Title:	Flexible Multidimensional Scaling and 'smacof' Extensions
Version:	1.21-1
Maintainer:	Thomas Rusch <thomas.rusch@wu.ac.at>
Description:	Flexible multidimensional scaling (MDS) methods and extensions to the package 'smacof'. This package contains various functions, wrappers, methods and classes for fitting, plotting and displaying a large number of different flexible MDS models. These are: Torgerson scaling (Torgerson, 1958, ISBN:978-0471879459) with powers, Sammon mapping (Sammon, 1969, <doi:10.1109/T-C.1969.222678>) with ratio and interval optimal scaling, Multiscale MDS (Ramsay, 1977, <doi:10.1007/BF02294052>) with ratio and interval optimal scaling, s-stress MDS (ALSCAL; Takane, Young & De Leeuw, 1977, <doi:10.1007/BF02293745>) with ratio and interval optimal scaling, elastic scaling (McGee, 1966, <doi:10.1111/j.2044-8317.1966.tb00367.x>) with ratio and interval optimal scaling, r-stress MDS (De Leeuw, Groenen & Mair, 2016, https://rpubs.com/deleeuw/142619) with ratio, interval, splines and nonmetric optimal scaling, power-stress MDS (POST-MDS; Buja & Swayne, 2002 <doi:10.1007/s00357-001-0031-0>) with ratio and interval optimal scaling, restricted power-stress (Rusch, Mair & Hornik, 2021, <doi:10.1080/10618600.2020.1869027>) with ratio and interval optimal scaling, approximate power-stress with ratio optimal scaling (Rusch, Mair & Hornik, 2021, <doi:10.1080/10618600.2020.1869027>), Box-Cox MDS (Chen & Buja, 2013, https://jmlr.org/papers/v14/chen13a.html), local MDS (Chen & Buja, 2009, <doi:10.1198/jasa.2009.0111>), curvilinear component analysis (Demartines & Herault, 1997, <doi:10.1109/72.554199>), curvilinear distance analysis (Lee, Lendasse & Verleysen, 2004, <doi:10.1016/j.neucom.2004.01.007>), nonlinear MDS with optimal dissimilarity powers functions (De Leeuw, 2024, https://github.com/deleeuw/smacofManual/blob/main/smacofPO(power)/smacofPO.pdf), sparsified (power) MDS and sparsified multidimensional (power) distance analysis aka extended curvilinear (power) component analysis and extended curvilinear (power) distance analysis (Rusch, 2024, <doi:10.57938/355bf835-ddb7-42f4-8b85-129799fc240e>). Some functions are suitably flexible to allow any other sensible combination of explicit power transformations for weights, distances and input proximities with implicit ratio, interval, splines or nonmetric optimal scaling of the input proximities. Most functions use a Majorization-Minimization algorithm. Currently the methods are only available for one-mode two-way data (symmetric dissimilarity matrices).
Depends:	R (≥ 3.5.0), smacof (≥ 1.10-4)
Imports:	MASS, minqa, plotrix, ProjectionBasedClustering, weights, vegan
License:	GPL-2 | GPL-3
LazyData:	true
URL:	https://r-forge.r-project.org/projects/stops/
BugReports:	https://r-forge.r-project.org/tracker/?atid=5375&group_id=2037&func=browse
RoxygenNote:	7.3.2
Encoding:	UTF-8
NeedsCompilation:	no
Packaged:	2025-07-07 09:13:09 UTC; trusch
Author:	Thomas Rusch [aut, cre], Jan de Leeuw [aut], Lisha Chen [aut], Patrick Mair [aut]
Repository:	CRAN
Date/Publication:	2025-07-07 10:30:02 UTC

smacofx: Flexible multidimensional scaling methods and SMACOF extensions

Description

Flexible multidimensional scaling (MDS) methods centered around the Majorization algorithm. The package contains various functions, wrappers, methods and classes for fitting, plotting and displaying a large number of different flexible MDS models such as Torgerson scaling, ratio, interval and nonmetric MDS with majorization, nonlinear MDS with optimal powers for dissimilarities, Sammon mapping with ratio and interval optimal scaling, multiscale MDS with ratio and interval optimal scaling, Alscal (s-stress) MDS with ratio and interval optimal scaling, elastic scaling with ratio and interval optimal scaling, r-stress MDS for ratio, interval and nonmetric scaling, power stress for interval and ratio optimal scaling, restricted power-stress with ratio and interval optimal scaling, approximate power-stress with ratio scaling, curvilinear component analysis with ratio, interval and ordinal optimal scaling, power curvilinear component analysis with ratio, interval and ordinal optimal scaling, Box-Cox MDS and local MDS. Some functions are suitably flexible to allow any other sensible combination of explicit power transformations for weights, distances and input proximities with implicit ratio, interval or ordinal optimal scaling of the input proximities. Most functions use a majorization algorithm.

Details

The package provides:

Models:

• alscal... ALSCAL (s-stress) MDS with ratio, interval optimal scaling

• elscal.. Elastic scaling MDS with ratio, interval optimal scaling

• multiscale... Multiscale MDSwith ratio, interval optimal scaling

• rstressMin .. R-Stress MDS with ratio, interval, ordinal optimal scaling

• powerStressMin... power stress MDS (POST-MDS) with ratio, interval optimal scaling

• apStressMin... approximate POST-MDS with ratio, interval optimal scaling

• rpowerStressMin... restricted POST-MDS with ratio, interval optimal scaling

• clca ... curvilinear component analysis with ratio optimal scaling

• clda ... curvilinear distance analysis with ratio optimal scaling (ie.e. clca with Isomap distances)

• bcmds ... Box-Cox MDS with ratio optimal scaling

• lmds... Local MDS with ratio optimal scaling

• sammonmap... Sammon mapping with ratio, interval optimal scaling

• eCLPA aka smdda ... sparsified multidimensional distance analsysis with ratio, interval, spline scaling (this is smds with Isomap distances)

• eCLPDA aka spmdda ... sparsified power multidimensional distance analsysis with ratio, interval, spline scaling (this is spmds with Isomap distances)

• eCLCA aka smds ... sparsified multidimensional scaling with ratio, interval, spline scaling (extended CLCA, so fitted distances larger than tau are weighted with 0)

• eCLPCA spmds ... sparsified multidimensional scaling with ratio, interval, spline scaling (extended CLPCA, so fitted distances larger than tau are weighted with 0)

• opmds ... nonlinear MDS with optimal power for the dissimilarities.

Classes and Methods: The objects are of classes that extend the S3 classes smacof and smacofB. For the objects returned by the high-level functions S3 methods for standard generics were implemented, including print, coef, residuals, summary, plot, plot3dstatic. Wrappers and convenience functions for the model objects:

• bootmds ... Bootstrapping an MDS model

• biplotmds ... MDS Biplots

• icExploreGen ... Expore initial configurations

• jackmds ... Jackknife for MDS

• multistart ... Multistart function for MDS

• permtest ... Permutation test for MDS

Wrappers:

• cmdscale ... stats::cmdscale but returns an S3 objects to be used with smacof classes

• sammon... MASS::sammon but returns S3 objects to be used with smacof classes

Author(s)

Maintainer: Thomas Rusch thomas.rusch@wu.ac.at (ORCID)

Authors:

• Jan de Leeuw

• Lisha Chen

• Patrick Mair mair@fas.harvard.edu (ORCID)

See Also

Useful links:

• https://r-forge.r-project.org/projects/stops/

• Report bugs at https://r-forge.r-project.org/tracker/?atid=5375&group_id=2037&func=browse

Examples

data(BankingCrisesDistances)

res<-rStressMin(BankingCrisesDistances[,1:69],type="ordinal",r=2)
res

summary(res)
plot(res)
plot(res,"transplot")
plot(res,"Shepard")

msres<- multistart(res)

res2<-msres$best
permtest(res2)

Banking Crises Distances

Description

Matrix of Jaccard distances between 70 countries (Hungary and Greece were combined to be the same observation) based on their binary time series of having had a banking crises in a year from 1800 to 2010 or not. See data(bankingCrises) in package Ecdat for more info. The last column is Reinhart & Rogoffs classification as a low (3), middle- (2) or high-income country (1).

Format

A 69 x 70 matrix.

Source

data(bankingCrises) in library(Ecdat)

ALSCAL - MDS via S-Stress Minimization

Description

An implementation to minimize s-stress by majorization with ratio and interval optimal scaling.

Usage

alscal(
  delta,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed untransformed dissimilarities

• tdelta: Observed explicitly transformed (squared) dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix (here NULL)

See Also

rStressMin

Examples

dis<-smacof::kinshipdelta
res<-alscal(as.matrix(dis),type="interval",itmax=1000)
res
summary(res)
plot(res)

Approximate Power Stress MDS

Description

An implementation to minimize approximate power stress by majorization with ratio or interval optimal scaling. This approximates the power stress objective in such a way that it can be fitted with SMACOF without distance transformations. See Rusch et al. (2021) for details.

Usage

apStressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

apowerstressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

apostmds(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

apstressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

apstressmds(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Tranformed configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix (here weightmat^nu)

Note

Internally we calculate the approximation parameters upsilon=nu+2*lambda*(1-(1/kappa)) and tau=lambda/kappa. They are not output.

References

Rusch, Mair, Hornik (2021). Cluster Optimized Proximity Scaling. JCGS <doi:10.1080/10618600.2020.1869027>

Examples

dis<-smacof::kinshipdelta
res<-apStressMin(as.matrix(dis),kappa=2,lambda=1.5,itmax=1000)
res
summary(res)
plot(res)
plot(res,"Shepard")
plot(res,"transplot")

Box-Cox MDS

Description

This function minimizes the Box-Cox Stress of Chen & Buja (2013) via gradient descent. This is a ratio metric scaling method. The transformations are not straightforward to interpret but mu is associated with fitted distances in the configuration and lambda with the dissimilarities. Concretely for fitted distances (attraction part) it is BC_{mu+lambda}(d(X)) and for the repulsion part it is delta^lambda BC_{mu}(d(X)) with BC being the one-parameter Box-Cox transformation.

Usage

bcmds(
  delta,
  mu = 1,
  lambda = 1,
  rho = 0,
  type = "ratio",
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  itmax = 2000,
  init = NULL,
  verbose = 0,
  addD0 = 1e-04,
  principal = FALSE,
  normconf = FALSE,
  acc = 1e-05
)

bcStressMin(
  delta,
  mu = 1,
  lambda = 1,
  rho = 0,
  type = "ratio",
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  itmax = 2000,
  init = NULL,
  verbose = 0,
  addD0 = 1e-04,
  principal = FALSE,
  normconf = FALSE,
  acc = 1e-05
)

bcstressMin(
  delta,
  mu = 1,
  lambda = 1,
  rho = 0,
  type = "ratio",
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  itmax = 2000,
  init = NULL,
  verbose = 0,
  addD0 = 1e-04,
  principal = FALSE,
  normconf = FALSE,
  acc = 1e-05
)

boxcoxmds(
  delta,
  mu = 1,
  lambda = 1,
  rho = 0,
  type = "ratio",
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  itmax = 2000,
  init = NULL,
  verbose = 0,
  addD0 = 1e-04,
  principal = FALSE,
  normconf = FALSE,
  acc = 1e-05
)

Arguments

Details

For numerical reasons with certain parameter combinations, the normalized stress uses a configuration as worst result where every d(X) is 0+addD0. The same number is not added to the delta so there is a small inaccuracy of the normalized stress (but negligible if min(delta)>>addD0). Also, for mu<0 or mu+lambda<0 the normalization cannot generally be trusted (in the worst case of D(X)=0 one would have an 0^(-a)).

Value

an object of class 'bcmds' (also inherits from 'smacofP'). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats)

• confdist: Configuration dissimilarities

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• ndim: Number of dimensions

• model: Name of MDS model

• type: Must be "ratio" here.

• niter: Number of iterations

• nobj: Number of objects

• pars: hyperparameter vector theta

• weightmat: 1-diagonal matrix. For compatibility with smacofP classes.

• parameters, pars, theta: The parameters supplied

• call the call

and some additional components

• stress.m: default stress is the explicitly normalized stress on the normalized, transformed dissimilarities

• mu: mu parameter (for attraction)

• lambda: lambda parameter (for repulsion)

• rho: rho parameter (for weights)

Author(s)

Lisha Chen & Thomas Rusch

Examples

dis<-smacof::kinshipdelta
res<-bcmds(dis,mu=2,lambda=1.5,rho=0)
res
summary(res)
plot(res)

Calculates the blended Chi-square distance matrix between n vectors

Description

The pairwise blended chi-distance of two vectors x and y is sqrt(sum(((x[i]-y[i])^2)/(2*(ax[i]+by[i])))), with originally a in [0,1] and b=1-a as in Lindsay (1994) (but we allow any non-negative a and b). The function calculates this for all pairs of rows of a matrix or data frame x.

Usage

bcsdistance(x, a = 0.5, b = 1 - a)

Arguments

Value

a symmetric n times n matrix of pairwise blended chi-square distance (between rows of x) with 0 in the main diagonal. It is an object of class distance and matrix with attributes "method", "type" and "par", the latter returning the a and b values.

References

Lindsay (1994). Efficiency versus robustness: the case for minimum Hellinger distance and related methods. Annals of Statistics, 22 (2), 1081-1114. <doi:10.1214/aos/1176325512>

S3 method for bcmds objects

Description

S3 method for bcmds objects

Usage

## S3 method for class 'bcmds'
biplotmds(object, extvar, scale = TRUE)

Arguments

Details

If a model for individual differences is provided, the external variables are regressed on the group stimulus space configurations. For objects returned from 'biplotmds' we use the plot method in biplotmds. In the biplot called with plot() only the relative length of the vectors and their direction matters. Using the vecscale argument in plot() the user can control for the relative length of the vectors. If 'vecscale = NULL', the 'vecscale()' function from the 'candisc' package is used which tries to automatically calculate the scale factor so that the vectors approximately fill the same space as the configuration. In this method vecscale should usually be smaller than the one used in smacof by a factor of 0.1. Note that in the biplot object, the configuration is always normalized (which it may not necessarily be in the bcmds object).

Value

Returns an object belonging to classes 'mlm' and 'mdsbi'. See 'lm' for details. R2vec: Vector containing the R2 values. See also biplotmds for the plot method.

S3 method for lmds objects

Description

S3 method for lmds objects

Usage

## S3 method for class 'lmds'
biplotmds(object, extvar, scale = TRUE)

Arguments

Details

If a model for individual differences is provided, the external variables are regressed on the group stimulus space configurations. For objects returned from 'biplotmds' we use the plot method in biplotmds. In the biplot called with plot() only the relative length of the vectors and their direction matters. Using the vecscale argument in plot() the user can control for the relative length of the vectors. If 'vecscale = NULL', the 'vecscale()' function from the 'candisc' package is used which tries to automatically calculate the scale factor so that the vectors approximately fill the same space as the configuration. In this method vecscale should usually be smaller than the one used in smacof by a factor of 0.1. Note that in the biplot object, the configuration is always normalized (which it may not necessarily be in the lmds object).

Value

Returns an object belonging to classes 'mlm' and 'mdsbi'. See 'lm' for details. R2vec: Vector containing the R2 values. See also biplotmds for the plot method.

S3 method for smacofP objects

Description

S3 method for smacofP objects

Usage

## S3 method for class 'smacofP'
biplotmds(object, extvar, scale = TRUE)

Arguments

Details

If a model for individual differences is provided, the external variables are regressed on the group stimulus space configurations. For objects returned from 'biplotmds' we use the plot method in biplotmds. In the biplot called with plot() only the relative length of the vectors and their direction matters. Using the vecscale argument in plot() the user can control for the relative length of the vectors. If 'vecscale = NULL', the 'vecscale()' function from the 'candisc' package is used which tries to automatically calculate the scale factor so that the vectors approximately fill the same space as the configuration. In this method vecscale should usually be smaller than the one used in smacof by a factor of 0.1.

Value

Returns an object belonging to classes 'mlm' and 'mdsbi'. See 'lm' for details. R2vec: Vector containing the R2 values. See also biplotmds for the plot method.

Examples

## see smacof::biplotmds for more
res <- powerStressMin(morse,kappa=0.5,lambda=2)
fitbi <- biplotmds(res, morsescales[,2:3])
plot(fitbi, main = "MDS Biplot", vecscale = 0.03)

MDS Bootstrap for smacofP objects

Description

Performs a bootstrap on an MDS solution. It works for derived dissimilarities only, i.e. generated by the call dist(data). The original data matrix needs to be provided, as well as the type of dissimilarity measure used to compute the input dissimilarities.

Usage

## S3 method for class 'smacofP'
bootmds(
  object,
  data,
  method.dat = "pearson",
  nrep = 100,
  alpha = 0.05,
  verbose = FALSE,
  ...
)

Arguments

Details

In order to examine the stability solution of an MDS, a bootstrap on the raw data can be performed. This results in confidence ellipses in the configuration plot. The ellipses are returned as list which allows users to produce (and further customize) the plot by hand. See bootmds for more.

Value

An object of class 'smacofboot', see bootmds. With values

• cov: Covariances for ellipse computation

• bootconf: Configurations bootstrap samples

• stressvec: Bootstrap stress values

• bootci: Stress bootstrap percentile confidence interval

• spp: Stress per point (based on stress.en)

• stab: Stability coefficient

Examples

##see ?smacof::bootmds for more 
data <- na.omit(smacof::PVQ40[,1:5])
diss <- dist(t(data))   ## Euclidean distances
fit <- rStressMin(diss,r=0.5,itmax=1000) ## 2D ratio MDS
set.seed(123)
resboot <- bootmds(fit, data, method.dat = "euclidean", nrep = 10) #run for more nrep
resboot
plot(resboot) #see ?smacof::bootmds for more on the plot method

Curvilinear Component Analysis (CLCA)

Description

A wrapper to run curvilinear component analysis via CCA and returning a 'smacofP' object. Note this functionality is rather rudimentary.

Usage

clca(
  delta,
  Epochs = 20,
  alpha0 = 0.5,
  lambda0,
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  acc = 1e-06,
  itmax = 10000,
  verbose = 0,
  method = "euclidean",
  principal = FALSE
)

Arguments

Details

This implements CCA as in Desmartines & Herault (1997). A different take on the ideas of curvilinear compomnent analysis is available in the experimental functions spmds and spmds.

Value

a 'smacofP' object. It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Configuration dissimilarities

• conf: Matrix of fitted configuration

• stress: Default stress (stress-1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of model

• niter: Number of iterations (training length)

• nobj: Number of objects

• type: Type of MDS model. Only ratio here.

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix; it is weightmat here.

Examples

dis<-smacof::morse
res<-clca(dis,lambda0=0.4)
res
summary(res)
plot(res)

Curvilinear Distance Analysis (CLDA)

Description

A function to run curvilinear distance analysis via CCA and returning a 'smacofP' object. Note this functionality is rather rudimentary.

Usage

clda(
  delta,
  Epochs = 20,
  alpha0 = 0.5,
  lambda0,
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  acc = 1e-06,
  itmax = 10000,
  verbose = 0,
  method = "euclidean",
  principal = FALSE,
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE
)

Arguments

Details

This implements CLDA as CLCA with geodesic distances. The geodesic distances are calculated via 'vegan::isomapdist', see isomapdist for a documentation of what these distances do. 'clda' is just a wrapper for 'clca' applied to the geodesic distances obtained via isomapdist.

Value

a 'smacofP' object. It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Configuration dissimilarities

• conf: Matrix of fitted configuration

• stress: Default stress (stress-1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of model

• niter: Number of iterations (training length)

• nobj: Number of objects

• type: Type of MDS model. Only ratio here.

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix; it is weightmat here.

Examples

dis<-smacof::morse
res<-clda(dis,lambda0=0.4,k=4)
res
summary(res)
plot(res)

Classical Scaling

Description

Classical Scaling

Usage

cmds(Do)

Arguments

Wrapper to cmdscale for S3 class

Description

Wrapper to cmdscale for S3 class

Usage

cmdscale(d, k = 2, eig = FALSE, ...)

Arguments

Details

overloads stats::cmdscale turns on the liosting and adds slots and class attributes for which there are methods.

Value

Object of class 'cmdscalex' and 'cmdscale' extending cmdscale. This wrapper always returns the results of cmdscale as a list, adds column labels to the $points and adds extra elements (conf=points, delta=d, confdist=dist(conf), dhat=d) and the call to the list, and assigns S3 class 'cmdscalex' and 'cmdscale'.

Examples

dis<-as.matrix(smacof::kinshipdelta)
res<-cmdscale(dis)

conf_adjust: a function to procrustes adjust two matrices

Description

conf_adjust: a function to procrustes adjust two matrices

Usage

conf_adjust(conf1, conf2, verbose = FALSE, eps = 1e-12, itmax = 100)

Arguments

Value

a list with 'ref.conf' being the reference configuration, 'other.conf' the adjusted coniguration and 'comparison.conf' the comparison configuration

Corpse Paint

Description

A matrix of gray scale images of people in "corpse paint", a black-and-white make-up, plus a surprise.

Format

A 8100 x 32 matrix

Details

The images are gray scale 8 bit, i.e., 0-255 unique gray values scaled to be between 0 and 1. There are 32 total images with pixel size of 90 x 90 that have been vectorized to 32 columns labeled as "F1" through "F32". An image i can be reconstructed with matrix(corpsepaint[,i],ncol=90,nrow=90)

Examples

oldpar<-par(no.readonly=TRUE)
par(mfrow=c(4,8))
for(i in 1:ncol(corpsepaint)){
p1<-matrix(corpsepaint[,i],ncol=90,nrow=90,byrow=FALSE) 
image(p1,col=gray.colors(256),main=colnames(corpsepaint)[i])
}
par(oldpar)

Double centering of a matrix

Description

Double centering of a matrix

Usage

doubleCenter(x)

Arguments

Value

the double centered matrix

Elastic Scaling SMACOF

Description

An implementation to minimize elastic scaling stress by majorization with ratio and interval optimal scaling. Uses a repeat loop.

Usage

elscal(
  delta,
  type = c("ratio", "interval"),
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from smacofB, see smacofSym). It is a list with the components

• delta: Observed untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformation configuration distances

• conf: Matrix of fitted configuration, NOT normalized

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point (based on stress.en)

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• tweightmat: transformed weighting matrix (here weightmat/delta^2)

• stress.m: Default stress (stress-1^2)

See Also

rStressMin

Examples

dis<-smacof::kinshipdelta
res<-elscal(as.matrix(dis),itmax=1000)
res
summary(res)
plot(res)

Explicit Normalization Normalizes distances

Description

Explicit Normalization Normalizes distances

Usage

enorm(x, w = 1)

Arguments

Value

a constant

Exploring initial configurations in an agnostic way

Description

Allows to user to explore the effect of various starting configurations when fitting an MDS model. This is a bit more general than the icExplore function in smacof, as we allow any PS model to be used as the model is either setup by call or by a prefitted object (for the models in cops and stops we do not have a single UI function which necessitates this). Additionally, one can supply their own configurations and not just random ones.

Usage

icExploreGen(
  object,
  mdscall = NULL,
  conflist,
  nrep = 100,
  ndim,
  returnfit = FALSE,
  min = -5,
  max = 5,
  verbose = FALSE
)

Arguments

Details

If no configuration list is supplied, then nrep configurations are simulated. They are drawn from a ndim-dimensional uniform distribution with minimum min and maximum max. We recommend to use the route via supplying a fitted model as these are typically starting from a Torgerson configuration and are likely quite good.

Value

an object of class 'icexplore', see icExplore for more. There is a plot method in package 'smacof'.

Examples

dis<-kinshipdelta

## Version 1: Using a fitted object (recommended)
res1<-rStressMin(dis,type="ordinal",itmax=100)
resm<-icExploreGen(res1,nrep=5)

## Version 2: Using a call object and supplying conflist
conflist<-list(res1$init,jitter(res1$init,1),jitter(res1$init,1),jitter(res1$init,1))
c1 <- call("smds",delta=dis,tau=0.2,itmax=100)
resm<-icExploreGen(mdscall=c1,conflist=conflist,returnfit=TRUE)

plot(resm)

MDS Jackknife for smacofP objects

Description

These functions perform an MDS Jackknife and plot the corresponding solution.

Usage

## S3 method for class 'smacofP'
jackmds(object, eps = 1e-06, itmax = 100, verbose = FALSE)

Arguments

Details

In order to examine the stability solution of an MDS, a Jackknife on the configurations can be performed (see de Leeuw & Meulman, 1986) and plotted. The plot shows the jackknife configurations which are connected to their centroid. In addition, the original configuration (transformed through Procrustes) is plotted. The Jackknife function itself returns also a stability measure (as ratio of between and total variance), a measure for cross validity, and the dispersion around the original smacof solution.

Value

An object of class 'smacofJK', see jackmds. With values

• smacof.conf: Original configuration

• jackknife.confboot: An array of n-1 configuration matrices for each Jackknife MDS solution

• comparison.conf: Centroid Jackknife configurations (comparison matrix)

• cross: Cross validity

• stab: Stability coefficient

• disp: Dispersion

• loss: Value of the loss function (just used internally)

• ndim: Number of dimensions

• call: Model call

• niter: Number of iterations

• nobj: Number of objects

Examples

dats <- na.omit(smacof::PVQ40[,1:5])
diss <- dist(t(dats))   ## Euclidean distances
fit <- rStressMin(diss,type="ordinal",r=0.4,itmax=1000) ## 2D ordinal MDS

res.jk <- jackmds(fit)
plot(res.jk, col.p = "black", col.l = "gray")
plot(res.jk, hclpar = list(c = 80, l = 40))
plot(res.jk, hclpar = list(c = 80, l = 40), plot.lines = FALSE)

Responses to the SCC scale and CSII scale (n=1013).

Description

1013 respondents answered the items of the consumer susceptibility top interpersonal influence (CSII) and the self-concept clarity (SCC) scale. The answers are on a five-point-Likert scale with 1 meaning "fully agree" and 5 meaning "fully disagree". Note that on scc the 10th item is reversed coded to align with the other items.

Format

A 1013 x 24 data frame.

Source

Koller, Floh, Zauner, Rusch (2013) "Persuasibility and the Self – Investigating Heterogeneity among Consumers". Australasian Marketing Journal, 21, 94-104.

Local MDS

Description

This function minimizes the Local MDS Stress of Chen & Buja (2006) via gradient descent. This is a ratio metric scaling method.

Usage

lmds(
  delta,
  k = 2,
  tau = 1,
  type = "ratio",
  ndim = 2,
  weightmat = 1 - diag(nrow(delta)),
  itmax = 5000,
  acc = 1e-05,
  init = NULL,
  verbose = 0,
  principal = FALSE,
  normconf = FALSE
)

Arguments

Details

Note that k and tau are not independent. It is possible for normalized stress to become negative if the tau and k combination is so that the absolute repulsion for the found configuration dominates the local stress substantially less than the repulsion term does for the solution of D(X)=Delta, so that the local stress difference between the found solution and perfect solution is nullified. This can typically be avoided if tau is between 0 and 1. If not, set k and or tau to a smaller value.

Value

an object of class 'lmds' (also inherits from 'smacofP'). See powerStressMin. It is a list with the components as in power stress

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats)

• confdist: Configuration dissimilarities

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• ndim: Number of dimensions

• model: Name of MDS model

• type: Is "ratio" here.

• niter: Number of iterations

• nobj: Number of objects

• pars: explicit transformations hyperparameter vector theta

• weightmat: 1-diagonal matrix (for compatibility with smacof classes)

• parameters, pars, theta: The parameters supplied

• call the call

and some additional components

• stress.m: default stress is the explicitly normalized stress on the normalized, transformed dissimilarities

• tau: tau parameter

• k: k parameter

Author(s)

Lisha Chen & Thomas Rusch

Examples

dis<-smacof::kinshipdelta
res<- lmds(dis,k=2,tau=0.1)
res
summary(res)
plot(res)

Auxfunction1

Description

only used internally

Usage

mkBmat(x)

Arguments

Take matrix to a power

Description

Take matrix to a power

Usage

mkPower(x, r)

Arguments

Value

a matrix

Multiscale SMACOF

Description

An implementation for maximum likelihood MDS aka multiscale that minimizes the multiscale stress by majorization with ratio and interval optimal scaling. Uses a repeat loop. Note that since this done via the route of r-sytress, the multiscale stress is approximate and only accuarte for kappa->0.

Usage

multiscale(
  delta,
  type = c("ratio", "interval"),
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  kappa = 0.01,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed dissimilarities

• tdelta: Observed explicitly transformed (log) dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix

• stress.m: Default stress (stress-1^2)

Warning

The input delta will internally get transformed to the log scale, so make sure that log(delta)>=0 otherwise it throws an error. It is often a good idea to use 1+delta in this case.

See Also

rStressMin

Examples

dis<-smacof::kinshipdelta
res<-multiscale(as.matrix(dis),type="interval",itmax=1000)
res
summary(res)
plot(res)

Multistart MDS function

Description

For different starting configurations, this function fits a series of PS models given in object or call and returns the one with the lowest stress overall. The starting configuirations can be supplied or are generated internally.

Usage

multistart(
  object,
  mdscall = NULL,
  ndim = 2,
  conflist,
  nstarts = 108,
  return.all = FALSE,
  verbose = TRUE,
  min = -5,
  max = 5
)

Arguments

Details

If no configuration list is supplied, then nstarts configurations are simulated. They are drawn from a ndim-dimesnional uniform distribution with minimum min and maximum max. We recommend to use the route via supplying a fitted model as these are typically starting from a Torgerson configuration and are likely quite good.

One can simply extract $best and save that and work with it right away.

Value

if 'return.all=FALSE', a list with the best fitted model as '$best' (minimal badness-of-fit of all fitted models) and '$stressvec' the stresses of all models. If 'return.all=TRUE' a list with slots

• best: The object resulting from the fit that had the overall lowest objective function value (usually stress)

• stressvec: The vector of objective function values

• models: A list of all the fitted objects.

Examples

dis<-smacof::kinshipdelta

## Version 1: Using a fitted object (recommended)
res1<-rStressMin(delta=dis,type="ordinal",itmax=100)
resm<-multistart(res1,nstarts=2)
## best model
res2<-resm$best
#it's starting configuration
res2$init

## Version 2: Using a call object and supplying conflist
conflist<-list(res2$init,jitter(res2$init,1))
c1 <- call("rstressMin",delta=dis,type="ordinal",itmax=100)
resm<-multistart(mdscall=c1,conflist=conflist,return.all=TRUE)

Nonlinear ratio MDS with optimal power of dissimilarities

Description

An implementation to minimize explicitly normalized stress over dissimilarities to a power by majorization with ratio optimal scaling in an alternating minimization algorithm. The optimal power transformation lambda of the dissimilarities is found by an inner optimization step via the Brent-Dekker method.

Usage

opmds(
  delta,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  itmax = 1000,
  acc = 1e-10,
  verbose = FALSE,
  principal = FALSE,
  interval = c(0, 4)
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed fitted configuration distances

• iord: optimal scaling ordering

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix (here NULL)

• pars, theta: The optimal transformation parameter lambda

See Also

See stops for a similar, more flexible idea.

Examples

dis<-smacof::kinshipdelta
res<-opmds(dis,itmax=1000)
res
summary(res)
plot(res)

Squared p-distances

Description

Squared p-distances

Usage

pdist(x, p)

Arguments

Value

squared Minkowski distance matrix

Permutation test for smacofP objects

Description

Performs a permutation test on an MDS solution. It works with a smacofP object alone and also for derived dissimilarities, i.e. generated by the call dist(data). The original data matrix needs to be provided, as well as the type of dissimilarity measure used to compute the input dissimilarities.

Usage

## S3 method for class 'smacofP'
permtest(
  object,
  data,
  method.dat = "pearson",
  nrep = 100,
  verbose = FALSE,
  ...
)

Arguments

Details

This routine permutes m dissimilarity values, where m is the number of lower diagonal elements in the corresponding dissimilarity matrix. For each sample a symmetric, nonmetric SMACOF of dimension 'ndim' is computed and the stress values are stored in ‘stressvec’. Using the fitted stress value, the p-value is computed. Subsequently, the empirical cumulative distribution function can be plotted using the plot method.

If the MDS fit provided on derived proximities of a data matrix, this matrix can be passed to the ‘permtest’ function. Consequently, the data matrix is subject to permutations. The proximity measure used for MDS fit has to match the one used for the permutation test. If a correlation similarity is provided, it is converted internally into a dissimilarity using 'sim2diss' with corresponding arguments passed to the ... argument.

Value

An object of class 'smacofPerm', see permtest for details and methods. It has values

• stressvec: Vector containing the stress values of the permutation samples

• stress.obs: Stress (observed sample)

• pval: Resulting p-value

• call: Model call

• nrep: Number of permutations

• nobj: Number of objects

Examples

##see ?smacof::permtest for more
## permuting the dissimilarity matrix (full)
#' data(kinshipdelta)
fitkin <- rStressMin(kinshipdelta, ndim = 2, r=0.5,itmax=10) #use higher itmax
set.seed(222)
res.perm <- permtest(fitkin,nrep=5) #use higher nrep in reality
res.perm
plot(res.perm)
## permuting the data matrix
GOPdtm[GOPdtm > 1] <- 1     ## use binary version
diss1 <- dist(t(GOPdtm[,1:10]), method = "binary")  ## Jaccard distance
fitgop1 <- alscal(diss1,type="interval",itmax=10) #use higher itmax
fitgop1
set.seed(123)
permtest(fitgop1, GOPdtm[,1:10], nrep = 5, method.dat = "binary")

S3 plot method for smacofP objects

Description

S3 plot method for smacofP objects

Usage

## S3 method for class 'smacofP'
plot(
  x,
  plot.type = "confplot",
  plot.dim = c(1, 2),
  bubscale = 1,
  col,
  label.conf = list(label = TRUE, pos = 3, col = 1, cex = 0.8),
  hull.conf = list(hull = FALSE, col = 1, lwd = 1, ind = NULL),
  shepard.x = NULL,
  identify = FALSE,
  type = "p",
  cex = 0.5,
  pch = 20,
  asp = 1,
  main,
  xlab,
  ylab,
  xlim,
  ylim,
  col.hist = NULL,
  legend = TRUE,
  legpos,
  loess = TRUE,
  shepard.lin = TRUE,
  ...
)

Arguments

Details

• Configuration plot (plot.type = "confplot"): Plots the MDS configuration.

• Residual plot (plot.type = "resplot"): Plots the dhats f(T(delta)) against the transformed fitted distances T(d(X)).

• (Linearized) Shepard diagram (plot.type = "Shepard"): Is shep.lin=TRUE a diagram with the transformed observed normalized dissimilarities (T(delta) on x) against the transformed fitted distance (T(d(X) on y) as well as a loess curve and a regression line corresponding to type (linear without intercept for ratio, linear for interval and isotonic for ordinal). If shep.lin=FALSE it uses the untransformed delta. Note that the regression line corresponds to the optimal scaling results (dhat) only up to a linear transformation.

• Transformation Plot (plot.type = "transplot"): Diagram with normalized observed dissimilarities (delta, light grey) and the normalized explicitly transformed dissimilarities (T(Delta), darker) against the untransformed fitted distances (d(X)) together with a nonlinear regression curve corresponding to the explicit transformation (fitted power transformation). This is most useful for ratio models with power transformations as the transformations can be read of directly. For other MDS models and stresses, it still gives a quick way to assess how the explicit transformations worked.

• Stress decomposition plot (plot.type = "stressplot"): Plots the stress contribution in of each observation. Note that it rescales the stress-per-point (SPP) from the corresponding function to percentages (sum is 100). The higher the contribution, the worse the fit.

• Bubble plot (plot.type = "bubbleplot"): Combines the configuration plot with the point stress contribution. The larger the bubbles, the worse the fit.

• histogram (‘plot.type = "histogram"’: gives a weighted histogram of the dissimilarities (weighted with tweightmat if exists else with weightmat). For optional arguments, see ‘wtd.hist’.

Value

no return value; just plots for class 'smacofP' (see details)

Examples

dis<-as.matrix(smacof::kinshipdelta)
res<-powerStressMin(dis)
plot(res)
plot(res,"Shepard")
plot(res,"resplot")
plot(res,"transplot")
plot(res,"stressplot")
plot(res,"bubbleplot")
plot(res,"histogram")

Power stress minimization by NEWUOA (nloptr)

Description

An implementation to minimize power stress by a derivative-free trust region optimization algorithm (NEWUOA). Much faster than majorizing as used in powerStressMin but perhaps less accurate.

Usage

powerStressFast(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed dissimilarities, not normalized

• obsdiss: Observed dissimilarities, normalized

• confdist: Configuration dissimilarities, NOT normalized

• conf: Matrix of fitted configuration, NOT normalized

• stress: Default stress (stress 1, square root of the explicitly normalized stress on the normalized, transformed dissimilarities)

• spp: Stress per point (based on stress.en)

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

and some additional components

• gamma: Empty

• stress.m: default stress for the COPS and STOP. Defaults to the explicitly normalized stress on the normalized, transformed dissimilarities

• stress.en: explicitly stress on the normalized, transformed dissimilarities and normalized transformed distances

• deltaorig: observed, untransformed dissimilarities

• weightmat: weighting matrix

See Also

smacofSym

Examples

dis<-smacof::kinshipdelta
res<-powerStressFast(as.matrix(dis),kappa=2,lambda=1.5)
res
summary(res)
plot(res)

Power Stress SMACOF

Description

An implementation to minimize power stress by majorization with ratio or interval optimal scaling. Usually more accurate but slower than powerStressFast. Uses a repeat loop.

Usage

powerStressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

powerstressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

postmds(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

pstressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

pStressMin(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

pstressmds(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  type = "ratio",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed fitted configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighthingmatrix (here weightmat^nu)

See Also

smacofSym

Examples

dis<-smacof::kinshipdelta
res<-powerStressMin(dis,type="ratio",kappa=2,lambda=1.5,itmax=1000)
res
summary(res)
plot(res)

procruster: a procrustes function

Description

procruster: a procrustes function

Usage

procruster(x)

Arguments

Value

a matrix

R stress SMACOF

Description

An implementation to minimize r-stress by majorization with ratio, interval, monotonic spline and ordinal optimal scaling. Uses a repeat loop.

Usage

rStressMin(
  delta,
  r = 0.5,
  type = c("ratio", "interval", "ordinal", "mspline"),
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

rstressMin(
  delta,
  r = 0.5,
  type = c("ratio", "interval", "ordinal", "mspline"),
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

rstressmds(
  delta,
  r = 0.5,
  type = c("ratio", "interval", "ordinal", "mspline"),
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

rstress(
  delta,
  r = 0.5,
  type = c("ratio", "interval", "ordinal", "mspline"),
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed fitted configuration distances

• iord: Optimally scaled disparities function

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• weightmat: Weighting matrix as supplied

• resmat: Residual matrix

• rss: Sum of residuals

• init: The starting configuration

• model: Name of MDS model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of optimal scaling

• call : the matched call

• stress.m: Default stress (stress-1^2)

• alpha: Alpha matrix

• sigma: Stress

• parameters, pars, theta: Optimal transformation parameter

• tweightmat: Transformed weighting matrix (here NULL)

See Also

smacofSym

Examples

dis<-smacof::kinshipdelta

## ordinal MDS
res<-rStressMin(as.matrix(dis), type = "ordinal", r = 1, itmax = 1000)
res
summary(res)
plot(res)

## spline MDS 
ress<-rStressMin(as.matrix(dis), type = "mspline", r = 1,
      itmax = 1000)
ress
plot(ress,"Shepard")

Two interlocking rings

Description

Artifical data of two interlocking rings in 3D space. The first three columns are the coordinates and the last column is a color designation.

Format

A 500 x 4 matrix.

Restricted Power Stress SMACOF

Description

An implementation to minimize restricted power stress by majorization with ratio or interval optimal scaling. Restricted means that the same power is used for both dissimilarities and fitted distances. Uses a repeat loop.

Usage

rpowerStressMin(
  delta,
  expo = 1,
  nu = 1,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

rpowerstressMin(
  delta,
  expo = 1,
  nu = 1,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

rpostmds(
  delta,
  expo = 1,
  nu = 1,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

rpstressMin(
  delta,
  expo = 1,
  nu = 1,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

rpStressMin(
  delta,
  expo = 1,
  nu = 1,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

rpstressmds(
  delta,
  expo = 1,
  nu = 1,
  type = "ratio",
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighthing matrix (here weightmat^nu)

• parameters, pars, theta: The parameter vector of the explicit transformations

Examples

dis<-smacof::kinshipdelta
res<-rpowerStressMin(as.matrix(dis),expo=1.7,itmax=1000)
res
summary(res)
plot(res)

Wrapper to sammon for S3 class

Description

Wrapper to sammon for S3 class

Usage

sammon(d, y = NULL, k = 2, ...)

Arguments

Details

Overloads MASS::sammon and adds new slots and class attributes for which there are methods.

Value

Object of class 'sammonx' inheriting from sammon. This wrapper adds an extra slot to the list with the call, adds column labels to the $points, adds slots conf=points, delta=d, dhat=normalized dissimilarities, confdist=distance between points in conf, stress.m=stress, stress=sqrt(stress.m) and assigns S3 classes 'sammonx', 'sammon' and 'cmdscalex'.

Examples

dis<-as.matrix(smacof::kinshipdelta)
res<-sammon(dis)

Sammon Mapping SMACOF

Description

An implementation to minimize Sammon stress by majorization with ratio and interval optimal scaling. Uses a repeat loop.

Usage

sammonmap(
  delta,
  type = c("ratio", "interval"),
  weightmat,
  init = NULL,
  ndim = 2,
  acc = 1e-06,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE
)

Arguments

Value

a 'smacofP' object (inheriting from smacofB, see smacofSym). It is a list with the components

• delta: Observed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Observed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point (based on stress.en)

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: default stress (stress-1^2)

• tweightmat: weighting matrix atfer transformation (here weightmat/delta)

See Also

rStressMin

Examples

dis<-smacof::kinshipdelta
res<-sammonmap(as.matrix(dis),itmax=1000)
res
summary(res)
plot(res)

Adjusts a configuration

Description

Adjusts a configuration

Usage

scale_adjust(conf, ref, scale = c("sd", "std", "proc", "none"))

Arguments

Value

The scale adjusted configuration.

Secular Equation

Description

Secular Equation

Usage

secularEq(a, b)

Arguments

Helper function to conduct jackknife MDS

Description

Function deletes every object row and columns once and fits the MDS in object and returns the configuration. The deleted row is set to 0 in the configuration. Is meant for smacofx functions, but should also work for every smacof models.

Usage

smacofxDeleteOne(
  object,
  delta,
  weightmat,
  init,
  ndim,
  type,
  verbose = FALSE,
  itmaxi = 10000
)

Arguments

Value

An array of size n with n coonfigurations

Clelia curve on a sphere

Description

Artifical data of data sampled along a Clelia curve on a sphere. The first three columns are the coordinates and the last column is a color designation (viridis palette).

Format

A 500 x 4 matrix.

Extended Curvilinear (Power) Distance Analysis (eCLPDA or eCLDA) aka Sparse (POST-)Multidimensional Distance Analysis (SPMDDA or SMDDA) either as self-organizing or not

Description

An implementation of a sparsified version of (POST-)MDS by pseudo-majorization with ratio, interval and ordinal optimal scaling for geodesic distances and optional power transformations. This is inspired by curvilinear distance analysis but works differently: It finds an initial weightmatrix where w_ij(X^0)=0 if d_ij(X^0)>tau and fits a POST-MDS with these weights. Then in each successive iteration step, the weightmat is recalculated so that w_ij(X^(n+1))=0 if d_ij(X^(n+1))>tau. Right now the zero weights are not found by the correct optimization, but we're working on that.

Usage

spmdda(
  delta,
  lambda = 1,
  kappa = 1,
  nu = 1,
  tau,
  type = "ratio",
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2,
  traceIt = FALSE
)

smdda(
  delta,
  tau = stats::quantile(delta, 0.9),
  type = "ratio",
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  traceIt = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_spmdda(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  tau = max(delta),
  epochs = 10,
  type = c("ratio"),
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_smdda(
  delta,
  tau = max(delta),
  epochs = 10,
  type = c("ratio"),
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eCLDA(
  delta,
  tau = stats::quantile(delta, 0.9),
  type = "ratio",
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  traceIt = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eCLPDA(
  delta,
  lambda = 1,
  kappa = 1,
  nu = 1,
  tau,
  type = "ratio",
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2,
  traceIt = FALSE
)

so_eCLPDA(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  tau = max(delta),
  epochs = 10,
  type = c("ratio"),
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_eCLDA(
  delta,
  tau = max(delta),
  epochs = 10,
  type = c("ratio"),
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eclpda(
  delta,
  lambda = 1,
  kappa = 1,
  nu = 1,
  tau,
  type = "ratio",
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2,
  traceIt = FALSE
)

eclda(
  delta,
  tau = stats::quantile(delta, 0.9),
  type = "ratio",
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  traceIt = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_eclpda(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  tau = max(delta),
  epochs = 10,
  type = c("ratio"),
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_eclda(
  delta,
  tau = max(delta),
  epochs = 10,
  type = c("ratio"),
  ties = "primary",
  epsilon,
  k,
  path = "shortest",
  fragmentedOK = FALSE,
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

Arguments

Details

In 'spmdda' the logic is that we first transform to geodesic distance, then apply the explicit power transformation and then the implicit optimal scaling. There is a wrapper 'smdda', 'eCLDA' where the exponents are 1, which is standard SMDDA or eCLPDA but extend to allow optimal scaling. The neighborhood parameter tau is kept fixed in 'spmdda', 'eCLPDA' and 'smdda', 'eCLDA'. The functions 'so_spmdda', 'so_eCLPDA' and 'so_smdda', 'so_eCLDA' implement a self-organising principle where the is repeatedly fitted for a decreasing sequence of taus.

The solution is found by "quasi-majorization", which mean that the majorization is only working properly after a burn-in of a few iterations when the assignment which distances are ignored no longer changes. Due to that it can be that in the beginning the stress may not decrease monotonically and that there's a chance it might never.

The geodesic distances are calculated via 'vegan::isomapdist', see isomapdist for a documentation of what these distances do. The functions of '(p)smdda' are just a wrapper for '(p)clca' applied to the geodesic distances obtained via isomapdist.

If tau is too small it may happen that all distances for one i to all j are zero and then there will be an error, so make sure to set a larger tau.

In the standard functions 'spmdda' and 'smdda' we keep tau fixed throughout. This means that if tau is large enough, then the result is the same as the corresponding MDS. In the orginal publication the idea was that of a self-organizing map which decreased tau over epochs (i.e., passes through the data). This can be achieved with our function 'so_spmdda' 'so_smdda' which creates a vector of decreasing tau values, calls the function 'spmdda' with the first tau, then supplies the optimal configuration obtained as the init for the next call with the next tau and so on.

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Configuration dissimilarities

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix; it is weightmat but containing all the 0s for the distances set to 0.

• trace: if 'traceIt=TRUE' a vector with the iteration progress

Examples

dis<-smacof::morse
res<-spmdda(dis,kappa=2,lambda=2,tau=0.4,k=5,itmax=500) #use higher itmax
res
#already many parameters 
coef(res)

res2<-smdda(dis,type="interval",tau=0.4,epsilon=1,itmax=500) #use higher itmax
#aliases:
resa<-eCLPDA(dis,kappa=2,lambda=2,tau=0.4,k=5,itmax=500) #use higher itmax
res2a<-eCLDA(dis,type="interval",tau=0.4,epsilon=1,itmax=500) #use higher itmax

res2
summary(res)
oldpar<-par(mfrow=c(1,2))
plot(res)
plot(res2)
par(oldpar)

##which d_{ij}(X) exceeded tau at convergence (i.e., have been set to 0)?
res$tweighmat
res2$tweightmat


## Self-organizing map style (as in the original publication)
#run the som-style (p)smdda 
sommod1<-so_spmdda(dis,tau=2,k=5,kappa=0.5,lambda=2,epochs=10,verbose=1)
sommod2<-so_smdda(dis,tau=2.5,epsilon=1,epochs=10,verbose=1)
sommod1
sommod2

Extended Curvilinear (Power) Component Analysis aka Sparsified (POST-) Multidimensional Scaling (SPMDS or SMDS) either as self-organizing or not

Description

An implementation of extended CLPCA which is a sparsified version of (POST-)MDS by quasi-majorization with ratio, interval and ordinal optimal scaling for dissimilarities and optional power transformations. This is inspired by curvilinear component analysis but works differently: It finds an initial weightmatrix where w_ij(X^0)=0 if d_ij(X^0)>tau and fits a POST-MDS with these weights. Then in each successive iteration step, the weightmat is recalculated so that w_ij(X^(n+1))=0 if d_ij(X^(n+1))>tau.

Usage

spmds(
  delta,
  lambda = 1,
  kappa = 1,
  nu = 1,
  tau,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2,
  traceIt = FALSE
)

smds(
  delta,
  tau,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  traceIt = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_spmds(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  tau = max(delta),
  epochs = 10,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_smds(
  delta,
  tau = max(delta),
  epochs = 10,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eCLCA(
  delta,
  tau,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  traceIt = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eCLPCA(
  delta,
  lambda = 1,
  kappa = 1,
  nu = 1,
  tau,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2,
  traceIt = FALSE
)

so_eCLPCA(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  tau = max(delta),
  epochs = 10,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_eCLCA(
  delta,
  tau = max(delta),
  epochs = 10,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eclca(
  delta,
  tau,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  traceIt = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

eclpca(
  delta,
  lambda = 1,
  kappa = 1,
  nu = 1,
  tau,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2,
  traceIt = FALSE
)

so_eclca(
  delta,
  tau = max(delta),
  epochs = 10,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

so_eclpca(
  delta,
  kappa = 1,
  lambda = 1,
  nu = 1,
  tau = max(delta),
  epochs = 10,
  type = "ratio",
  ties = "primary",
  weightmat = 1 - diag(nrow(delta)),
  init = NULL,
  ndim = 2,
  acc = 1e-08,
  itmax = 10000,
  verbose = FALSE,
  principal = FALSE,
  spline.degree = 2,
  spline.intKnots = 2
)

Arguments

Details

There are a wrappers 'smds' and 'eCLCA' where the exponents are 1. The neighborhood parameter tau is kept fixed in 'spmds', 'smds', 'eCLCA' and 'eCLPCA'. The functions 'so_spmds', 'so_eCLPCA' and 'so_smds', 'so_eCLCA' implement a self-organising principle, where the model is repeatedly fitted for a decreasing sequence of taus.

The solution is found by "quasi-majorization", which means that the majorization is only real majorization once the weightmat no longer changes. This typically happens after a few iterations. Due to that it can be that in the beginning the stress may not decrease monotonically and that there's a chance it might never.

If tau is too small it may happen that all distances for one i to all j are zero and then there will be an error, so make sure to set a larger tau.

In the standard functions 'spmds' and 'smds' we keep tau fixed throughout. This means that if tau is large enough, then the result is the same as the corresponding MDS. In the orginal publication the idea was that of a self-organizing map which decreased tau over epochs (i.e., passes through the data). This can be achieved with our function 'so_spmds' 'so_smds' which creates a vector of decreasing tau values, calls the function 'spmds' with the first tau, then supplies the optimal configuration obtained as the init for the next call with the next tau and so on.

Value

a 'smacofP' object (inheriting from 'smacofB', see smacofSym). It is a list with the components

• delta: Observed, untransformed dissimilarities

• tdelta: Observed explicitly transformed dissimilarities, normalized

• dhat: Explicitly transformed dissimilarities (dhats), optimally scaled and normalized

• confdist: Transformed configuration distances

• conf: Matrix of fitted configuration

• stress: Default stress (stress 1; sqrt of explicitly normalized stress)

• spp: Stress per point

• ndim: Number of dimensions

• model: Name of smacof model

• niter: Number of iterations

• nobj: Number of objects

• type: Type of MDS model

• weightmat: weighting matrix as supplied

• stress.m: Default stress (stress-1^2)

• tweightmat: transformed weighting matrix; it is weightmat but containing all the 0s for the distances set to 0.

• trace: if 'traceIt=TRUE' a vector with the iteration progress

Examples

dis<-smacof::morse
res<-spmds(dis,type="interval",kappa=2,lambda=2,tau=0.3,itmax=100) #use higher itmax
res2<-smds(dis,type="interval",tau=0.3,itmax=500,traceIt=TRUE) #use higher itmax
#Aliases
resa<-eCLPCA(dis,type="interval",kappa=2,lambda=2,tau=0.3,itmax=100) #use higher itmax
res2a<-eCLCA(dis,type="interval",tau=0.3,itmax=500,traceIt=TRUE) #use higher itmax

res
res2
summary(res)
oldpar<-par(mfrow=c(1,2))
plot(res)
plot(res2)
par(oldpar)

##which d_{ij}(X)^kappa exceeded tau at convergence (i.e., have been set to 0)?
res$tweightmat
res2$tweightmat

# We use Quasi-Majorization
res2$trace



## Self-organizing map style (as in the clca publication)
#run the som-style (p)smds 
sommod1<-so_spmds(dis,tau=1,kappa=0.5,lambda=2,epochs=10,verbose=1)
sommod2<-so_smds(dis,tau=1,epochs=10,verbose=1)
sommod1
sommod2

Calculating stress per point

Description

Calculating stress per point

Usage

spp(dhat, confdist, weightmat)

Arguments

Value

a list

Squared distances

Description

Squared distances

Usage

sqdist(x)

Arguments

Value

squared distance matrix

Noisy Data on a U Fold

Description

Artifical data of data on a 3D U fold with noise that increases towards the edges. The first three columns are the coordinates and the last column is a color designation (viridis palette).

Format

A 500 x 4 matrix.