Type:	Package
Title:	Multidimensional Scaling Development Kit
Version:	0.1.5
Author:	Frank M.T.A. Busing [aut, cre], Juan Claramunt Gonzalez [aut, com]
Maintainer:	Frank M.T.A. Busing <busing@fsw.leidenuniv.nl>
Description:	Multidimensional scaling (MDS) functions for various tasks that are beyond the beta stage and way past the alpha stage. Currently, options are available for weights, restrictions, classical scaling or principal coordinate analysis, transformations (linear, power, Box-Cox, spline, ordinal), outlier mitigation (rdop), out-of-sample estimation (predict), negative dissimilarities, fast and faster executions with low memory footprints, penalized restrictions, cross-validation-based penalty selection, supplementary variable estimation (explain), additive constant estimation, mixed measurement level distance calculation, restricted classical scaling, etc. More will come in the future. References. Busing (2024) "A Simple Population Size Estimator for Local Minima Applied to Multidimensional Scaling". Manuscript submitted for publication. Busing (2025) "Node Localization by Multidimensional Scaling with Iterative Majorization". Manuscript submitted for publication. Busing (2025) "Faster Multidimensional Scaling". Manuscript in preparation. Barroso and Busing (2025) "e-RDOP, Relative Density-Based Outlier Probabilities, Extended to Proximity Mapping". Manuscript submitted for publication.
Depends:	R (≥ 3.5.0)
Imports:	graphics, stats
Suggests:	ggplot2, microbenchmark
License:	BSD_2_clause + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.2
LazyData:	true
NeedsCompilation:	yes
Packaged:	2025-06-02 13:14:03 UTC; prueba
Repository:	CRAN
Date/Publication:	2025-06-03 07:00:01 UTC

Additive Constant Function for Classical Multidimensional Scaling

Description

addconst returns the smallest additive constant which, added to the dissimilarities, makes the data true Euclidean distances. Note: NA's are not allowed.

Usage

addconst(delta, faster = FALSE, error.check = FALSE)

Arguments

Value

additive constant

Author(s)

Frank M.T.A. Busing

References

Cailliez (1983)

Asymmetry Function

Description

asymmetry return statistics from asymmetry analyses.

Usage

asymmetry(delta, z, error.check = FALSE, echo = FALSE)

Arguments

Value

ssaf skew-symmetric sum-of-squares-accounted-for

vaf skew-symmetric variance-accounted-for

drifts drift vectors starting from coordinates in z (n by p matrix)

radii n vector with circle radii

Author(s)

Frank M.T.A. Busing

References

Gower (1968).

Color data

Description

Dissimilarities are one minus the original ratings. 31 subjects rated 91 combinations for similarity on a 5-point scale (0-4). The ratings were averaged and divided by 4 to obtain similarities between 0 and 1. The 14 colors had wavelengths 434, 445, 465, 472, 490, 504, 537, 555, 54, 600, 610, 628, 651, and 674.

Usage

colors

Format

14 x 14 dissimilarity matrix

• V1: dissimilarities for V1.

• V2: dissimilarities for V2.

• V3: dissimilarities for V3.

• V4: dissimilarities for V4.

• V5: dissimilarities for V5.

• V6: dissimilarities for V6.

• V7: dissimilarities for V7.

• V8: dissimilarities for V8.

• V9: dissimilarities for V9.

• V10: dissimilarities for V10.

• V11: dissimilarities for V11.

• V12: dissimilarities for V12.

• V13: dissimilarities for V13.

• V14: dissimilarities for V14.

References

Ekman (1954). Dimensions of color vision. Journal of Psychology, 38, 467-474.

Repeated Cross-Validation Penalized Restricted Multidimensional Scaling Function

Description

cv.fastmds performs repeated cross-validation for a penalized restricted multidimensional scaling model.

Usage

cv.fastmds(
  delta,
  w = NULL,
  p = 2,
  q = NULL,
  b = NULL,
  lambda = 0,
  alpha = 1,
  grouped = FALSE,
  NFOLDS = 10,
  NREPEATS = 30,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

mserrors mean squared errors for different values of lambda.

stderrors standard errors for mean squared errors.

varnames labels of independent row variables.

coefficients list with final h by p matrices with regression coefficients (lambda order).

lambda sorted regularization penalty parameters.

alpha elastic-net parameter (default = 1.0: lasso only).

grouped boolean for lasso penalty (default = FALSE: ordinary lasso).

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser,W. J. (1987a). Joint ordination of species and sites: The unfolding technique. In P. Legendre and L. Legendre (Eds.), Developments in numerical ecology (pp. 189–221). Berlin, Heidelberg: Springer-Verlag.

Busing, F.M.T.A. (2010). Advances in multidimensional unfolding. Unpublished doctoral dissertation, Leiden University, Leiden, the Netherlands.

Dilation

Description

dilation returns dilation or scale factor, such that || factor * source - target ||^2_weights is minimal.

Usage

dilation(source, weights = NULL, target = NULL, error.check = FALSE)

Arguments

Value

dilation factor

Author(s)

Frank M.T.A. Busing

References

Gower (1968). Commandeur (1991)

Explain method for all fmds objects

Description

explain fits variables to coordinate configuration

Usage

## S3 method for class 'fmds'
explain(
  x,
  q,
  w = NULL,
  level = c("absolute", "ratio", "linear", "ordinal", "nominal"),
  MAXITER = 1024,
  FCRIT = 1e-08,
  error.check = FALSE,
  echo = FALSE,
  ...
)

Arguments

Value

data data

weights weights

transformed.data transformed data

approach approach

degree zero

ninner zero

iknots NULL

anchor anchor

knotstype zero

coordinates NULL

coefficients coefficients

distances distances

last.iteration last.iteration

last.difference last.difference

n.stress n.stress

stress.1 stress.1

call call

Facial Expressions Data

Description

Dissimilarities represent the correspondence in facial expressions. 30 students rated the dissimilarity between 13 female portraits (photographs) on a 9-point scale. The dissimilarities are the means of the re-scaled values obtained by the method of successive intervals.

Usage

expressions

Format

13 x 16 matrix. The first 13 x 13 matrix is a dissimilarity matrix

• V1: dissimilarities for V1.

• V2: dissimilarities for V2.

• V3: dissimilarities for V3.

• V4: dissimilarities for V4.

• V5: dissimilarities for V5.

• V6: dissimilarities for V6.

• V7: dissimilarities for V7.

• V8: dissimilarities for V8.

• V9: dissimilarities for V9.

• V10: dissimilarities for V10.

• V11: dissimilarities for V11.

• V12: dissimilarities for V12.

• V13: dissimilarities for V13.

• P1: Property 1.

• P2: Property 2.

• P3: Property 3.

References

Abelson and Sermat (1962). Multidimensional scaling of facial expressions. Journal of experimental psychology, 63(6), 546-554. Diederich, Messick, and Tucker (1957). A general least squares solution for successive intervals. Psychometrika, 22(2), 159-173. Woodworth (1938). Experimental psychology. New York, Holt. Engen, Levy, and Schlosberg (1958). The dimensional analysis of a new series of facial expressions. Journal of Experimental Psychology, 55(5), 454-458.

Box-Cox Multidimensional Scaling Function

Description

fastlinearmds performs Box-Cox multidimensional scaling. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables. The dissimilarities are optimally Box-Cox transformed.

Usage

fastboxcoxmds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

faster if a faster procedure has been used.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Stochastic Iterative Majorization Multidimensional Scaling Function

Description

fastermds performs multidimensional scaling using a stochastic iterative majorization algorithm. The data are either dissimilarities (full or only lower triangular part) or multivariate data. The dissimilarities and the weights may not contain negative values. The configuration is either unrestricted or (partly) fixed. Local multidimensional scaling is performed when a boundary is provided. Interval multidimensional scaling is performed with a full dissimilarity matrix, using the lower triangular part for the lower bound and the upper triangular part for the upper bound.

Usage

fastermds(
  delta = NULL,
  lower = NULL,
  data = NULL,
  w = NULL,
  p = 2,
  z = NULL,
  fixed = NULL,
  linear = FALSE,
  boundary = NULL,
  interval = FALSE,
  NCYCLES = 32,
  MINRATE = 0.01,
  error.check = FALSE,
  test = 0
)

Arguments

Details

One of the following three data formats need to be specified:

Value

n by p matrix with final coordinates.

Author(s)

Frank M.T.A. Busing

References

Agrafiotis, and others, and Busing

Examples

n <- 1000
m <- 10
delta <- as.matrix( dist( matrix( runif( n * m ), n, m ) ) )
p <- 2
zinit <- matrix( runif( n * p ), n, p )
# r <- fastermds( delta = delta, p = p, z = zinit, error.check = TRUE )

Faster Stress Function

Description

fasterstress calculates stochastic normalized stress. Neither data nor distances based on z are optimally scaled.

Usage

fasterstress(data = NULL, z = NULL, nsamples = 100, samplesize = 30)

Arguments

Value

n.stress normalized stress, mean over samples and observations

se standard error of se, standard deviation over samples

Author(s)

Frank M.T.A. Busing

References

agrafiotis, and others, and busing

Examples

n <- 10000
m <- 10
data <- matrix( runif( n * m ), n, m )
p <- 2
zinit <- matrix( runif( n * p ), n, p )
# r <- fastermds( data = data, p = p, z = zinit )
# s <- fasterstress( data = data, z = r )

Linear Multidimensional Scaling Function

Description

fastlinearmds performs linear multidimensional scaling. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables. The dissimilarities are optimally linearly transformed.

Usage

fastlinearmds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  anchor = TRUE,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

anchor whether an intervept was used or not.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

faster if a faster procedure has been used.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Examples

data( "colors" )
delta <- as.matrix( colors^3 )
n <- nrow( delta )
w <- 1 - diag( n )
p <- 2
z <- matrix( runif( n * p ), n, p )
#r <- fastlinearmds( delta, w, p, z, echo = TRUE )

Multidimensional Scaling Function

Description

fastmds performs multidimensional scaling. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables, penalized or not.

Usage

fastmds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  anchor = 0,
  lambda = 0,
  alpha = 1,
  grouped = FALSE,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

lambda (optimal) penalty parameter.

alpha elastic-net penalty parameter.

grouped common or grouped lasso penalty.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Examples

data( "colors" )
delta <- as.matrix( ( colors )^3 )
n <- nrow( delta )
w <- 1 - diag( n )
p <- 2
zinit <- matrix( runif( n * p ), n, p )
#r <- fastmds( delta, w, p, z = zinit, echo = TRUE )

Ordinal Multidimensional Scaling Function

Description

fastordinalmds performs ordinal multidimensional scaling. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables. The dissimilarities are optimally monotone transformed.

Usage

fastordinalmds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  approach = 1,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

approach apporach to ties: 1 = untie ties, 2 = keep ties tied.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

faster if a faster procedure has been used.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Power Multidimensional Scaling Function

Description

fastpowermds performs power multidimensional scaling. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables. The dissimilarities are optimally power transformed.

Usage

fastpowermds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

faster if a faster procedure has been used.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Polynomial Multidimensional Scaling Function

Description

fastpolynomialmds performs monotone polynomial multidimensional scaling. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables. The dissimilarities are optimally monotone transformed following a polynomial function.

Usage

fastsplinemds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  anchor = TRUE,
  ninner = 0,
  degree = 2,
  knotstype = c("none", "uniform", "percentile", "midpercentile"),
  iknots = NULL,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

anchor whether an intercept was used or not.

degree spline degree.

ninner number of interior knots.

knotstype type of procedure creating the interior knot sequence.

iknots interior knots sequence.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

faster if a faster procedure has been used.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Fast Stress Function

Description

faststress calculates value for normalized stress with different input parameters. Distances d are optimally scaled.

Usage

faststress(
  lower = NULL,
  delta = NULL,
  data = NULL,
  w = NULL,
  z = NULL,
  d = NULL
)

Arguments

Value

n.stress normalized stress value

Author(s)

Frank M.T.A. Busing

Food data

Description

38 subjects places 45 food items in categories based on similarity. The dissimilarities are the proportions of combinations NOT placed in the same category.

Usage

food

Format

45 x 45 dissimilarity matrix

• V1: dissimilarities for V1.

• V2: dissimilarities for V2.

• V3: dissimilarities for V3.

• V4: dissimilarities for V4.

• V5: dissimilarities for V5.

• V6: dissimilarities for V6.

• V7: dissimilarities for V7.

• V8: dissimilarities for V8.

• V9: dissimilarities for V9.

• V10: dissimilarities for V10.

• V11: dissimilarities for V11.

• V12: dissimilarities for V12.

• V13: dissimilarities for V13.

• V14: dissimilarities for V14.

• V15: dissimilarities for V15.

• V16: dissimilarities for V16.

• V17: dissimilarities for V17.

• V18: dissimilarities for V18.

• V19: dissimilarities for V19.

• V20: dissimilarities for V20.

• V21: dissimilarities for V21.

• V22: dissimilarities for V22.

• V23: dissimilarities for V23.

• V24: dissimilarities for V24.

• V25: dissimilarities for V25.

• V26: dissimilarities for V26.

• V27: dissimilarities for V27.

• V28: dissimilarities for V28.

• V29: dissimilarities for V29.

• V30: dissimilarities for V30.

• V31: dissimilarities for V31.

• V32: dissimilarities for V32.

• V33: dissimilarities for V33.

• V34: dissimilarities for V34.

• V35: dissimilarities for V35.

• V36: dissimilarities for V36.

• V37: dissimilarities for V37.

• V38: dissimilarities for V38.

• V39: dissimilarities for V39.

• V40: dissimilarities for V40.

• V41: dissimilarities for V41.

• V42: dissimilarities for V42.

• V43: dissimilarities for V43.

• V44: dissimilarities for V44.

• V45: dissimilarities for V45.

References

Ross and Murphy (1999). Food for thought: Cross-classification and category organization in a complex real-world domain. Cognitive psychology, 38(4), 495-553. Brusco and Stahl (2000). Using Quadratic Assignment Methods to Generate Initial Permutations for Least-Squares Unidimensional Scaling of Symmetric Proximity Matrices. Journal of Classification, 17(2).

Full Multidimensional Scaling Function

Description

fullmds performs a complete multidimensional scaling analysis. The function follows algorithms given by de Leeuw and Heiser (1980). The data, dissimilarities and weights, are either symmetric or asymmetric. The dissimilarities are may contain negative values, the weights may not. The configuration is either unrestricted, (partly) fixed, or a linear combination of independent variables. The dissimilarities may be transformed by different functions, with related parameters.

Usage

fullmds(
  delta,
  w = NULL,
  p = 2,
  z = NULL,
  r = NULL,
  b = NULL,
  level = c("none", "linear", "power", "box-cox", "spline", "ordinal"),
  anchor = TRUE,
  ninner = 0,
  degree = 2,
  knotstype = c("none", "uniform", "percentile", "midpercentile"),
  iknots = NULL,
  approach = 1,
  lambda = 0,
  alpha = 1,
  grouped = FALSE,
  MAXITER = 1024,
  FCRIT = 1e-08,
  ZCRIT = 1e-06,
  rotate = TRUE,
  faster = FALSE,
  error.check = FALSE,
  echo = FALSE
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

anchor whether an intercept was used or not.

degree spline degree.

ninner number of interior knots.

knotstype type of procedure creating the interior knot sequence.

iknots interior knots sequence.

approach apporach to ties: 1 = untie ties, 2 = keep ties tied.

coordinates final n by p matrix with coordinates.

restriction either the fixed coordinates or the independent variables.

coefficients final h by p matrix with regression coefficients.

lambda (optimal) penalty parameter.

alpha elastic-net penalty parameter.

grouped common or grouped lasso penalty.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

rotate if solution is rotated to principal axes.

faster if a faster procedure has been used.

Author(s)

Frank M.T.A. Busing

References

de Leeuw, J., and Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 5, pp. 501–522). Amsterdam, The Netherlands: North-Holland Publishing Company.

Heiser, W.J. (1991). A generalized majorization method for least squares multidimensional scaling of pseudo-distances that may be negative. Psychometrika, 55, pages 7-27.

Busing, F.M.T.A. (submitted). Node Localization by Multidimensional Scaling with Iterative Majorization: A Psychometric Perspective. Signal Processing, Elsevier.

Kabah data

Description

Data description

Usage

kabah

Format

17 x 17 dissimilarity matrix

Mixed Measurement Level Euclidean Distances Function

Description

fastmixed returns Euclidean distances for variables from mixed measurement levels.

Usage

mdist(
  data,
  level = rep("numeric", ncol(data)),
  scale = FALSE,
  error.check = FALSE
)

Arguments

Value

'dist' object with Euclidean distances between objects.

Author(s)

Frank M.T.A. Busing

References

Busing (2025). A Consistent Distance Measure for Mixed Data: Bridging the Gap between Euclidean and Chi-Squared Distances. Manuscript in progress.

Classical Multidimensional Scaling Function

Description

pcoa performs classical multidimensional scaling or principal coordinates analysis. The function uses an eigenvalue decomposition on a Gramm matrix. The data are supposed to be distances, but often dissimilarities will do fine. The data matrix contains nonnegative values, is square, symmetric, and hollow. NA's are not allowed. An additive constant may be provided, which is added to the dissimilarities. This constant might be obtained optimally with the function fastaddconst(). Error checking focuses on the data requirements.

Usage

pcoa(
  delta = NULL,
  lower = NULL,
  data = NULL,
  p = 2,
  k = NULL,
  ac = 0,
  q = NULL,
  faster = FALSE,
  error.check = FALSE
)

Arguments

Value

either n by p coordinates matrix (if q == NULL) or h by p coefficients matrix b (if q != NULL), in which case z = qb

Author(s)

Frank M.T.A. Busing

References

Young and Householder (1938) Torgerson (1952, 1958) Gower (1966) Carroll, Green, and Carmone (1976) De Leeuw and Heiser (1982) Ter Braak (1992) Borg and Groenen (2005)

Mark-Recapture Population Size Estimator

Description

petersen returns the estimated population size based on two independent equally-sized samples

Usage

petersen(first, second)

Arguments

Value

population size estimate

Author(s)

Frank M.T.A. Busing

References

Busing (2025). A Simple Population Size Estimator for Local Minima Applied to Multidimensional Scaling.

Visualisation of an fmds object

Description

Plot function for a fmds object. The plot shows the result of fmds.

Usage

## S3 method for class 'fmds'
plot(
  x,
  type = c("configuration", "transformation", "fit", "residuals", "shepard", "stress",
    "biplot", "dendrogram", "threshold", "neighbors"),
  markers = NULL,
  labels = NULL,
  ...
)

Arguments

Value

none

Predict method for all fmds objects

Description

predict provides locations for additional objects. Function is under construction.

Usage

## S3 method for class 'fmds'
predict(
  object,
  delta,
  w = NULL,
  level = c("absolute", "ratio", "linear", "ordinal"),
  MAXITER = 1024,
  FCRIT = 1e-08,
  error.check = FALSE,
  echo = FALSE,
  ...
)

Arguments

Value

data original n by n matrix with dissimilarities.

weights original n by n matrix with weights.

transformed.data final n by n matrix with transformed dissimilarities.

approach apporach to ties: 1 = untie ties, 2 = keep ties tied.

degree spline degree.

ninner number of interior knots.

iknots interior knots sequence.

anchor whether an intervept was used or not.

knotstype type of procedure creating the interior knot sequence.

coordinates final n by p matrix with coordinates.

coefficients final h by p matrix with regression coefficients.

distances final n by n matrix with Euclidean distances between n rows of coordinates.

last.iteration final iteration number.

last.difference final function difference used for convergence testing.

n.stress final normalized stress value.

Author(s)

Frank M.T.A. Busing

Print method for all fmds objects

Description

print produces output for fmds object.

Usage

## S3 method for class 'fmds'
print(x, ...)

Arguments

Value

none

Author(s)

Frank M.T.A. Busing

Relative Density-based Outlier Probabilities Function

Description

rdop returns the relative density-based outlier probabilities according to Barroso and Busing (2025).

Usage

rdop(data, k = 0, lambda = 3, extended = FALSE, alpha = 0.2, beta = 0.25)

Arguments

Value

if ( extended == FALSE ): outlier scores; else: weights matrix

Author(s)

Frank M.T.A. Busing

References

Barroso and Busing (2025).

Rotation

Description

rotation returns rotation matrix, such that || rotation * source - target ||^2_weights is minimal.

Usage

rotation(source, weights = NULL, target = NULL, error.check = FALSE)

Arguments

Value

rotation matrix

Author(s)

Frank M.T.A. Busing

References

Gower (1968). Commandeur (1991).

Summary method for all fmds objects

Description

summary produces an output summary for fmds objects.

Usage

## S3 method for class 'fmds'
summary(object, ...)

Arguments

Value

none

Tortula data

Description

Morphological data on 10 taxa of Tortula labeled C, CC, CCS, I, I, N, N, RF, RH, RS, R, R, V, V.

Usage

tortula

Format

Multivariate data set with 14 cases and 7 variables.

• hydroids: (nominal)

• leaves: (nominal)

• hairs: (ordinal)

• apex: (ordinal)

• length: (numerical)

• diameter: (numerical)

• papillae: (numerical)

References

Podani (1999). Extending Gower's general coefficient of similarity to ordinal characters. Taxon, 48(2), 331-340.