mdscale
Nonclassical multidimensional scaling
Syntax
Y = mdscale(D,p)
[Y,stress] = mdscale(D,p)
[Y,stress,disparities] = mdscale(D,p)
[...] = mdscale(D,p,'Name',value)
Description
Y = mdscale(D,p) performs nonmetric multidimensional
scaling on the n-by-n dissimilarity
matrix D, and returns Y, a configuration
of n points (rows) in p dimensions
(columns). The Euclidean distances between points in Y approximate
a monotonic transformation of the corresponding dissimilarities in D.
By default, mdscale uses Kruskal's normalized stress1
criterion.
You can specify D as either a full n-by-n matrix,
or in upper triangle form such as is output by pdist.
A full dissimilarity matrix must be real and symmetric, and have zeros
along the diagonal and non-negative elements everywhere else. A dissimilarity
matrix in upper triangle form must have real, non-negative entries. mdscale treats NaNs
in D as missing values, and ignores those elements. Inf is
not accepted.
You can also specify D as a full similarity
matrix, with ones along the diagonal and all other elements less than
one. mdscale transforms a similarity matrix to
a dissimilarity matrix in such a way that distances between the points
returned in Y approximate sqrt(1-D).
To use a different transformation, transform the similarities prior
to calling mdscale.
[Y,stress] = mdscale(D,p) returns the minimized
stress, i.e., the stress evaluated at Y.
[Y,stress,disparities] = mdscale(D,p) returns
the disparities, that is, the monotonic transformation of the dissimilarities D.
[...] = mdscale(D,p,'Name',value) specifies
one or more optional parameter name/value pairs that control further
details of mdscale. Specify Name in
single quotes. Available parameters are
Criterion— The goodness-of-fit
criterion to minimize. This also determines the type of scaling, either
non-metric or metric, that mdscale performs. Choices
for non-metric scaling are:
'stress' — Stress normalized
by the sum of squares of the inter-point distances, also known as
stress1. This is the default.
'sstress' — Squared stress,
normalized with the sum of 4th powers of the inter-point distances.
Choices for metric scaling are:
'metricstress' — Stress,
normalized with the sum of squares of the dissimilarities.
'metricsstress' — Squared
stress, normalized with the sum of 4th powers of the dissimilarities.
'sammon' — Sammon's nonlinear
mapping criterion. Off-diagonal dissimilarities must be strictly positive
with this criterion.
'strain' — A criterion equivalent
to that used in classical multidimensional scaling.
Weights — A matrix or vector
the same size as D, containing nonnegative dissimilarity
weights. You can use these to weight the contribution of the corresponding
elements of D in computing and minimizing stress.
Elements of D corresponding to zero weights are
effectively ignored.
Note
When you specify weights as a full matrix, its diagonal elements
are ignored and have no effect, since the corresponding diagonal elements
of D do not enter into the stress calculation.
Start — Method used to choose
the initial configuration of points for Y. The choices are
'cmdscale' — Use the classical
multidimensional scaling solution. This is the default. 'cmdscale' is
not valid when there are zero weights.
'random' — Choose locations
randomly from an appropriately scaled p-dimensional normal
distribution with uncorrelated coordinates.
An n-by-p matrix
of initial locations, where n is the size of the matrix D and p is
the number of columns of the output matrix Y. In
this case, you can pass in [] for p and mdscale infers p from
the second dimension of the matrix. You can also supply a 3-D array,
implying a value for 'Replicates' from the array's
third dimension.
Replicates — Number of times
to repeat the scaling, each with a new initial configuration. The
default is 1.
Options — Options for the
iterative algorithm used to minimize the fitting criterion. Pass in
an options structure created by statset.
For example,
opts = statset(param1,val1,param2,val2, ...);
[...] = mdscale(...,'Options',opts)
The choices of statset parameters are
'Display' — Level of display
output. The choices are 'off' (the default), 'iter',
and 'final'.
'MaxIter' — Maximum number
of iterations allowed. The default is 200.
'TolFun' — Termination tolerance
for the stress criterion and its gradient. The default is 1e-4.
'TolX'— Termination tolerance
for the configuration location step size. The default is 1e-4.
Examples
load cereal.mat
X = [Calories Protein Fat Sodium Fiber ...
Carbo Sugars Shelf Potass Vitamins];
% Take a subset from a single manufacturer.
X = X(strcmp('K',cellstr(Mfg)),:);
% Create a dissimilarity matrix.
dissimilarities = pdist(X);
% Use non-metric scaling to recreate the data in 2D,
% and make a Shepard plot of the results.
[Y,stress,disparities] = mdscale(dissimilarities,2);
distances = pdist(Y);
[dum,ord] = sortrows([disparities(:) dissimilarities(:)]);
plot(dissimilarities,distances,'bo', ...
dissimilarities(ord),disparities(ord),'r.-');
xlabel('Dissimilarities'); ylabel('Distances/Disparities')
legend({'Distances' 'Disparities'},'Location','NW');
% Do metric scaling on the same dissimilarities.
figure
[Y,stress] = ...
mdscale(dissimilarities,2,'criterion','metricsstress');
distances = pdist(Y);
plot(dissimilarities,distances,'bo', ...
[0 max(dissimilarities)],[0 max(dissimilarities)],'r.-');
xlabel('Dissimilarities'); ylabel('Distances')
Introduced before R2006a