WCCI 2012 IEEE World Congress on Computational Intelligence
June, 10-15, 2012 - Brisbane, Australia
FUZZ IEEE
Backward Fuzzy Interpolation and Extrapolation
with Multiple Multi-antecedent Rules
Shangzhu Jin, Ren Diao, and Qiang Shen
Department of Computer Science
Aberystwyth University
Aberystwyth, SY23 3DB, UK
Email: {shj1, rrd09, qqs}@aber.ac.uk
Abstract—Fuzzy rule interpolation is well known for reducing
the complexity of fuzzy models and making inference possible
in sparse rule-based systems. However, in practical applications
with inter-connected subsets of rules, situations may arise when
a crucial antecedent of observation is absent, either due to
human error or difficulty in obtaining data, while the associated
conclusion may be derived according to alternative rules or even
observed directly. If such missing antecedents were involved in
the subsequent interpolation process, the final conclusion would
not be deduced using conventional means. However, missing
antecedents may be related to certain conclusion and therefore,
may be inferred or interpolated using the known antecedents and
conclusion. For this purpose, this paper presents a novel approach
termed backward fuzzy rule interpolation and extrapolation.
In particular, the approach supports both interpolation and
extrapolation which involve multiple intertwined fuzzy rules,
with each having multiple antecedents. An algorithm is given
to implement the approach via the use of the scale and move
transformation-based fuzzy interpolation. The algorithm makes
use of trapezoidal fuzzy membership functions. Realistic application examples are provided to demonstrate the efficacy of the
approach.
The second category is based on the analogical reasoning mechanism [5]. These approaches [1], [11], [12] first
interpolate an artificially created intermediate rule, such that
the antecedents of the intermediate rule are “closer” to the
given observation. Then, a conclusion can be deduced by
firing this intermediate rule through analogical reasoning. The
shape distinguishability between the resulting fuzzy set and
the consequence of the intermediate rule, is then analogous to
the shape distinguishability between the observation and the
antecedent of the created intermediate rule. In particular, the
scale and move transformation-based approach (T-FIR) [11],
[12] offers a flexible means to handle both interpolation and
extrapolation involving multiple fuzzy rules. T-FIR guarantees
the uniqueness as well as normality and convexity of the
interpolated fuzzy sets. It is also able to handle interpolation
of multiple antecedent variables with different types of fuzzy
membership function.
I. I NTRODUCTION
Fuzzy Rule Interpolation (FRI) was originally proposed in
[19], [20]. It is of particular significance for reasoning in the
presence of insufficient knowledge or sparse rule bases. When
a given observation has no overlap with antecedent values, no
rule can be invoked in classical fuzzy inference, and therefore
no consequence can be derived. The techniques of FRI may
not only support inference in such situations, but also help
in reducing the complexity of fuzzy models. A number of
important interpolation approaches have been proposed in the
literature, including [1], [7], [11], [12], [21], [17], [23], most
of which can be categorised into two classes with several
exceptions (e.g. type II fuzzy interpolation [6]).
The first category of approaches directly interpolates rules
whose antecedents match the given observation. The consequence of the interpolated rule is thus the logical outcome.
Most typical approaches in this group [18], [19], [20] are based
on the use of α -cuts (α ∈ (0, 1]). The α -cut of the interpolated
consequent fuzzy set is calculated from the α -cuts of the
observed antecedent fuzzy sets, and those of all the fuzzy sets
involved in the rules used for interpolation. Having found the
consequent α -cuts for all α ∈ (0, 1], the consequent fuzzy set
is then assembled by applying the Resolution Principle.
U.S. Government work not protected by U.S. copyright
Fig. 1.
A system structure that may benefit from backward fuzzy rule
interpolation
Despite the numerous approaches developed, FRI techniques are relatively rarely applied in practice [14]. One of
the main reasons is that many applications require multipleinput and multiple-output systems. The rules involved may be
irregular in nature, and could be arranged in an inter-connected
mesh, where observations and conclusions in between different
subsets of rules could be overlapped, and yet not directly
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related. For such complex systems, any missing values in a
given set of observations may leed to failure in interpolation.
For instance, in Fig. 1, the final conclusion Bn of rule Rn can
not be interpolated in a straightforward manner, because the
three missing observations Anp , Anr and Anm cannot be deduced
by conventional means.
To address such problems, a novel approach termed backward fuzzy rule interpolation and extrapolation (B-FRIE)
is proposed in this paper. The work enables the unknown
antecedents to be interpolated, if all other antecedents and
the conclusion are known. Being a beneficial addition to FRI,
B-FRIE effectively achieves indirect interpolative reasoning.
Using the earlier example in Fig. 1, the unknown antecedents
Anr and Anp can be backward interpolated according to rules
R j and Ri , where the conclusions A1j , Aiq and other antecedent
terms are already known. The last missing antecedent Anm can
then be interpolated using R1 , and subsequently Bn can also
be deduced, as now all required antecedents are known for
forward interpolation. In this work, it is assumed that fuzzy
sets are represented by trapezoidal membership functions.
Since situations may arise in practice where the closest rules
do not flank a given observation (intertwined rules). This
research also attempts to cope with fuzzy extrapolation. The
resulting algorithm is implemented using T-FIR with multiple
multi-antecedent rules.
The rest of this paper is organised as follows. Section II
reviews the general concepts of T-FIR. Section III presents the
B-FRIE approach in detail, including both interpolation and
extrapolation methods, along with worked examples. Section
IV provides an application example through the use of B-FRIE,
demonstrating the efficacy of the proposed approach. Section
V concludes the paper and suggests future enhancements.
Fig. 2.
The representative value of trapezoidal fuzzy set
antecedent variables:
s
M
∑ dA2k ,
d=
dAk =
k=1
d(Rep(Aik ), Rep(A∗k ))
rangeAk
(2)
where rangeAk = minAk − maxAk is the domain of the variable
xk . This normalises the otherwise absolute distance measure
into the range [0, 1], so that distances are compatible with each
other over different domains. The N (N ≥ 2) rules which have
the least distance measurements, with regard to the observed
values A∗k and the conclusion B∗ , are then chosen to perform
interpolation.
B. Construct the Intermediate Rule
Suppose these N rules are represented as Ri , i ∈ {1, · · · , N},
each having M antecedents Aik , k ∈ {1, · · · , M}. Let the normalised displacement factor ωAi , as shown in Eqn. 3, denote
k
the weight of the kth antecedent of the ith rule:
′
II. T RANSFORMATION -BASED F UZZY I NTERPOLATION
This section provides an outline of the existing T-FIR
method, where the essential formulae are presented. A more
detailed explanation can be found in the original work [12].
Consider the following rules Ri and an observation O:
Ri : IF xk is Aik , k ∈ {1, · · · , M}, THEN y is Bi
O: A∗1 , · · · , A∗k , · · · , A∗M
where Aik = (a0 , a1 , a2 , a3 ) is the linguistic term of the ith rule,
which is defined on the domain of the antecedent variable k,
k ∈ {1, · · · , M}, where M is the total number of antecedents.
The observed fuzzy set of variable xk is denoted by A∗k . A
key notion of T-FIR is the representative value Rep(Aik ) of
the trapezoidal fuzzy set Aik , as illustrated in Fig. 2, which is
defined as:
2
a0 + a1 +a
2 + a3
Rep(A) =
3
ωAi =
k
ωAi
′
k
′
∑Ni=1 ωAi
,
∗
i
ωAi = exp−d(Ak ,Ak )
(3)
k
k
′′
The so-called intermediate fuzzy terms Ak are constructed
′
from the rule antecedents. These are shifted to Ak in order
for them to have the same representative values of A∗k :
′
′′
Ak = Ak + δAk rangeAk ,
′′
N
Ak = ∑ ωAi Aik
i=1
k
(4)
′
where δAk is the bias between A∗k and Ak on the kth input
domain:
′
δAk
Rep(A∗k ) − Rep(Ak )
=
rangek
(5)
′
(1)
Similar to Eqn. 4, the shifted intermediate consequence Bk
can also be computed, where the parameters ωBi and δB are
′
aggregated from the values of Ak , such that:
A. Determine Closest N Rules for Observation
The distance d between a rule and an observation can then
be determined by computing the aggregated distance of all
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ωBi =
1 M
∑ ωAik ,
M k=1
δB =
1 M
∑ δAk
M k=1
(6)
C. Scale Transformation
then be used for B-FRIE:
For each trapezoidal antecedent fuzzy set of the N chosen
rules, the scale transformation calculates two scale rates sAk
and sAk :
A∗l = fB−FRIE (B∗ , A∗1 , · · · , A∗l−1 , A∗l+1 , · · · , A∗M ), (Ri , · · · , Rt ))
(12)
where fB−FRIE denotes the entire process of backward interpolation/extrapolation, using N closest rules Ri to Rt , with regard
to the observed (or predicted) values from (M − 1) antecedents
and their corresponding conclusion B∗ . A∗l is the unknown
observation that is to be backward interpolated.
′
s=
′
′
a2 − a1
,
a2 − a1
′
s=
a3 − a0
a3 − a0
′
′
(7)
′
′
which rescale the top (a1 , a2 ) and bottom (a0 , a3 ) supports
′
′′
of Ak with respect to the observation A∗k , resulting in Ak .
The corresponding bottom scale rate sB , and the intermediate
′
top scale rate sB ′ of the intermediate conclusion B , are
obtained by averaging those of the antecedents. To maintain
the convexity of the rescaled fuzzy sets, sB ′ is then further
modified according to the scale ratio S:
 a ′ −a ′ a −a
2
1 − 2 1

a3 ′ −a0 ′ a3 −a0


i f s′ ≥ s ≥ 0 , S ∈ [0, 1]

 1− a2 ′ −a1 ′
a3 ′ −a0 ′
S = a2 ′ −a1 ′ a2 −a1
(8)

′ −a ′ − a3 −a0

a
 3 0′ ′

i f s ≥ s′ ≥ 0 , S ∈ [−1, 0]

a2 −a1
a3 ′ −a0 ′
which represents the actual increase of the ratios between the
top and the bottom support. The final sB becomes:
(
sB ∗S ′
′
sB − sB ∗ S + sB , sB ≥ sB ≥ 0
sB =
(9)
sB ∗ S, sB ≥ sB ′ ≥ 0
D. Move Transformation
′′
Using the move rate mAk as given in Eqn. 10, Ak is moved
so that the final transformed fuzzy set matches the exact shape
of the observed value A∗k . From this, mB for the conclusion can
be calculated such that:
(
′′
′′
0 −a0 )
mAk = 3(a
1 M
a1 ′′ −a0 ′′ , a0 ≥ a0
mB =
mAk ,
(10)
′′
∑
M k=1
mAk = 3(a′′0 −a0 ′′ ) , otherwise
A. The Approach
A close examination of the T-FIR algorithm reveals that
the parameters that are required to calculate the consequence
variable are ωBi , δB , sB , sB and mB . These parameters are algebraic averages of the parameters from individual antecedent
terms according to Eqns. 3, 5, 7 and 10. Thus, it has an
intuitive appeal to assume that, in order to perform backward
interpolation, the consequent variable should be treated with
a biased weight that is the sum of all antecedent weights.
The parameters for the missing antecedent should then be
calculated by subtracting parameter values of the known
antecedents from the conclusion. The following summarises
the proposed B-FRIE algorithm that reflects this intuition.
1) Determination of Closest Rules
In reference to the earlier definition of the B-FRIE process in Eqn. 12, when B∗ , (A∗1 , · · · , A∗l−1 , A∗l+1 , · · · , A∗M )
are given, in order to interpolate/extrapolate the unknown antecedent A∗l , the discovery of the closest rules
Ri to Rt are required. In contrast to the plain distance
measure introduced in 2, a modified scheme is proposed
which reflects the biased consideration towards the consequent variable (as per the intuition indicated above):
v
u
u
d = twB dB 2 +
a3 −a2
The final interpolated result B∗ can now be estimated by
′
applying the scale and move transformation to B , using the
parameters sB , sB , and mB .
(wAk dAk 2 )
In implementing B-FRIE, without sufficient expert
knowledge on the relative level of importance of different antecedents, all antecedents are treated equally:
wB =
∑ wAk = 1,
wA1 = wAk = wAM =
k=1
In this section, B-FRIE is developed for the general case involving multiple multi-antecedent rules, with trapezoidal fuzzy
sets. If a conventional fuzzy rule interpolation/extrapolation
can be represented as follows:
(11)
where fFRIE denotes the interpolation/extrapolation process
from M observed values, using N rules Ri to Rt that are closest
with respect to A∗l , l ∈ {1, · · · , M}, and B∗ is the interpolated
conclusion. Without losing generality, the following form can
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(13)
k=1, k6=l
M
III. BACKWARD F UZZY RULE I NTERPOLATION AND
E XTRAPOLATION
B∗ = fFRIE (A∗1 , · · · , A∗l , · · · , A∗M , (Ri , · · · , Rt ))
M
∑
1
M
(14)
The above formula can therefore be simplified to:
v
u
M
u
1
ˆ
d = tdB 2 +
∑ dAk 2
M k=1,
k6=l
(15)
Note that in choosing the closest rules, the square
root used in the original distance measure becomes
unnecessary, as only the ordering information is needed.
2) Construct the Intermediate Fuzzy Terms
For better illustration of the later interpolation procedures, the N closest rules Ri and the observation O are
represented as follows:
Ri : IF xk is Aik , k ∈ {1, · · · , M}, THEN y is Bi
O: A∗1 , · · · , A∗l−1 , A∗l+1 , · · · , A∗M , B∗
The first step of the actual interpolation process is
to create so-called intermediate fuzzy terms for each
′
antecedent and the consequent variable, including Al for
the missing l th antecedent. For this, ωAi needs to be
l
calculated. Recall that ωAi is part of the ∑M
k=1 ωAik on the
l
right hand side of Eqn. 3. Now that B∗ is already known,
ωBi along with ωAi for the other known antecedents Aik
k
can be calculated similar to Eqns. 3.
It is then possible to deduce ωAi by the use of:
l
l
∑
ωAi
k=1, k6=l
(16)
k
′′
and further allows the intermediate fuzzy term Al of the
missing term A∗l to be calculated:
N
′′
Al = ∑ (ωAi Ail )
∑
k=1, k6=l
and sAl ′ , SAl , and mAl can be calculated in exactly the
same manner. Note that to guarantee the transformed
fuzzy sets to be convex, sAl ′ should also be adjusted
into sAl in terms of the scale ratio SAl similar to Eqn. 9.
′
4) Finally, the scale and move transformation on Al is performed using the parameters sAl , sAl , and mAl , resulting
in the backward interpolated value A∗l .
Extrapolation [12] is a special case of interpolation, when
all of the closest rules chosen lie on one side of the given
observation. Determining the closest rules and constructing
the intermediate rule are carried out in the same way as those
for interpolation.
Example 3.1: This example illustrates backward interpolation.The observation and the four closest rules are given in
Table I and Fig. 3 (the sub-process of selecting the closet
rules is a straightforward application of Eqn. 15 to the sparse
rule base). Here, A∗3 is the missing antecedent which is to be
inferred.
µ
A 11 A 41
A 12
A *1
rule 1
rule 2
rule 3
rule 4
observation
X1
A 31
′
The intermediate fuzzy terms Ak , k = 1, · · · , M, k 6= l and
′′
B is now computed according to Eqn. 4, where δAk is
computed by Eqn. 5, and δB by:
δB =
(21)
sAk
(17)
l
i=1
M
sAl = M × sB −
B. Worked Examples
M
ωAi = M × ωBi −
derived as per Eqns. 15 and 16, such that:
′
Rep(B∗ ) − Rep(B )
rangey
0 1
µ
2 3
4 5
A 22 A 2*
6
7
8 9 10 11 12 13 14 15
A 32
A 12
A 42
(18)
X2
0 1
′
µ
The bias δAl between A∗l and Al is thus:
2
3 4 5
A 13
6
7
A 43
8 9 10 11 12 13 14 15
A 23
A *3
A 33
M
δAl = M × δB −
∑
δAk
X3
(19)
0 1
k=1, k6=l
µ
′
The shifted fuzzy term Al which has the same represen′′
tative value as A∗l , can be obtained from Al using:
′
′′
Al = Al + δAl rangeAl
(20)
where rangeAl = maxAl −minAl , is the range between the
maximal and minimal domain values of variable xl .
3) Scale and Move Transformation
Having the intermediate (shifted) fuzzy terms, the scale
rates s, s, scale ratio S, and move rate m for each of
the known antecedents Ak and conclusion B can now
be computed using Eqns. 7 to 10. Following the same
reasoning steps as those of dˆ and ωAi , by reversing
l
the forward transformation procedure, the generalised
formulae for parameters such as sAl can be similarly
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2
3 4
5
A 43
A *4
2
3 4 5
6
7
A 44
8 9 10 11 12 13 14 15
A 42
A 41
X4
0 1
µ
6
B3 B*
7
B1
8 9 10 11 12 13 14 15
B2
B4
Y
0 1
2
3
Fig. 3.
4 5
6
7
8 9 10 11 12 13 14 15
Example of B-FRI with multiple antecedents
1) Construct Intermediate Fuzzy Terms:
The normalised weights shown in Table II for the
antecedents and observed conclusion are obtained according to Eqn. 3. The parameter for the missing
TABLE I
F OUR CLOSEST RULES FOR OBSERVATION
Observation
O
Rule
R1
R2
R3
R4
Antecedents
A∗1 = (8.2, 9.5, 10.5, 11.0) A∗2 = (3.1, 3.2, 3.5, 4.3)
A∗3 = missing A∗4 = (3.6, 4.4, 4.8, 5.4)
Consequence
Antecedents
A11 = (0.2, 1.1, 2.2, 2.7) A12 = (5.5, 6.0, 7.0, 8.0)
A13 = (0.0, 1.0, 2.0, 3.0) A14 = (10.1, 12.0, 12.5, 13.3)
A21 = (4.3, 5.4, 6.4, 7.2) A22 = (1.5, 2.0, 2.5, 3.0)
A23 = (2.5, 3.5, 4.2, 4.5) A24 = (6.8, 8.1, 9.3, 10.0)
A31 = (10.5, 11.5, 12.5, 13.1) A32 = (7.5, 8.5, 9.2, 10.3)
A33 = (12.4, 13.5, 14.0, 14.5) A34 = (0.1, 1.5, 2.1, 2.5)
A41 = (2.0, 2.3, 2.5, 3.4) A42 = (10.0, 11.2, 12.3, 13.0)
A43 = (8.0, 8.5, 9.5, 10.0) A44 = (5.5, 6.2, 7.5, 8.5)
Consequence
TABLE II
T HE NORMALISED WEIGHTS OF THE KNOWN ANTECEDENTS
Observation
R1
0.21
R2
0.09
R3
0.62
R4
0.07
Antecedent 1
Antecedent 2
Antecedent 4
0.12
0.23
0.12
0.26
0.54
0.22
0.48
0.14
0.29
0.14
0.09
0.37
µ
B1 = (5.0, 6.1, 7.0, 8.0)
B2 = (9.5, 10.0, 11.3, 12.5)
B3 = (0.2, 2.0, 2.5, 3.0)
B4 = (12.0, 13.0, 13.5, 14.2)
A *1
A 11
A 31
rule1
rule2
rule3
observation
A 12
X1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
µ
observation ωAi , i = 1, 2, 3, 4 can then be calculated
3
using Eqn. 16: ωA1 = 0.37, ωA2 = −0.64, ωA3 =
3
3
3
1.58, ωA4 = −0.31, and the intermediate fuzzy set
′′
B∗ = (2.0, 2.8, 3.3, 4.0)
A 22
A 12
A 32
X2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
µ
3
A3 = (15.51, 16.82, 17.22, 18.04) can be obtained according to Eqn. 17. Then, the bias δA3 between A∗3
′
and A3 is calculated by Eqn. 19, δA3 = −0.24. The
′
shifted fuzzy term A3 which has the same represen∗
tative value as A3 , can be obtained from Eqn. 20,
′
A3 = (10.74, 12.05, 12.46, 13.27).
′
2) Scale and Move from A3 to A∗3 :
The individual scale and move parameters can be calculated according to Eqns. 7 to 10, resulting in sB =
0.71, sB = 0.76, mB = 0.22. The scale ratio SB∗ = 0.02
is obtained similar to Eqn. 8. Similarly, the relevant
parameters sAk , sAk , mAk of antecedents A∗1 , A∗2 , A∗4 can
be obtained. From equations similar to Eqn. 21, it
can be calculated that sA3 = 0.51, sA3 = 2.38 and
′′
mA3 = 0.38. The second intermediate fuzzy term A3 can
be computed to be (11.43, 11.63, 12.60, 12.72). Finally,
the transformed A∗3 = (11.46, 11.58, 12.54, 12.75) can be
obtained.
A *2
A 33
A *3
A 31
A 23
X3
0 1 2 3 4 5
µ
6 7 8 9 10 11 12 13 14 15
B*
B1
B2
B3
Y
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Fig. 4. Example of backward fuzzy rule extrapolation with multiple multiantecedent rules
Example 3.2: This example explains the process of backward fuzzy rule extrapolation (B-FRE), which in essence is
the same as that for B-FRI. Suppose that the observation and
the three closest rules as given in Table III and Fig. 4 are used
for extrapolation, where all rules lie on the right side of the
observation.In this example, A∗2 is the missing antecedent.
1) Construct Intermediate Fuzzy Terms:
The normalised weights associated with the observed
antecedents and conclusion are listed in Table IV. The
parameters for the missing observation ωAi , i = 1, 2, 3
2
can then be calculated using Eqn. 16 such that ωA1 =
2
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TABLE IV
T HE NORMALISED WEIGHTS OF THE KNOWN ANTECEDENTS
Observation
R1
0.55
R2
0.27
R3
0.18
Antecedent 1
Antecedent 3
0.53
0.18
0.19
0.25
0.28
0.57
0.94, ωA2 = 0.37, ωA3 = −0.31, and the intermediate
2
′′
2
fuzzy set A2 = (4.96, 5.96, 6.96, 7.96) can be obtained
according to Eqn. 17. Then, the bias δA2 between A∗2
′
and A2 is calculated by Eqn. 19, δA2 = 0.23. The
′
shifted fuzzy term A2 which has the same represen∗
tative value as A2 , can be obtained from Eqn. 20,
′
A2 = (1.53, 2.53, 3.53, 4.53).
′
2) Scale and Move from A3 to A∗3 :
TABLE III
T HREE CLOSEST RULES FOR OBSERVATION
Observation
O
Antecedents
A∗1 = (0, 1, 2, 3) A∗2 = missing A∗3 = (0, 0.5, 1.5, 2.5)
Consequence
B∗ = (2.0, 2.9, 3.75, 4.25)
Rule
R1
R2
R3
Antecedents
A11 = (3.5, 5, 6, 7) A12 = (7, 8, 9, 10) A13 = (10, 11, 12, 13)
A21 = (11, 12, 13, 14) A22 = (4, 5, 6, 7) A23 = (7, 8, 9, 11)
A31 = (7.5, 8, 9, 10) A32 = (10, 11, 12, 13) A33 = (3, 4, 5, 6)
Consequence
B1 = (4.6, 5.8, 6.8, 7.9)
B2 = (8, 9, 10, 11)
B3 = (11, 12, 13, 14)
The individual scale and move parameters can be calculated with respect to Eqns. 7 to 10, resulting in
sB = 0.71, sB = 0.85, mB = −0.27. The scale ratio
SB∗ = 0.09 is obtained according to Eqn. 8. Similarly, the
relevant parameters sAk , sAk , mAk of antecedents A∗1 , A∗3
can be obtained. sA2 = 0.40, sA2 = 0.50 and mA2 = −1.11
are calculated using equations similar to Eqn. 21. The
′′
second intermediate fuzzy term A2 can then be computed: (2.43, 2.78, 3.28, 3.64). Finally, the transformed
A∗2 = (2.30, 3.05, 3.54, 3.51) is obtained.
IV. E XPERIMENTATION AND D ISCUSSION
A. Experimental Scenario
To resolve the puzzle of a given crime from a set of available
evidence, investigators and forensic analysts aim to disclose
the scenario that has actually taken place and to determine
efficient strategies to proceed with the investigation [15]. The
effectiveness of such an analytical process depends critically
on investigators’ ability to articulate plausible scenarios and
to identify course of actions needed for differentiating them
[9].
The bottleneck of accomplishing this task is the fact that
humans are relatively inefficient at hypothetical reasoning.
A fuzzy decision-support system may assist investigators in
generating plausible scenarios and analysing them objectively.
Examples of such serious crimes detection systems have been
reported in the literature, including trafficking and terrorism
[3], [4].
In this paper, a simplified practical scenario that involves the
possible detection of a terrorist bombing threat is considered.
The likelihood of an explosion may be directly affected by
the number of people in the area, a crowded place is usually
more likely to attract terrorists’ attention. The number of
public warning signs displayed in the area may also affect
the potential outcome. With many eye-catching warning signs,
people may raise their awareness of the surroundings, and may
promptly report suspicious individuals or items, giving less
opportunities for the terrorists to attack. Given such concepts
regarding the antecedent variables Crowdedness, Warning
level, and the consequent variable Explosion Likelihood, a
set of rules can be established with example rules listed in
Table V.
Additionally, there exists another rule set that focuses on
the prediction of crowdedness. The number of people in an
area is directly related to the Popularity of the place, but
it can also be affected by the level of Travel Convenience.
A well known place that is hard to get to by usual means,
may not have more people than an attraction that is less wellknown, but very easy to reach. Moreover, considering the
current scenario, the crowdedness may also change in relation
to the Warning level. The less brave individuals may shy away
from places that are considered dangerous or high risk, judged
from the amount of explosion alerts in the surrounding areas.
Summarising the above information, the second rule set may
be derived. Table VI displays a subset of these rules.
It is easy to identify that both rule sets involved are fuzzy
and sparse. Fuzziness is naturally introduced by the presence
of linguistic terms used to describe the domain variables. From
the linguistic terms that are represented by fuzzy sets, it can be
deduced that the rule base consists of two subsets of rules, both
contain substantial gaps amongst the underlying domains of
the variables concerned. The terms Low, Moderate, and High
although provide reasonable coverage of the entire domain,
intermediate values such as Moderate Low, Moderate High
are not captured.
For traditional forward interpolative reasoning, in order to
interpolate Explosion Likelihood, the observed values for
the level of Crowdedness and the Warning level are both
required. The fuzzy value of the antecedent variable Warning
is particularly important for it is required by both sets of the
rules. Without it, no matter what other information is available,
even with the Crowdedness known, as illustrated in Table VII,
forward interpolation would still fail.
In order to interpolate Explosion Likelihood, it is essential
to first determine the value of Warning using B-FRIE. Following the steps detailed previously, the three closest rules are
first selected using the consequence-biased distance measure
as per Eqn. 15. The resulting closest rules with respect to the
observation given in Table VII are shown in Table VIII and
Fig. 5.
′
The
intermediate
fuzzy
term
Awarning =
(8.95, 9.57, 10.82, 11.45) can be derived from the three
neighbouring rules, by using the relative displacement factor
ωwarning calculated from Eqn. 16. It is then scaled and moved
using swarning , swarning and mwarning according to equations
such as Eqns. 21, producing the backward interpolated value
A∗warning = (8.88, 9.38, 10.91, 11.58).
Now with both antecedents A∗crowdedness = (6.3, 7.1, 8.0, 8.9)
and A∗warning = (8.88, 9.38, 10.91, 11.58) present, the second set
of rules can be evoked to produce the final interpolation result,
B∗explosion = (4.56, 5.80, 5.68, 7.21), this time using the standard
T-FIR (whose calculation is omitted here).
Note that if the normal distance measure in Eqn. 2 is
used, the rules will no longer be selected with biased focus
1175
TABLE V
E XAMPLE RULES FOR E XPLOSION L IKELIHOOD
Crowdedness
Very Low
Moderate
Moderate
High
C1
C2
C3
C4
Warning
Moderate
Moderate
High
Moderate
Explosion Likelihood
Very Low
Low
Low
Moderate
TABLE VI
E XAMPLE RULES FOR C ROWDEDNESS
C1
C2
C3
C4
C5
C6
C7
C8
Popularity
Very Low
Very Low
Very Low
Moderate
Moderate
High
High
High
Travel
Very Low
Very High
High
Moderate
High
Very Low
High
High
Warning
Very High
Very Low
High
High
Low
High
Moderate
Very Low
Crowdedness
Very Low
Low
Low
Low
Moderate
Very Low
Moderate
High
TABLE VII
O BSERVATION
Popularity
(5.8, 6.8, 7.4, 8.2)
Moderate High
Travel
(2.6, 3.4, 4.0, 5.2)
Moderate Low
Warning
N/A
N/A
Crowdedness
(6.3,7.1,8.0,8.9)
Moderate High
TABLE VIII
T HREE C LOSEST RULES USING B IASED D ISTANCE M EASURE
Popularity
(7.82,8.45,9.70,10.32)
(3.22,3.84,5.09,5.72)
(9.40,10.03,11.28,11.90)
µ
A 12
A *1 A 11
A 31
Travel
(4.46,5.09,6.34,6.96)
(0.95,1.57,2.82,3.45)
(0.08,0.35,1.60,2.22)
rule1
rule2
rule3
observation
Popularity
0 1
µ
2 3
4 5
6
A 32A 22
A *2
A 12
2
3 4 5
6
7
8 9 10 11 12
7
8 9 10 11 12
Travel
0 1
µ
A 13 A 23 A 33
A *3
Warning
0 1
µ
2
3 4
5
6
B1
B3 B2
7
8 9 10 11 12
B*
Warning
(0.59,1.21,2.46,3.09)
(0.86,1.48,2.73,3.36)
(2.40,3.03,4.28,4.90)
Crowdedness
(4.91,5.53,6.78,7.41)
(1.52,2.15,3.40,4.02)
(1.41,2.04,3.29,3.91)
on the consequence, instead treating everything with equal
weight. The rules shown in Table IX will then be selected.
Following the backward interpolation process again, a different
outcome of A∗warning = (1.17, 1.73, 3.29, 3.97) will be derived,
followed by B∗explosion = (2.96, 4.26, 4.13, 5.66) interpolated
using T-FIR on the second set of rules. Looking back at
the original observation, given the two antecedent values
∗
A∗popularity = (5.8, 6.8, 7.4, 8.2) and Atravel
= (2.6, 3.4, 4.0, 5.2),
the intuitive deduction of Crowdedness should be quite high,
as the place is both moderately high in popularity and is moderately convenient to reach. The only reason that the observed
Crowdedness is having a moderate value of (6.3, 7.1, 8.0, 8.9)
may well have been caused by a reasonable level of Warning.
Intuitively the outcome A∗warning = (8.88, 9.38, 10.91, 11.58)
from the biased distance measure is therefore more agreeable
than A∗warning = (1.17, 1.73, 3.29, 3.97) from the plain distance measure. In-depth experiments show that, if A∗warning =
(1.17, 1.73, 3.29, 3.97) and the two observed antecedent values
were used, T-FIR method would interpolate Crowdedness as
(2.89, 4.29, 3.89, 5.49), which would be much further than the
original observation.
Crowdedness
0 1
2
3
Fig. 5.
4 5
6
7
V. C ONCLUSION
8 9 10 11 12
Calculate the value of Warning using B-FRI
This paper has presented a novel extension to traditional
fuzzy rule interpolation by introducing methods to perform
1176
TABLE IX
T HREE C LOSEST RULES USING P LAIN D ISTANCE M EASURE
Popularity
(4.88,5.51,6.76,7.38)
(7.19,7.82,9.07,9.69)
(9.70,10.33,11.58,12.20)
Travel
(2.83,3.45,4.70,5.33)
(3.72,4.34,5.59,6.22)
(6.70,7.33,8.58,9.20)
backward interpolation and extrapolation. The proposed technique builds upon the scale and move transformation-based
fuzzy interpolation mechanism, in order to support intertwined multiple multi-antecedent rules involving trapezoidal
membership functions. The proposed method supports flexible
interpolation when certain antecedents are missing from the
observation, where traditional approach fails. Two worked
examples have been provided to illustrate the operation of this
approach, and a practical application case has been included
to demonstrate its applicability to real-world problems.
It would be useful to have a generalised approach that can
be implemented using other type of interpolation method (e.g.
GM [1], FIVE [13], or IMUL [22]), with results compared.
The proposed method may also be further extended and combined with the adaptive fuzzy interpolation technique which
ensures inference consistency [23]. Furthermore, an intelligent
rule selection procedure may be adopted by identifying the
more relevant attributes [2] so that more appropriate rules can
be used for interpolation while reducing the dimensionality of
the problem. For this, techniques developed in feature selection
can be used [8]. Intuitively, to further evaluate the potential of
this research, it may be interesting to apply the technique to
different problem domains, such as medical diagnosis [10],
network intrusion detection [16] and oil exploration [22].
Finally, it may also be beneficial to investigate how the
proposed work might be used to form a theoretical basis, upon
which a hierarchical fuzzy interpolation mechanism would be
developed.
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