A CRITERION MATRIX
FOR THE SECOND ORDER DESIGN
OF CONTROL NETWORKS
Fabio CROSILLA
In: BORRE, Kai / WELSCH, Walter (Eds.) [1982]:
International Federation of Surveyors - FIG Proceedings Survey Control Networks
Meeting of Study Group 5B, 7th - 9th July, 1982, Aalborg University Centre, Denmark
Schriftenreihe des Wissenschaftlichen Studiengangs Vermessungswesen der Hochschule der Bundeswehr München, Heft 7, S. 143-157
ISSN: 0173-1009
A CRITERION MATRIX FOR THE SECOND ORDER DESIGN OF CONTROL NETWORKS
Fabio CROSILLA
Trieste, Italy
ABSTRACT
This paper proposes an original method of constructing a criterion matrix
for the optimal design of control networks by means of the contraction
of the eigenvalues and the rotation of the eigenvectors of a covariance
matrix. A Second Order Design problem is then resolved, that is the
optimization of the precision of the observations of a local free distance
network to be constructed for the study of recent crustal movements
in the seismogenetic area of Friuli (Italy).
1. Introduction.
Sprinsky
(1981),
(1978)
of
suggested
constructing
a
a
method,
criterion
subsequently
matrix
by
developed
means
of
by
the
Wimmer
reduction
of the trace of the covariance matrix Qxx of the adjusted net coordinates
vector x. It involves proceeding to a singular value decomposition of
a
a
(mxm)
covariance
diagonal
matrix
Qxx = V 𝛬 V'
matrix
whose
terms
with
correspond
in
r (Qxx ) ≤ m;
descending
where
order
𝛬
to
is
the
eigenvalues of the matrix Qxx, and V corresponds to the orthonormalized
eigenvectors.
The
matrix
can
Qxx
be
interpreted
geometrically
as
an
m-dimensional error ellipsoid in which the length and direction of the
semi-axes correspond to the square root of the eigenvalues and the eigenvectors respectively.
Reducing the dimensions of this ellipsoid by a contraction of the greater
eigenvalues, it is possible to obtain a higher global precision. Wimmer
(1981)
thus
�SVD = V 𝛬̃ V'
Q
proposed
obtained
to
from
consider
the
as
a
criterion
covariance
matrix
matrix,
Qxx
the
whose
matrix
greater
eigenvalues (𝜆i )·are reduced by a parameter of contraction t
𝜆̃i = 𝜆i - t (𝜆i - 𝜆r)
, 0 ≤ t ≤ 1
and where
r = r (Qxx)
(1)
Sprinsky mentioned the possibility of a redistribution of the allocated
variances
through
a
rotation
of
the
143
m-dimensional
error
ellipsoid
by
means of a change of the orthonormal basis of the vector· space. The
rotation procedures imply the definition of a class of covariance matrices similar to the matrix of the contracted eigenvalues 𝛬̃.
First, this paper presents two methods of rotation of the error ellipsoid
for the construction of a criterion matrix for control networks. This
criterion matrix is then used for the solution of a S.O.D. problem in a
network to be constructed in the seismogenetic area of Friuli. Finally
the limits to the application of this criterion matrix are defined.
2. Definition of a type of criterion matrix for control networks
Let d be defined as the deformation vector of the net coordinates characterized by a covariance matrix Qdd and given by d = x1 - x2, where
x1 and x2 are the vectors of the coordinates relative to two different
periods
their
of
measurement.
precision
are
the
On
the
same
hypothesis
for
both
that
periods,
the
observations
according
to
the
and
law
of variance propagation the covariance matrix of the deformation vector
d will be given by:
r
r
Qdd = 2Qxx = 2(V 𝛬 V') = 2 �� 𝜆i vi v'i � = � 2𝜆i vi v'i
i=1
(2)
i=1
where vi are the eigenvectors related to the 𝜆i eigenvalues.
The matrices Qxx and Qdd have the same eigenvectors and the eigenvalues
are different by a factor of 2.
Regarding the possibility of a rotation of the error ellipsoid mention
has been made above of a class of covariance matrices similar to that
of
the
contracted
eigenvalues
𝜆̃.
In
this
class
could
be
considered
as a criterion matrix the matrix for which the components of the "essential eigenvectors" relative to the pairs of variables xi, yi (i = l...n)
(Pelzer, 1976, 1980) (Dupraz and Niemeier, 1979) are disposed in a direction as orthogonal as possible with respect to the predicted deformation.
This criterion should be satisfied in particular by the components of the
essential eigenvector, that is the eigenvector relative to the greatest
eigenvalue.
It
mi-axis
the
of
therefore
error
represents
ellipsoid
of
the
direction
the
coordinate
144
of
the
vector
greatest
x
and,
sefrom
what has been said above, of the deformation vector d. In the direction
defined by the essential eigenvector possible deformations d thus cannot
be established with any great precision.
Furthermore, in the case of non circular error ellipses, there is generally an isodirectionality between the greatest semi-axis of the ellipse
and the components xi, yi (i = l...n) of the essential eigenvector. The
greater the difference between the semi-axes of the ellipse, the more
evident this isodirectionality becomes. From this it follows that considering
as
essential
a
criterion
eigenvector
matrix
the
components
covariance
which
are
as
matrix
characterised
orthogonal
as
by
possible
to the direction of the predicted deformation, error ellipses are obtained in such a way that their greatest semi-axes are also disposed in a
direction as orthogonal as possible to that of the predicted deformation.
The probability P (𝜒2) that the real values of the coordinates are contained in the area defined by the dimension of the ellipse, may be applied to all error ellipses. From this derives the necessity that the
greatest semi-axis of the ellipse be orthogonal to the predicted direction of deformation.
3. Construction of the criterion matrix
The
singular
r (Qxx ) ≤ m,
value
relating
decomposition
to
a
free
net
of
the
in
its
covariance
eigenvalues
matrix
and
Qxx
(mxm),
eigenvectors
may be considered thus:
Qxx = � V
|
|
|
𝛬
U � �--0
|
+
|
𝛬̃
U � �--0
|
+
|
0
V'
---� �---�
0
U'
(3)
The criterion matrix proposed by Wimmer (1981) follows from:
�SVD = � V
Q
|
|
|
V'
0
�
�
�
----U'
0
(4)
where: 𝛬̃ is a matrix of eigenvalues 𝜆̃i (i = l...r);
𝜆̃i are calculated in formula (1);
V (mxr) is a matrix of eigenvectors relative to 𝛬 and 𝛬̃;
U (mxm-r) is a matrix of eigenvectors relative to m-r
eigenvalues = 0.
The orthogonality of the essential eigenvector components with respect
145
to the predicted direction of deformation may be achieved by rotating
the eigenvector matrix V in an orthonormal matrix Vo in which the elements of the first column correspond to the components required by the
essential eigenvector. The construction of the matrix Vo can be carried
out in two different ways:
- by means of independent rotations of all the r eigenvector component
pairs
vxij,
vyij
(j = 1...r)
relative
to
the
n
pairs
of
variables
xi, yi (i = 1...n);
- by means of an orthogonal procrustean transformation of the matrix V.
In
the
vyij
(j
first
case
any
pair
of
components
of
the
eigenvectors
vxij ,
1...r) relative to the variables xi, yi is rotated through an
angle 𝜑xi yi by means of
voxij
⎡ cos�𝜑xi yi � sin�𝜑xi yi �⎤ vxij
⎥�
�
�=⎢
�
⎢
⎥ vyij
voyij
⎣-sin�𝜑xi yi � cos�𝜑xi yi �⎦
(5)
where the angle 𝜑xi yi is understood as > 0 if the direction is clockwise.
In this way the introduction of new components voxij , voyij in the original
matrix of the eigenvectors V does not invalidate the properties of normality and orthogonality of the matrix itself. The criterion matrix derived
from
this
method
of
rotation
of
the
eigenvector
components
relative
to the variables xi, yi is given by:
�IR = VoIR 𝛬̃ Vo'
Q
IR
(6)
The construction of the matrix Vo can also be achieved through an orthogonal procrustean transformation of the matrix V. In the factor analysis
"procrustean transformation" is understood to mean any linear transformation which under certain specified conditions allows the transformation
of a given matrix into a matrix as near as possible equal to a preconstructed one.
For example, let V and Vo be two matrices (mxr) m ≥ r
where: V is an original orthonormal eigenvector matrix;
Vo is a preconstructed matrix, not necessarily orthonormal, containing the estimates of k (1 ≤ k ≤ r) rotated eigenvectors.
For k = 1 the rotated eigenvector estimate corresponds to the required essential eigenvector one.
The problem is to find an orthogonal transformation matrix T (rxr) such
146
that
where
the
approximation
VT ≅ Vo
VT ≅ Vo
is
a
(7)
least
square
approximation
and
the matrix VT is an orthonormal matrix. To calculate the matrix T which
transforms the matrix V into a least squares approximation of Vo, the
sum of the squares of the elements of the matrix E = (Vo - VT) that is
tr (E' E) = tr ((Vo - VT)' (Vo - VT))
must
be
minimal
under
the
condition
of
(8)
orthogonality
of
T
(TT' - I) = 0
so
that
(9)
Now in accordance with Lagrange's method the minimum condition of tr(E'E)
under the condition that (TT' - I) = 0 is given by
∂ tr(E' E) ∂ �trΘ (TT' -I)�
+
=0
∂T
∂T
where Θ is a matrix of Lagrangian multipliers.
Developing
the
matrix
derivatives
of
the
matrix
(10)
traces
(Schonemann,
1965) the following equation is obtained
2 V'VT - 2 V'Vo + 2 Θ T = 0
(11)
V'V + Θ = V'VoT'
(12)
Dividing both parts by 2 and multiplying both parts by T' the result
is
Now given that V'V and Θ are both symmetrical V'VoT' must also be symV'VoT' = TVo'V
metrical. It thus follows that
(13)
V'Vo = TVo'VT
that is
Let R = V'Vo
(15),
(14)
therefore R = TR'T
(16)
Let two orthonormal matrices be defined P and Q (rxr) calculated from
the singular value decomposition of RR' and R'R
RR' = PDP'
(17a)
R'R = QMQ'
(17b)
where D and M are the diagonal matrices of the eigenvalues of RR' and
R'R respectively.
Now
RR'
and
R'R
have
the
same
eigenvalues
(Johnson,
1963),
therefore
D = M. Putting the terms of (16) in (17a) and using (17b) it results that
RR' = TQDQ'T' = PDP'
clearly TQ = P, that is to say
T = PQ'
(18)
(19)
This method is also valid for the case in which the matrices V and Vo
are not of full rank. Moreover, in the particular case where the matrix
D contains roots equal to each other and different from zero the matrices
147
P and Q will not be unique and consequently T will not be unique either.
To satisfy the condition that the trace tr (E'E) = minimum, the matrices
P and Q will also have to satisfy the following condition (Schonemann,
P'RQ = D1⁄2
1966):
(20)
where R = V'Vo and D1⁄2 is the matrix of the square root of the eigenvalues of D. Now let H = Po'RQo
(21)
where Po and Qo are two orthonormal matrices which satisfy only (17).
The matrix H (Schonemann, Bock, Tucker, 1965) is diagonal with diagonal
elements of D1⁄2 exept for the blocks Hj of elements of order njxnj along
the diagonal, corresponding to multiple roots 𝜆j of multiplicity nj in
D. Since the matrices Po and Qo satisfy (17a) and (l7b) the matrix Hj
has the property that H'jHj = Dj and in turn H'H = D.
Each of these square blocks Hj can be considered proportional by a scalar 𝜆j to an orthonormal matrix Wj (njxnj). Hj can therefore be decomposed
⁄2
Hj = 𝜆1j
into
Wj = D1j
⁄2
Wj
(22)
Let P = Po in (20). To find a matrix Q which satisfies (20), on the
basis of (22), matrix H is multiplied by a diagonal matrix K where
-1⁄2
kii = �𝜆i
0
Wo = KH
𝜆i >0
𝜆i =0
(i = 1…r)
(23)
from which is obtained a matrix Wo which can in its turn be transformed
into an orthonormalised matrix W inserting 1s in the positions of the
principal diagonal of Wo which contain 0s.
Finally the matrix Q can be obtained from Q = Qo W'
This
general
solution
to
the
problem
of
(24)
procrustean
transformation,
even in the case where the matrices V and Vo are of rank r < m, gives
a notable flexibility to the definition of the preconstructed matrix Vo.
In this regard it is sufficient to define the estimates of k eigenvectors
(k ≥ 1)
and
to
insert
zeros
in
the
residual
r-k
columns
of
Vo.
For
k = 1 the estimates of the essential eigenvector components correspond.
The matrix VT which results from the orthogonal procrustean transformation of V to approximate to Vo, will contain estimated orthonormalised
components of the essential eigenvector in its first column and orthonormal r - 1 vectors in the residual ones.
The criterion matrix resulting from this method of procrustean transfor-
148
mation of the eigenvector matrix V is finally given by
�PT = VoPT 𝛬̃ VoPT
Q
'
(25)
4. Solution of a Second Order Design problem for a control network
The
S.O.D.
problem
consists
in
designing
a
matrix
of
the
observation
weights P (sxs) in such a way that the covariance matrix of the coordinates Qxx (mxm) resulting from (A'PA)-, where A (sxm) is the design
matrix and ()- is a generalised inverse, is equivalent to a criterion
matrix Qxx (Baarda, 1973; Grafarend, 1972; Grafarend, Schaffrin, 1979).
The generalised inverse corresponds to a Cayley inverse ()-1 for constrained
nets
and
to
a
Moore
Penrose
()+
inverse
for
a
minimum
norm
solution of a free net. For the solution of the S.O.D. problem the Kroneker product (Bossler et al., 1973) was proposed for correlated observations and the Katri Rao product (Rao, Mitra, 1971) for uncorrelated observations.
Starring
from
observation
the
matrix
weights
equation
matrix
P
Katri
Rao
(A' PA)+ = Qxx
the
extraction
all
a
of
the
requires
first
of
decomposition
product
gives
(K ⨀ K) vecd P+ = vec Qxx
K = (A'PA)+ A'P.
This,
applying
where
(K ⨀ K)
vec Qxx
is
of
the
is
m2xl
of
m2xs
dimension,
dimension.
Since
vecd P+
the
is
matrix
of
sxl
dimension
and
Qxx
is
symmetrical
it
is sufficient to consider only its lower or upper triangle.
The reduced matrix equation is given by (K ⨀ K) vecd P+ = vech Qxx with
(K ⨀ K) of gxs dimension, vech Qxx of gxl dimension and where
m(m+1)�
g=
2. Let 0 be a gxg diagonal weight matrix in which off = 0.5(f=1...g)
if the fth component of vech Qxx corresponds to a diagonal element of Qxx;
off = 1 otherwise (Wimmer, 1978). The solution for vecd P+, whenever g > s,
is given by
vecd P+ = ((K ⨀ K)'0 (K ⨀ K))+ (K ⨀ K)'0 vech Qxx
(26)
from which vecd P = ( vecd P+)+.
This calculation method is iterative (Wimmer, 1981) since the matrix of
the weights is contained in the decomposition K. The calculation must
therefore be repeated until the matrix of the weights no longer varies.
149
Fig. 1
Control network design to be constructed in the seismogenetic area of
Friuli (Lake Cavazzo valley). The original essential eigenvector components and the predicted direction of ground deformation are also
shown.
150
The net for which a S.O.D. problem is solved is a pure trilateration
net made up of 16 vertices and 42 distances, to be constructed in the
Lake Cavazzo Valley for the study of recent crustal movements in the
seismogenetic area of Friuli (Crosilla, Marchesini, 1982).
Fig. 1 shows the design of the net and the essential eigenvector components relative to the coordinates of each point obtained from the singular
Qxx
value
decomposition
calculated
with
the
of
the
unit
covariance
weight
matrix
matrix
of
of
observations
the
the
coordinates
.
The
predicted direction of ground deformation is also shown.
Previous considerations have shown that in the control nets the components of the essential eigenvector must be orthogonal with respect to
the direction of predicted movement. It can thus be observed from fig.
1 that in the case of this net it is necessary to rotate the essential
eigenvector components of two distinct groups of points.
The first group comprises points 2, 3, 4, 5 and the second group points
13, 14, 15. The essential eigenvector components relative to the points
of the first group were rotated each time by -10 gon, -20 gon, -30 gon,
-40 gon, and those of the second group of points by +10 gon, +20 gon,
+30 gon, +40 gon. Two criterion matrices were constructed for each of the
5 rotations.
other
by
One
means
was
of
by
means
procrustean
of
independent
transformation
rotation
�PT ,
Q
after
�IR
Q
and
the
contracting
the greatest eigenvalue of the covariance matrix Qxx through the parameter of contraction t = 0.5 (Wimmer, 1981). A S.O.D. problem was then
resolved in each case with the iterative method reported above.
The precision of the observations obtained after one rotation of -10 gon
of
the
essential
eigenvector
components
of
the
first
group
of
points
and +10 gon of the components of the second group of points are shown
�IR and Q
�PT respectively.
in figs. 2a and 2b for Q
The error ellipses (P(χ2 ) > 0.99) for the 16 points of the net are shown
in fig. 3 and 4. They were obtained from a covariance matrix calculated
with a unit weight matrix of the observations (dotted line), from the
�IR
criterion matrix Q
to
a
rotation
�PT
in fig. 3 and Q
of ± 10 gon,
and
from
151
in fig. 4 (thin line) relative
the
covariance
matrix
calculated
Fig. 2a Weight distribution of the observations calculated by the Second Order
�IR as a criterion matrix.
Design considering the matrix Q
152
Fig. 2b Weight distribution of the observations calculated by the Second Order
�PT as a criterion matrix.
Design considering the matrix Q
153
with the weights resulting from the S.O.D. (thick line). As can be seen
in figs. 3 and 4 there is a high level of correspondence between the
error ellipses postulated by the criterion matrix and the ones obtained.
It can also be seen that the ellipses of the points 2, 3, 4, 5 and 13, 14, 15,
actually undergo the rotation imposed on the eigenvectors in the construction of the criterion matrix.
Finally it can be seen that the results obtained with the two methods
of rotation are substantially identical.
The solution to the S.O.D. problem for criterion matrices obtained with
rotations of ± 20 gon, ± 30 gon, ± 40 gon of the essential eigenvector
components does not give satisfactory results. In fact the weights of
many observations are often prone to be negative and weights which are
very different from each other often result in the other observations.
In
the
case
of
the
rotation
of
± 40 gon,
for
example,
7 observations
with negative weights are obtained from the criterion matrix
a ratio pmax/pmin equal to 21.16/0.10 = 211.6.
�IR
Q
with
5. Conclusion
The results obtained from the solution of this S.O.D. problem confirm
the validity of the method, suggested by Sprinsky and used by Wimmer,
for
the
construction
contraction
parameter
of
the
criterion
matrix
of
the
eigenvalues
of
�SVD ,
Q
the
provided
covariance
that
matrix
the
is
not taken to be too high.
The criterion of rotation of the essential eigenvector components introduced
in
this
paper
also
make
it
possible
to
improve
the
definition
of a criterion matrix for control nets. The results have made it clear,
however, that rotations of this type must be limited. Rotations of great
amplitude give physically unreal criterion matrices.
Finally, the two methods of rotation here proposed, that is independent
rotation
and
procrustean
transformation,
give
results for limited rotations of the eigenvectors.
154
substantially
identical
Fig. 3
Error Ellipses obtained from a unit weight matrix of the observations
�IR (thin line) and the S.O.D. co(dotted line), criterion matrix Q
variance matrix (thick line).
155
Fig. 4
Error Ellipses obtained from a unit weight matrix of the observations
�PT (thin line) and the S.O.D. co(dotted line), criterion matrix Q
variance matrix (thick line).
156
REFERENCES
BAARDA, W., 1973: S-transformations and criterion matrices, Neth. Geod.
Comm., Publ. on Geod., New Series 5, n° 1, Delft (1973).
BOSSLER, J., GRAFAREND, E., KELM, R., 1973: Optimal design of
nets II, Journal Geophysical Research, 78 (1973), pp. 5887-5897.
CROSILLA, F., MARCHESINI, C., 1982: Geodetic
the detection of crustal movements using a
Geodesia e Sc. Affini, to appear.
geodetic
network optimization for
mekometer, Bollettino di
DUPRAZ, H., NIEMEIER, H., 1979: Un critère pour l'analyse des
géodésiques de controle, Mensuration, Photogrammetrie, Génie
4 ( 1979), pp. 70-76.
GRAFAREND, E., 1972: Genauigkeitsmaße geodätischer
dätische Kommission, Reihe A, n° 73 (1972).
GRAFAREND, E., SCHAFFRIN, B., 1979: Kriterion
für Vermessungswesen, 104 (1979), pp. 133-149.
Netze,
Matrizen
résaux
rural,
Deutsche GeoI,
Zeitschrift
JOHNSON, R.M., 1963: On a theorem stated by Eckart and Young, Psychometrika, 28 (1963), pp. 259-263.
PELZER, H., 1976: Genauigkeit und Zuverlässigkeit geodätischer
Tagung, Mathematische Probleme der Geodäsie, Oberwolfach (1976).
Netze.
PELZER, H., 1980: Some criteria for the accuracy and the reliability
of networks, Deutsche Geodätische Kommission, Reihe B, n° 252 (1980).
RAO, C.R., MITRA, S.K., 1971: Generalized inverse of
applications. New York/London/Sydney/Toronto, (1971).
matrices
SCHONEMANN, P.H., 1965: On the formal differentiation of traces
and determinants. Research Memorandum, n° 27, Chapel Hill,
Psychometric Laboratory, (1965).
and
its
N.C.:
The
SCHONEMANN, P.H., 1966: The generalized solution of the orthogonal procrustes problem. Psychometrika, n° 31, (1966), pp. 1-16.
SCHONEMANN, P.H., BOCK, R.D., TUCKER, L.R., 1965: Some notes on a theorem
by Eckart and Young. Research Memorandum, n° 25, Chapel Hill, N.C.: The
Psychometric Laboratory, (1965).
SPRINSKY, W.H., 1978: Improvement of parameter accuracy by choice
quality of observation, Bollettin géodésique, 52 (1978), pp. 269-279.
and
WIMMER, H., 1978: Eine Bemerkung zur günstigsten Gewichtsverteilung bei
vorgegebener Dispersionsmatrix der Unbekannten. Allgemeine VermessungsNachrichten, 85 (1978), pp. 375-377.
WIMMER, H., 1981: Second Order Design of geodetic networks by an iterative
approximation of a given criterion matrix. Paper presented at the
Symposium on Geodetic Networks and Computations, Munich, August 31 September 5, (1981).
157