Figure 8. East residual for tracks 1-5 
Figure 9. North residual for tracks 1-5 
Figure 10. Horizontal residual for tracks 1-5 
From the graphs presented, it is visually apparent that the train 
starts on track 2 and after 55 s changes to track 3. The large 
change in the east, north and horizontal residuals of 
approximately 2 m indicates the change of track. Therefore, the 
data was broken into two sections and each section was analysed 
individually. The change of track is evident in the LOM test, 
whereby the null hypothesis for track 2 is rejected after 55 s and 
the null hypothesis for track 3 is accepted. This is also confirmed 
in the GOM test, where within 4 s, it is confirmed at 90% 
probability that the train has changed track (Table 3). 
Table 2 contains the number of times each predetermined 
probability was met, divided by the number of points within that 
section of track. When Table 2 and Table 3 are analysed together, 
it can be seen that once a certain probability is met, this 
probability is retained for the remainder of the section. 
Probability % 
90 
95 
99 
99.9 
99.99 
99.999 
Section 1 
51/55 
50/55 
48/55 
45/55 
43/55 
41/55 
Section 2 
43/46 
42/46 
41/46 
40/46 
38/46 
37/46 
Table 2. The number of times each predetermined probability 
was met, divided by the number of points in the section 
Probability % 
90 
95 
99 
99.9 
99.99 
99.999 
Section 1 
5 
6 
9 
11 
13 
15 
Section 2 
4 
5 
6 
7 
9 
10 
Table 3. Time (s) taken to achieve predetermined probabilities 
The residuals in Figure 8 and Figure 9 (55 - 101 s) that appear 
parabolic in shape, are due to inadequate modelling of the track. 
It is assumed that track between two nodes is straight, however, 
in practice this is seldom the case. The parabolic errors associated 
with generalising the track to a series of nodes are a maximum 
half way between two nodes and a minimum near the map nodes. 
To reduce the magnitude of these errors, the track needs to be 
mapped at more regular intervals (Figure 11). However, one must 
consider the increase in file size associated with a decrease in the 
distance between two map nodes. 
Figure 11. The effect of reducing the distance between two 
nodes. Note the reduction in parabolic errors using option b). 
In Figure 8, the east residual for all five tracks (88 - 101 s) 
appears to converge. To determine the residual between the 
filtered position and the track, a perpendicular offset from each 
track is calculated using simple trigonometry. From this offset, a 
map matched position on the track is determined. This assumes 
that there is no long-track error associated within the filtered 
position. Therefore, when the train is travelling in an easterly 
(88 - 101 s) or westerly direction, there will be minimal, if any, 
east or west residual. Likewise, when the train is travelling in a 
northerly or southerly direction, the respective residuals will be 
minimal. However, when the train is travelling between two 
cardinals, i.e. in a north-easterly direction, a residual in both the 
east and north component would be expected (0 - 55 s). As the 
train travels closer to a cardinal, that cardinals residual will 
diminish and the other cardinals residual will increase (55 - 88 s).