Inclusion degree measure
Parameters of membership functions of linguistic variable Probability values
𝜇𝑉𝑆
𝜇𝑆𝑀
𝜇𝐴𝑉
𝜇𝐵𝐺
𝜇𝑉𝐵
a
10-20
10-8
10-6
10-4
10-2
b
10-20
10-7
10-5
10-3
10-1
c
10-8
10-6
10-4
10-2
1
d
10-7
10-5
10-3
10-1
1
Very small
Small
Big
Very big
Average
0
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Fig. 2. Linguistic variable Probability in logarithmic form
In order to calculate the value of linguistic variable Probability for fuzzy probability P(K), similarity measure for
two fuzzy sets is required. In paper (Rajati et al., 2011), in order to find a value of linguistic probabilities,
Jacard’s similarity measure has been used. This measure for fuzzy sets A, B with membership functions μA, μB is
given by (Rajati et al., 2011):
𝑠𝐽 (𝐴, 𝐵) =
∫𝑋 𝑚𝑖𝑛(𝜇𝐴 (𝑥),𝜇𝐵 (𝑥)) 𝑑𝑥
(1)
∫𝑋 𝑚𝑎𝑥(𝜇𝐴 (𝑥),𝜇𝐵 (𝑥)) 𝑑𝑥
It has linear nature.
Let us consider the proposed logarithmic scale and fuzzy set A expressed by trapezoidal membership function
with parameters 𝑎 = 10−7 , 𝑏 = 10−6 , 𝑐 = 10−5 , and 𝑑 = 10−4 . This set and values SM, AV are given at Fig. 3.
Small
Average
Set A
1
0
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Fig. 3. Fuzzy set A and values “small”, “average” of linguistic variable Probability
We would like to have such a measure that has the following property 𝑀(𝐴, 𝑆𝑀) = 𝑀(𝐴, 𝐴𝑉). This is not true
for Jacard’s similarity. In order to satisfy reaching this requirement, one can define the logarithmic version of
Jacard’s similarity as follows:
𝑠𝑙𝑜𝑔𝐽 (𝐴, 𝐵) =
∫𝑋 𝑚𝑖𝑛(𝜇𝐴 (𝑙𝑜𝑔10 𝑥),𝜇𝐵 (𝑙𝑜𝑔10 𝑥)) 𝑑𝑥
∫𝑋 𝑚𝑎𝑥(𝜇𝐴 (𝑙𝑜𝑔10 𝑥),𝜇𝐵 (𝑙𝑜𝑔10 𝑥)) 𝑑𝑥
This measure satisfies the requirement 𝑀(𝐴, 𝑆𝑀) = 𝑀(𝐴, 𝐴𝑉). However, it causes the following difficulties. Let
A be the fuzzy set with membership function μA , and B be the membership function μB of a value of linguistic
variable Probability.
Let us suppose that
(2)
(∀𝑥 ∈ 𝑋)(𝜇𝐴 (𝑥) ≤ 𝜇𝐵 (𝑥))
(3)
and
∫𝑋 𝜇𝐴 (𝑙𝑜𝑔10 𝑥) 𝑑𝑥 ≪ ∫𝑋 𝜇𝐵 (𝑙𝑜𝑔10 𝑥) 𝑑𝑥
(4)
i.e. graphical picture of membership function of A is included in B’s membership function picture, and the
surface of the A is much smaller. In such a case, Jacard’s similarity has small value (<<1) what is inconsistent
with expectation that A should have the value B of the linguistic variable (the similarity should be close to 1).
Therefore, new measure called logarithmic inclusion degree
𝑠𝑙𝑜𝑔𝐼𝐷 (𝐴, 𝐵) =
∫𝑋 𝑚𝑖𝑛(𝜇𝐴 (𝑙𝑜𝑔10 𝑥),𝜇𝐵 (𝑙𝑜𝑔10 𝑥)) 𝑑𝑥
∫𝑋 𝜇𝐴 (𝑙𝑜𝑔10 𝑥)𝑑𝑥
where A is the fuzzy set, B is the value of linguistic variable Probability is introduced. This measure is
denoted by log ID.
(5)