Conjonctive rules

There are two main families of conjunctive fuzzy rules:

  1. Mamdani type. The rule conclusion is a fuzzy set.
    The rule is written as:
    $\displaystyle IF  x_1  is  A_1^i  AND  x_2  is  A_2^i  \ldots  AND  x_p  is  A_p^i$      
    $\displaystyle THEN  y_1  is  C_1^i  \ldots  AND  y_q  is  C_q^i$      

    where $A_j^i  and  C_j^i$ are fuzzy sets defining the input and output space partitioning.

  2. Takagi-Sugeno type. The rule conclusion is a crisp value.
    The conclusion of the $i_{th}$ rule for the $j_{th}$ output is calculated as a linear function of the input values: $y_j^i = b_{jo}^i + b_{j1}^i x_1 + b_{j2}^i x_2 + \cdots + b_{jp}^i x_p$, also denoted $y_j^i = f_j^i(x)$.

    In FisPro, for interpretability reasons, the conclusion is limited to a constant $b_{jo}^i$.