Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 276

SFF , ! 0 x X , PP x x z 0.

12 F F 1 2

According to this criterion, two overlapping fuzzy sets F 1 and F 2 should be
assigned a non-zero degree of similarity and should not be regarded as a
totally non-equal.

2. Only two equal fuzzy sets should have a similarity value s =1:

SFF , 1 P x P x , x X .

12 F F 1 2

This criterion assures that the equality is a special case of similarity, in the
same way as the crisp sets can be considered as a special case of fuzzy sets.

3. Non-overlapping fuzzy sets should be totally non-equal, i.e. s = 0:

SFF , 0 PP x x 0, x X .

12 F F

1 2

This assures that dissimilar (non-overlapping) fuzzy sets are excluded from
the set of similar fuzzy sets. Various degrees of similarity between distinct
fuzzy sets are related to the distance between them, and can be quantified
by a distance measure.

4. Similarity between two fuzzy sets should not be influenced by scaling or

shifting the domain on which they are defined:

c c

SFF , SFF , , P l kx P x ,

12 12 F c F

1 1

P l x P x kl , , ! \ , k .

F c F

2 2

This criterion is required for a fair comparison of similarities in the rule
base, as a similarity measure that satisfies this criterion is not influenced by
the numerical values of the domain variables.

Many methods have been proposed to assess the similarity or compatibility of
fuzzy concepts. A comparative analysis of different measures using human
subjects was reported by Zwick et al. (1987) and the mathematical relations
between the various measures were studied by Cross (1993). Later, Setnes (1995)
investigated the usefulness of various measures for fuzzy modelling.
According to the taxonomy presented by Cross (1993), the compatibility
measures can be divided into three broad classes: set-theoretical, logic-based, and
distance-based measures. Zwick et al . (1987) and Setnes (1995) used the term
similarity measures as a general description for methods of comparing fuzzy sets.
Unlike in the taxonomy by Cross, the term similarity is not reserved for a subclass
of measures, and all measures are divided into two main groups:

x geometric similarity measures