,
Jan de Leeuw [aut],
Lisha Chen [aut],
Patrick Mair [aut] | Title: | Flexible Multidimensional Scaling and 'smacof' Extensions |
|---|---|
| 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and classes for fitting, plotting and displaying a large number of different flexible MDS models (some as of yet unpublished). 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 and non-metric 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>), non-linear MDS with optimal dissimilarity powers functions (De Leeuw, 2024, <https://github.com/deleeuw/smacofManual/blob/main/smacofPO/smacofPO.pdf>), sparsified (power) MDS and sparsified multidimensional (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 or non-metric optimal scaling of the input proximities. Most functions use a Majorization-Minimization algorithm. Currently the methods are only available for one-mode data (symmetric dissimilarity matrices). |
| Authors: | Thomas Rusch [aut, cre] ,
Jan de Leeuw [aut],
Lisha Chen [aut],
Patrick Mair [aut] |
| Maintainer: | Thomas Rusch <[email protected]> |
| License: | GPL-2 | GPL-3 |
| Version: | 1.71-1 |
| Built: | 2024-10-05 13:26:27 UTC |
| Source: | https://github.com/r-forge/stops |
An implementation to minimize s-stress by majorization with ratio and interval optimal scaling.
alscal( delta, type = "ratio", weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )alscal( delta, type = "ratio", weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
type |
what type of MDS to fit. Currently one of "ratio" or "interval". Default is "ratio". |
weightmat |
a matrix of finite weights |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If ‘TRUE’, principal axis transformation is applied to the final configuration |
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)
dis<-smacof::kinshipdelta res<-alscal(as.matrix(dis),type="interval",itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-alscal(as.matrix(dis),type="interval",itmax=1000) res summary(res) plot(res)
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.
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 )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 )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
kappa |
power of the transformation of the fitted distances; defaults to 1 |
lambda |
the power of the transformation of the proximities; defaults to 1 |
nu |
the power of the transformation for weightmat; defaults to 1 |
type |
what type of MDS to fit. Only "ratio" currently. |
weightmat |
a binary matrix of finite nonegative weights. |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
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)
Internally we calculate the approximation parameters upsilon=nu+2*lambda*(1-(1/kappa)) and tau=lambda/kappa. They are not output.
Rusch, Mair, Hornik (2021). Cluster Optimized Proximity Scaling. JCGS <doi:10.1080/10618600.2020.1869027>
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")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")
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).
A 69 x 70 matrix.
data(bankingCrises) in library(Ecdat)
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 and for the repulsion part it is with BC being the one-parameter Box-Cox transformation.
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 ) 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 ) 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 ) 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 )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 ) 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 ) 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 ) 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 )
delta |
dissimilarity or distance matrix, dissimilarity or distance data frame or 'dist' object |
mu |
mu parameter. Should be 0 or larger for everything working ok. If mu<0 it works but I find the MDS model is strange and normalized stress tends towards 0 regardless of fit. Use normalized stress at your own risk in that case. |
lambda |
lambda parameter. Must be larger than 0. |
rho |
the rho parameter, power for the weights (called nu in the original article). |
type |
what type of MDS to fit. Only "ratio" currently. |
ndim |
the dimension of the configuration |
weightmat |
a matrix of finite weights. Not implemented. |
itmax |
number of optimizing iterations, defaults to 2000. |
init |
initial configuration. If NULL a classical scaling solution is used. |
verbose |
prints progress if > 3. |
addD0 |
a small number that's added for D(X)=0 for numerical evaluation of worst fit (numerical reasons, see details). If addD0=0 the normalized stress for mu!=0 and mu+lambda!=0 is correct, but will give useless normalized stress for mu=0 or mu+lambda!=0. |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
normconf |
normalize the configuration to sum(delta^2)=1 (as in the power stresses). Default is FALSE. Note that then the distances in confdist do not match manually calculated ones. |
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)).
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)
Lisha Chen & Thomas Rusch
dis<-smacof::kinshipdelta res<-bcmds(dis,mu=2,lambda=1.5,rho=0) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-bcmds(dis,mu=2,lambda=1.5,rho=0) res summary(res) plot(res)
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.
bcsdistance(x, a = 0.5, b = 1 - a)bcsdistance(x, a = 0.5, b = 1 - a)
x |
an n times p numeric matrix or data frame. Note that the valeus of x must be non-negative. |
a |
first blending weight. Must be non-negative and should be in [0,1] if a blended chi-square distance as in Lindsay (1994) is sought. Defaults to 0.5. |
b |
second blending weight. Must be non-negative and should be 1-a if a blended chi-square distance as in Lindsay (1994) is sought. Defaults to 1-a. |
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.
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 smacofP objects
## S3 method for class 'smacofP' biplotmds(object, extvar, scale = TRUE)## S3 method for class 'smacofP' biplotmds(object, extvar, scale = TRUE)
object |
An object of class smacofP |
extvar |
Data frame with external variables. |
scale |
if 'TRUE' external variables are standardized internally. |
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.
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.
## 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)## 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)
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.
## S3 method for class 'smacofP' bootmds( object, data, method.dat = "pearson", nrep = 100, alpha = 0.05, verbose = FALSE, ... )## S3 method for class 'smacofP' bootmds( object, data, method.dat = "pearson", nrep = 100, alpha = 0.05, verbose = FALSE, ... )
object |
Object of class smacofP if used as method or another object inheriting from smacofB (needs to be called directly as bootmds.smacofP then). |
data |
Initial data (before dissimilarity computation). |
method.dat |
Dissimilarity computation used as MDS input. This must be one of "pearson", "spearman", "kendall", "euclidean", "maximum", "manhattan", "canberra", "binary". |
nrep |
Number of bootstrap replications. |
alpha |
Alpha level for condfidence ellipsoids. |
verbose |
If 'TRUE', bootstrap index is printed out. |
... |
Additional arguments needed for dissimilarity computation as specified in |
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.
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
##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##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
Classical Scaling
cmds(Do)cmds(Do)
Do |
dissimilarity matrix |
cmdscale for S3 classWrapper to cmdscale for S3 class
cmdscale(d, k = 2, eig = FALSE, ...)cmdscale(d, k = 2, eig = FALSE, ...)
d |
a distance structure such as that returned by 'dist' or a full symmetric matrix containing the dissimilarities |
k |
the maximum dimension of the space which the data are to be represented in |
eig |
indicates whether eigenvalues should be returned. Defaults to TRUE. |
... |
additional parameters passed to cmdscale. See |
overloads stats::cmdscale turns on the liosting and adds slots and class attributes for which there are methods.
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'.
dis<-as.matrix(smacof::kinshipdelta) res<-cmdscale(dis)dis<-as.matrix(smacof::kinshipdelta) res<-cmdscale(dis)
conf_adjust: a function to procrustes adjust two matrices
conf_adjust(conf1, conf2, verbose = FALSE, eps = 1e-12, itmax = 100)conf_adjust(conf1, conf2, verbose = FALSE, eps = 1e-12, itmax = 100)
conf1 |
reference configuration, a numeric matrix |
conf2 |
another configuration, a numeric matrix |
verbose |
should adjustment be output; default to FALSE |
eps |
numerical accuracy |
itmax |
maximum number of iterations |
a list with 'ref.conf' being the reference configuration, 'other.conf' the adjusted coniguration and 'comparison.conf' the comparison configuration
Double centering of a matrix
doubleCenter(x)doubleCenter(x)
x |
numeric matrix |
the double centered matrix
An implementation to minimize elastic scaling stress by majorization with ratio and interval optimal scaling. Uses a repeat loop.
elscal( delta, type = c("ratio", "interval"), weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )elscal( delta, type = c("ratio", "interval"), weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
type |
what type of MDS to fit. Currently one of "ratio" and "interval". Default is "ratio". |
weightmat |
a matrix of finite weights |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
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)
dis<-smacof::kinshipdelta res<-elscal(as.matrix(dis),itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-elscal(as.matrix(dis),itmax=1000) res summary(res) plot(res)
Explicit Normalization Normalizes distances
enorm(x, w = 1)enorm(x, w = 1)
x |
numeric matrix |
w |
weight |
a constant
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.
icExploreGen( object, mdscall = NULL, conflist, nrep = 100, ndim, returnfit = FALSE, min = -5, max = 5, verbose = FALSE )icExploreGen( object, mdscall = NULL, conflist, nrep = 100, ndim, returnfit = FALSE, min = -5, max = 5, verbose = FALSE )
object |
A fitted object of class 'smacofP', 'smacofB' or 'smacof'. If supplied this takes precedence over the call argument. If given this is added to the output and may be the optimal one. |
mdscall |
Alternatively to a fitted object, one can pass a syntactically valid call for any of the MDS functions cops, stops or smacof that find a configuration (not the ones that do parameter selection like pcops or stops). If object and call is given, object takes precedence. |
conflist |
Optional list of starting configurations. |
nrep |
If conflist is not supplied, how many random starting configurations should be used. |
ndim |
Dimensions of target space. |
returnfit |
Should all fitted MDS be returned. If FALSE (default) none is returned. |
min |
lower bound for the uniform distribution to sample from |
max |
upper bound for the uniform distribution to sample from |
verbose |
If >0 prints the fitting progress. |
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.
an object of class 'icexplore', see icExplore for more. There is a plot method in package 'smacof'.
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$init,1)) c1 <- call("smds",delta=dis,tau=0.2,itmax=100) resm<-icExploreGen(mdscall=c1,conflist=conflist,returnfit=TRUE) plot(resm)dis<-kinshipdelta ## Version 1: Using a fitted object (recommended) res1<-rStressMin(dis,type="ordinal",itmax=100) resm<-icExploreGen(res1,nrep=5) ## Version 2: Using a call object and supplying conflist conflist<-list(res1$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)
These functions perform an MDS Jackknife and plot the corresponding solution.
## S3 method for class 'smacofP' jackmds(object, eps = 1e-06, itmax = 100, verbose = FALSE)## S3 method for class 'smacofP' jackmds(object, eps = 1e-06, itmax = 100, verbose = FALSE)
object |
Object of class smacofP if used as method or another object inheriting from smacofB (needs to be called directly as jackmds.smacofP then). |
eps |
Convergence criterion |
itmax |
Maximum number of iterations |
verbose |
If 'TRUE', intermediate stress is printed out. |
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.
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
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)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)
This function minimizes the Local MDS Stress of Chen & Buja (2006) via gradient descent. This is a ratio metric scaling method.
lmds( delta, k = 2, tau = 1, type = "ratio", ndim = 2, weightmat = 1 - diag(nrow(delta)), itmax = 5000, init = NULL, verbose = 0, principal = FALSE, normconf = FALSE )lmds( delta, k = 2, tau = 1, type = "ratio", ndim = 2, weightmat = 1 - diag(nrow(delta)), itmax = 5000, init = NULL, verbose = 0, principal = FALSE, normconf = FALSE )
delta |
dissimilarity or distance matrix, dissimilarity or distance data frame or 'dist' object |
k |
the k neighbourhood parameter |
tau |
the penalty parameter (suggested to be in [0,1]) |
type |
what type of MDS to fit. Only "ratio" currently. |
ndim |
the dimension of the configuration |
weightmat |
a matrix of finite weights. Not implemented. |
itmax |
number of optimizing iterations, defaults to 5000. |
init |
initial configuration. If NULL a classical scaling solution is used. |
verbose |
prints info if > 0 and progress if > 1. |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
normconf |
normalize the configuration to sum(delta^2)=1 (as in the power stresses). Note that then the distances in confdist do not match the manually calculated ones. |
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.
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
Lisha Chen & Thomas Rusch
dis<-smacof::kinshipdelta res<- lmds(dis,k=2,tau=0.1) res summary(res) plot(res)dis<-smacof::kinshipdelta res<- lmds(dis,k=2,tau=0.1) res summary(res) plot(res)
Take matrix to a power
mkPower(x, r)mkPower(x, r)
x |
matrix |
r |
numeric (power) |
a matrix
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.
multiscale( delta, type = c("ratio", "interval"), weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, kappa = 0.01, principal = FALSE )multiscale( delta, type = c("ratio", "interval"), weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, kappa = 0.01, principal = FALSE )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances. Warning: these will get transformed to the log scale, so make sure that log(delta)>=0. |
type |
what optimal scaling type of MDS to fit. Currently one of "ratio" or "interval". Default is "ratio". |
weightmat |
a matrix of finite weights |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
kappa |
As this is not exactly multiscale but an r-stress approximation, we have multiscale only for kappa->0. This argument can therefore be used to make the approximation more accurate by making it smaller. Default is 0.1. |
principal |
If ‘TRUE’, principal axis transformation is applied to the final configuration |
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)
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.
dis<-smacof::kinshipdelta res<-multiscale(as.matrix(dis),type="interval",itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-multiscale(as.matrix(dis),type="interval",itmax=1000) res summary(res) plot(res)
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.
multistart( object, mdscall = NULL, ndim = 2, conflist, nstarts = 108, return.all = FALSE, verbose = TRUE, min = -5, max = 5 )multistart( object, mdscall = NULL, ndim = 2, conflist, nstarts = 108, return.all = FALSE, verbose = TRUE, min = -5, max = 5 )
object |
A fitted object of class 'smacofP', 'smacofB' or 'smacof'. If supplied this takes precedence over the call argument. If given this is added to the output and may be the optimal one. |
mdscall |
Alternatively to a fitted object, one can pass a syntactically valid call for any of the MDS functions cops, stops or smacof that find a configuration (not the ones that do parameter selection like pcops or stops). If object and call is given, object takes precedence. |
ndim |
Dimensions of target space. |
conflist |
Optional list of starting configurations. |
nstarts |
If conflist is not supplied, how many random starting configurations should be used. The default is 108, which implies that at least one of the stress is within the lowest 1 percent of all stresses with probability of 1/3 or within the lowest 5 percent of stresses with probability 0.996 |
return.all |
Should all fitted MDS be returned. If FALSE (default) only the optimal one is returned. |
verbose |
If >0 prints the fitting progress. |
min |
lower bound for the uniform distribution to sample from |
max |
upper bound for the uniform distribution to sample from |
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 $best and save that and work with it right away.
if 'return.all=FALSE', a list with the best fitted model as '$best' (minimal badness-of-fit of all fitted models) and '$stressvec' the stresses of all models. If 'return.all=TRUE' a list with slots
best: The object resulting from the fit that had the overall lowest objective function value (usually stress)
stressvec: The vector of objective function values
models: A list of all the fitted objects.
dis<-smacof::kinshipdelta ## Version 1: Using a fitted object (recommended) res1<-rStressMin(delta=dis,type="ordinal",itmax=100) resm<-multistart(res1,nstarts=2) ## best model res2<-resm$best #it's starting configuration res2$init ## Version 2: Using a call object and supplying conflist conflist<-list(res2$init,jitter(res2$init,1)) c1 <- call("rstressMin",delta=dis,type="ordinal",itmax=100) resm<-multistart(mdscall=c1,conflist=conflist,return.all=TRUE)dis<-smacof::kinshipdelta ## Version 1: Using a fitted object (recommended) res1<-rStressMin(delta=dis,type="ordinal",itmax=100) resm<-multistart(res1,nstarts=2) ## best model res2<-resm$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)
Squared p-distances
pdist(x, p)pdist(x, p)
x |
numeric matrix |
p |
p>0 the Minkoswki distance |
squared Minkowski distance matrix
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.
## S3 method for class 'smacofP' permtest( object, data, method.dat = "pearson", nrep = 100, verbose = FALSE, ... )## S3 method for class 'smacofP' permtest( object, data, method.dat = "pearson", nrep = 100, verbose = FALSE, ... )
object |
Object of class smacofP if used as method or another object inheriting from smacof (needs to be called directly as permtest.smacofP then). |
data |
Optional: Initial data; if provided permutations are performed on the data matrix (see details) |
method.dat |
Dissimilarity computation used as MDS input. This must be one of "pearson", "spearman", "kendall", "euclidean", "maximum", "manhattan", "canberra", "binary". If data is provided, then this must be provided as well. |
nrep |
Number of permutations. |
verbose |
If TRUE, bootstrap index is printed out. |
... |
Additional arguments needed for dissimilarity computation as specified in |
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.
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
##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")##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
## 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, ... )## 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, ... )
x |
an object of class smacofP |
plot.type |
String indicating which type of plot to be produced: "confplot", "resplot", "Shepard", "stressplot","transplot", "bubbleplot" (see details) |
plot.dim |
dimensions to be plotted in confplot; defaults to c(1, 2) |
bubscale |
Scaling factor (size) for the bubble plot |
col |
vector of colors for the points |
label.conf |
List with arguments for plotting the labels of the configurations in a configuration plot (logical value whether to plot labels or not, label position, label color) |
hull.conf |
Option to add convex hulls to a configuration plot. Hull index needs to be provided. |
shepard.x |
Shepard plot only: original data (e.g. correlation matrix) can be provided for plotting on x-axis |
identify |
If 'TRUE', the 'identify()' function is called internally that allows to add configuration labels by mouse click |
type |
What type of plot should be drawn (see also 'plot') |
cex |
Symbol size. |
pch |
Plot symbol |
asp |
Aspect ratio; defaults to 1 so distances between x and y are represented accurately; can lead to slighlty weird looking plots if the variance on one axis is much smaller than on the other axis; use NA if the standard type of R plot is wanted where the ylim and xlim arguments define the aspect ratio - but then the distances seen are no longer accurate |
main |
plot title |
xlab |
label of x axis |
ylab |
label of y axis |
xlim |
scale of x axis |
ylim |
scale of y axis |
col.hist |
Color of the borders of the histogram. |
legend |
Flag whether legends should be drawn for plots that have legends |
legpos |
Position of legend in plots with legends |
loess |
if TRUE a loess fit (by Tukey's rescending M-Estimator) of configuration distances explained by delta is added to the Shepard plot |
shepard.lin |
Shepard plot only: if TRUE the Shepard plot is linearized so d^kappa~delta^lambda. If FALSE d~delta^lambda |
... |
Further plot arguments passed: see 'plot.smacof' and 'plot' for detailed information. |
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’.
no return value; just plots for class 'smacofP' (see details)
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")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")
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.
powerStressFast( delta, kappa = 1, lambda = 1, nu = 1, weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE )powerStressFast( delta, kappa = 1, lambda = 1, nu = 1, weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
kappa |
power of the transformation of the fitted distances; defaults to 1 |
lambda |
the power of the transformation of the proximities; defaults to 1 |
nu |
the power of the transformation for weightmat; defaults to 1 |
weightmat |
a matrix of finite weights |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
The smallest value of the trust region radius that is allowed. If not defined, then 1e-6 will be used. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
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
dis<-smacof::kinshipdelta res<-powerStressFast(as.matrix(dis),kappa=2,lambda=1.5) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-powerStressFast(as.matrix(dis),kappa=2,lambda=1.5) res summary(res) plot(res)
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.
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 )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 )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
kappa |
power of the transformation of the fitted distances; defaults to 1 |
lambda |
the power of the transformation of the proximities; defaults to 1 |
nu |
the power of the transformation for weightmat; defaults to 1 |
type |
what type of MDS to fit. One of "ratio" or "interval". Default is "ratio". |
weightmat |
a matrix of finite weights or dist object |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
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)
dis<-smacof::kinshipdelta res<-powerStressMin(dis,type="ratio",kappa=2,lambda=1.5,itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-powerStressMin(dis,type="ratio",kappa=2,lambda=1.5,itmax=1000) res summary(res) plot(res)
procruster: a procrustes function
procruster(x)procruster(x)
x |
numeric matrix |
a matrix
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.
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 )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 )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
expo |
power of the transformation of the fitted distances and dissimilarities; defaults to 1 |
nu |
the power of the transformation for weightmat; defaults to 1 |
type |
what type of MDS to fit. One of "ratio" or "interval". Default is "ratio". |
weightmat |
a matrix of finite weights |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
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
dis<-smacof::kinshipdelta res<-rpowerStressMin(as.matrix(dis),expo=1.7,itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-rpowerStressMin(as.matrix(dis),expo=1.7,itmax=1000) res summary(res) plot(res)
An implementation to minimize r-stress by majorization with ratio, interval and ordinal optimal scaling. Uses a repeat loop.
rStressMin( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE ) rstressMin( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE ) rstressmds( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE ) rstress( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )rStressMin( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE ) rstressMin( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE ) rstressmds( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE ) rstress( delta, r = 0.5, type = c("ratio", "interval", "ordinal"), ties = "primary", weightmat = 1 - diag(nrow(delta)), init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
r |
power of the transformation of the fitted distances (corresponds to kappa/2 in power stress); defaults to 0.5 for standard stress |
type |
what type of MDS to fit. Currently one of "ratio", "interval" or "ordinal". Default is "ratio". |
ties |
the handling of ties for ordinal (nonmetric) MDS. Possible are "primary" (default), "secondary" or "tertiary". |
weightmat |
a matrix of finite weights. |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
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 weighting matrix (here NULL)
dis<-smacof::kinshipdelta res<-rStressMin(as.matrix(dis),type="ordinal",r=1,itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-rStressMin(as.matrix(dis),type="ordinal",r=1,itmax=1000) res summary(res) plot(res)
sammon for S3 classWrapper to sammon for S3 class
sammon(d, y = NULL, k = 2, ...)sammon(d, y = NULL, k = 2, ...)
d |
a distance structure such as that returned by 'dist' or a full symmetric matrix. Data are assumed to be dissimilarities or relative distances, but must be positive except for self-distance. This can contain missing values. |
y |
An initial configuration. If NULL, |
k |
The dimension of the configuration |
... |
Additional parameters passed to |
Overloads MASS::sammon and adds new slots and class attributes for which there are methods.
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'.
dis<-as.matrix(smacof::kinshipdelta) res<-sammon(dis)dis<-as.matrix(smacof::kinshipdelta) res<-sammon(dis)
An implementation to minimize Sammon stress by majorization with ratio and interval optimal scaling. Uses a repeat loop.
sammonmap( delta, type = c("ratio", "interval"), weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )sammonmap( delta, type = c("ratio", "interval"), weightmat, init = NULL, ndim = 2, acc = 1e-06, itmax = 10000, verbose = FALSE, principal = FALSE )
delta |
dist object or a symmetric, numeric data.frame or matrix of distances |
type |
what type of MDS to fit. Currently one of "ratio" and "interval". Default is "ratio". |
weightmat |
a matrix of finite weights |
init |
starting configuration |
ndim |
dimension of the configuration; defaults to 2 |
acc |
numeric accuracy of the iteration. Default is 1e-6. |
itmax |
maximum number of iterations. Default is 10000. |
verbose |
should iteration output be printed; if > 1 then yes |
principal |
If 'TRUE', principal axis transformation is applied to the final configuration |
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)
dis<-smacof::kinshipdelta res<-sammonmap(as.matrix(dis),itmax=1000) res summary(res) plot(res)dis<-smacof::kinshipdelta res<-sammonmap(as.matrix(dis),itmax=1000) res summary(res) plot(res)
Adjusts a configuration
scale_adjust(conf, ref, scale = c("sd", "std", "proc", "none"))scale_adjust(conf, ref, scale = c("sd", "std", "proc", "none"))
conf |
a configuration |
ref |
a reference configuration (only for scale="proc") |
scale |
Scale adjustment. "std" standardizes each column of the configurations to mean=0 and sd=1, "sd" scales the configuration by the maximum standard devation of any column, "proc" adjusts the fitted configuration to the reference |
The scale adjusted configuration.
Secular Equation
secularEq(a, b)secularEq(a, b)
a |
matrix |
b |
matrix |
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, 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.
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, interval, ordinal optimal scaling
pclca ... power curvilinear component analysis with ratio, interval, ordinal optimal scaling
bcmds ... Box-Cox MDS with ratio optimal scaling
lmds... Local MDS with ratio optimal scaling
sammonmap... Sammon mapping with ratio, interval optimal scaling
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 and 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
Authors: Thomas Rusch, Jan de Leeuw, Lisha Chen, Patrick Mair
Maintainer: Thomas Rusch
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)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)
Calculating stress per point
spp(dhat, confdist, weightmat)spp(dhat, confdist, weightmat)
dhat |
a dist object or symmetric matrix of dissimilarities |
confdist |
a dist object or symmetric matrix of fitted distances |
weightmat |
dist objetc or symmetric matrix of weights |
a list
Squared distances
sqdist(x)sqdist(x)
x |
numeric matrix |
squared distance matrix