- the spatial distribution of the fixed points, that must be 
homogeneous over the entire network; 
- the spatial density of the fixed points, that must satisfy the 
technical specifications of the Cadastre for a second order 
network; 
- the number of polygons having in common the same fixed 
points. 
According to these conditions, three configurations of fixed 
points have been identified: 
Configuration. l: constituted by 9 fixed points, that is 1.59 
points per km? , partially distributed along the perimeter of the 
network and partially | in the internal part; 
Configuration 2: constituted by 7 fixed points, that is 1.19 
points per km”, distributed along the perimeter of the network; 
Configuration 3 : constituted by 4 points, that is 0.70 points per 
km”, distributed along the perimeter of the network. 
4.2 Error attribution for the simulated network 
Before the adjustment, particular error models have been 
applied to the original file containing the true coordinates of the 
various polygons: in this way a new set of data files simulating 
the effective measurements has been obtained. According to the 
kind of errors applied, five different classes and subclasses of 
analysis have been generated: 
1) measurements affected only by systematic errors (S): 
2) measurements contemporarily affected by systematic and 
random errors (S and R), which can be partitioned into three 
different subclasses: 
2a) prevalence of the systematic error component with 
respect to the random one (S>R) 
2b) equivalence between the systematic, and the random 
error components (S=R) 
2c) prevalence of the random error component with respect 
to the systematic one (R>S) 
3) measurements affected only by random error components 
(R). 
The systematic errors have been generated by scaling the single 
fiducial polygons of a random amount. The scale variations are 
randomly distributed in the maximum range of +100 ppm. 
These systematic components have only a local effect: 
considering the general network their overall averaged 
contribution becomes practically negligible. 
For each class and subclass of analysis, three different error 
simulations have been executed. The total number of datasets is 
therefore equal to 15; in this way a significant number of 
possible situations has been originated. Every dataset has been 
then adjusted applying both the traditional and the Procrustes 
methods, considering, for every computation, the scheme of 
fixed points characterising one of the three mentioned 
configurations. In this way, 90 different computations have been 
performed, comparing the obtained results with the original true 
values, for each of the two adjustment procedures (Clerici, 
2002). 
4.3 Numerical results 
The numerical experiments have been executed with the aim to 
verify the capability of the Procrustes algorithm to compensate 
for the various kinds of error. To verify the capability of this 
original method, it is necessary to compare the obtained results 
with those furnished by the classical least squares adjustment of 
the individual polygon sides. For this purpose, the mean values, 
the medians and the root mean square errors for a specific error 
simulation referring to each of the fixed point configurations 
have been collected and compared. Analogous investigations 
have been carried out for the other two cases of error 
generation. 
34 
An analysis of the results, in particular for what is referred to 
the Configuration 1, allows to conclude that for the Procrustes 
analysis the mean and the median values for the x and y 
coordinate components never differ for values greater than 6 
mm in absolute value. For the traditional adjustment technique 
this difference reaches 5 cm; this fact indicates a better 
symmetry of the residual distribution when the Procrustes 
algorithm is applied. Furthermore, it is possible to observe that 
in the case of the only presence of systematic errors, Procrustes 
adjustment leads to mean and median values equal to zero for 
all of the three configurations of fixed points. In this case, the 
residuals follow a perfectly symmetrical distribution, without 
any systematic error component; vice versa, by applying the 
traditional adjustment procedure, these values vary between 2 
and 11 cm. When applying the Procrustes algorithm, the root 
mean square error is always near to zero, while for the classical 
adjustment procedure this has a magnitude of 10 centimetres. 
However, due to the particular systematic local random error 
generation, the residual values after the adjustment of the whole 
network are always contained within the rms error. Finally, if 
only random errors are present, the values of the differences 
between means and medians are always comparable for the two 
methods; this remains true also for the values of the root mean 
square error, but is limited to the configurations of fixed points 
I and 2 that consider a sufficient number of fixed points for the 
whole network. 
The graphs reported show, for the Configuration 1, first, the 
differences between the true values of the point coordinates and 
the values obtained by applying the Procrustes algorithm (see 
proper symbol), and second, the differences between the true 
coordinate values and those ones obtained by a traditional 
adjustment procedure of the polygon sides (see proper symbol). 
Figure 6 refers to the case in which the systematic error is only 
present. The results obtained by applying the Procrustes 
technique are almost corresponding to the true values, while the 
classical adjustment leads to an error dispersion of 20-30 cm. 
  
X COORDINATE DIFFERENCES 
E €^099999097v^o99909^99 EHE 
Point ID 
  
  
Y COORDINATE DIFFERENCES 
E | 00900.0000600000000000000000, 
Point ID 
  
  
  
Figure 6. Residuals for pure systematic error conditions 
(Procrustes €, traditional . ) 
  
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