NAG Library Routine Document
G03FCF performs non-metric (ordinal) multidimensional scaling.
|SUBROUTINE G03FCF (
||TYP, N, NDIM, D, X, LDX, STRESS, DFIT, ITER, IOPT, WK, IWK, IFAIL)
||N, NDIM, LDX, ITER, IOPT, IWK(N*(N-1)/2+N*NDIM+5), IFAIL
||D(N*(N-1)/2), X(LDX,NDIM), STRESS, DFIT(2*N*(N-1)), WK(15*N*NDIM)
For a set of
objects, a distance or dissimilarity matrix
can be calculated such that
is a measure of how ‘far apart’ the objects
have been recorded for each observation this measure may be based on Euclidean distance,
, or some other calculation such as the number of variables for which
. Alternatively, the distances may be the result of a subjective assessment. For a given distance matrix, multidimensional scaling produces a configuration of
points in a chosen number of dimensions,
, such that the distance between the points in some way best matches the distance matrix. For some distance measures, such as Euclidean distance, the size of distance is meaningful, for other measures of distance all that can be said is that one distance is greater or smaller than another. For the former metric scaling can be used, see G03FAF
, for the latter, a non-metric scaling is more appropriate.
For non-metric multidimensional scaling, the criterion used to measure the closeness of the fitted distance matrix to the observed distance matrix is known as STRESS
is given by,
is the Euclidean squared distance between points
is the fitted distance obtained when
is monotonically regressed on
, that is
is monotonic relative to
and is obtained from
with the smallest number of changes. So STRESS
is a measure of by how much the set of points preserve the order of the distances in the original distance matrix. Non-metric multidimensional scaling seeks to find the set of points that minimize the STRESS
An alternate measure is squared STRESS
in which the distances in STRESS
are replaced by squared distances.
In order to perform a non-metric scaling, an initial configuration of points is required. This can be obtained from principal coordinate analysis, see G03FAF
. Given an initial configuration, G03FCF uses the optimization routine E04DGF/E04DGA
to find the configuration of points that minimizes STRESS
. The routine E04DGF/E04DGA
uses a conjugate gradient algorithm. G03FCF will find an optimum that may only be a local optimum, to be more sure of finding a global optimum several different initial configurations should be used; these can be obtained by randomly perturbing the original initial configuration using routines from Chapter G05
Chatfield C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press
- 1: TYP – CHARACTER(1)Input
: indicates whether STRESS
is to be used as the criterion.
- STRESS is used.
- is used.
- 2: N – INTEGERInput
On entry: , the number of objects in the distance matrix.
- 3: NDIM – INTEGERInput
On entry: , the number of dimensions used to represent the data.
- 4: D() – REAL (KIND=nag_wp) arrayInput
: the lower triangle of the distance matrix
stored packed by rows. That is
is missing then set
; for further comments on missing values see Section 8
- 5: X(LDX,NDIM) – REAL (KIND=nag_wp) arrayInput/Output
th row must contain an initial estimate of the coordinates for the
th point, for
. One method of computing these is to use G03FAF
On exit: the
th row contains coordinates for the th point, for .
- 6: LDX – INTEGERInput
: the first dimension of the array X
as declared in the (sub)program from which G03FCF is called.
- 7: STRESS – REAL (KIND=nag_wp)Output
: the value of STRESS
at the final iteration.
- 8: DFIT() – REAL (KIND=nag_wp) arrayOutput
: auxiliary outputs.
, the first
elements contain the distances,
, for the points returned in X
, the second set of
contains the distances
ordered by the input distances,
, the third set of
elements contains the monotonic distances,
, ordered by the input distances,
and the final set of
elements contains fitted monotonic distances,
, for the points in X
corresponding to distances which are input as missing are set to zero.
If , the results are as above except that the squared distances are returned.
Each distance matrix is stored in lower triangular packed form in the same way as the input matrix .
- 9: ITER – INTEGERInput
: the maximum number of iterations in the optimization process.
- A default value of is used.
- A default value of (the default for E04DGF/E04DGA) is used.
- 10: IOPT – INTEGERInput
: selects the options, other than the number of iterations, that control the optimization.
- Default values are selected as described in Section 8. In particular if an accuracy requirement of is selected, see Section 7.
- The default values are used except that the accuracy is given by where .
- The option setting mechanism of E04DGF/E04DGA can be used to set all options except Iteration Limit; this option is only recommended if you are an experienced user of NAG optimization routines. For further details see E04DGF/E04DGA.
- 11: WK() – REAL (KIND=nag_wp) arrayWorkspace
- 12: IWK() – INTEGER arrayWorkspace
- 13: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|or|| or ,|
|On entry,||all elements of .|
The optimization has failed to converge in ITER
function iterations. Try either increasing the number of iterations using ITER
or increasing the value of
, given by IOPT
, used to determine convergence. Alternatively try a different starting configuration.
The conditions for an acceptable solution have not been met but a lower point could not be found. Try using a larger value of
, given by IOPT
The optimization cannot begin from the initial configuration. Try a different set of points.
The optimization has failed. This error is only likely if
. It corresponds to
After a successful optimization the relative accuracy of STRESS
should be approximately
, as specified by IOPT
The optimization routine E04DGF/E04DGA
used by G03FCF has a number of options to control the process. The options for the maximum number of iterations (Iteration Limit
) and accuracy (Optimality Tolerance
) can be controlled by ITER
respectively. The printing option (Print Level
) is set to
to give no printing. The other option set is to stop the checking of derivatives (
) for efficiency. All other options are left at their default values. If however
is used, only the maximum number of iterations is set. All other options can be controlled by the option setting mechanism of E04DGF/E04DGA
with the defaults as given by that routine.
Missing values in the input distance matrix can be specified by a negative value and providing there are not more than about two thirds of the values missing the algorithm may still work. However the routine G03FAF
does not allow for missing values so an alternative method of obtaining an initial set of coordinates is required. It may be possible to estimate the missing values with some form of average and then use G03FAF
to give an initial set of coordinates.
The data, given by Krzanowski (1990)
, are dissimilarities between water vole populations in Europe. Initial estimates are provided by the first two principal coordinates computed.
9.1 Program Text
Program Text (g03fcfe.f90)
9.2 Program Data
Program Data (g03fcfe.d)
9.3 Program Results
Program Results (g03fcfe.r)