"M rV
(D ^ 1
wv?
calculation of the inner product matrix R =(ry) constituted
by the position vectors taking the gravity point as the
origin (Young and Householder's transformation). The
inner product of points ; andj is given by
sxd[iseH uds; 2-2
P SN nr fij A i (1)
Assume that the points can be distributed in Euclidean
space and the rank of the inner product matrix R is r.
From the Young and Householder's theorem, R can be
decomposed as
R = DD' (2)
and each row vector of D shows the coordinates of
the point concerned in r -dimensional space. Since
D isa real symmetric matrix and the rank is r , R is
diagonalized by the orthogonal matrix X as follows;
X'RX zA (3)
Aq 0
end
where À1,.., À are the positive eigen values of R and X
is the matrix in which the column vectors are composed
by the normalized eigen vectors x;,.,x, , ie., xj 2 1,
corresponding to 41,..,4,. Since the diagonal components
of A is all positive,
R =XAX' (4)
=(x AY?) (X Any
Thus,
DzXA!?
= Varn... Vor, | (5)
X11 Xn 11
= Aq 0009 Ar
X1n Xrn
This is the outline of Torgerson' MDS. Next, we discuss
the relationship between Torgerson's MDS and the least
squares MDS. Consider the configuration of points on
two-dimensional space from the point of view of the
application into time-space mapping. Define the (u,v)
coordinates of points in the time-space by the first and
Second column of D , that is, the coordinates of points
i andj are given by
U; = Va dn v;= YA xs
(6)
uj = FI Vj = YÀ2xz;.
The inner product of points i andj |, ny , is
ry = Axx + A2X 3X aj- (7)
Hence the mean squares error for all components of the
inner product matrix, m?, is
nta
= > Y (Axa + AaX qiX 4). : eru)
Tj
22 2. 22.2 2 212
=> Sess + Max aiX dj AXES)
3
N X (risesspeait d. 0) (8)
i 2545 tava) 3 sfr] E.
BAM... «A.
Therefore, if A; 24; 2...24,, the configuration of points
by Torgerson's MDS is said to be optimal in the sense of
approximating the components of the inner product matrix
in the least squares criterion. However, we should not
forget that Torgerson's MDS approximates not the given
time-distances but the inner product matrix. If the given
time-distances have errors, these two criteria have the
following differences according to the error propagation
law;
- If two time-distances are same both in magnitude and
in direction, the precision of the time-distance which is far
from the gravity point is higher than another.
- If two time-distances are same both in magnitude and
in the distance between the center point of two points and
the gravity point, the precision of the time-distance which
is in the radial direction from the gravity point is higher
than another.
Accordingly the variance-covariance matrix of the
estimated coordinates is affected not only by the time-
distance itself but also by its direction and distance from
the gravity point. Thus itis concluded that the configuration
of points based on Torgerson's MDS is not optimal in the
restrictive sense of consistence with the given time-
distances. It is requested to employ the MDS that
approximates directly the given time-distances in the least
squares criterion.
3. BASIC FORMULATION OF LEAST SQUARES MDS
The least squares MDS is basically equivalent to the error
adjustment problem of the trilateration. The physical
793
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
