mmds Metric (classical) Multidimensional Scaling (a.k.a Principal Coordinate Analysis) of a (Euclidean) Distance Matric
Perform an MMDS of a (Euclidean) distance matrix measured between a set of weighted objects.
MMDS Give factor scores that make it possible to draw a map of the objects such that the distances between objects on the map best approximate the original distances between objects.
Method: Transform the distance matrix into a (double centered)
covariance matrix which is then analyze via its eigen-decomposition.
The factor score of each dimension are scaled such that their variance
(i.e., the sum ot their weighted squared factor scores) is
equal to the eigen-value of the corresponding dimension.
Note that if the masses vector is absent, equal masses (i.e. 1 divided by number of objects)
are used.
mmds(DistanceMatrix,masses=NULL)
DistanceMatrix |
. A squared (assumed to be Euclidean) distance matrix |
masses |
A vector of masses (i.e.,
non negative numbers with a sum of 1)
of same dimensionality as number of rows of
|
Sends back a list
LeF |
factor scores for the objects. |
eigenvalues |
the eigenvalues for the factor scores (ie.a variance). |
tau |
the percentage of explained variance of each dimension. |
Contributions |
give the proporion of explained variance by an object for a dimension. |
Herve Abdi
The procedure and references are detailled in: Abdi, H. (2007). Metric multidimensional scaling. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 598–605.
(Paper available from www.utdallas.edu/~herve).
# An example of MDS from Abdi (2007)
# Discriminability of Brain States
# Table 1.
# 1. Get the distance matrix
D <- matrix(c(
0.00, 3.47, 1.79, 3.00, 2.67, 2.58, 2.22, 3.08,
3.47, 0.00, 3.39, 2.18, 2.86, 2.69, 2.89, 2.62,
1.79, 3.39, 0.00, 2.18, 2.34, 2.09, 2.31, 2.88,
3.00, 2.18, 2.18, 0.00, 1.73, 1.55, 1.23, 2.07,
2.67, 2.86, 2.34, 1.73, 0.00, 1.44, 1.29, 2.38,
2.58, 2.69, 2.09, 1.55, 1.44, 0.00, 1.19, 2.15,
2.22, 2.89, 2.31, 1.23, 1.29, 1.19, 0.00, 2.07,
3.08, 2.62, 2.88, 2.07, 2.38, 2.15, 2.07, 0.00),
ncol = 8, byrow=TRUE)
rownames(D) <- c('Face','House','Cat','Chair','Shoe','Scissors','Bottle','Scramble')
colnames(D) <- rownames(D)
# 2. Call mmds
BrainRes <- mmds(D)
# Note that compared to Abdi (2007)
# the factor scores of mmds are equal to F / sqrt(nrow(D))
# the eigenvalues of mmds are equal to \Lambda *{1/nrow(D)}
# (ie., the normalization differs but the results are proportional)
# 3. Now a pretty plot with the prettyPlot function from prettyGraphs
prettyPlot(BrainRes$FactorScore,
display_names = TRUE,
display_points = TRUE,
contributionCircles = TRUE,
contributions = BrainRes$Contributions)
# 4. et Voila!Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.