.999278 
iter 700 stress = 1.996048 
iter 800 stress = 1.979463 
iter (final) 859 stress = 1.975457 
converged