Stochastic Approach
3.5 Continuous method
3.5.4 Some examples
To see how the three methods work, some small examples are introduced here.
Example 3.10
We consider five variablesXiwithi∈{1 . . . 5}. Their multivariate normal distribution is characterized by the following covariance matrix
Σ=
3.5 Continuous method 55 Since all the nondiagonal elements of the matrix are different from zero, the
Covari-ance Graph will connect every couple of variables.
The Full Conditional Independence Graph will give instead as a result a sort of “fork”, connecting just the nodeX1with all the other variables, since the corresponding pre-cision matrix is:
The CG and the FCIG are represented in Figure 3.3.
Figure 3.3: Example without cycles in the Full Conditional Independence Graph To apply the Tri-graph method, we recall Definitions (3.3) and (3.8) and we compute the value of the pairwise partial correlation coefficients:
(Xi,Xj) Xk ωi j|k Edge
Result:The TG coincides with the FCIG.
C This is an excellent result, and we would like to get such a one-to-one correspondence between TG and FCIG every time, however this is not always the case. Let us consider another example.
Example 3.11
We slightly change the precision matrix of the previous example:
Ω0=
The covariance matrix becomes
The corresponding graphs for the CG and the FCIG are shown in Figure 3.4.
Figure 3.4: Example with cycles in the Full Conditional Independence Graph Evaluating the pairwise partial correlation coefficients we get:
(Xi,Xj) Xk ωi j|k Edge Result:The TG coincides with the CG.
C In both examples the Full Conditional Independence Graph is a subgraph of the Co-variance Graph, but this is not always the case.
Example 3.12
We consider five variablesXiwithi∈{1 . . . 5}. Their multivariate normal distribution is characterized by the following covariance matrix
Σ=
The resulting Covariance Graph is a fork connecting the first variable, X1, with all other four variables (see Figure 3.5). The corresponding precision matrix is dense.
3.5 Continuous method 57
Hence in the Full Conditional Independence Graph all possible edges are drawn.
Figure 3.5: Example of Covariance Graph as subgraph of Full Conditional Indepen-dence Graph
If we apply the Tri-graph procedure we get the following results:
(Xi,Xj) Xk ωi j|k Edge
Result: the TG does coincide with the CG and both graphs contain less edges than
the FCIG. C
Remark 3.13 The Tri-graph is always a subgraph of the Covariance Graph.
Before we move on investigating deeply the question in which instances the TG co-incides with the FCIG, we examine if these theoretical examples correspond to real cases of the railway network.
Example 3.14
Two stations vandu are connected by a single track line without any overlapping point. Two trainstandsare scheduled to travel in different directions along this line.
Figure 3.6 shows the Activity-on-arc Project Network of the problem.
We suppose that train tleaves the stationvwith some delay (either due to a source delay at the stationv, e.g. longer time for the boarding and deboarding of the passen-gers, or due to previous forced delays). Trainshas to wait at stationuuntil the arrival
Figure 3.6: single track line
of traint(to receive the green light to proceed) since along the line there are no points where the two trains can travel simultaneously. Trainswill have a forced delay, that might spread out along its journey.
We consider four variables corresponding to the delays at:
• Y1the departure of traintfrom stationv;
• Y2the arrival of traintat stationu;
• Y3the departure of trainsfrom stationu;
• Y4the arrival of trainsat stationv.
The following covariance matrix refers to such a situation with four events correspond-ing to a scorrespond-ingle track line (between Seesen and Salz-Rcorrespond-ingelheim) in the data file that the Deutsche Bahn put on our disposal to test the Tri-graph procedure. More details about the source file and the considered events are given in Chapter 4.
Σ=
Accordingly the precision matrix of the problem is:
Ω=
For the Tri-graph method, we compute the value of the pairwise partial correlation coefficients: Since we are considering real delay measurement, we used a Maximum Likelihood test (see Appendix 7) to check if the elements of the covariance/precision matrix and the pairwise partial correlation coefficient are zero. The quantile was set to1%.
3.5 Continuous method 59
Figure 3.7: Tri-graph coincides with FCIG
The results of the three methods are shown in Figure 3.7.
Result:The TG coincides with the FCIG.
C Example 3.15
We consider three stationsv,u,wamong which three trainsr, sandtare traveling, such that the corresponding Activity-on-arc Project Network is the one shown in Fig-ure 3.8.
Figure 3.8: Triangular connection These three trains are pairwise connected at the three stations:
• randsat stationv;
• sandtat stationu;
• tandrat stationw.
Passengers have two possibilities to travel between vandw: either with the direct train or using the connection between the other two trains of the system. E.g. to travel from stationvtowa passenger can take either trainror the connection betweensand t at stationu. In reality such a situation occurs when it is necessary to have a high frequency of trains among some stations but it is not possible to increase the number of vehicles on duty.
We consider six variables corresponding to the delays at:
• Y1the departure of trainrfrom stationv;
• Y2the departure of trainsfrom stationv;
• Y3the arrival of trainsat stationu;
• Y4the departure of traintfrom stationu;
• Y5the arrival of traintat stationw;
• Y6the arrival of trainrat stationw.
Supposing that the precision matrixΩobtained from the delay data of the three trains is
the corresponding covariance matrix will be dense, thus the CG will give the complete set of possible edges among the six variables as result, while the FCIG contains just the edges of a cycle.
Evaluating the pairwise partial correlation coefficients for the Tri-graph we get:
(Xi,Xj) Xk ωi j|k Edge
The three corresponding graphs are shown in Figure 3.9.
Result:The TG coincides with the CG.
In this case, the Tri-graph method identifies all possible dependencies without being able to highlight the most important ones, as the FCIG does. C