## 3 Results
### Examples
#### 3.1.2 Example No. 2: A Res \(\to\) Dt Change Detector Applied to a Homologous RET Transition  
To make full use of the synthetic TS sequence given in the top band of Table 2 and to provide a deeper insight into the proposed CD method, let now the ego-car start its virtual journey from the right.  
As a side note, we can reverse such a virtual journey fairly easily in a table. In real life and in real traffic, however, it would be much more problematic as one might encounter very different TSs on the way back (if at all the reversed route is permitted by the TSs installed). Furthermore, the TSs that are facing us now are located on the opposite side of the road as in the first journey. However, even in case of this widely virtual journey, some modifications in the table, namely, with respect to the TS counts, are necessary.  
These new TS counts, i.e., the counts for right-to-left virtual trips, are given in the bottom band of the table. In this reversed case, the ego-car is driven from an intended Res area to an intended Dt area, i.e., a Res \(\to\) Dt transition is expected.  
Again, as in the first example, we wish to form functions \(g_{T}\) and \(h_{T}\) that signal the RET transition that is actually expected. Accordingly, these functions are now denoted by \(g_{l}^{Res\to Dt}\) and \(h_{l}^{Res\to Dt}\), respectively.  
In order to detect such a transition, the roles of the process parameters \(\lambda_{k}\) and \(\mu_{k}\)--given in the fourth and the fifth column of Table 1, respectively--need to be swapped in Equations (8) and (9), as now, it is supposed that the first road environment is a Res area, rather than a Dt area. Again, the unified counts are to be used, namely, the TS counts given in bottom band.  
Functions \(g_{l}^{Res\to Dt}\) and \(h_{l}^{Res\to Dt}\) are given in Equations (14) and (15), respectively.  
\[g_{l}^{Res\to Dt} \approx -2.80\text{ km}^{-1}\cdot l+1.74\cdot N^{NS}(l)+ \tag{14}\] \[+1.92\cdot N^{PL}(l)-0.13\cdot N^{GW}(l)-0.69\cdot N^{SL}(l)\] \[h_{l}^{Res\to Dt} =g_{l}^{Res\to Dt}-\inf_{s\leq l}g_{s}^{Res\to Dt}. \tag{15}\]  
Figure 1: A downtown (Dt) \(\to\) residential (Res) change detector applied to a homologous road environment type (RET) transition showing up in the traffic sign (TS) sequence given in the top band of Table 2: (a) Function \(g_{l}^{Dt\to Res}\) for the considered TS sequence and (b) Function \(h_{\to l}^{Dt\to Res}\).  
In Figure 2, the functions \(g^{Res\to Dt}_{l\leftarrow}\) and \(h^{Res\to Dt}_{l\leftarrow}\) have been plotted. The corresponding virtual journey had started from the right in the band, as indicated by the \"\(\leftarrow\)\" in the lower indices. The diagrams presented herein corresponding to virtual journeys from the right are presented in green.  
Again, threshold \(\delta=3.0\) would be an appropriate choice based on the diagram in Figure 2. With this threshold, the RET change is detected at about the path-length 3 km. Since the first signs of the Dt area begin to appear at the path-length 2.8 km, the detection lag is 0.2 km long.  
Comparing the diagrams of \(g^{Dt\to Res}_{l\leftarrow}\) and \(g^{Res\to Dt}_{l\leftarrow}\) shown in Figures 1 and 2, respectively, one notices the symmetry between these. This symmetry can be traced back to two facts: first, from Equations (8) and (9)--as used in Example 1, and by swapping the roles of parameters \(\lambda_{k}\) and \(\mu_{k}\) for equations corresponding to Example 2--furthermore, from Equations (12) and (14), it follows that  
\[g^{Res\to Dt}_{l}\equiv-1.0\cdot g^{Dt\to Res}_{l}.\]  
and second, to the fact that the same route--with its TSs--was covered from opposite directions.  
There will be further symmetries perceptible between the respective \(g^{\leftarrow\rightarrow\rightarrow\rightarrow}_{l\leftarrow}\) diagrams. However, similar symmetries do not show up amongst the respective \(h^{\rightarrow\rightarrow\rightarrow}_{l\leftarrow}\) functions.