smacofRect: Nonmetric unfolding

In smacof: Multidimensional Scaling

Nonmetric unfolding

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)

Arguments

delta	Data frame or matrix of preferences, ratings, dissimilarities.
ndim	Number of dimensions.
type	MDS type: "interval", "ratio", "ordinal", or "mspline".
conditionality	A single transformations are applied for the entire matrix "unconditional", or for each row separately "row".
lambda	Penalty strength balancing the loss contribution of stress and the penalty (see details).
omega	Penalty width determines for what values of the variation coefficient the penalty should become active (see details).
circle	If "column", the column configurations are restricted to be on a circle, if "row", row configurations are on a circle, if "none", there are no restrictions on row and column configurations
weightmat	Optional matrix with dissimilarity weights.
init	Optional list of length two with starting values for the row coordinates (first element) and column coordinates (second element).
fixed	Do external unfolding by fixing the row coordinates, column coordinate, or choose none (default) to do normal unfolding. Even fixed coordinates are uniformly scaled by a constant to fit the loss function.
fixed.coord	Matrix with fixed coordinates of the appropriate size.
ties	Tie specification for ordinal transformations: primary unties the ties and secondary keeps the ties tied.
verbose	If TRUE, intermediate stress is printed out.
relax	If TRUE, block relaxation is used for majorization after 100 iterations. It tends to reduce the number of iterations by a factor 2.
itmax	Maximum number of iterations.
eps	Convergence criterion.
spline.degree	Degree of the spline for an "mspline" transformation.
spline.intKnots	Number of interior knots of the spline for a "mspline" transformation.
parallelize	Tries to parallelize the computations when conditionality = "row".

Details

Unfolding tries to match a rectangular matrix delta of dissimilarities between row and column objects by Euclidean distances between row and column points. Badness of fit is measured by raw Stress as the sum of squared differences between delta and the Euclidean distances. Instead of dissimilarities optimal transformations (dhats) can be found. The dhats should be a function of the original delta restricted to be "ratio", "interval", "ordinal", or "mspline". These transformations can be the same for the entire matrix (conditionality = "unconditional") of data, or different per row (conditionality = "row"). To avoid a degenerate solution with all dhats and distances equal to 1, the prefscal penalty is is used.

A penalty is added based on the variation coefficient of the dhats (mean dhat divided by the standard deviation of the dhats). The penalty width (omega) weights the penalty and determines from what value of the variation coefficient of the dhats the penalty should become active. The penalty strength (lambda) is needed to ensure that the penalty can be strong enough. Busing et al. (2005) suggest to use λ = 0.5 and ω = 1.0 (for unconditional solutions ω can be lowered to a value as low as 0.1).

External unfolding can be done by specifying fixed = "row" or "column" and providing the fixed coordinates in fixed.coord. Then, either the rows or columns are fixed up to a uniform constant.

Creates an object of class smacofR.

Value

obsdiss	Observed dissimilarities, corresponds to delta
confdist	Configuration dissimilarities
dhat	Matrix with optimal transformation of size delta
iord	List of size 1 for matrix conditional and size nrow(delta) for row conditional with the index that orders the dhats. Needed for the Shepard plot
conf.row	Matrix of final row configurations
conf.col	Matrix of final column configurations
stress	Final, normalized stress value
pstress	Penalized stress value (the criterion that is minimized)
spp.row	Stress per point, rows
spp.col	Stress per point, columns
congvec	Vector of congruency coefficients
ndim	Number of dimensions
model	Type of smacof model
niter	Number of iterations
nind	Number of individuals (rows)
trans	Transformation
conditionality	Conditionality of the transformation
nobj	Number of objects (columns)

Author(s)

Patrick Groenen, 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

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

Busing, F. M. T. A., Groenen, P. J. F., & Heiser, W. J. (2005). Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70, 71-98.

See Also

plot.smacof, smacofConstraint, smacofSym, smacofIndDiff, smacofSphere

Examples

## Ratio unfolding
res <- unfolding(breakfast)
res

## various configuration plots
plot(res)
plot(res, type = "p", pch = 25)
plot(res, type = "p", pch = 25, col.columns = 3, 
     label.conf.columns = list(label = TRUE, pos = 3, col = 3), 
     col.rows = 8, label.conf.rows = list(label = TRUE, pos = 3, col = 8))

## Shepard plot
plot(res, "Shepard")

## Stress decomposition chart
plot(res, "stressplot")

## Not run: 
## Ordinal unfolding, row-conditional
## Note that ordinal unfolding may need many iterations (several thousands)
res <- unfolding(breakfast, type = "ordinal", conditionality = "row", omega = 0.1, itmax = 3000)
res
plot(res, "Shepard")      ## Shepard plot
plot(res)

## End(Not run)

smacof documentation built on May 6, 2022, 9:05 a.m.