Incorporating Data

The effect of Data

In the previous example we ignored the test data (partially because our prior information about check valves was so strong). However, Martz and Waller [1990] provide data about 240 tests for each components in the system. Let us look at another component in which the prior information is not so strong: the pumps. Figure 5 shows the data for the pumps.

Figure 5. Data for Pumps.

The pumps were created specially for this reactor. There are exactly four of them. Although we have some equivalent experience from similar designs and engineering knowledge, this is only the equivalent of 191 tests. Thus the 240 test on each test represent relatively strong data about both the individual pumps and the average behavior of all pumps.

Figure 6. Inheritance graph for Pumps.

In Graphical-Belief we use a hierarchical model to describe this rather complex phenomenon. Both the class of all pumps and each instance of a pump (Figure 6) have a failure rate associated with them. Both the average failure rate for all pumps and all the individual pump failure rates are uncertain, so both are controlled by laws. The average failure rate for all pumps (an uncertain value) is a hyperparameter for each of the individual failure rate laws. When presented with the test data, Graphical-Belief updates both the average failure rate for all pumps and the individual failure rate for each pump. The column "P" in Figure 5 gives the failure rates for all of the pumps.

How does Graphical-Belief know how much to attribute to the class of all pumps and the individual pump? This is controlled by the "similarity" parameter. If the similarity is high, then the instance behaves a lot like is prototype. If the similarity is low, the individual pumps will behave differently. Figures 7 and 8 show the effects of the similarity parameter on the pump data. In this model, similarity doesn't have much effect on the system failure rate (it remains unchanged at 4.0e-6).

Figure 7. Data for Pumps with low similarity.

Figure 8. Data for Pumps with high similarity.

Parameter Networks and Object Oriented Model Construction

Note that the hierarchical model would be very difficult to use without Graphical-Belief's object--oriented model construction facility. In order to implement the hierarchical model for the pumps, we must build forward links from the class of all pumps to each individual pump and backwards links from the individual pumps to the class. As you can imagine, the formulae linking the parameters of these rules are quite complex. Almond [1995] describe an even more complex example which does not have the symmetry of this example.

The object--oriented model construction procedure makes this complexity transparent to the user. Each time the analyst adds a new pump to the model, they create an instance of the pump class. Graphical-Belief takes care of making all the connections and setting up all the formulae. This means that the analyst can draw on the knowledge of a statistician or reliability engineer when building a model. It is even easy to make a new class of components which follow a hierarchical Poisson process, the analyst just creates a new instance of the Poisson process rule.

But wait a minute. One of the fundamental assumptions of the graphical model is that all of the dependencies are indicated in the graph. When two rules share a parameter (through inheritance or some other mechanism), it violates that assumption. This is called common parameter dependence and Graphical-Belief has special techniques for working with model with common parameter dependence.

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