ul 2004 
| to be 
ressure, 
ne data 
network 
fluence, 
d in the 
he data 
of data, 
ince the 
ly basis, 
with the 
] to. be 
assumed 
g hours. 
| in the 
CC 
s both in 
to each 
network 
rs is due 
position 
and the 
pt of the 
ally done 
) account 
ork. First 
'ease the 
ay fit the 
rfitting is 
'edictions 
elds poor 
s training 
se in the 
be found 
platform 
nt hourly 
| network 
yer feed- 
umber of 
'alidation 
training 
1e content 
ture. The 
> external 
  
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
parameters of the network such as the learning rate has been 
determined by trials (Heine, 1999, Miima et al., 2001). 
The optimization of the networks has been done by using 
Levenberg-Marquardt optimization method. The threshold 
values for the cost function was selected due to the mean 
standard errors obtained from the position accuracies 
determined by the adjustment of GPS measurements. The 
training was cut where the mean standard approximation errors 
reach minimum for both training and test data sets in order to 
avoid overtraining. 
After successful training, the resulting weights for each signal 
were obtained as an intrinsic representation of the mapping 
function between inputs and output for the vertical motions of 
the bridge platform for respective time interval of each 
observation day. To validate the modelling process, a residual 
analysis of the modelling errors was performed. For this end, 
the error mean yp and the coefficient of determination r^ for 
each model prediction were calculated for the residual signals 
as follows, respectively (Chatfield, 1975). Further on, the 
frequency content in residual signals is investigated by Fast 
Fourier Transform (FFT) in order to prove the randomness of 
the residuals. 
nm 
H- F6) — yi(k)) 
i-l 
(4) 
n 
Y», 00 - y! qo 
S ees] ze iz] 
"(5,09 -r' up} 
i=} 
r (5) 
where y;(K )and m denotes the mean of the actual output 
vector and its size, respectively. For a perfect approximation, 
the mean error and the coefficient of determination should be 0 
and |. 
S. SAMPLE RESULTS 
Modelling results for the vertical motion of the platform in July 
2 2001 between 7:20 — 8:20 and 10:15 — 11:15 are given in Fig. 
5 and Fig. 6, respectively. 
observed and predicted changes in height on the platform 
0.2 i 
0.1 i 
! | 
E 0 1 th ni 
z pi 1 
5 i HP" íi d 
0.1 i Ï 1 
-0.2 
0 50 100 150 200 250 300 350 
epoch number 
0.2 
E o1 
5 | li, 
S lei a i 
S 0 ^ | 
S ijj i 
S 
$ 01 
a 
0.2 
0 50 100 150 200 250 300 350 
epoch number 
Figure 5. Actual (black) and predicted (grey) height changes 
(top) and corresponding prediction errors in July 2 2001, 
between 7:20-8:20 
705 
observed and predicted changes in height on the platiorm 
02i 
0.1} i 
| 
S ita, 
E 0 : 
= 
© i | 
-0.1 à 
0.2 
0 50 100 
0.2; 
E 01 
$ pi 
5 ib 
c 0: AM 
S PU i 
5 Hi 
$01 
a 
0.2 
0 50 100 
150 200 
epoch number 
150 200 
epoch number 
Figure 6. Actual (black) and predicted (grey) height changes 
(top) and corresponding prediction errors in July 2 2001, 
between 10:15-11:15 
Some information about the prediction quality for the time 
spans given in Fig. 5 and Fig. 6 are summarized in Table 1. 
  
  
  
  
  
  
  
  
  
July 2, 2001 
7:20 —8:20 10:15— 11:15 
u (m) 0.000 0.000 
r 0.853115 0.840395 
Mean abs. error (m) 0.013 0.018 
Standard deviation (m) 0.020 0.027 
Max. error (m) 0.086 0.118 
Min. error (m) -0.066 -0.102 
Table 1. Quality measures of the prediction results for July 2, 
observed and predicted changes in height on the platform 
0.2 
2001. 
  
0.1} 
a n |, d 5s! 
f 6 M ao. Aou 
Lom EI i Ï i 
-0.1 Uu apt | 
0.2} 
0 50 100 150 200 250 
epoch number 
02 
prediction error [m] 
o 
  
150 200 250 
epoch number 
Fig. 7 and Fig. 8 show the prediction results for the date July 9, 
2001 between 7:20 — 8:20 and 10:15 — 11:15, respectively. 
300 350 
Figure 7. Actual (black) and predicted (grey) height changes 
(top) and corresponding prediction errors in July 9 2001, 
between 7:20-8:20