AisAgyeeeensh are the nonvanishing singular values of A, and
r is its rank.
It is known that, when some singular values are exactly zeros,
the matrix A is not full rank; in this case we say that A contains
exact linear dependencies whose number is exactly equal to the
number of null singular values.
Therefore, since a null singular value is an indication of exact
linear dependency, Kendall and Silvey (Belsley et al, 1980)
have extended this idea to say that, the existence of a small
singular value is indicative of a near dependency; which means
that, there will be as many near dependencies as there are small
singular values.
2.2 Condition Number and Condition Indices
The condition number is one of the most popular stability
indicators. The condition number is defined as:
x(A) = [A]
A7 (2.3)
or in terms of the singular value decomposition of A as the
ratio of the largest to the smallest singular values as:
À
max
À
K=
min (2.4)
In a similar fashion, the i condition index can be defined as
the ratio of the largest singular value to the ji" singular value :
À max
Ki“ Fp 2.3)
Hence there will be as many condition indices as there are
nonzero singular values.
2.3 Variance-Decomposition Proportions
It is well known to all least square users that the variance-
covariance matrix of the adjusted parameters (if we assume a
unit weight matrix P — I, and unit a priori variance of unit
weight cl = 1) is given by:
Xx - (ATA)! (2.6)
Expressing this in terms of the singular values of the design
matrix A as in Eq.(2.1 ) we get:
X, -(VDU'UDV?)' (2.7)
- Vp?vt
Therefore, the variance of the if^ parameter x; may be written
as:
2 ; vi
Cx, = > Cy) (2.8)
which means that, the variance of any parameter decomposes
into a sum of components, each of which is associated with one
of the singular values A;. For instance, the component of the
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
variance of the if^ parameter associated with the j? singular
value is given by:
2
Vii
(61); = — (2.9)
Aj
The proportion of the variance the parameter x; associated
with the j" singular value is given as:
2 .
pa oe (2.10)
Oo
Xi
The decomposition of the variances of all parameters with
respect to all singular values gives what is called the variance-
decomposition proportions matrix.
2.4 Identification of Correlated Parameters
Since a null singular value is an indication of an exact linear
dependency, a small singular value is indicative of a near
dependency. Therefore, there will be as many near
dependencies as there are small singular values.
On the other hand, since in Equation(2.9) A;'s appear in the
denominator of the expression of the components of the
variance, components associated with small singular values will
be large compared to the other components; which will lead to
high proportions.
It follows that , two or more parameters can be said to be
involved in a near dependeny when a high proportion of their
variances is associated with the same small singular value (or
same high condition index).
Hence, the method of variance decomposition will enable
identify:
- The number of near dependencies (multiple
correlation) affecting the system as the number of high
condition indices (small singular values).
- The parameters involved in these multiple
correlations as those that have a large proportion of their
variances associated with the same high condition indiex.
It remains however to decide on what should be considered as a
large proportion of the variance and what should be considered
as high condition index.
In this matter no standard exists on which to base this decision.
Concerning the threshold for the proportion of the variance,
Belsley et al (1980) considered a proportion to be large when it
accounts for more than 50% of the variance of a parameter.
The threshold for the condition index is however more
complicated, because what can be considered a high condition
index inducing ill-conditioning for a particular application ,
may not be a source of ill-conditioning for other type of
applications.
Hence in the testing that follows, all condition indices will be
considered until a conclusion can be reached during the testing
on threshold to consider as harmfull.
3. CASES
In-flight ca
problem
perturbatio:
The mode
equations
orientation
F(x) = (x-
{BE +2;
Fy) = L(y:
bris
where:
X'-agq0
Y'-a410
Z'=az10
- [2
r-yx +
with:
231p 81».
angles def
coordinate
X, Y,Z.C
Xo Voss.
X, y : Obst
Xp» yp : Pr
c : Camer
K,,K;,K
distortion.
P, P, P5 |
In this m
recovered
with exte
the prob
demonstre
Mrchant,
1969). X
has been
partial de
parameter
In this pa]
to data pe
3.1 Desc
To allow
was gene
resulting
reasonabl
space co:
