Multidimensional least-squares scaling etc.
LSCAL <distance_matrix>,<initial_coordinates>,L
computes on the basis of a given n*n dissimilarity or <distance matrix>
a configuration of n points in an m dimensional space. The distances of
points in that configuration should be as close to the given distances
as possible. <initial coordinates> gives initial estimates of the
configuration as a matrix file. # of columns in <initial coordinates>
gives the dimension m.
The result CSCAL.M of classical multidimensional scaling obtained by
the sucro /CSCAL (or /CSCAL2 when n is very large (m>1000) and m is
small, typically m=2) is often a good choice for <initial coordinates>.
The initial solution is improved iteratively by using the least squares
(or other) criteria. Thus by default, the squared sum of differences
between the true distances and distances given by estimated coordinates
should be minimized. Since the object function generally has many local
minima, several initial coordinates should be employed.
The squared sum of distance differences can be weighted by a n*n weight
matrix given by WEIGHTS=<matrix_name>. For example, weights could be
inverses of given distances. By default, weights are 1.
In the default case (no weights, least squares criterion, no additive
constant) a conjugate gradient method is used.
Otherwise Powell's method (without analytic gradient vector) is used.
Options for optimization:
METHOD=1 conjugate gradient method
METHOD=12 conjugate gradient method (weights 1/d^2)
METHOD=13 conjugate gradient method (weights 1/d)
METHOD=2 Powell's method
METHOD=3 polytope algorithm of Nelder and Mead
An additive constant C for transforming the original distances D -> D+C
can also be estimated by giving CONSTANT=C .
Missing values in the <distance matrix> are given as negative numbers.
Default metrics for distances is Euclidean. Another metrics is selected
by METRICS=Lp where p=1 means city-block distance and p=2 Euclidean
distance.
For general p>0, corresponding Minkowski metrics is used.
METRICS=MAD and METRICS=ABS are equivalent to METRICS=L1.
METRICS=MAX implies maximum difference in coordinate values to be used
as a distance.
The goodness-of-fit measure for comparing given and estimated distances
is the ordinary least squares criterion. This can be replaced by a
CRITERION specification with the same alternatives as METRICS.
For example, CRITERION=L2 is default.
LSCAL gives its results as matrix files
LSCAL.M estimated configuration matrix,
LSDIST.M reproduced distances.
LSCAL.M is centered to the origin and rotated to principal axes.
By default, it is assumed that the distance matrix is symmetric.
If it is not, set SYMMETRIC=0.
1 = More information on additional multivariate operations
M = More information on multivariate analysis
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