PUZZ'Y
sets and systems ELSEVIER
Fuzzy Sets and Systems 80 (1996) 23 35
Control of error in fuzzy logic modeling Thomas Whalen*' 1, Brian Schott, Gwangyong Gim Department of Decision Sciences, Georgia State University, Atlanta, Georgia 303033083, USA Received June 1995
Abstract
This paper analyzes some of the major sources of error which account for variations in performance among the 72 logical systems studied in [2, 4, 5]. One of the most important of these sources is discretization error. We show that a new approach to implementing logical systems based on level sets is a powerful and efficient way to control discretization error for material implication relations, though it produces mixed results under Mamdani logic.
Keywords: Implication operators; Error; Level sets; Fuzzy logic modeling
1. Introduction Kiszka, Kochanski and Sliwinska [4, 5] (hereafter referred to as KKS) investigated the different effects of 36 techniques for defining fuzzy implications and two versions of the sentence connective ALSO (MIN and MAX) on the accuracy of a fuzzy model of a DC series motor. The exact relation between the current I and the rotating speed N in a DC motor was defined by a crisp (nonfuzzy) function N = f ( I ) based on physical measurement. This function was approximated by six subrules which described verbally the relation between the current I and the rotation speed N (see Table 1). These linguistic variables were fuzzy sets over the sets of all possible values of the corresponding physical variables I and N.
KKS then used this linguistic approximation to simulate the behavior of the motor. First, they constructed a fuzzy relation matrix for each of the six subrules based on the definition of each of the 36 implication operators. Second, they aggregated the six relation matrices using either M~N or MAX to produce a single aggregate implication relation. Finally, the crisp value of the rotating speed N was calculated according to the mean of mode defuzzification operator. They compared these final
* Corresponding author. Supported in part by Research Program Committee, College of Business Administration, Georgia State University. 01650114/96/$15.00 ~" 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5  0 1 1 4 ( 9 5 ) 0 0 2 8 0  4
Table 1 Benchmark fuzzy subrules
Also Also Also Also Also
If I if I if I if I if I if I
= = = = = =
null zero small medium large very large
then then then then then then
N N N N N N
= = = = = =
very large, large, medium, small, zero, zero.
where I is the linguistic variable for motor current and N is the linguistic variable for motor rotation.
24
T. Whalen et al. / Fuzzy Sets and Systems 80 (1996) 23 35
values for the 72 fuzzy relations with the actual rotation speed based on physical observation using criteria such as the mean square error and the computational complexity. KKS found that the five fuzzy relation operators had the smallest mean square error: 2
KKS
Dubois and Prade
ClaSST. . . . .
2* 4* 5" 27* 28*
GainesRescher Goguen Lukasiewicz G6delBrouwer Modifiedquotient
R 1/ I1
R2 13
They recommend two of these five (2* and 27*) on the basis of computational complexity, which is an issue if linguistic rules are to be evaluated on line, but not when the fuzzy system is compiled into an inputoutput table for actual deployment. Cao and Kandel [2] (hereafter referred to as C&K), reported the results of an expanded study. A major concern of C&K is the robustness of an implication system. Accordingly, they modeled not just one system, but seven bivariate systems. They also expanded the membership function shapes to include examples that were subnormal, examples that were nearly regular but with slight errors from regular, and examples that represent empirically constructed irregular memberships. Using criteria of accuracy and robustness, C&K recommended five of the 72 KKS fuzzy relations as having good applicability, viz. KKS
Dubois and Prade
Classy. . . . .
5" 22* 8 25 31
Lukasiewicz Reichenbach Mamdani
R 1/11 12 T3 T2
T1
2 Classes: ' T ' for strong implication, "R" for residuated implication as in I6], and "Q" for quantum implication. Subscript usage follows [1, 10].
The first two fuzzy relations they recommended, Lukasiewicz implication and Reichenbach implication, were material implication type logics. The second group of three logics, Mamdani, product, and Lukasiewicz Tnorm, were Mamdanitype conjunction logics. This is a striking contrast from KKS' results in that all of the KKS "best" logics were of the material implication type. Neither set of authors, however, give a detailed analysis of the source of error they detect.
2. Logical analysis KKS and C&K generate their 72 inference methods by using each of their 36 fuzzy implication operators twice; once with ALSOoperationalized as MIN and once with ALSO operationalized as MAX. However, most of these 72 inference methods are inadmissible on logical grounds; this is borne out by the numerical results reported by KKS and C&K. Any fuzzy logic, to merit the name of logic, must include classical twovalued logic as a special case when each membership grade is either zero or 100%. Consider a collection of rules like those in C&K's benchmark, but built from antecedents and consequents which are ordinary sets whose membership grades are restricted to zero and 100%. Each rule defines a relation on the Cartesian product of the antecedent and consequent universes of discourse. Every point in this Cartesian product space specifies whether or not the logical relation in question permits a particular value of the antecedent variable(s) to be paired with a particular value for the consequent variable. The 36 candidate implication operators can be classified into ten categories on the basis of their behavior when the antecedent and consequent membership grades are both crisp (zero or 100%). Nine of the 36 candidate implication operators return a value of zero when the antecedent and consequent memberships are both 100%, indicating that values perfectly satisfying the antecedent and consequent of a rule are to be totally excluded from the implication. This is obviously contrary to the meaning of the rules in Table 1, which explains why none of these operators are among those that KKS
T. Whalen et aL / Fuzzy Sets and Systems 80 (1996) 2335
or C&K report as successful. The operators reduce, in the crisp case, to the logical relations XOR, NAND, NOR, a ANDNOT b, b ANDNOV a, and CONTRADIC'rION. Of the remaining 27 candidate operators, another six reduce in the crisp case to the logical relation OR or to tautologies. They return a value of 100% when the antecedent membership grade is 100% and the consequent membership grade is zero. This indicates that a situation where the antecedent is perfectly satisfied but the consequent is absolutely contradicted, nevertheless satisfies the rule perfectly. Again, this is obviously contrary to the intent of the rules in Table 1, explaining the poor performance of these operators in the findings of KKS and C&K. The remaining 21 candidate implication operators fall into three categories: eleven material implications, four conjunctions, and six equivalences. All of these operators have two essential properties in common. The first property is that when the antecedent is perfectly satisfied and the consequent is perfectly satisfied, then the implication is perfectly satisfied. The second property is that when the antecedent is perfectly satisfied and the consequent is absolutely contradicted, then the implication is absolutely contradicted. In other words, when the antecedent base value belongs 100% to the antecedent set, these operators permit any consequent base value that belongs to the consequent set and reject any consequent base value that is outside the consequent set [9]. Having narrowed the original list of 36 candidate operators to 21, we now turn our attention to the choice of an aggregation operator to represent ALSO. Consider the set of points in the Cartesian product space that correspond to antecedent and consequent base values that have a membership grade of zero in the corresponding (crisp or fuzzy) antecedent and consequent sets defining the rule. Intuitively, these points are totally outside the scope of the rule in question. If the operator is a material implication or an equivalence, these irrelevant points belong 100% to the implication relation because the rule provides no evidence against them. If the operator is a conjunction, the irrelevant points have a membership of zero in the implication relation because the rule provides no evidence in their favor.
25
The two aggregation operators studied by C&K reduce to the two classical operators AND and OR when membership grades are restricted to zero and 100%. Suppose the implication operator in use is one which assigns membership grades of 100% to points not satisfying the antecedent of the rule. When several individual rules of this sort are combined, any information contained in one rule will be lost if that rule is aggregated by OR with another rule whose antecedent differs from that of the first rule. Similarly, if the implication operator assigns membership grades of zero to points not satisfying the antecedent, then information will be lost when rules are aggregated using AND. If the rule set is sufficiently diverse that each possible input satisfies the antecedent of some rules but not of others, no information whatsoever will be preserved in the crisp case if the wrong aggregation operator is used. (In the fuzzy case, some residual information may survive the aggregation process, but as shown numerically by C&K's own results, this information is of much less benefit than the larger quantity of information preserved by a better aggregation operator.) Thus, a little analysis immediately reduces the KKS and C&K collection of 72 logical systems down to 21; each is defined by one of their generalized implication operators paired with the appropriate aggregation operator. In addition to the MIN and MAXaggregation operators used by C&K, Schott and Whalen [7, 8] suggested other candidate generalizations of AND and OR, such as Tnorms (Table 2) for the 5 R and Simplications. (T2 :Goguen (4*), Tl:Lukasiewicz (5*), T2:Reichenbach (22*), T3:G6del Brouwer (27*), Ta:KleeneDienes (6*)). These particular Tnorms have natural, mathematical kinships with particular implication operators. For example, the Lukasiewicz Tnorm (T1) would function more compatibly with the Lukasiewicz implication operator (R1) than the MN Tnorm (T3) used by KKS and C&K would. In the same way, the Product Tnorm (T2) is theoretically allied with the Reichenbach 12 and Goguen R2 implication operators. But the MIN Tnorm (T3) would correspond better with the KleeneDienes implication operator ($3) and the G6delBrouwer implication operator ( R 3 ) because KleeneDienes and G6delBrouwer use M~N as their natural Tnorm.
26
T. Whalen et al. / Fuzzy Sets and Systems 80 (1996) 2335 Table 2 Material implication subrules, modus ponens detachment KKS
Dubois and Prade
1" 2* 3* 4* 5" 6* 7* 22* 27* 28* 29*
Standard Sharp GainesRescher Goguen Lukasiewicz KleeneDienes Early Zadeh Reichenbach G6delBrouwer Modified quotient Yager
Conjunction subrules, disjunctive syllogism detachment Classr.....
R2 R1/11 13 Q3 12 R3
Slight improvements in benchmark performance were noted in some cases when the "natural" combinations (summarized below) were used.
KKS Dubois and Prade 4* 5" 6* 22* 27*
Goguen Lukasiewicz KleeneDienes Reichenbach G6delBrouwer
Classv
.....
Complementary Tnorm
R2
T2
R 1/I1 I3/Q1 12 13
T1 T3 T2 T3
3. Knowledge base psychosis Another source of error for some rules was a form of "knowledge base psychosis" [10]. This occurred when a row of the aggregate rule matrix had the same value for all consequent base values. This was shown, for example, under C&K's Mfl [2] using the KleeneDienes (6*) implication operator when two subrules both had a membership grade 0.5 for a given current. The best performing operators in KKS and in C&K did not encounter this rulebase failure.
KKS
Dubois and Prade
ClassT .....
8 10 25 31
Mamdani
T3 To T2 Tl
Equivalence subrules 15" 16" 17" 18" 19" 26*
4. Membership function representation Essentially all systems of fuzzy inference depend on directly or indirectly comparing membership grades. Very often, the most critical region in the Cartesian product space for the overall behavior of a logical system is in the neighborhood of points for which the membership grades in the antecedent and consequent are equal. However, the standard practice in the study and application of fuzzy inference up to now (including C&K's study) is to represent a linguistic value by a systematic sample of (base value, membership grade) pairs. They construct their samples by choosing convenient, evenly spaced values of the base variable and pairing them with their membership grades, which may or may not be evenly spaced convenient values. This approach may be called the "membership function representation" approach since it implies the use of a membership function #(x) to map from the selected base values to their membership grades. A subrule in the benchmark fuzzy model takes the form of a matrix. Each row of the matrix corresponds to a value from the antecedent domain (current), and each column corresponds to a value from the consequent domain (speed). The elements in the matrix result from applying the implication
12 Whalen et al. / Fuzzy Sets" and Systems 80 (1996) 23 35
operator to the antecedent and consequent membership grades. The six subrule matrices then combine into a single aggregate rule matrix via cell by cell minimization in the case of material implication operators or maximization in the case of conjunction operators. The benchmark fuzzy model takes crisp inputs. It is reasonable to assume that these crisp inputs will be rounded off to the same level of precision (0.5 amp) used in specifying the model. Thus, the intermediate (fuzzy) output of the system is simply the row of the aggregate rule matrix corresponding to the crisp input. The final output of the system is the defuzzified value of the consequent variable inferred from the crisp value of the antecedent variable given as input. For this reason, our discussion will focus on the numeric value of the implication operator for various values of the consequent when the antecedent is held constant; the antecedent value corresponds to a row of the implication relation matrix in the standard representation.
5. Modal interval
Many of the candidate implication operators that KKS and C&K examine yield their maximum value throughout an interval of consequent membership grades for a given antecedent membership grade. For example, consider the row of the subrule implication matrix corresponding to an antecedent base value (current) x whose membership grade in the antecedent fuzzy set ~b is ~: / ~ ( x ) = ~. (x ~ {0 amp,0.5 amp . . . . ,10 amp}.) This row specifies the fuzzy set of consequent values compatible with the antecedent value x, by giving the membership of each sample consequent base value. Any Rimplication function (such as R1 Lukasiewicz, R2 Goguen, R3 G6delBrouwer) produces an implication membership grade of 100% for each value y of the consequent variable whose membership grade B = I~o(y) in the consequent fuzzy set 0 is greater than or equal to the antecedent membership grade 2. In terms of the DC motor benchmark, the membership of a (currentspeed) pair in the R implication relation, l(x,y), equals 100% for all current x and speed y such that ~to(y) >>,#4,(x). Furthermore, the value of an Rimplication function is
27
always less than 100% for a (current, speed) pair when /~o(Y)< It,/,(x). The modal level set is the 100% level set. Similarly, the Mamdani operator T3 produces an implication membership grade of/z~/,(x) whenever l*o(y))l14,(x), and an implication membership grade less than ll4,(x) whenever/~0(Y) < ll,,,(x). Thus, the highest (modal) consequent level set corresponding to an antecedent base value whose membership in the antecedent level set equals /z,~,(x) consists of the l,~/,(x) level set of the consequent fuzzy set. The KleeneDienes Simplication operator I3 and the Reichenbach Simplication operator I2 reach their peak value of 100% at only a single point except in the trivial case in which ~ = 0 and all values of the consequent have 100% membership in the implication. This is because their implication functions equal 100% if and only if either the antecedent membership is zero or the consequent membership is 100%. When an implication operator attains its maximum over an interval of values of the consequent base variable, let us refer to that interval as the modal interval of the consequent base variable corresponding to the antecedent membership grade. For simplicity, we say that a fuzzy set that reaches its maximum at a unique point has a degenerate modal interval of length zero. Next we consider modal intervals in matrices representing aggregated rules. For material implication type operators, the aggregate rule is formed by minimization. For any given antecedent (amperage) base value x, most of the subrules will have an antecedent membership grade iL,/,(x) equal to zero, which produces a subrule implication membership of 100% for all the consequent values (a "trivial" modal interval). Typically, there are two subrules with nontrivial implication values for a given antecedent base value. If the modal intervals of these two nontrivial subrules for the antecedent base value in question do not overlap, then the aggregate implication membership for that base value may or may not be unimodal. But if the modal intervals of the subrules overlap, the aggregate rule will always have a modal interval containing the intersection of the two subrule modal intervals, with a membership grade equal to the
28
T. Whalen et al. / Fuzzy Sets and Systems 80 (1996) 2335
Composite Rule Calculations (Mr2) Mamdani Logic
3.5 amps Disjoint modal intervals with unequal heights 0.9 0.8 0.7 0.6 0 .o 0.5
! //
~ 0.4
/I/ J
0.3 0.2
'1////
0.1 0 4O0
I 600
800
I 1000
1200
/
1400
1600
1800
2000
RPM Fig. 1.
lesser of the two modal membership grades in the subrules. For conjunction type implication operators, the aggregate rule matrix is formed by maximizing across subrules. In this case, the modal interval of the composite rule will be the modal interval of that subrule whose mode has highest membership (it will be the union of multiple modal intervals in the case of ties).
6. Discretization error
KKS use a collection of fuzzy membership functions that contains only five membership grades: 0, 0.25, 0.5, 0.75, and 1.0. C&K also use this collection; they refer to it as Mfl. Consider any one of the sample pairs in the representation of an antecedent fuzzy set ~b. This pair associates a base value of the antecedent variable, x ~ [0,10] amps, with the membership of x amps in ~b, / ~ ( x ) = ~. Similarly, the consequent fuzzy set 0 is modeled by a set of sample pairs associating a base value of the consequent fuzzy variable, y ~ [400, 2000] rpm, with the membership of y rpm in 0, #o(Y) = 13. The implica
tion matrix in turn specifies a set of sample points each of which associates a (current, speed) pair with its degree of membership in the implication relation, I(~,13). In a single subrule using the R1, R2, R3, or T3 implication operator, the modal interval of the consequent variable corresponding to a value of x amps for the antecedent variable is the ~ level set of the consequent fuzzy set 0. In the Mfl membership functions, the endpoints of this level set are always found among the sample points representing 0. Because the Mfl collection of membership functions completely specifies the ~ level set of the consequent variable for any membership grade that occurs in the representation of the antecedent variable, there is no distortion in the representation of any subrule. However, this is not the case in general. In addition to the KKS membership functions Mfl, C&K also introduce five more collections of membership functions, Mf2 through Mf6, to represent the same fuzzy subsets of current and speed. In these collections, the set of sample points for the consequent fuzzy set often does not include any point whose membership grade matches that of a given sample point from the antecedent fuzzy set.
T. Whalen et al.
29
/Fuzzv Sets and Systems 80 (1996) 2335 Composite Rule Calculations (Mr2)
3.5a m p s
Lukasiewicz Logic
/
0.9 0.8
0.7 ~3 0 . 6 .2 0.5
~.
Overlapping~ ~ \ withequal \
0.4
m o d a l intervals
0.3
heights
0.2 0.1 0
~ 400
600
I 800
I 1000
As a result, the set of sample points for the implication relation does not contain the true endpoints of the modal interval corresponding to the base value of the antecedent sample point in question. This misspecification of the modal interval generates discretization error, which is a major contributor to the error levels documented by C&K. For conjunction type operators, the fact that the aggregate rule is formed by maximization implies that correct specification of the modal intervals for the subrules guarantees correct specification of the modal interval of the aggregate rule (see Fig. 1). For material implication type operators, there are three cases to consider. If the modal intervals of the nontrivial subrules overlap and have equal membership grades, the aggregate rule will have a modal interval (see Fig. 2). Whenever the modal intervals of the subrules are correctly specified, the aggregate modal interval will also be correct. Similarly, if the modal intervals have equal membership grades and touch at just one point, this point is the unique modal value of the aggregate rule for the antecedent base value in question (see Fig. 3). Again, if the modal intervals are correctly specified, their common point will be as well. The third case is
I
1200 RPM Fig.2.
I
~
I
q
1400
1600
1800
2000
when the modal intervals are disjoint (as in Fig. 4), or have unequal membership grades. In this situation, the modal value of the aggregate rule is not fully specified by the modal intervals of the subrules, so correct specification of those intervals is clearly not a universal guarantee that the aggregate mode will be correct. One obvious way to address the problem is to increase the number of (rpm, membership) pairs used to define each consequent fuzzy set. If enough such pairs are implemented, then the subrule matrices will come arbitrarily close to precise representations of the implication relation for a given antecedent value. However, this can be extremely demanding computationally since each additional consequent sample point creates an additional column in every subrule matrix. To ensure that the subrule modal intervals are correctly identified for each operator, the representation of the consequent fuzzy set 0 must include a pair (Y,I~o(Y)) for every y that belongs to/~ 1(el), for any ei in the set of pairs (x~, 7~) defining any antecedent fuzzy set ~b. Even this does not guarantee precision in the cases where the modal intervals of nontrivial subrules are disjoint, so even more columns must be added for these cases.
30
"1~ Whalen et al. / Fuzzy Sets and Systems 80 (1996) 2335 Composite Rule Calculations (Mfl) Lukasiewicz Logic
3.5 amps 1
..................
\\\\\\\
0.9 0.8 0.7 L~ 0.6 .2 0.5 Modal intervals with one point in
~. 0.4
\
common
0.3 0.2 0.1 0 400
I
I
I
I
I
I
I
I
600
800
1000
1200
1400
1600
1800
2000
RPM Fig. 3. Composite Rule Calculations (Mf2) 3.0 amps
Lukasiewicz Logic
~ /1 ..................... .\
1
\
0.9
0.8 0.7 0
\
k k \
k \k \
0.6
\ \\
.o 0.5 ~ . O.4
intervals with equal heights
0.3 0.2 0.1 0 4O0
I 60O
I 800
I 1000
I
I
I
I
I
1200
1400
1600
1800
2O0O
RPM Fig. 4.
7. Level
set approaches
It is possible to reduce or eliminate this form of diseretization error by using a method based on
the level set representation of the consequent fuzzy sets. Level sets are an alternative representation for fuzzy sets that specify a set of base values for selected membership grades instead of
T. Whalen et al. / Fuzzy Sets and Systems 80 (1996) 2335
specifying a membership grade for selected base values) Consider the inverse membership function of the fuzzy set 0, tLot(g). The image of the mapping #ol(B) is the set of base values y for which #0(Y) = 13. For unimodal linguistic variables,/*o 1(g) will produce two base values y' and y" for 0 < B < 1 unless 0 is an "extreme" fuzzy set whose mode is an endpoint of the domain of the base variable. In the first instance, the B level set of 0 is the interval between y' and y"; in the case of an "extreme" fuzzy set, the B level set is the interval from the single value of/iol(B) to the appropriate endpoint of the domain. The "B level set" method for fuzzy logic modeling or control uses three matrices to represent each subrule. The first matrix is an inverse of the implication operator. Specifically, each row of this matrix corresponds to a base value x of the antecedent variable, as in the membership function method. Each column, however, corresponds to a membership grade ~ in the implication relation. The cells of this matrix contain the lowest membership grade B of the consequent variable whose implication membership is greater than or equal to the membership ~ specified by the column header. Consider the matrix for a subrule whose antecedent fuzzy set is q5 and whose implication operator is I(~,B). Symbolically, where the row for the base value x intersects with the column for the implication membership ~, the entry in the cell equals B(x, ~):= min ~B: l(14/,(x),B) >~ ~}. The corresponding elements of the second and third matrices in this approach contain, respectively, the minimum rpm and maximum rpm endpoints of B level sets of of the subrule's consequent fuzzy set, taking/3 from the corresponding element of the first matrix. Linear interpolation is used to find rpm base values whose membership grades are not represented in the sample of pairs defining the consequent membership function. The aggregate implication rule is defined in terms of two matrices. As in the subrules, the rows of the matrices correspond to base values of the
3 See [9] for a different approach to fuzzy inference using level sets.
31
antecedent variable and the columns correspond to implication membership grades ~. For material implication type operators, the maximin matrix contains the maximum of all the corresponding cells in the matrices representing the minimum rpm endpoints of subrule level sets. The minimax matrix contains the minimum of all the corresponding cells in the matrices representing maximum rpm endpoints. The ~ level sets of the aggregates rule can be read from these two matrices. If the entry in the maximin matrix is less than or equal to the corresponding entry in the minimax matrix, these two values define the level set specified by the column header when the antecedent variable equals the row header. If the maximin entry is greater than the minimax entry, the corresponding level set is empty, and the mode of the implication must be sought at a lower implication membership level of ~. The information in the minimax and maximin matrices is summarized in another matrix, the "midpoint matrix". For each (x, ~) pair, if the ~ level set is nonempty the matrix contains the midpoint of the level set; otherwise it contains a flag value indicating the level set does not exist.
8. Example We now illustrate the effects of discretization error and compare several methods for controlling it. We use the DC motor benchmark model from KKS and C&K (Table 1). The "true" membership functions are piecewise linear, based on C&K's Mf2 set of membership functions. For each amperage level, we found the rotational speed according to the model without discretization error by using the 13level set approach with 501 implication membership grades (~): 0, 0.002, 0.004 . . . . . 0.998, 1. This level of resolution is sufficient to reduce discretization error to less than 1 rpm over a range from 400 to 2000 rpm. The B level set approach requires three matrices per subrule (memberships, lower bounds, and upper bounds in the consequent), and three aggregate matrices (greatest lower bounds, least upper bounds, and midpoints of nonempty level sets). Each matrix has one row per antecedent base value (x), and one column per implication membership
32
T. Whalen et aL / Fuzzy Sets and Systems 80 (1996) 2335
grade (~). Thus, with six subrules the criterion calculation required 3 x 7 x 501 or 10 521 virtual processors (spreadsheet cells) for each antecedent base value (plus some additional input and housekeeping cells that were essentially the same for all realizations of the model). We defined discretization error in terms of deviation from the 501level B level set in order to evaluate five other realizations of the same model. The first realization used the membership function approach just as KKS and C&K did, evaluating implication memberships for each of the 17 consequent base values (steps of 100 rpm) in the sampled representation. The membership function approach requires one matrix per subrule plus three aggregate matrices; each matrix has one column per consequent base value considered. Thus, for six subrules and 17 base values, the number of virtual processors required per antecedent base value is 9 x 17 or 153. This approach produces a mean squared discretization error of 255.4 under Lukasiewicz logic, 823.1 under G6del Brouwer logic, 634.4 under Mamdani logic, and 243.7 under Reichenbach logic (in units of squared rpm). One way to attempt to control discretization error is to constrain the membership grades to regular values. If the membership grades in C&K's Mf2 are rounded to the nearest quartiles for the 21 antecedent base values and 17 consequent base values, the result is the original (Mfl) set of membership grades studied by KKS. This approach was another of the five we evaluated; it also uses 153 virtual processors per antecedent base value. The discretization error was 265.5 under Lukasiewicz logic, 170.2 under G6delBrouwer logic, 8470.6 under Mamdani logic, and 163.5 under Reichenbach logic. An obvious way to decrease discretization error is to sample at more points; we evaluated this "brute force" approach at two levels. The first level reduced the step size from 100 rpm to 20 rpm. This resulted in the nine matrices each having 81 columns, requiring 729 virtual processors per antecedent base value. The discretization error was 18.5 under Lukasiewicz logic, 53.5 under G6delBrouwer logic, and 15.2 under Mamdani logic. The second level reduced step size to 10 rpm, requiring 9 x 161 or 1449 virtual processors per antecedent
base value. In this case, the discretization error was 7.1 under Lukasiewicz logic, 8.4 under G6delBrouwer logic, 3.2 under Mamdani logic, and 11.2 under Reichenbach logic. Finally, we evaluated a reduced version of the g level set approach. Instead of 501 implication membership grades, we selected seven. This choice resulted in a system that required 3 x 7 x 7 or 147 virtual processors per antecedent base value, which is fewer than the number required by the standard membershipfunction approach with the 17 consequent base values considered by KKS and by C&K. Consequent membership grades below the minimum membership grade employed in the original Mf2 membership function approach of KKS and C&K were not relevant to this problem under the four logics (Lukasiewicz, G6delBrouwer, Mamdani, and Reichenbach) that we examined, and probably should be avoided whenever possible. Between this lowest necessary implication membership value and 100% we selected evenly spaced membership values which partitioned the range into six intervals. The resulting seven selected membership grades are shown in Table 3. Discretization errors were 0.1 under Lukasiewicz logic, 41.3 under G6delBrouwer logic, 1 862.1 under Mamdani logic, and 147.5 under Reichenbach logic. Fig. 5 shows the discretization error at each antecedent base value for the five realizations of the model using Lukasiewicz implication. The vertical axis, labeled "error in rpm', denoted the difference between the indicated model realization and the criterion model based on 501 implication membership grades under the g level set approach. Note the close correspondence between the 13 level set with seven membership grades and the criterion. Figs. 68 show the same analysis as Fig. 5, applied to G6delBrouwer logic, Mamdani logic, and Reichenbach, respectively. Table 4 summarizes the effectiveness and efficiency of the five realizations under the four logics. Under Lukasiewicz logic, the g level set approach with seven implication membership grades and 147 virtual processors per antecedent base value is the most accurate of the five models even though it is the smallest. Under G6delBrouwer logic, the same
T. Whalen et al. / Fuzzy Sets and Systems 80 (1996) 2335
33
Table 3 Levels to evenly partition the reduced version of the g level set approach Partition number
kukasiewicz
G6del Brouwer
Mamdani
Reichenbach
1 2 3 4 5 6 7
0.84 0.87 0.89 0.92 0.95 0.97 1
0.49 0.58 0.66 0.75 0.83 0.92 1
0.51 0.59 0.67 0.76 0.84 0.92 1
0.7 0.75 0.8 0.85 0.9 0.95 1
Discretization Error: C&K Mf2 50
Lukasiewiczlogic
Discretization Error: C&K Mr2 50
ffi
I
GodelBrouwerLogic
I
30
30 iI
1o
~. 10
._=
' ...~,~, •"~, , I •
,~1
•
,
,, 10
t
/,,
"
• ',/~ "~L...~
__ fKb,'~,, o ~
,
,
~
I
~ ~
\/
.=_
",,/
IX*" I4
/,_,
.,
\

~
~,~
~:'
', ",,i!/ I%:"1
~
'\i
~':~" "al/
IV , ~  +   +       +
10
I t
~ W
&'
/i
,,5
iI
/
30
', "/
~V/,
!,,
v
/
30
50 0
I
I
I
I
I
2
4
6
8
10
Amperes
Fig. 5. Discretization error at each antecedent base value for the five realizations of the model using Lukasiewicz logic. ~ : Original 17 step membership function:  ~   : Rounded 17 step membership function: . . . . . x . . . . : Interpolated 81 step membership function: . . . . . : Interpolated 161 step membership function;  +    : Betacut, 7 implication levels. The curves in Figs. 68 also correspond to the same functions.
13 level set approach is second only to the largest "brute force" model with steps of 10 rpm and 1 449 virtual processors per antecedent base value. The Mamdani logic, on the other hand, can only be
50
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4
6
8
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Amperes
Fig. 6. The same analysis as in Fig. 5 applied to G6del Brouwer logic.
improved by adding base values in the membership function approach; switching from the 153 processor original membership function realization to the 147 processor 13 level set realization actually increases the Mamdani discretization error. It is interesting to note that increasing the number of processors from 147 to 168 by adding one column
34
7". Whalen et al. I Fuzzy Sets and Systems 80 (1996) 2335 I
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Discretization
' " ' Error:~&I~Mf2 ~is~.etlzatlon ] '
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Amperes Fig. 7. The same analysis as in Fig. 5 applied to Mamdani logic.
Fig. 8. The same analysis as in Fig. 5 applied to Reichenbach logic.
to the B level set approach yields an MSE of only 29.1. However, the instability of the MSE under the 13 level set approach only underscores the well known fit between Mamdani logic and the member
ship function representation. The Reichenbach logic is relatively unresponsive to our minimalist B level set attempts to reduce discretization error. For example the reduced B level set realization
Table 4 Comparison of realizations Model realization
Virtual processors per antecedent
MSE Lukasiewicz
MSE G6delBrouwer
MSE Mamdani
MSE Reichenbach
Original Mf2 17 step membership function
153
255.4
823.7
634.4
243.7
Rounded ( Mfl ) 17 step membership function
153
265.6
170.8
8470.6
163.5
Interpolated Mf2 81 step membership function
729
18.5
52.9
15.2
27.3
Interpolated Mf2 161 step membership function
1449
7.1
8.5
3.2
11.2
41.3
1862.1
147.5
B level set 7 consequent membership grades
147
0.01
7". Whalen et al. / Fuz;v Sets and Systems 80 (1996) 23 35
reduces the MSE for the Reichenbach logic only slightly more than does the rounded 17 step membership function method.
9. Conclusions We have given a more complete theoretical discussion of discretization error issues in fuzzy logic modeling and a comparative evaluation of the techniques using the C&K membership functions and benchmark fuzzy model. For the residuated material implication fuzzy logic systems we have studied, the g level set approach produces improved accuracy relative to using additional consequent pairs in a membership function approach, while requiring fewer processors in a parallel algorithm (simulated using a spreadsheet). On the other hand, the Reichenbach logic (based on strong implication) did not benefit as much from the 13 level set approach and the Mamdani logic system was confounded by it.
References [1] P. Bonissone, Summarizing and propagating uncertain information with triangular norms, Internat. J. Approx. Reasoninq 1(1)(1987)71 102.
35
[2] Z. Cao and A. Kandel, Applicability of some fuzzy implication operators, Fuzzy Sets and Systems 31 (1989J 151186. [3] D. Dubois and H. Prade, Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions, Fuzzy Sets and Systems 40(1) (1991) 143 202. [4] J. Kiszka, M. Kochanski and D. Sliwinska, The influence of some fuzzy implication operators on the accuracy of a fuzzy model: Part I, Fuzzy Sets and Systems 15 (1985) 111 128. [5] J. Kiszka, M. Kochanski and D. Sliwinska, The influence of some fuzzy implication operators on the accuracy of a fuzzy model: Part 11, Fuzz), Sets and Systems 15 (1985) 223 240. [6] B. Schott and T. Whalen, Sources of error in fuzzy inference, Proc. 1990 North American Fuzz)" lrformation Processin9 Society Conf. (1990) 223 226. [7] B. Schott and T. Whalen, Analysis of error in fuzzy inference, Proc. 199l North American Fuzz)" hformation Processin9 Society Conf. (1991) 78 81. [8] E. Trillas and L. Valverde, On mode and inference in approximate reasoning, in: M. Gupta et al., Eds., Approximate Reasonimj in Expert Systems (Elsevier, Amsterdam, 1985). [9] K. Uehara and M. Fujise, Fuzzy inference based on families of :~level sets, IEEE Trans. Fuzzy Systems 1 (2) (1993) 111 124. [10] T. Whalen, G. Gim and B. Schott, Discretization error in fuzzy logic modeling, Proc. FiJth Internat. Fuzz)' Systems Association World Congress (1993) 183 186. [11] T. Whalen and B. Schott, Presumption and prejudice in logical inference, Internat. J. Approx. Reasoninq 3(5) (1989) 359 382.