Summarization of similar nodes - Evaluating the graph reduction techniques

6.2 Evaluating the graph reduction techniques

6.2.5 Summarization of similar nodes

This technique identifies two or more nodes with similar attribute profiles and summarizes these nodes into a single node. Thus, this technique preserves all the original attribute infor- mation while reducing the structural complexity and, thus, the complexity of the mathematical

operations applied to the graph data. The methodology used for evaluating this technique is as follows:

Generate the attributed graph data for the normal condition.

Summate the physical and chemical exergy attribute values for each node in the attributed graph.

For each node in the attributed graph, calculate the difference between the summated attribute value of that node and the summated attribute value of all the other nodes in the graph.

Identify two nodes with similar attribute size profiles and, thus, a relatively small difference between them. One of these nodes is selected as the primary node and will remain in the graph, while the other node is the secondary node and will be removed from the graph once the summarization process is complete.

Add the physical and chemical exergy attribute values of the secondary node to the primary node’s physical and chemical exergy attribute values, respectively.

Take all the links between the secondary node and all the other nodes in the graph and connect those links to the primary node. If there exists a link between the primary and secondary nodes, simply omit it from the new summarized graph.

Remove the secondary node and all the links that connect to that node from the attributed graph to form the new summarized attributed graph.

Repeat this process until there are no more nodes with similar attribute profiles.

The overall detection and isolation rates of the FDI methods relative to the percentage of attributes removed from the graph data can be seen in Figures 6.9 & 6.10, respectively.

The overall detection rate of all three FDI methods remained relatively constant as more attributes were reduced with Technique 5. However, the final overall detection rate of the eigendecomposition FDI method was slightly lower than the initial rate. Conversely, the final overall detection rate was slightly higher than the initial rate for both the distance and residual-based FDI methods.

Figure 6.9: The overall detection rates after applying Tech- nique 5.

Figure 6.10: The overall isolation rates after applying Tech- nique 5.

The overall isolation rates of the three FDI methods also remained relatively stable as more attributes were reduced. The initial and final overall isolation rates were very close for the eigendecomposition FDI method. The final overall isolation rate of the distance FDI method was slightly lower than the initial rate, while the final overall isolation rate of the residual- based FDI method was slightly higher than the initial rate.

When looking at the specific isolation rates of the FDI methods as recorded in Table 6.5, it can be seen that the specific isolation rates of both the eigendecomposition and residual-based FDI methods remained relatively high and even improved, as more node summarizations were conducted. Only one of the distance FDI method’s specific isolation rates dropped slightly below 50 %. This drop occurred when the third pair of nodes were summarized.

This is, however, not a significant drop since the specific isolation rate before any nodes were summarized was 59 %.

The relatively consistent performance of all three FDI methods after being reduced by Tech- nique 5 can be ascribed to the fact that no attributes are discarded from the graph data.

Instead, Technique 5 simply reorganizes the location of attributes within the structure of the process graph data.

Table 6.1: Performance of all three FDI methods using the attributed graph data as reduced by Technique 1.

Distance FDI Method

Percentiles 0th 10th 20th 30th 40th 50th 60th 70th 80th 90th Overall Detection Rate (%) 90.00 89.85 92.15 92.15 92.74 92.78 92.72 93.43 93.66 94.46

Overall Isolation Rate (%) 19.90 19.65 16.95 16.95 15.40 15.45 15.30 12.10 10.35 10.25 Attribute Reduction (%) 0.00 12.90 27.42 27.42 59.68 62.90 69.35 77.42 87.10 96.77 Isolation rate of Fault 1 (%) 75.00 74.00 70.00 70.00 4.00 4.00 3.00 8.00 16.00 75.00 Isolation rate of Fault 2 (%) 85.00 87.00 80.00 80.00 79.00 79.00 78.00 84.00 32.00 10.00 Isolation rate of Fault 6 (%) 59.00 59.00 56.00 56.00 79.00 79.00 79.00 38.00 55.00 14.00

Eigen FDI Method

Percentiles 0th 10th 20th 30th 40th 50th 60th 70th 80th 90th Overall Detection Rate (%) 96.28 96.01 97.29 97.29 87.84 95.38 91.78 86.58 92.51 91.90

Overall Isolation Rate (%) 17.55 15.25 16.80 16.80 8.90 13.20 10.85 8.35 14.90 9.55 Attribute Reduction (%) 0.00 12.90 27.42 27.42 59.68 62.90 69.35 77.42 87.10 96.77 Isolation rate of Fault 1 (%) 90.00 87.00 94.00 94.00 8.00 11.00 9.00 8.00 63.00 78.00 Isolation rate of Fault 2 (%) 75.00 26.00 50.00 50.00 64.00 62.00 89.00 92.00 79.00 18.00 Isolation rate of Fault 6 (%) 100.00 99.00 99.00 99.00 10.00 87.00 9.00 10.00 78.00 20.00

Residual FDI Method

Percentiles 0th 10th 20th 30th 40th 50th 60th 70th 80th 90th Overall Detection Rate (%) 98.86 98.41 98.66 98.66 98.95 98.73 98.30 98.66 100.00 0.00 Overall Isolation Rate (%) 20.15 19.50 18.85 18.85 9.40 14.50 12.40 13.20 8.65 0.00 Attribute Reduction (%) 0.00 12.90 27.42 27.42 59.68 62.90 69.35 77.42 87.10 96.77 Isolation rate of Fault 1 (%) 80.00 83.00 89.00 89.00 23.00 52.00 60.00 80.00 99.00 0.00 Isolation rate of Fault 2 (%) 61.00 72.00 79.00 79.00 45.00 58.00 80.00 93.00 0.00 0.00 Isolation rate of Fault 6 (%) 88.00 87.00 80.00 80.00 0.00 85.00 0.00 0.00 0.00 0.00

Table 6.2: Performance of all three FDI methods using the attributed graph data as reduced by Technique 2.

Distance FDI Method