Package 'smacof'

Description

Regresses external variables on a MDS configuration which results in a MDS biplot.

Usage

## S3 method for class 'smacof'
biplotmds(object, extvar, scale = TRUE)
## S3 method for class 'mdsbi'
plot(x, vecscale = NULL, plot.dim = c(1,2), sphere = TRUE, col = 1, 
                     label.conf = list(label = TRUE, pos = 3, col = 1, cex = 0.8), 
                     vec.conf = list(col = 1, cex = 0.8, length = 0.1), 
                     identify = FALSE, type = "p", pch = 20, 
                     asp = 1, main, xlab, ylab, xlim, ylim, ...)

Arguments

Details

If a model for individual differences is provided, the external variables are regressed on the group stimulus space configurations. In the biplot only the relative length of the vectors and their direction matters. Using the scale argument 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.

Value

Returns an object belonging to classes "mlm" and "mdsbi". See lm for details.

References

Greenacre, M. (2010). Biplots in Practice. Fundacion BBVA, Bilbao, Spain.

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

See Also

plot.smacof

Examples

## morse code data with external scales
res <- mds(morse)
fitbi <- biplotmds(res, morsescales[,2:3])
plot(fitbi, main = "MDS Biplot", vecscale = 0.5)

## wish data with external economic development factor
diss <- sim2diss(wish, method = 7)
res <- mds(diss, type = "ordinal")
ecdev <- data.frame(ecdev = c(3,1,3,3,8,3,7,9,4,7,10,6))
fitbi <- biplotmds(res, ecdev)
plot(fitbi, main = "MDS Biplot", vecscale = 1)
plot(fitbi, main = "MDS Biplot", vecscale = 0.5, xlim = c(-1, 1), 
vec.conf = list(col = "red", length = 0.05))

## Ekman's color data (by Michael Friendly)
require(colorspace)
wavelengths <- attr(ekman, "Labels")
colors <- c("#2600F0", "#0028FF", "#0092FF", "#00B2FF", "#00FFFF", "#00FF61", "#77FF00", "#B3FF00",
            "#FFF200", "#FFBE00", "#FF9B00", "#FF5700", "#F60000", "#D60000")
ekmanD <- sim2diss(ekman)
res <- mds(ekmanD, type = "ordinal")
RGB <- t(col2rgb(colors)) / 255
HCL <- as(hex2RGB(colors), "polarLUV")
HCL <- slot(HCL, "coords")
fit <- biplotmds(res, cbind(RGB, HCL))
fit
plot(fit, vecscale = 0.5, cex = 6, col = colors,
    label.conf=list(cex = 1, pos = ifelse(wavelengths < 560, 2, 4)),
    vec.conf = list(cex = 1.2), main = "Ekman configuration and color properties" )

Description

Performs a bootstrap on a SMACOF solution. It works for derived dissimilarities only. The original data matrix needs to be provided, as well as the type of dissimilarity measure used to compute the input dissimilarities.

Usage

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

## S3 method for class 'smacofboot'
plot(x, plot.dim = c(1,2), col = 1, 
label.conf = list(label = TRUE, pos = 3, cex = 0.8), 
ell = list(lty = 1, lwd = 1, col = "gray"), main, xlab, ylab, xlim, ylim, 
asp = 1, type = "p", pch = 20, ...)

Arguments

Details

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

Value

References

Jacoby, W. G., & Armstrong, D. A. (2014). Bootstrap confidence regions for multidimensional scaling solutions. American Journal of Political Science, 58, 264-278.

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

See Also

jackmds

Examples

## Example using Euclidean distances
data <- na.omit(PVQ40[,1:5])
diss <- dist(t(data))   ## Euclidean distances 
fit <- mds(diss)        ## 2D interval MDS

set.seed(123)
resboot <- bootmds(fit, data, method.dat = "euclidean", nrep = 50)
resboot
plot(resboot)

## Example using Pearson correlations 
sim <- cor(data)
diss <- sim2diss(sim, method = 1)  ## subtract from 1 (method needs to be passed to bootmds)
fit <- mds(diss, type = "ratio", ndim = 3)        ## 3D ratio MDS

set.seed(123)
resboot <- bootmds(fit, data, method.dat = "pearson", nrep = 50, alpha = 0.1, method = 1)
resboot
## plot 1st against 3rd dimension
ellipses <- plot(resboot, plot.dim = c(1,3), ell = list(lty = 2, col = "gray", lwd = 0.8))
str(ellipses)  ## list of ellipse coordinates for each object

Description

The data set is described in Bro (1998). The raw data consist of ratings of 10 breads on 11 different attributes carried out by 8 raters. Note that the bread samples are pairwise replications: Each of the 5 different breads, which have a different salt content, was presented twice for rating.

Usage

data(bread)

Format

A list of length 8 with elements of class "dist". The attributes are bread odor, yeast odor, off-flavor, color, moisture, dough, salt taste, sweet taste, yeast taste, other taste, and total taste.

References

Bro, R. (1998). Multi-way Analysis in the Food Industry: Models, Algorithms, and Applications. Ph.D. thesis, University of Amsterdam (NL) & Royal Veterinary and Agricultural University (DK).

Examples

bread

Description

42 individuals were asked to order 15 breakfast items due to their preference.

Usage

data(breakfast)

Format

Data frame with students in the rows and breakfast items in the columns.

toast: toast pop-up

butoast: buttered toast

engmuff: English muffin and margarine

jdonut: jelly donut

cintoast: cinnamon toast

bluemuff: blueberry muffin and margarine

hrolls: hard rolls and butter

toastmarm: toast and marmalade

butoastj: buttered toast and jelly

toastmarg: toast and margarine

cinbun: cinnamon bun

danpastry: Danish pastry

gdonut: glazed donut

cofcake: coffee cake

cornmuff: corn muffin and butter

References

Green, P. E. & Rao, V. (1972). Applied multidimensional scaling. Hinsdale, IL: Dryden.

Examples

breakfast

Description

We took Canadian newspapers that appeared in the time period between June and September 2009 and searched for articles that contained the word "aboriginal". A total of 92 articles was found. In these articles, we determined the frequencies of other meaningful words (e.g., tribal, moose, arctic, and health). The data are organized as word co-occurrence matrix.

Usage

data(CanadaNews)

Format

Matrix with word co-occurrence counts.

References

Borg, I., Groenen, P. J. F., & Mair, P. (2017). Applied Multidimensional Scaling and Unfolding. New York: Springer.

Examples

str(CanadaNews)

Description

Computes pseudo-confidence ellipses for symmetric and individual difference MDS fits.

Usage

## S3 method for class 'smacofID'
confEllipse(object)

## S3 method for class 'confell'
plot(x, eps = 0.05, plot.dim = c(1,2), col = 1, 
label.conf = list(label = TRUE, pos = 3, cex = 0.8), 
ell = list(lty = 1, lwd = 1, col = 1), main, xlab, ylab, xlim, ylim, 
asp = 1, type = "p", pch = 20, ...)

Arguments

Details

The confEllipse function normalizes the dissimilarities and performs a few more iterations to optimize the configuration and the individual diffierence weights. This result is then passed to a function that computes the stress derivatives which are the basis of the ellipses in the plot function. This function works for ratio scaled versions only.

Value

Returns an object belonging to classes "confell".

References

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

See Also

plot.smacofboot

Examples

## Simple ratio MDS fit 
delta <- sim2diss(cor(PVQ40agg))
res <- mds(delta, ndim = 3)
cres <- confEllipse(res)
plot(cres, plot.dim = c(1,2))
plot(cres, plot.dim = c(1,3))
plot(cres, plot.dim = c(2,3))


## INDSCAL on Helm data
fit1 <- indscal(helm)
cfit1 <- confEllipse(fit1)
plot(cfit1, ell = list(col = "gray", lty = 2), ylim = c(-0.04, 0.04))

## IDIOSCAL on Helm data
fit2 <- idioscal(helm)
cfit2 <- confEllipse(fit2)
plot(cfit1, ell = list(col = "gray", lty = 2), ylim = c(-0.04, 0.04))

Description

Correlations of crime rates in 50 US states.

Usage

data(crimes)

Format

Crime correlation matrix.

References

Borg, I., Groenen, P. J. F., & Mair, P. (2017). Applied Multidimensional Scaling and Unfolding. New York: Springer.

Examples

crimes

Description

This dataset collects rankings of 100 individual on 5 topics that reflect social responsibilties on corporations.

Usage

data(csrranking)

Format

A data frame where each individual ranked prevention of environmental pollution (Environment), waste prevention (Waste Prevention), selling organic products (Organic Products), participating on charity programs (Charity), and fair treatment of employees (Employee) according to its own preferences. A value of 1 corresponds to highest importance, 5 to lowest importance.

Examples

csrranking

Description

Compute weights as a function of the dissimilarities.

Usage

dissWeights(delta, type = c("unif", "knn", "power", "unifpower"), 
           k = NULL, power = 0)

Arguments

Details

The weights are computed as a function of the dissimilarities depending on type.

- "unif" Compute weights such that the weighted empirical distribution (histogram) of the dissimilarities is uniform. Particularly if the number of objects is large, the dissimilarities that occuur most often will start to dominate the solution. This option de-emphasizes often occuring dissimilarities such that the weighted empirical distribution (the weighted histogram) becomes approximately uniform.

- "knn" Per row of the dissimilarity matrix the k smallest dissimilarities obtain a weight of 1 and the others a 0.

- "power" The weights are set to the delta^power. If power is small (e.g., power = -5) then the smaller dissimilarities will be better fitted. If power is large (e.g., power = 5) then the larger dissimilarities will be better fitted.

- "unifpower" First weights are determined by the "unif" option and then multiplied by the weights obtained by the "power" option. If the dissimilarity matrix is large, then this option is needed to see an effect of the "power" option on the MDS solution.

Value

Author(s)

Patrick Groenen

Examples

## mds solution for kinship data with uniform weights
res <- mds(kinshipdelta, weightmat = dissWeights(kinshipdelta, type = "unif"))
par(mfrow = c(2,2))
plot(res, main = "uniform weights")
plot(res, plot.type = "Shepard")
plot(res, plot.type = "histogram")

## mds solution for kinship data with knn weights
res <- mds(kinshipdelta, weightmat = dissWeights(kinshipdelta, type = "knn", k = 5))
par(mfrow = c(1,2))
plot(res, main = "knn weights with k=5")
plot(res, plot.type = "Shepard")

## mds solution for kinship data with power weights emphasizing large dissimilarities
res <- mds(kinshipdelta, weightmat = dissWeights(kinshipdelta, type = "power", power = 5))
par(mfrow = c(2,2))
plot(res, main = "Power = 5 weights")
plot(res, plot.type = "Shepard")
plot(res, plot.type = "histogram")

## mds solution for kinship data with power weights emphasizing small dissimilarities
res <- mds(kinshipdelta, weightmat = dissWeights(kinshipdelta, type = "power", power = -5))
par(mfrow = c(2,2))
plot(res, main = "Power = -5 weights")
plot(res, plot.type = "Shepard")
plot(res, plot.type = "histogram")

## mds solution for kinship data with power weights emphasizing large dissimilarities 
## while correcting for nonuniform dissimilarities
res <- mds(kinshipdelta, weightmat = dissWeights(kinshipdelta, type = "unifpower", power = 5))
par(mfrow = c(2,2))
plot(res, main = "Uniform power = 5 weights")
plot(res, plot.type = "Shepard")
plot(res, plot.type = "histogram")

## mds solution for kinship data with power weights emphasizing small dissimilarities
## while correcting for nonuniform dissimilarities
res <- mds(kinshipdelta, weightmat = dissWeights(kinshipdelta, type = "unifpower", power = -5))
par(mfrow = c(2,2))
plot(res, main = "Uniform power = -5 weights")
plot(res, plot.type = "Shepard")
plot(res, plot.type = "histogram")

Description

Takes an asymmetric dissimilarity matrix and decomposes it into a symmetric and a skew-symmetric part. Fits an MDS on the symmetric part and computes drift vectors for the skew-symmetric portion. This model makes it possible to see how these two components are related to each other. It is limited to two dimensions only.

Usage

driftVectors(data, type = c("ratio", "interval", "ordinal","mspline"), 
             weightmat = NULL, init = "torgerson", ties = "primary",  verbose = FALSE, 
             relax = FALSE, modulus = 1, itmax = 1000, eps = 1e-6, 
             spline.degree = 2, spline.intKnots = 2)
             
## S3 method for class 'driftvec'
plot(x, adjust = 1, main, xlim, ylim, 
                        xlab = "Dimension 1", ylab = "Dimension 2", pch = 20, asp = 1, 
                        col.conf = "black", col.drift = "lightgray", 
                        label.conf = list(label = TRUE, pos = 3, col = "black", 
                        cex = 0.8), ...)

Arguments

Details

The skew-symmetric values are embedded into the MDS representation of the symmetrized data by drawing errors (drift vectors) from each point $i$ to each point $j$ in the configuration so that these vectors correspond in length and direction to the values of row $i$ of the skew-symmetric matrix.

Value

Author(s)

Patrick Mair

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

See Also

smacofSym

Examples

## simple example
P <- matrix(c(0,  4,  6, 13, 
              5,  0, 37, 21, 
              4, 38,  0, 16, 
              8, 31, 18,  0), nrow = 4, ncol = 4, byrow = TRUE) 
D <- sim2diss(P, method = 40) 
res <- driftVectors(D, type = "interval")
plot(res)
plot(res, adjust = 0.1)    ## shorten drift vectors 

## Morse code data
fit.drift <- driftVectors(morse2, type = "ordinal")
fit.drift
plot(fit.drift)

Description

The DurationRaw dataset contains the duration rating of 76 subjects on 24 situations. Subjects were asked to rate the duration on a 7 point scale (1 ... substantially shorter, 7 ... substantially longer). The Duration data file contains the corresponding correlations between the 24 situations including some information about the facets.

Usage

data(Duration)
data(DurationRaw)

Format

Data frame 24 correlations based on duration ratings:

S1-S24: situation

F1: pleasant (1), neutral (2), unpleasant (3)

F2: variable (1), monotonous (2)

F3: difficult (1), easy (2)

F3: many (1), few (2)

structuple: the facet structure written as a tuple

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

Examples

ddiss <- sim2diss(Duration[,paste0("S", 1:24)])
fit <- mds(ddiss, type = "ordinal", ndim = 4)
plot(fit)

Description

Ekman dissimilarities

Usage

data(ekman)

Format

Object of class dist

Details

Ekman presents similarities for 14 colors which are based on a rating by 31 subjects where each pair of colors was rated on a 5-point scale (0 = no similarity up to 4 = identical). After averaging, the similarities were divided by 4 such that they are within the unit interval. Similarities of colors with wavelengths from 434 to 674 nm.

References

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

Examples

ekman

Description

Intercorrelations of 13 working values for former West (first list element) and East Germany.

Usage

data(EW_eng)

Format

Object of class dist

Details

Note that in EW_ger the labels are given in German. For smacof, the data must be converted into a dissimilarity matrix by applying the sim2diss() function to each list element.

References

ALLBUS 1991, German General Social Survey.

Borg, I., Groenen, P. J. F., & Mair, P. (2010). Multidimensionale Skalierung. Muenchen: Hampp Verlag.

Borg, I., Groenen, P. J. F., & Mair, P. (2012). Multidimensional Scaling. New York: Springer, forthcoming.

Examples

data(EW_eng)
data(EW_ger)

Description

Dissimilarity matrix of 13 facial expressions (Abelson & Sermat, 1962). The external scales are taken from Engen et al. (1958) reflecting the following three perceptual dimensions: pleasant-unpleasant (PU), attention-rejection (AR), and tension-sleep (TS).

Usage

data(FaceExp)
data(FaceScale)

Format

Symmetric dissimilarity matrix and data frame with 3 perceptual dimensions

References

Abelson, R. P., & Sermat, V. (1962). Multidimensional scaling of facial expressions. Journal of Experimental Psychology, 63, 546-554.

Engen, B., Levy, N., & Schlossberg, H. (1958). The dimensional analysis of a new series of facial expressions. Journal of Experimental Psychology, 55, 454-458.

Examples

str(FaceExp)
str(FaceScale)

Description

Utility function for fitting a circle into 2D point configurations.

Usage

fitCircle(x, y)

Arguments

Value

References

Pratt, V. (1987). Direct least-squares fitting of algebraic surfaces. Computer Graphics, Vol. 21, pages 145-152.

Examples

## Dataset on Schwartz values:
require(plotrix)
valsD <- 1 - cor(indvalues)
fit <- mds(valsD)
plot(fit, main = "MDS Value Circle")
circle <- fitCircle(fit$conf[,1], fit$conf[,2])
draw.circle(circle$cx, circle$cy, radius = circle$radius, border = "gray")

Description

Document-term matrix based on statements by Republican voters.

Usage

data(GOPdtm)

Format

Document-term matrix with statements in the rows and terms (keywords) in the columns

Details

This dataset emerges from statements of Republican voters scraped from the official GOP website. They were asked to complete the sentence "I am a Republican because ...". We have selected the 37 most frequent words and created a document-term matrix.

References

air, P., Rusch, T. & Hornik, K. (2014). The Grand Old Party - A party of values? SpringerPlus, 3(697), https://springerplus.springeropen.com/articles/10.1186/2193-1801-3-697

Examples

data(GOPdtm)
GOPdtm

Description

Computes the dissimilarities using a gravity model based on co-occurrences.

Usage

gravity(X, lambda = 1)

Arguments

Details

The first step in this function is to compute the co-occurences. Based on the binarized data matrix $Y$ we compute $Y'Y$ which leads to the co-occurence matrix. We then use the gravity model to compute the gravity dissimilarities. In order to give more (or less) structure to the MDS solution, the tuning parameter (which defines a power transformation) can be increased (or decreased). In addition, a weight matrix is created that sets cells with no co-occurences to 0. The corresponding weight matrix for blanking out these cells is established automatically in mds().

Value

Author(s)

Patrick Mair

References

Mair, P., Rusch, T. & Hornik, K. (2014). The Grand Old Party - A party of values? SpringerPlus, 3(697), https://springerplus.springeropen.com/articles/10.1186/2193-1801-3-697

See Also

mds

Examples

data(GOPdtm)
gravD <- gravity(GOPdtm, lambda = 2)
res <- mds(gravD$gravdiss)  
res$weightmat  ## NA's were blanked out when fitting the model
plot(res)

Description

Distances (in km) among French Departments in 1830.

Usage

data(Guerry)

Format

Symmetric matrix with distances.

References

Friendly, M. (2007). A. M. Guerry's Moral Statistics of France: Challenges for Multivariate Spatial Analysis, Statistical Science, 2007, 22(3), 368-399.

Examples

Guerry

Description

The first dataset (Guttman1991) contains similarities and facets for Guttman's 3D cylindrical intelligence structure as published in Guttman (1991). The second dataset (Guttman1965) contains similarities and structural intelligence facets from Guttman (1965).

Usage

data(Guttman1991)
data(Guttman1965)

Format

List with two elements: The first element contains the similarity matrix, the second element the facets labels.

References

Guttman, L. & Levy, S. (1991). Two structural laws for intelligence tests. Intelligence, 15, 79-103.

Guttman, L. (1965). The structure of interrelations among intelligence tests. In C. W. Harris (Ed.), Proceedings of the 1964 Invitational Conference on Testing Problems (pp. 23-36). Princeton: ETS.

Examples

Guttman1991[[1]]        ## similarity matrix
Guttman1991[[2]]        ## facets

Guttman1965[[1]]        ## similarity matrix
Guttman1965[[2]]        ## facets

Description

Contains dissimilarity data for individual difference scaling from an experiment carried out by Helm (1959).

Usage

data(helm)

Format

List containing objects of class dist

Details

A detailed description of the experiment can be found in Borg and Groenen (2005, p. 451) with the corresponding Table 21.1. containing distance estimates for color pairs. There were 14 subjects that rated the similarity of colors, 2 of whom replicated the experiment. 10 subjects have a normal color vision (labelled by N1 to N10 in our list object), 4 of them are red-green deficient in varying degrees. In this dataset we give the dissimilarity matrices for each of the subjects, including the replications. They are organized as a list of length 16 suited for smacofIndDiff computations.

The authors thank Michael Friendly and Phil Spector for data preparation.

References

Helm, C. E. (1959). A multidimensional ratio scaling analysis of color relations. Technical Report, Princeton University and Educational Testing Service. Princeton, NJ.

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling: Theory and Applications (2nd edition). New York: Springer.

Examples

helm

Description

Allows to user to explore the effect of various random starting configurations when fitting an MDS model.

Usage

icExplore(delta, nrep = 100, returnfit = FALSE, ndim = 2, 
type = c("ratio", "interval", "ordinal","mspline"), weightmat = NULL, ties = "primary",	
verbose = FALSE, relax = FALSE, modulus = 1, itmax = 1000, eps = 1e-6, 
spline.degree = 2, spline.intKnots = 2)

Arguments

Details

This function generates a large set of MDS solutions using random initial configurations, matches them all by Procrustean fittings, computes the inter-correlations of their point coordinates, and finally runs an interval MDS of these inter-correlations. It can be used to explore local minima.

In the plot function the number reflects the index of corresponding MDS fit, the size reflects the stress value: the larger the font, the larger the stress (i.e., the worse the solution). The size is associated with a corresponding color shading (the smaller the size the darker the color).

Value

References

Borg, I. and Mair, P. (2017). The choice of initial configurations in multidimensional scaling: local minima, fit, and interpretability. Austrian Journal of Statistics, 46, 19-32. doi:10.17713/ajs.v46i2.561

See Also

mds

Examples

## simple example with 20 random starts
diss <- sim2diss(wish, method = 7)
set.seed(123)
res <- icExplore(diss, type = "ordinal", nrep = 20, returnfit = TRUE)
res
plot(res) 

res$mdsfit[[14]]  ## bad fitting solution
res$mdsfit[[3]]   ## better fitting solution

Description

Responses from a sample in Britain were collected varying in value measures of the Schwartz value theory. The instrument used was the Schwartz Value Survey (SVS).

Usage

data(indvalues)

Format

Data frame with 327 persons in the rows and psychological values in the columns.

Details

The data were centered (row-wise) and converted from preferences into dissimilarities.

References

Borg, I., Bardi, A., & Schwartz, S. H. (2017). Does the value circle exist within persons or only across persons? Journal of Personality, 85(2), 151-162.

See Also

PVQ40

Examples

str(indvalues)

Description

Contains intercorrelations of 8 intelligence tests, together with two facets. In addition, a hypothesized restriction matrix for the intercorrelations is provided. The proximities for items with the same structuples, such as p(NA1,NA2) and p(GA1,GA3), all are set to the value 5. The proximities that correspond to the immediate neighborhood relations are set to the value 4, since none of these distances should be larger than any distance between definitionally equivalent items. Finally, the large distances between the groups NI, GA and the groups NA, GI are set to 3. The intelligence tests are coded on the follwing facets: format (N = numerical, G = geometrical) and requirement (A = application, I - inference).

Usage

data(intelligence)

Format

Data frame of 8 intelligence tests: facets, intercorrelations, and restrictions

Test: Test number

Languange: numerical, geometrical

Requirement: application, inference

T1-T8: intercorrelations

R1-R8: restrictions

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

Examples

idiss <- sim2diss(intelligence[,paste0("T", 1:8)])
fit <- mds(idiss)
plot(fit)

Description

These methods perform a SMACOF Jackknife and plot the corresponding solution.

Usage

## S3 method for class 'smacofB'
jackmds(object, eps = 1e-6, itmax = 100, verbose = FALSE)

## S3 method for class 'smacofJK'
plot(x, plot.dim = c(1,2), hclpar = list(c = 50, l = 70), 
col.p, col.l, plot.lines = TRUE, legend = FALSE, inset = c(-0.2, 0), cex.legend = 0.7,
main, xlab, ylab, xlim, ylim, asp = 1, ...)

Arguments

Details

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

Value

Author(s)

Jan de Leeuw and Patrick Mair

References

De Leeuw, J., & Meulman, J. (1986). A special jackknife for multidimensional scaling. Journal of Classification, 3, 97-112.

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

See Also

bootmds

Examples

## symmetric smacof
data <- na.omit(PVQ40[,1:5])
diss <- dist(t(data))   ## Euclidean distances 
fit <- mds(diss) 
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)

Description

Percentages of how often 15 kinship terms were not grouped together by college students including three external scales.

Usage

data(kinshipdelta)

data(kinshipscales)

Format

Dissimilarity matrix of 15 kinship terms and data frame with the following external scales:

Gender (1 = male, 2 = female)

Generation (-2 = two back, -1 = one back, 0 = same generation, 1 = one ahead, 2 = two ahead)

Degree (1 = first, 2 = second, 3 = third, 4 = fourth)

References

Rosenberg, S. & Kim, M. P. (1975). The method of sorting as a data gathering procedure in multivariate research. Multivariate Behavioral Research, 10, 489-502.

Examples

kinshipdelta
kinshipscales

Description

Contains correlations of eight test items of the Kennedy Institute Phonics Test (KIPT), a test for reading skills.

Usage

data(KIPT)

Format

An 8 times 8 correlation matrix. Items:

Nonsense word production: NP

Long vowel production: LVP

Short vowel production: SVP

Consonant cluster production: CCP

Nonsense word recognition: NR

Single letter production: SLP

Consonant cluster recognition: CCR

Initial letter recognition: ILR

References

Guthrie, J. T. (1973). Models of reading and reading disability. Journal of Educational Psychology, 65, 9-18.

Examples

KIPT
sim2diss(KIPT)

Description

Performance of managers: 3 criteria ("traits") and 3 methods. Traits: T1 = Quality of output, T2 = Ability to generate output, T3 = Demonstrated effort to perform. Methods: M1 = Rating by superior, M2 = Peer rating, M3 = Self-rating.

Usage

data(Lawler)

Format

Symmetric matrix (trait-method combinations) with inter-correlations.

References

Lawler, E. E. (1967). Management performance as seen from above, below, and within. In Evaluation of executive performance. Princeton, New Jersey. Educational Testing Service.

Examples

Lawler

Description

Confusion percentages between Morse code signals. The scores are derived from confusion rates on 36 Morse code signals (26 for the alphabet; 10 for the numbers 0,...,9). Each Morse code signal is a sequence of up to five 'beeps'. The beeps can be short (0.05 sec) or long (0.15 sec), and, when there are two or more beeps in a signal, they are separated by periods of silence (0.05 sec).

Rothkopf asked 598 subjects to judge whether two signals, presented acoustically one after another, were the same or not. The values are the average percentages with which the answer 'Same!' was given in each combination of row stimulus i and column stimulus j, where either i or j was the first signal presented. The values are 1 minus the symmetrized confusion rates and are thus dissimilarities.

Usage

data(morse)
data(morse2)

Format

Symmetric and asymmetric dissimilarity matrices of 36 morse codes

Details

The first dataset (morse) contains a symmetric version, the second dataset (morse2) the original asymmetric version.

References

Rothkopf, E. Z. (1957). A measure of stimulus similarity and errors in some paired-associate learning. Journal of Experimental Psychology, 53, 94-101.

Examples

morse
morse2

Description

Two properties of Morse code signals. Each Morse code signal is a sequence of up to five 'beeps'. The beeps can be short (0.05 sec) or long (0.15 sec), and, when there are two or more beeps in a signal, they are separated by periods of silence (0.05 sec). The two external variables are: Signal type (1 = all short beeps, 2 = more short than long beeps, 3 = same short and long beeps, 4 = more long than short beeps, 5 = all long beeps) and Signal length (in seconds; 1 = .05, 2 = .15, 3 = .25, 4 = .35, 5 = .45, 6 = .55, 7 = .65, 8 = .85, 9 = .95).

Usage

data(morsescales)

Format

Matrix of 36 morse codes by 2 properties. The first column contains the morse code letters.

References

Rothkopf, E. Z. (1957). A measure of stimulus similarity and errors in some paired-associate learning. Journal of Experimental Psychology, 53, 94-101.

Examples

morsescales

Description

Contains similarities (correlations) of 54 OCP (see O'Reilly, Chatman, and Caldwell, 1991) items. The last three columns contain the facet assigned by Bilsky and Jehn (2002) as well as the external variables for regional restrictions.

Usage

data(OCP)

Format

Data frame with OCP item correlations and facet:

i1-i54: OCP item correlations

facet: factor with facets

z1, z2: external constraints

References

Bilsky, W. & Jehn, K. (2002). Organizational Culture and Individual Values: Evidence for a Common Structure. In M. Myrtek (Ed.), The Person in Biological and Social Context, pp. 211-228. Goettingen, Germany: Hogrefe Press.

Examples

ocpD <- sim2diss(OCP[,1:54])
fit <- mds(ocpD, type = "ordinal")
plot(fit)

Description

Artificial dataset containing the judges in the rows and the parties in the columns.

Usage

data(partypref)

Format

Matrix of party preferences.

References

Borg, I., Groenen, P. J. F., & Mair, P. (2010). Multidimensionale Skalierung. Muenchen: Hampp Verlag.

Examples

partypref

Description

42 subjects are assigned to two groups of 21 persons. 120 stimulus pairs of rectangles are presented. For the first group (width-height; WH), the rectangles were constructed according to a design as given in rect_constr. For the second group (size-shape; SS) the rectangles were constructed according to a grid design, which is orthogonal in the dimensional system reflecting area (size), and width/height (shape). All subjects had to judge the similarity of the rectangles on a scale from 0 to 9.

Usage

data(perception)

Format

List of subject dissimilarities for WH (first element) and SS group (second element).

References

Borg, I. & Leutner, D. (1983). Dimensional models for the perception of rectangles. Perception and Psychophysics, 34, 257-269.

See Also

rectangles

Examples

perception
rect_constr

Description

These methods perform a permutation test for a symmetric or an unfolding SMACOF model.

Usage

## S3 method for class 'smacof'
permtest(object, data,  method.dat = "pearson", nrep = 100, verbose = TRUE, ...)
## S3 method for class 'smacofR'
permtest(object, data = NULL,  method.dat = "rows", nrep = 100, verbose = TRUE, ...)
## S3 method for class 'smacofPerm'
plot(x, alpha = 0.05, main, xlab, ylab, ...)

Arguments

Details

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

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

Value

Author(s)

Patrick Mair and Ingwer Borg

References

Mair, P., Borg, I., and Rusch, T. (2016). Goodness-of-fit assessment in multidimensional scaling and unfolding. Multivariate Behavioral Research, 51, 772-789.

Mair, P, Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

See Also

jackmds, bootmds

Examples

## permuting the dissimilarity matrix (full)
data(kinshipdelta)
fitkin <- mds(kinshipdelta, ndim = 2, type = "interval")
set.seed(222)
res.perm <- permtest(fitkin)
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 <- mds(diss1, type = "ordinal")
fitgop1
set.seed(123)
permtest(fitgop1, GOPdtm[,1:10], nrep = 10, method.dat = "binary")

rmat <- cor(GOPdtm[,1:10], method = "kendall")  ## Kendall correlation
diss2 <- sim2diss(rmat, method = 1)
fitgop2 <- mds(diss2, type = "ordinal")
fitgop2
set.seed(123)
permtest(fitgop2, GOPdtm[,1:10], nrep = 10, method.dat = "kendall", method = 1)

## unfolding permutation
data(breakfast)
res.unfolding <- unfolding(breakfast, ndim = 2)
set.seed(123)
permtest(res.unfolding, nrep = 20, method.dat = "rows")

Description

This dataset contains statistical information about Plato's seven works. The underlying problem to this dataset is the fact that the chronological order of Plato's works is unknown. Scholars only know that Republic was his first work, and Laws his last work. For each work, Cox and Brandwood (1959) extracted the last five syllables of each sentence. Each syllable is classified as long or short which gives 32 types. Consequently, we obtain a percentage distribution across the 32 scenarios for each of the seven works.

Usage

data(Plato7)

Format

Data frame containing syllable percentages of Plato's 7 works.

References

Cox, D. R. & Brandwood, L. (1959). On a discriminatory problem connected with the work of Plato. Journal of the Royal Statistical Society (Series B), 21, 195-200.

Examples

Plato7

Description

These methods provide various 2D plots for SMACOF models.

Usage

## S3 method for class 'smacof'
plot(x, plot.type = "confplot", plot.dim = c(1,2), sphere = TRUE, 
                      bubscale = 1, col = 1, 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", pch = 20, cex = 0.5, asp = 1, main, xlab, ylab, 
                      xlim, ylim, col.hist = NULL, ...)

## S3 method for class 'smacofR'
plot(x, plot.type = "confplot", what = c("both", "columns", "rows"), 
                       plot.dim = c(1,2), col.rows = hcl(0), col.columns = hcl(240), 
                       label.conf.rows = list(label = TRUE, pos = 3, 
                       col = hcl(0, l = 50), cex = 0.8), 
                       label.conf.columns = list(label = TRUE, pos = 3, 
                       col = hcl(240, l = 50), cex = 0.8),  
                       shepard.x = NULL, col.dhat = NULL, type = "p", pch = 20,
                       cex = 0.5, asp = 1, main, xlab, ylab, xlim, ylim, ...)

## S3 method for class 'smacofID'
plot(x, plot.type = "confplot", plot.dim = c(1,2), bubscale = 1, 
                        col = 1, label.conf = list(label = TRUE, pos = 3, col = 1, 
                        cex = 0.8), identify = FALSE, type = "p", pch = 20,  cex = 0.5, 
                        asp = 1, plot.array, main, xlab, ylab, xlim, ylim, ...)

Arguments

Details

mds() and smacofSym() create an object of class "smacof", unfolding(), prefscal(), and smacofRect() produce "smacofR", and smacofIndDiff() generates "smacofID".

Plot description:

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

- Residual plot (plot.type = "resplot"): Plots the disparities (d-hats) distances against the fitted distances.

- Shepard diagram (plot.type = "Shepard"): Diagram with the observed dissimilarities against the fitted distances including (isotonic) regression line.

- 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 smacof function to percentages (sum is 100). The higher the contribution, the worse the fit.

- Bubble plot (plot.type = "bubbleplot", not available for rectangular SMACOF): 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. For optional arguments, see wtd.hist.

For smacofIndDiff() the residual plot, Shepard diagram, and stress plot are based on the sum of the residuals across individuals/ways. The configuration plot represents the group stimulus space (i.e., joint configuration). If plot.array is not specified, it produces a Shepard plot of the distances summed across subjects, if plot.array = 0 it produces a sqrt(nsubjects) times sqrt(nsubjects) array of graph panels, if plot.array = 3 it produces 3x3 arrays of graph panels, if plot.array = c(2, 3) it produces 2x3 arrays of graph panels, and if plot.array = c(3, 2, 5) produces 3x2 arrays of panels (only the first two values are used).

See Also

plot.procr

Examples

## 2D plots for simple MDS
data(trading)
res <- mds(trading)
plot(res, plot.type = "confplot")
plot(res, plot.type = "confplot", label.conf = list(pos = 5)) ## avoid overlapping labels
plot(res, plot.type = "Shepard")
plot(res, plot.type = "stressplot")
plot(res, plot.type = "resplot")
plot(res, plot.type = "bubbleplot")
plot(res, plot.type = "histogram")

## Add convex hulls to configuration plot
r <- cor(PVQ40, use = "pairwise.complete.obs")
diss <- sim2diss(r, method = "corr") 
res <- mds(delta = diss, type = "ordinal") 
codes <- substring(colnames(PVQ40), 1, 2)  ## supplementary variable
plot(res, hull.conf = list(hull = TRUE, ind = codes, col = "coral1", lwd = 2))

## Shepard plots
ekmanD <- sim2diss(ekman)
fit1 <- mds(ekmanD, type = "ordinal")
plot(fit1, plot.type = "Shepard")
plot(fit1, plot.type = "Shepard", shepard.x = ekman)  ## original data on x-axis

## Joint configuration plot and row/column stressplots for unfolding
data(breakfast)
res <- unfolding(breakfast)
plot(res, plot.type = "confplot")
plot(res, plot.type = "stressplot")

Description

Solves the Procrustean problem of fitting one (MDS) configuration (testee) to another (target) MDS configuration.

Usage

Procrustes(X, Y)

## S3 method for class 'procr'
plot(x, plot.type = "jointplot", plot.dim = c(1,2), main, xlab, ylab, 
                     xlim, ylim, asp = 1, pch = 20, col.X = "cadetblue", 
                     col.Y = "gray", col.Yhat = "coral1", 
                     label.conf = list(label = TRUE, pos = 3, cex = 0.8), 
                     arrows = TRUE, length = 0.10, 
                     legend = list(plot = TRUE, labels = c("Target", "Testee"), 
                                   pos = "bottomright"), ...)

Arguments

Details

Y is going to be modified by finding an optimal dilation factor, an optimal translation and rotation for Y such that it is a similar as possible to X. X remains untouched.

Value

Returns an object of class procr with:

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling and Unfolding (2nd ed.). Springer.

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

Borg, I. and Mair, P. (2022). A note on Procrustean fittings of noisy configurations. Austrian Journal of Statistics, 51, 1-9. doi:10.17713/ajs.v51i4.1423

Examples

## artificial example:
X <- matrix(c(1, -1, -1, 1, 2, 2, -2, -2), ncol = 2)
Y <- matrix(c(0.07, 0.93, 1.93, 1.07, 2.62, 3.12, 1.38, 0.88), ncol = 2)
op <- par(mfrow = c(1,2))
plot(X[,1], X[,2], xlim = c(-3, 3), ylim = c(-2, 3.5), asp = 1, xlab = "", ylab = "")
rect(-1, -2, 1, 2)
points(Y[,1], Y[,2], xlim = c(-3, 3), col = "gray")
polygon(Y[,1], Y[,2], border = "gray")
fitp <- Procrustes(X, Y)
plot(fitp$Yhat[,1], fitp$Yhat[,2], col = "red", xlim = c(-3, 3), ylim = c(-2, 3.5), 
     asp = 1, xlab = "", ylab = "")
polygon(fitp$Yhat[,1], fitp$Yhat[,2], border = "red")
par(op)

## MDS example:
eastD <- sim2diss(EW_eng$east)
attr(eastD, "Labels") <- abbreviate(attr(eastD, "Labels"))
fit.east <- mds(eastD, type = "ordinal")
westD <- sim2diss(EW_eng$west)
attr(westD, "Labels") <- abbreviate(attr(westD, "Labels"))
fit.west <- mds(westD, type = "ordinal", init = torgerson(eastD))

fit.proc <- Procrustes(fit.east$conf, fit.west$conf)
fit.proc

## Configuration plots; Procrustes plots.
plot(fit.east, main = "MDS East Germany")   ## MDS plot East Germany
plot(fit.west, main = "MDS West Germany")   ## MDS plot West Germany

## Procrustes configurations (X and Yhat)
plot(fit.proc, ylim = c(-1, 1),  col.X = "cadetblue", col.Yhat = "brown", pch = 19, 
    legend = list(pos = "topleft", labels = c("East Germany", "West Germany"))) 
    
## Procrustes transformations (Y and Yhat)
plot(fit.proc, plot.type = "transplot", length = 0.05, ylim = c(-1,1), 
     legend = list(pos = "bottomright", 
     labels = c("West Germany (untransformed)", "West Germany (transformed)")))

Description

The PVQ40 (Schwartz et al., 1999) consists of 40 items, each a short portrait of one person. For example, to measure power, the PVQ includes two portraits (male/female versions): It is important to him to be rich. He wants to have a lot of money and expensive things. It is important to him to get respect from others. He wants people to do what he says. Respondents indicate on 6-point bipolar rating scale (1 ... not at all like me, 6 ... very much like me) the degree to which the description also fits himself/herself. Gender and age of the participants are added as attributes.

Usage

data(PVQ40)

Format

PVQ40 data of 151 adults from various states in the USA:

sd1-sd4: self-direction

po1-po3: power

un1-un6: universalism

ac1-ac4: achievement

se1-se5: security

st1-st3: stimulation

co1-co4: conformity

tr1-tr4: tradition

he1-he3: hedonism

be1-be4: benevolence

Age and Gender are added as attributes.

Details

PVQ40agg is an aggregated version of PVQ40 where the item scores belonging to the same value are averaged. Abbreviations: power (PO), achievement (AC), hedonism (HE), stimulation (ST), self-direction (SD), universalism (UN), benevolence (BE), tradition (TR), conformity (CO), security (SE).

References

Borg, I., Bardi, A., & Schwartz, S. H. (2017). Does the value circle exist within persons or only across persons? Journal of Personality, 85(2), 151-162.

See Also

indvalues

Examples

str(PVQ40)
head(PVQ40)
attr(PVQ40, "Gender")
attr(PVQ40, "Age")
str(PVQ40agg)

Description

Creates random dissimilarity matrices (n objects), fits an MDS, and returns the stress values of each MDS fit.

Usage

randomstress(n, ndim, nrep = 100, type = c("ratio", "interval", "ordinal", "mspline"))

Arguments

Details

The random dissimilarities are drawn from a U(0,1) distribution.

Value

Returns a vector with stress values.

References

Spence I., Ogilvie, J.C. (1973). A table of expected stress values for random rankings in nonmetric multidimensional scaling. Multivariate Behavioral Research, 8, 511-517.

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2. Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

Examples

## 8 objects, 2 dimensions, interval MDS (50 replications)
stressvec <- randomstress(n = 8, ndim = 2, nrep = 50, type = "interval")
mean(stressvec)

Description

These data are based on an experiment by Borg and Leutner (1983). They constructed rectangles on the basis of the grid design (see rect_constr). Each point in this grid defines a rectangle. Rectangle 16, for example, had a width of 4.25 cm and a height of 1.25 cm; rectangle 4 was 3.00 cm wide and 2.75 cm tall. A total of 21 persons rated (twice) the similarity of each pair of these 16 rectangles (on a 10-point scale ranging from from 0 = equal/identical to 9 = very different) The means of these ratings over persons and replications are given in rectangles. A second dataset (rectangles2) is constructed based on area and shape of the rectangles.

Usage

data(rectangles)
data(rect_constr)
data(rectangles2)

Format

The rectangles are object of class dist, the constraints are given as matrix

References

Borg, I., & Leutner, D. (1983). Dimensional models for the perception of rectangles. Perception and Psychophysics, 34, 257-269.

Borg, I., Groenen, P. J. F., & Mair, P. (2017). Applied Multidimensional Scaling and Unfolding. New York, Springer.

Examples

rectangles
rect_constr
rectangles2

Description

Computes the residuals by subtracting the configuration dissimilarites from the observed dissimilarities.

Usage

## S3 method for class 'smacof'
residuals(object, ...)
## S3 method for class 'smacofR'
residuals(object, ...)
## S3 method for class 'smacofID'
residuals(object, ...)

Arguments

Examples

res <- mds(kinshipdelta)
residuals(res)

Description

Data from RockHard Magazine: In this German Heavy Metal Magazine around 50 records are rated by the writers on a scale from (0 ... worst to 10 ... best) each month. The dataset contains all ratings from 2013.

Usage

data(RockHard)

Format

Data frame with raters in the colums, bands/albums in the rows.

References

Mair, P., de Leeuw, J., & Wurzer, M. (2015). Multidimensional Unfolding. Wiley StatsRef: Statistics Reference Online. New York: Wiley.

Examples

head(RockHard)

Description

Utility function for converting similarities into dissimilarities. Different methods are provided.

Usage

sim2diss(s, method = "corr", to.dist = FALSE)

Arguments

Details

The conversion formulas for the various methods can be found in the package vignette.

Value

Returns dissimiarities either as matrix or as dist object.

References

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2. Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

Examples

## Convert crimes data (correlations)
data(crimes)
crimeD <- sim2diss(crimes, method = "corr", to.dist = TRUE)

## Convert Wish data (similarities) by subtracting from 7
data(wish)
wishD <- sim2diss(wish, method = 7, to.dist = TRUE)

## Convert Ekman data (similarities) into dissimilarities
data(ekman)
ekmanD <- sim2diss(ekman, method = "confusion", to.dist = TRUE)

## Convert album ratings (rectangular similarities) by reversing the ratings
data(RockHard)
rockD1 <- sim2diss(RockHard[,5:18], method = "reverse")

Description

SMACOF with internal constraints on the configurations.

Usage

smacofConstraint(delta, constraint = "unrestricted", external, ndim = 2, 
                 type = c("ratio", "interval", "ordinal", "mspline"), weightmat = NULL,
                 init = NULL, ties = "primary", verbose = FALSE, modulus = 1, 
                 itmax = 1000, eps = 1e-6, spline.intKnots = 4, spline.degree = 2, 
                 constraint.type = c("ratio", "interval", "ordinal", "spline", 
                 "mspline"), constraint.ties = "primary", 
                 constraint.spline.intKnots = 2, constraint.spline.degree = 2)

Arguments

Details

The argument external is mandatory to specify and requires a data frame (or matrix) of dimension (n x q). Alternatively, for simplex fitting the user can specify a list of the following structure: external = list("simplex", dim2) with dim2 denoting the dimension of the simplex with dim2 < n. For a circumplex fitting, the list has to be of the following form: external = list("circumplex", dim2, k1, k2) with $1 \leq k1 \leq k2 \leq n$ (see also examples section). k1 and k2 denote the circumplex width.

In constraint smacof, the configuration matrix $X$ is subject to a constraint based on the external scales (predictors $Z$ specified using external) of the following linear form: $X = ZC$. The type of constraint in $C$ can be specified using the constraint argument. We provide the following standard setting:

For constraint = "unrestricted", $C$ is unrestricted. Note that "linear" still works as well for backward compatibility.

The same for constraint = "diagonal" where $X$ needs to be of dimension $(n x q)$ where $q$ is the number of columns of the external scale matrix (and thus number of dimensions). Here, $C$ is restricted to be diagonal.

For constraint = "unrestricted" or "diagonal", the external covariates $Z$ can be optimally transformed as specified by constraint.type. Choosing the number of covariates equal to the number of dimensions together with constraint.type = "ordinal", constraint.ties = "primary" will effectively restrict the configuration to parallel regions defined by the categories of the covariates. Note that missing values of the covariates are estimated by the model.

For constraint = "unique" we get the Bentler-Weeks uniqueness model. Hence $X$ is of dimension $(n x (n + p))$. This implies that we fit a certain number of dimensions p and, in addition we extract n additional dimensions where each object is scored on a separate dimension. More technical details can be found in the corresponding JSS article (reference see below).

In addition, the user can specify his own constraint function with the following arguments: configuration matrix with starting values (init) (mandatory in this case), matrix $V$ (weightmat; based on the weight matrix, see package vignette), external scale matrix (external). The function must return a matrix of resulting configurations.

If no starting configuration is provided, a random starting solution is used. In most applications, this is not a good idea in order to find a well fitting model. The user can fit an exploratory MDS using mds() first, and use the resulting configurations as starting configuration for smacofConstraint(). Alternatively, if the user has starting configurations determined by some underlying theory, they can be used as well.

Value

References

De Leeuw, J. & Mair, P. (2009). Multidimensional scaling using majorization: The R package smacof. Journal of Statistical Software, 31(3), 1-30, doi:10.18637/jss.v031.i03

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2. Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

De Leeuw, J., & Heiser, W. (1980). Multidimensional scaling with restrictions on the configurations. In P. R. Krishnaiah (eds.), Multivariate Analysis V, pp. 501-522. North-Holland.

Borg, I., & Lingoes, J. C. (1980). A model and algorithm for multidimensional scaling with external constraints on the distances. Psychometrika, 45, 25-38.

See Also

smacofSym, smacofRect, smacofIndDiff, smacofSphere

Examples

## theoretical grid restrictions (rectangles; keep covariate ties tied)
fit.rect1 <- mds(rectangles, type = "ordinal", init = rect_constr) 
fit.rect2 <- smacofConstraint(rectangles, type = "ordinal", ties = "secondary",
                        constraint = "diagonal", init = fit.rect1$conf, 
                        external = rect_constr, constraint.type = "ordinal")
plot(fit.rect2)

## regional restrictions morse code data (signal length, strength)
fitMorse1 <- mds(morse, type = "ordinal")
fitMorse1
fitMorse2 <- smacofConstraint(morse, type = "ordinal", constraint = "unrestricted",
                              external = morsescales[,2:3], 
                              constraint.type = "ordinal", 
                              init = fitMorse1$conf)
fitMorse2
plot(fitMorse2)

## facial expression data I (axial restriction, C diagonal)
Delta <- FaceExp
attr(Delta, "Labels") <- NULL            
fitFace <- mds(Delta, type = "ordinal")   ## starting solution
Z <- FaceScale[, c(1,3)]                  ## external variables
fitFaceC1 <- smacofConstraint(Delta, type = "ordinal", 
  constraint = "diagonal", external = Z, constraint.type = "ordinal", 
  init = fitFace$conf)
fitFaceC1$C 
plot(fitFaceC1, xlab = "Pleasant-Unpleasant", ylab = "Tension-Sleep", 
  main = "Face Expression (Diagonal Restriction)")

## facial expression data II (C unrestricted)
fitFaceC3 <- smacofConstraint(Delta, type = "ordinal", 
  constraint = "unrestricted", external = Z, constraint.type = "ordinal", 
  init = fitFace$conf)
fitFaceC3$C   
plot(fitFaceC3, main = "Face Expression (C Unrestricted, Ordinal Transformation)")

Description

Performs smacof for individual differences also known as Three-Way smacof on a list of dissimilarity matrices. Various restrictions decompositions and restrictions on the weight matrix are provided. The most prominent models are INDSCAL and IDIOSCAL.

Usage

smacofIndDiff(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"), 
              constraint = c("indscal", "idioscal", "identity"),
              weightmat = NULL, init = "torgerson", ties = "primary",  
              verbose = FALSE, modulus = 1, itmax = 1000, eps = 1e-6,
              spline.degree = 2, spline.intKnots = 2)
              
indscal(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"), 
        weightmat = NULL, init = "torgerson", ties = "primary", 
        verbose = FALSE, modulus = 1, itmax = 1000, eps = 1e-6,
        spline.degree = 2, spline.intKnots = 2)

idioscal(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"), 
         weightmat = NULL, init = "torgerson", ties = "primary", 
         verbose = FALSE, modulus = 1, itmax = 1000, eps = 1e-6,
         spline.degree = 2, spline.intKnots = 2)

Arguments

Details

If the constraint is "indscal", INDSCAL is performed with configuration weight matrices restricted to be diagonal. indscal() is a corresponding wrapper function that can be used instead of smacofIndDiff() with "indscal" constraints.

IDIOSCAL can be computed using the "idioscal" argument. The weight matrices are then unconstrained. idioscal() is a corresponding wrapper function that can be used instead of smacofIndDiff() with "idioscal" constraints.

Addtional weight restrictions can be imposed with "identity" which restricts the configurations across individuals/replications/ways to be equal.

Value

Author(s)

Jan de Leeuw and Patrick Mair

References

De Leeuw, J., & Mair, P. (2009). Multidimensional scaling using majorization: The R package smacof. Journal of Statistical Software, 31(3), 1-30, doi:10.18637/jss.v031.i03

See Also

smacofConstraint, smacofSym, smacofRect, smacofSphere

Examples

## Example 1: rectangle perception data
res.diag <- indscal(perception, type = "ordinal")        ## INDSCAL
res.diag$cweights
plot(res.diag)
plot(res.diag, type = "p", pch = 25, col = 4, label.conf = list(label = TRUE, pos = 3, col = 4))

res.idio <- idioscal(perception, type = "ordinal")  ## IDIOSCAL
Wk <- res.idio$cweights
G <- res.idio$gspace
G
G

## identity restricted weights
res.id <- smacofIndDiff(perception, type = "ordinal", constraint = "identity")     
summary(res.id)
res.id$cweights
plot(res.id)
plot(res.id, type = "p", pch = 25, col = 4, label.conf = list(label = TRUE, pos = 3, col = 4))


## Example 2: Helm's color data 
res.helm <- indscal(helm, type = "interval")
plot(res.helm, plot.type = "confplot")
barplot(sort(res.helm$sps, decreasing = TRUE), main = "Stress per Subject", cex.names = 0.8) 
plot(res.helm, plot.type = "bubbleplot")
plot(res.helm, plot.type = "stressplot")
plot(res.helm, plot.type = "Shepard")

## idioscal and indscal with random starting configuration:
set.seed(123)
startconf <- matrix(rnorm(20), 10, 2)
idioscal(helm, init = startconf, type = "interval")
indscal(helm, init = startconf, type = "interval")

Description

Dual and primal approach for spherical SMACOF.

Usage

smacofSphere(delta, ndim = 2, type = c("ratio", "interval", "ordinal","mspline"), 
             algorithm = c("dual", "primal"), weightmat = NULL, 
             init = "torgerson", ties = "primary", verbose = FALSE, penalty = 100, 
             relax = FALSE, modulus = 1, itmax = 1000, eps = 1e-6,
             spline.degree = 2, spline.intKnots = 2)

Arguments

Details

For large scale problems it is suggested to use the dual algorithm. Using the penalty parameter (dual algorithm), the user allow for slight point deviations from the circle (the higher the penalty, the stricter the algorithm is in terms of placing points in the sphere, see examples section below).

Value

Author(s)

Jan de Leeuw and Patrick Mair

References

De Leeuw, J. & Mair, P. (2009). Multidimensional scaling using majorization: The R package smacof. Journal of Statistical Software, 31(3), 1-30, doi:10.18637/jss.v031.i03

See Also

smacofRect, smacofIndDiff, smacofSym,smacofConstraint

Examples

## spherical SMACOF solution for trading data
## dual algorithm
res <- smacofSphere(trading, type = "ordinal")  
res
plot(res)

## lower penalty
res <- smacofSphere(trading, penalty = 20, type = "ordinal")  
res
plot(res)

## primal algorithm, interval
res <- smacofSphere(trading, type = "interval", algorithm = "primal")  
res

Description

Multidimensional scaling on a symmetric dissimilarity matrix using SMACOF.

Usage

smacofSym(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"), 
          weightmat = NULL, init = "torgerson", ties = "primary", principal = FALSE, 
          verbose = FALSE, relax = FALSE, modulus = 1, itmax = 1000, eps = 1e-06, 
          spline.degree = 2, spline.intKnots = 2)

mds(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"), 
    weightmat = NULL, init = "torgerson", ties = "primary", principal = FALSE, 
    verbose = FALSE, relax = FALSE, modulus = 1, itmax = 1000, eps = 1e-06, 
    spline.degree = 2, spline.intKnots = 2)

Arguments

Details

The function mds() is a wrapper function and can be used instead of smacofSym(). It reports the Stress-1 value (normalized). The main output are the coordinates in the low-dimensional space (configuration; conf; see also plot.smacof).

Four types of MDS can be fitted: ratio MDS (no dissimilarity transformation), interval MDS (linear transformation), ordinal MDS (ordinal transformation with various options for handling ties), and spline MDS (monotone spline transformation). Shepard plots in plot.smacof give insight into this transformation.

Setting principal = TRUE is useful for interpretatbility of the dimensions, or to check hypotheses about the dimensions.

In case of missing input dissimilarities, the weightmat is computed internally so that missings are blanked out during optimization.

Value

Author(s)

Jan de Leeuw, Patrick Mair, and Patrick Groenen

References

De Leeuw, J. & Mair, P. (2009). Multidimensional scaling using majorization: The R package smacof. Journal of Statistical Software, 31(3), 1-30, doi:10.18637/jss.v031.i03

Mair, P, Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2. Journal of Statistical Software, 102(10), 1-47. doi:10.18637/jss.v102.i10

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

Borg, I., Groenen, P. J. F., & Mair, P. (2018). Applied Multidimensional Scaling and Unfolding (2nd ed.). Springer.

See Also

smacofConstraint, smacofRect, smacofIndDiff, smacofSphere, plot.smacof

Examples

## simple SMACOF solution (interval MDS) for kinship data
res <- mds(kinshipdelta, type = "interval")
res
summary(res)
plot(res)
plot(res, type = "p", label.conf = list(label = TRUE, col = "darkgray"), pch = 25, col = "red")

## ratio MDS, random starts
set.seed(123)
res <- mds(kinshipdelta, init = "random")
res

## 3D ordinal SMACOF solution for trading data (secondary approach to ties)
data(trading)
res <- mds(trading, ndim = 3, type = "ordinal", ties = "secondary")
res

## spline MDS 
delta <- sim2diss(cor(PVQ40agg))
res <- mds(delta, type = "mspline", spline.degree = 3, spline.intKnots = 4)
res
plot(res, "Shepard")

Description

A distance matrix for the 10 brightest stars in each of the 12 zodiac signs was computed. Astronomers measure the projected positions of objects on the celestial sphere in two angles, i.e. right ascension $\alpha$ and declination $\delta$. For every zodiac sign, the projected distances on the sky between individual stars $S_{i}$ and $S_{j}$ have been calculated in decimal degrees by means of the Pythagorean theorem

$d_{i,j} = \sqrt {\left( {\alpha _i - \alpha _j } \right)2 + \left( {\delta _i - \delta _j } \right)2 }$

assuming planar geometry. Since the zodiac signs are relatively small compared to the whole celestial sphere and the computation is only done for illustrative purposes, such a simplified assumption is appropriate.

Usage

data(stardist)

Format

A dist object contaning the star distances.

Note

Thanks to Paul Eigenthaler, Department of Astronomy, University of Vienna for calculating the distances.

Examples

stardist

Description

Computes the stress for 0 iterations based on a starting configuration provided by the user.

Usage

stress0(delta, init, type = c("interval", "ratio", "ordinal", "mspline"), 
        weightmat = NULL, ties = "primary", spline.degree = 2, spline.intKnots = 2)

Arguments

Details

Computes stress-1 for a particular starting configuration the user needs to provide. It can also be helpful if the user wants to force a particular theoretical configuration (as specified in inits) on the data.

Value

See Also

mds

Examples

## rectangle starting solution
rect_constr
stress0(rectangles, init = rect_constr)

## torgerson starting solution
tstart <- torgerson(rectangles)
stress0(rectangles, init = tstart)

Description

Print and summary methods for objects of class smacofB, smacofR (rectangular), and smacofID (individual differences).

Usage

## S3 method for class 'smacofB'
summary(object, ...)
## S3 method for class 'smacofB'
print(x, ...)
## S3 method for class 'smacofR'
summary(object, ...)
## S3 method for class 'smacofR'
print(x, ...)
## S3 method for class 'smacofID'
summary(object, ...)
## S3 method for class 'smacofID'
print(x, ...)

Arguments

Examples

data(kinshipdelta)
res <- smacofSym(kinshipdelta)
res
summary(res)

Description

Plots 2D MDS configuration including facets as determined by an SVM.

Usage

svm_mdsplot(mds_object, svm_object, class, legend1 = TRUE, legend2 = TRUE, 
            inset = c(-0.2, 0.5), plot.dim = c(1,2), by = 0.01, 
            main, xlab, ylab, xlim, ylim, ...)

Arguments

Details

Using the SVM implementation of e1071 one can determine facets in an MDS configuration based on an SVM fit. This function plots the resulting facets on top of the 2D MDS configuration. Note that this function is work in progress.

References

Mair, P., Cetron, J. S., and Borg, I. (2022). Using support vector machines for facet partitioning in multidimensional scaling. Multivariate Behavioral Research. Forthcoming; doi:10.1080/00273171.2022.2035207

See Also

svm, tune.svm

Examples

## Guttman intelligence data
Delta <- sim2diss(Guttman1965[[1]])
class <- Guttman1965[[2]]

## ordinal MDS fit
mds_gut <- mds(Delta, ndim = 2, type = "ordinal")  
mds_gut
cols <- rainbow_hcl(4)[as.numeric(class)]
plot(mds_gut, col = cols, label.conf = list(col = cols))
legend("bottomright", legend = levels(class), cex = 0.7, col = rainbow_hcl(4), pch = 19)

## radial SVM fit
X <- mds_gut$conf                    ## extract configuration
dat <- data.frame(class = class, X)  ## merge with class vector
costvec <- 2^seq(-4, 4)              ## tuning parameter grid
gamma <- seq(0.01, 0.5, 10)

set.seed(111)
svm_gut <- tune.svm(class ~ D1 + D2, data = dat, kernel = "radial", 
                    cross = 10, cost = costvec)$best.model
svm_gut
preds <- predict(svm_gut, data = dat)  ## predicted classes
table(obs = class, pred = preds)       ## confusion matrix

svm_mdsplot(mds_gut, svm_gut, dat$class, inset = c(-0.3, 0.5))

Description

Additive decomposition of an asymmetric, square proximity matrix into a symmetric matrix and an skew-symmetric matrix

Usage

symdecomp(P)

Arguments

Details

Performs the decompositon $P = M + N$ (M and N are orthogonal).

Value

Returns the following matrices:

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

Examples

P <- matrix(c(92,5,4,8,4,84,38,62,6,37,87,17,13,31,17,88), ncol = 4)
symdecomp(P)

Description

Classical MDS aka Torgerson Scaling

Usage

torgerson(delta, p)

Arguments

Value

Returns an $n \times p$ matrix of configurations

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

Examples

fit <- torgerson(Guerry)

Description

Data from the New Geographical Digest (1986) analyzed in Cox and Cox (2001). For 20 countries their main trading partners were dichotomously scored (1 = trade performed, 0 = trade not performed). Based on this dichotomous matrix the dissimilarities were computed using the Jaccard coefficient.

Usage

data(trading)

Format

Object of class "dist" with dissimilarities of the following countries:

Arge: Argentina

Aust: Australia

Braz: Brazil

Cana: Canada

Chin: China

Czec: Czechoslovakia

Egyp: Egypt

E.Ge: East Germany

Fran: France

Hung: Hungary

Indi: India

Ital: Italy

Japa: Japan

N.Ze: New Zealand

Pola: Poland

Swed: Sweden

USA

USSR: Soviet Union

U.K.: United Kingdom

W.Ge: West Germany

References

Cox, T.F., Cox, M.A.A. (1991). Multidimensional scaling on a sphere. Communications in Statistics: Theory and Methods, 20, 2943-2953.

Examples

data(trading)

Description

Utility functions for optimal scaling calls (used internally)

Usage

transPrep(x, trans = "ordinals", spline.intKnots = 4, spline.degree = 2, missing = "none")
transform(Target, x, w = rep(1,length(x$x)), normq = 0)

Arguments

Description

Variant of smacof for rectangular matrices (typically ratings, preferences) that allows for nonmetric transformations. Also known as nonmetric unfolding.

Usage

unfolding(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"),
         conditionality = "unconditional", lambda = 0.5, omega = 1, 
         circle = c("none", "row", "column"), weightmat = NULL, init = NULL, 
         fixed = c("none", "row", "column"), fixed.coord = NULL,
         ties = c("primary", "secondary"), verbose = FALSE, relax = TRUE, 
         itmax = 10000, eps = 1e-6, spline.degree = 2, spline.intKnots = 2,
         parallelize = FALSE)

smacofRect(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"),
         conditionality = "unconditional", lambda = 0.5, omega = 1, 
         circle = c("none", "row", "column"), weightmat = NULL, init = NULL, 
         fixed = c("none", "row", "column"), fixed.coord = NULL,
         ties = c("primary", "secondary"), verbose = FALSE, relax = TRUE, 
         itmax = 10000, eps = 1e-6, spline.degree = 2, spline.intKnots = 2,
         parallelize = FALSE)

prefscal(delta, ndim = 2, type = c("ratio", "interval", "ordinal", "mspline"),
         conditionality = "unconditional", lambda = 0.5, omega = 1, 
         circle = c("none", "row", "column"), weightmat = NULL, init = NULL, 
         fixed = c("none", "row", "column"), fixed.coord = NULL,
         ties = c("primary", "secondary"), verbose = FALSE, relax = TRUE, 
         itmax = 10000, eps = 1e-6, spline.degree = 2, spline.intKnots = 2,
         parallelize = FALSE)