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Appendices

Appendix A: Effect of the Choice of Architecture Set on the Decision Sensitivity Metric

Chapter 15 introduced the sensitivity of a metric to a decision, a numerical value that tells us how sensitive a metric is to a given decision, or, equivalently, the average change in a metric when we change the value of a decision. We provided two formulations for the sensitivity metric. The first is based on main effects and is valid only for binary decisions:

Main effect (Decision i, Metric M) \equiv \frac{1}{N_{1}} \sum_{{x | x_{i} = 1}} M (x) - \frac{1}{N_{0}} \sum_{{x | x_{i} = 0}} M (x)

$Main effect (Decision i, Metric M) \equiv \frac{1}{N_{1}} \sum_{{x | x_{i} = 1}} M (x) - \frac{1}{N_{0}} \sum_{{x | x_{i} = 0}} M (x)$

where N0 and N1 are the number of architectures for which xi = 0 and 1 respectively.

The second formulation extends this concept to k > 2 decisions:

Sensitivity (Decision i, Metric M) \equiv \frac{1}{K_{1}} \sum_{k \in K} | \frac{1}{N_{1, k}} \sum_{{x | x_{i} = k}} M (x) - \frac{1}{N_{0, k}} \sum_{{x | x_{i} \neq k}} M (x) |

$Sensitivity (Decision i, Metric M) \equiv \frac{1}{K_{1}} \sum_{k \in K} | \frac{1}{N_{1, k}} \sum_{{x | x_{i} = k}} M (x) - \frac{1}{N_{0, k}} \sum_{{x | x_{i} \neq k}} M (x) |$

where K is the set of options for decision i, and N0,k and N1,k are the number of architectures for which xi eq k and xi = k respectively.

Both equations require the definition of a set of architectures of interest, namely {x}, and the choice of this group may lead to different results. The following numerical example supports this claim.

Consider the dummy dataset provided in Figure 1. We have three binary decisions (D1 = {Y, N}, D2 = {1, 2}, and D3 = {A, B}) and two metrics (M1 and M2).

A table and a graph plot points for a numerical data set. — Figure 1 Dummy dataset: Numerical data and tradespace plot.

Figure 1 Full Alternative Text

Five out of eight architectures are non-dominated.

We computed the main effects of the three decisions for the two metrics with two different sets of architectures: the entire tradespace and the Pareto front. The results are given in Table 1.

A table — Table 1 | Main effects for the three decisions and two metrics for the entire tradespace and for the Pareto front only

In the case of the entire tradespace, the main effect for metric M1 and decision D1 (0.33) is greater in absolute value than the main effects for metric M1 decisions D2 and D3 (0.17), which are identical. Therefore, D2 and D3 have identical sensitivities for metric M1. However, when we restrict the analysis to the Pareto front, the main effect for metric M1 and decision D2 (–0.28) is greater in absolute value than the main effect for D3 (0.17), which suggests that in this case, M1 is more sensitive to D2 than to D3.

This means that although the sensitivity of metric M1 to D2 and D3 is the same when all architectures are considered, it is actually greater to D2 than to D3 when only good architectures are considered.

Thus, as explained in Chapter 15, it is important to choose the group of architectures that most closely matches our goals.

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