Gianni Bellocchi - Model and Method Validation

Multiple-metric Aggregation

The procedure based on the multi-valued fuzzy set introduced by Professor Lofti Zadeh, follows the Sugeno method of fuzzy inference (Information Sciences, volume 36, pages 59–83 ). Three membership classes are basically defined for all metrics used in the validation work, according to an expert judgment, i.e. Favorable (F), Unfavorable (U), and partial (or fuzzy) membership, using S-shaped curves as transition possibilities in the range F to U:

$S(x;a;b)=\begin{cases}0 & x \le a\\ \frac{\left(x-a\right)^2\times 2}{\left(b-a\right)^2} & a \le x \le c\\ 1-\frac{\left(b-x\right)^2\times 2}{\left(b-a\right)^2} & c \le x \le b\\ 1 & b\le x\end{cases}$

where: x = the value of the basic input; a = the lower bound of the transition interval ; b = the upper bound of the transition interval ; c = (a + b) /2. According to the equation, if a = F, then x ≤ a means x = F, and S(x;a;b) gives the degree of membership of the index value x to the set U. Its complement, 1 - S(x;a;b), gives the degree of membership of the index value x to the set F.

A two-stage design of a fuzzy-based rules' inferring system is applied where firstly inputs with similar characteristics are aggregated into modules and then, using the same procedure, the modules can be aggregated into a second level integrated index called indicator. Both modules and indicator range from 0 to 1.

The control rules for estimating module values are based on logic relationships between inputs and outputs, expressed in linguistic terms by 'if-then' statements. For example, when two input variables (validation metrics) are aggregated four rules are required, formalized as:

____PREMISE____CONCLUSION

if x₁ is F and x₂ is F then y_i is B₁

if x₁ is F and x₂ is U then y₂ is B₂

if x₁ is U and x₂ is F then y₃ is B₃

if x₁ is U and x₂ is U then y₄ is B₄

where x_i is an input variable, y_i is an output variable and B_i is a conclusion (or expert weight). The value of each conjunction (… and …) is the minimum of the quantified fuzzy groups, which are obtained from complementary S-shaped distribution curves.

The output fuzzy sets for all the rules are then aggregated into a single fuzzy set. This group encompasses a range of output values, and is de-fuzzified in order to resolve a single crisp output value from the group (i.e. a value between 0 and 1). This approach uses the centroid method to obtain the representative non-fuzzy value for the output, as commonly adopted in the Sugeno-type systems. The expert reasoning runs as follows: if all input variables are F, the value of the module is 0 (good response according to all metrics used); if all indices are U, the value of the module is 1 (bad response according to all inputs used), while all the other combinations assume intermediate values. Limits F and U may come from experience, may be extracted from literature, or may be set by law. The weights can be chosen based on the analyst own experience in handling each input.

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Gianni Bellocchi - Model and Method Validation - Multiple-metric Aggregation