Ontological Fuzzy Rule System (OFRS)

weights is that T₁ is T₃ to the extent of 0.9, but T₃ is T₂ only to the extent of 0.4.

The weight values (0.4 and 0.9) were chosen arbitrarily (as in [1]), but they can be learned from a corpus based on term co-occurrences [17].

Example 6.1 Consider the ontology snippet from Figure 6.3 with two terms, T₁ and T₂, that have a common parent T₃. There are two ways of getting from T₁ to T₂, (T₁,T₃,T₂) and (T₁,T₃,T₄,T₃,T₂), hence the similarity between T₁ and T₂ is according to (1) s₁₂=max{w(T₁→T₃) × w(T₃→T₂), w(T₁→T₃) × w(T₃→T₄) × w(T₄→T₃) × w(T₃→T₂)} = max{0.9 × 0.4, 0.9 × 0.9 × 0.4 × 0.4}=0.36.

A more general case is when C₁ is an object that has some properties described by a set of terms (such as T₁ and T₂). The question is then to compute the similar-ity, s(C₁,C₂), of the object C₁ to another object C₂, described by terms from the same ontology. The resulting similarity is interpreted as the object C₂ is a C₁ to the extent of s(C₁,C₂). One method for computing the similarity between two objects C₁={T₁₁,..., T_1n} and C₂={T₂₁, ...,T_2m}, described by ontology terms, is the

where s(t_1i, T_2j) is computed using (6.1). The normalization of the average ensures that s(C₁,C₁)=1. Other possible similarity measures are described in Chapter 2.

Example 6.2 Let us compute the similarity between two objects, C₁={T₁, T₂} and C₂={T₃, T₄}. First, observe that if we do not consider the ontological relations be-tween the terms T_i, i∈{1,2,3,4}, the similarity between the two objects is 0. Assume that, using some term-similarity measure, such as Formula (1) in some ontology, we have: s(T₁,T₂) = 0.2, s(T₁,T₃) = 0.3, s(T₁,T₄) = 0.4, s(T₂,T₃) = 0.5, s(T₂,T₄) = 0.6, s(T₃,T₄) = 0.7, and s(T_i,T_i) = 1 for any i∈{1,2,3,4}. Then, using (6.2), we ob-tain s_a(C₁,C₂) = 0.45, s_a(C₁,C₁) = 0.8, s_a(C₂,C₂) = 0.85 and, fi nally, s(C₁,C₂)=0.45/

max{0.8, 0.85} = 0.53.

An interesting method for defi ning the distance between entities in fi rst-order logic (FOL) was found in [3]. The method was based on comparing the predicates used to describe two objects, and it was used in the conceptual clustering system, KBG.

6.3 Ontological Fuzzy Rule System (OFRS)

We defi ne an ontological fuzzy rule system (OFRS) by analogy to a Mamdani fuzzy rule system (FRS). A typical Mamdani FRS with n inputs and 1 output variable has the following form [6]:

where G_ij and P_i, i∈ [1,m], and j∈ [1,n], are fuzzy sets, {x_i} is the inputs variable, and y is the output variable, respectively. The fuzzy sets G_ij are possible values for the variable x_i, while P_i is a possible value for the output y. G_ij and P_i are usually represented using membership functions of the type shown in Figure 6.1. A Mam-dani FRS for m = 2 and n = 2 is shown in Figure 6.4.

The computation of the FRS output, y₀ ∈ R, for two inputs x₁₀, x₂₀ ∈ R, is performed as follows:

The memberships, w_ij=G_ij(x_i0), are computed for each rule i and input j, i,j∈{1,2}.

Second, the activation, a_i, of rule i is computed as a_i = w_i1⊕w_i2, where ⊕ is an AND type operator. In most applications ⊕ is minimum (that is, a_i = min{w_i1,w_i2}), but other choices are possible (see [6]). It is also possible for the variables in the rule antecedent (left-hand side) to be joined by an OR type operator. In this case, the typical operator employed is maximum, that is, a_i = max{w_i1,w_i2}. After computing the activation, the output of each rule is computed as a_iP_i,which is the shaded area of each output membership P_i shown in Figure 6.4. The fuzzy output of the system, P_sum, is computed by aggregating the individual rule outputs as P_sum = max{a₁P₁, a₂P₂}. Here, too, more choices of aggregating operators, other than max, are pos-sible. The fi nal step of the computation is called defuzzifi cation, and it consists of reducing the output fuzzy set P_sum to a number. Among the most used defuzzifi -cation procedures are the center of gravity (COG, y₀ in Figure 6.4) and mean of maximum (MOM, y₁ in Figure 6.4). Using the COG procedure, the output of the FRS, y₀, is computed as the center of the area under the membership function P_sum. By employing the MOM algorithm, the output y₁ is computed as the center of the region where the membership P_sum is the maximum.

An OFRS is similar to the FRS described above, except Some/all input fuzzy sets

1. G_ij are replaced by ontology terms or by objects described by ontology terms. The variables related to these terms or objects are called symbolic variables. As opposed to the numeric variables, for

Figure 6.4 A Mamdani fuzzy rule system with n = 2 rules and m = 2 inputs. The inputs and the output of the system are real numbers.

6.3 Ontological Fuzzy Rule System (OFRS) 119

which the values are fuzzy sets represented by functions, the symbolic variables have values that consist of ontology terms or objects represented by ontology terms. Similarly, the output fuzzy sets P_i are replaced with terms from the same ontology (output ontology) in order to allow the assessment of relatedness (similarity). The output of the OFRS consists of a term or a set of terms from the output ontology.

The membership,

2. w_ij, of the input x_i in the G_ij is now computed, based on the similarity of the two terms (objects), using (6.1) or (6.2) as w_ij = s(x_i, G_ij).

The aggregation of the rule output has to take into account that the fi nal 3.

output of the OFRS should be a term or a set of terms. Consequently, the defuzzifi cation procedure has to summarize the output of the set of rules in few terms. Similar to the defuzzifi cation procedures described above for FRS, we mention two summarization procedures. The fi rst procedure, somewhat similar to MOM, chooses as output the term P_k, with the maximum activation, where k is given by

1,

{ }

arg max _i

i m

k a

= = (6.4)

The wining rule activation a_k may be used as the confi dence of the OFRS output.

This fi rst case does not use the term similarity as mentioned in item 2. The second procedure, denoted here as ontological COG (OCOG), tries to fi nd the

“center”of all output terms P_i. In this case, the output term P_k is chosen as

( )

The OCOG procedure has two desired properties. First, the ancestors of a term contribute to the term’s importance, and hence, to its chance of winning in (6.5).

Second, since the similarity relation is nonsymmetrical, that is s(ancestor, child) <

s(child, ancestor) (as explained in Figure 6.3), the child does not contribute as much to its ancestor. Hence, the OCOG procedure tends to choose the more specifi c term as the fi nal output. However, the average similarity value in (6.5) has a tendency of producing low membership values. A possible solution for this problem is to use an OWA-type operator [19], such as summing only the fi rst n’ most similar terms where n´ < n.

Example 6.3 Consider the following medical OFRS:

rule 1: IF x₁=“back aches” AND x₂=“high fever” THEN P₁=“spinal meningitis”

rule 2: IF x₁=“aches” AND x₂=“moderate fever” THEN P₂=“fl u”

In the above OFRS, x₁ is a symbolic variable (a symptom), and x₂ is a numeric one (the fever in degrees Fahrenheit). Assume, as in Figure 6.3, that s(“back aches,”

“aches”) = 0.9, s(“aches,” “back aches”) = 0.4, s(“fl u,” “spinal meningitis”) = 0.1, s(“spinal meningitis,” “fl u”) = 0.2, and all self-similarities s(T_i, T_i) = 1. Moreover,

assume that the memberships for fever (“moderate” and “high”) are similar to the ones from Figure 6.1, but adapted to the range of the fever. For the purpose of this example, we assume the following memberships: w(“moderate,” 100) = 0.9 and w(“high,”100) = 0.5.

For an input x = {x₁ = “aches,” x₂ = 100}, rule 1 will have an activation a₁ = min{0.9,0.5} = 0.5 and rule 2 will have a₂ = min{1,0.9}=0.9. Using (6.4), the output of the system is given by argmax{0.5, 0.9} = 2, that is, P₂ = “fl u,” with confi dence 0.9. Note, that the activation of the fi rst rule is also signifi cant (0.5).

If we use (6.5) for the same input, we get argmax{(0.5*1+0.5*0.2)/2, (0.9*1+0.9*0.1)/2} = argmax{0.3, 0.49} = 2; hence, the output is P₂, with confi -dence 0.49. Here we see the tendency of the average operator from (6.5) to produce low confi dence values (0.49 is much smaller than 0.9, which was obtained in the fi rst case.