W =(B' 5; B|B'z; s. (27)
In addition, the variance-covariance matrix of the estimated
parameters is given by
I, =o, (s: =, aJ. (28)
where 02 is the variance of unit weight parameter and
can be get by
1
t
c2 XT (29)
2n -k
The function form of the interpolation functions needs to
be in general derived after atrial and error process.
6. CONCLUSION
Torgerson's MDS, which has been most commonly used
as a metric MDS technique, is not an optimal for
time-space mapping. It is because Torgerson's method
fits the point configuration not directly on given time-
distances but on the inner products derived from the given
time-distances, and brings the distortions of estimates of
coordinates when there exist the errors in MDS process
onto two-dimensional space. The least squares MDS
should be applied into time-space mapping.
The least squares MDS is basically equivalent to the
error adjustment problem of trilateration. There is need
to fix at least three coordinates in order to get a set of
coordinates. With this, the configuration of points can be
obtained by the ordinary least squares method. The
least squares MDS based on trilateration adjustment can
be utilized as a metric MDS technique. However, such a
method does not provide the accurate variance-covariance
of the adjusted coordinates because it is dependent on
which coordinates are fixed. This is a fatal problem as a
MDS technique for time-space mapping, since it has need
toemploy the interpolation procedure based onthe adjusted
coordinates. In the interpolation process, the variance-
covariance of coordinates should be dealt as weights in
the least squares criterion. The least squares MDS is
required to fairly evaluate the variance-covariance of the
adjusted coordinates.
796
Free network adjustment of trilateration using Moore-
Penrose generalized inverse enables us to obtain the
points configuration without fixing any coordinates.
Furthermore, we are able to fairly evaluate the variance-
Applying this
free network MDS, time-space mapping procedure can
covariance of the adjusted coordinates.
be systematically represented on the theoretical
framework of the least squares method.
It is needless to say the proposed free network MDS is
significant as a metric MDS technique itself. There have
been so far scarcely any studies which aim to discuss the
accuracy of the coordinates of configured points. MDS is
a general-purpose technique to visually present the
complicated structure and analyze its basic structure.
There have been a wide variety of applications in the
fields including psychology, sociology, geography, and
The free network MDS may be
potentially a new powerful weapon in these fields.
regional science.
REFERENCES
Ewing, G. and Wolfe, R.,1977. Surface feature interpolation
on two-dimensional time-space map. Environment and
Planning A, Vol.9, pp.419-437.
Mittermayer, E. , 1972. A generalization of the least-
squares method for the adjustment of free networks. Bull.
Géodésique, No.104, pp.139-157.
Rao, C. R. and Mitra, S. K., 1971. Generalized inverse of
matrices and its applications. John Wiley and Sons, New
York.
Shimizu, E., 1992. Time-space mapping based on
topological transformation of physical map. Selected Proc.
the Sixth World Conference on Transport Research, Vol.1,
pp.219-230.
Spiekermann, K. and Wegener, M., 1993. New time-space
maps of Europe. Working Paper 132, Dortmund: Institut
für Raumplanung, Universität Dortmund.
Torgerson, W. S., 1952. Multidimensional scaling: |. Theory
and method. Psychometrika, Vol.17, pp.401-419.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
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