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For conjunctive outputs, whatever the output nature, crisp or fuzzy, two aggregation operators, max and sum, are available (see part II, section 1
For conjunctive fuzzy outputs, several defuzzification operators are available: weighted area, mean-max and the Sugeno operator.
The weighted area defuzzification is analogous to centroid defuzzification, the only difference being that areas shared by two fuzzy sets are counted twice.
The mean-max operator returns the middle point of the segment corresponding to the alpha cut which has the greatest matching degree. In case of ambiguous maxima, a warning message is displayed.
The Sugeno operator returns as the rule conclusion the middle point of the corresponding fuzzy set kernel.
For crisp outputs, two defuzzification operators can be used: MaxCrisp, a mean of maximum adaptation for crisps outputs, and the Sugeno operator. In this case, the output is limited to a constant value, to ensure interpretability. The Sugeno operator returns an output equal to the weighted sum of the rule conclusions, the weight being the matching degree.
Note on defuzzification of fuzzy outputs
For all fuzzy output defuzzification operators, a correction is applied to the upper bound (resp. lower) of the upper semi-trapezoidal MF (resp. lower), located at the fuzzy partition edge. This allows to infer extreme values (output range boundaries). This correction is automatically applied when building a whole partition: regular or irregular grid, or when generating a FIS without rules.
If these bounds are edited by hand, the correction is launched by a change in the choice of the defuzzification operator.
This correction can also be applied when loading a previously saved FIS. To do so, one needs to check the checkbox labelled Ensure the output bounds to be inferred.
Alarm threshold parameter
The threshold parameter is a tolerance threshold associated to a warning message at the defuzzification stage (see the alarm paragraph in section 2).
Classification option
If the classification option is chosen:
Note: Discrete MFs
Discrete MFs in a fuzzy output are not compatible with a weighted area defuzzification.
New output - output removal
As for inputs, a new output cannot be added to a FIS if a data file is currently opened. The data file must be closed first.
Implicative rules
The choice between implicative rule systems and conjunctive ones is done through the output paramers. Indeed the input variable partitioning, as well as the computation of the rule matching degree for given input data, are independent of the rule nature.
Implicative rules are available using an Impli checkbox in the output window. If this option is chosen, the only available defuzzification operator is called Impli, and the choice of the implication operator is done by selecting the rule agregation method (Disjunction field). Three operators are available: Resher-Gaines, Goguen et Gödel. The possibility distributions inferred using one of these operators all have the same kernel, and only differ by their supports.
The agregation method specific characteristic of implicative rules imposes a particular output partition if one wants to avoid having too many conflicting rules and empty inferred outputs. We advise to use the quasi-standard partitions (QSF) derived from the standard fuzzy partitions (SFP). They are automatically built when using the menu option Regular Grid.
The choice of implicative rules forbids the use of a Crisp
output.
The Regular Grid option builds a QSP partition, which is
adapted to implicative rules.
When transforming a conjunctive output into an implicative one, by checking the impli checkbox, the partition modification from SFP to QSP is systematically proposed.
Conversely, when transforming an implicative output into a conjunctive one by unchecking the Impli checkbox, the current QSP partition will be proposed to be transformed into a SFP one.
The Classification option has no effect on implicative rules.