G03FAF performs a principal coordinate analysis also known as classical metric scaling.
For a set of
objects a distance matrix
can be calculated such that
is a measure of how ‘far apart’ are objects
and
(see
G03EAF for example). Principal coordinate analysis or metric scaling starts with a distance matrix and finds points
in Euclidean space such that those points have the same distance matrix. The aim is to find a small number of dimensions,
, that provide an adequate representation of the distances.
The principal coordinates of the points are computed from the eigenvectors of the matrix where
with denoting the average of over the suffix , etc.. The eigenvectors are then scaled by multiplying by the square root of the value of the corresponding eigenvalue.
Provided that the ordered eigenvalues,
, of the matrix
are all positive,
shows how well the data is represented in
dimensions. The eigenvalues will be non-negative if
is positive semidefinite. This will be true provided
satisfies the inequality:
for all
. If this is not the case the size of the negative eigenvalue reflects the amount of deviation from this condition and the results should be treated cautiously in the presence of large negative eigenvalues. See
Krzanowski (1990) for further discussion. G03FAF provides the option for all eigenvalues to be computed so that the smallest eigenvalues can be checked.
Gower J C (1966) Some distance properties of latent root and vector methods used in multivariate analysis Biometrika 53 325–338
- 1: ROOTS – CHARACTER(1)Input
On entry: indicates if all the eigenvalues are to be computed or just the
NDIM largest.
- All the eigenvalues are computed.
- Only the largest NDIM eigenvalues are computed.
Constraint:
or .
- 2: N – INTEGERInput
On entry: , the number of objects in the distance matrix.
Constraint:
.
- 3: D() – REAL (KIND=nag_wp) arrayInput
On entry: the lower triangle of the distance matrix stored packed by rows. That is must contain for .
Constraint:
, for .
- 4: NDIM – INTEGERInput
On entry: , the number of dimensions used to represent the data.
Constraint:
.
- 5: X(LDX,NDIM) – REAL (KIND=nag_wp) arrayOutput
On exit: the th row contains coordinates for the th point, .
- 6: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which G03FAF is called.
Constraint:
.
- 7: EVAL(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if
,
EVAL contains the
scaled eigenvalues of the matrix
.
If
,
EVAL contains the largest
scaled eigenvalues of the matrix
.
In both cases the eigenvalues are divided by the sum of the eigenvalues (that is, the trace of ).
- 8: WK() – REAL (KIND=nag_wp) arrayWorkspace
- 9: IWK() – INTEGER arrayWorkspace
- 10: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL 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.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
G03FAF uses
F08JFF (DSTERF) or
F08JJF (DSTEBZ) to compute the eigenvalues and
F08JKF (DSTEIN) to compute the eigenvectors. These routines should be consulted for a discussion of the accuracy of the computations involved.
Alternative, non-metric, methods of scaling are provided by
G03FCF.
The relationship between principal coordinates and principal components, see
G03FCF, is discussed in
Krzanowski (1990) and
Gower (1966).
The data, given by
Krzanowski (1990), are dissimilarities between water vole populations in Europe. The first two principal coordinates are computed.