REDUCING THE NUMBER OF CLASSIFICATION RULES REQUIRED IN THE CONVENTIONAL INTERSECTION

RULE CONFIGURATION

We now illustrate a method of reducing the number of rules required by the conven-tional rule format in classification problems. In this method, we do not simply create rules with all combinations of input variables, such as fSlow, Medium, Fastg.

7.10 REDUCING THE NUMBER OF CLASSIFICATION RULES 135

Instead, we try to tailor our rules to a fuzzy set of output variables, such asfClass_1, Class_2, Class_3,. . .g.

Consider the well-known Iris classification problem, using the data of Fisher (1936). We have data on values of four attributes: petal length, petal width, sepal length, and sepal width, together with the correct classification of 150 samples as one of three species of the genus Iris: setosa, versicolor, and virginica.

The conventional approach is to define four discrete fuzzy sets (linguistic vari-ables), say PL, PW, SL, and SW, with several members each, say fSmall, Medium, andLargegas in done in Kasabov (1998), pp. 152 ff. We also define a discrete fuzzy set of classifications, say species, with three members: fsetosa, versicolor, virginicag. We then have a possible 3⁴¼81 rules of the type

IF (PL is Small AND PW is Medium AND SL is Large AND SW is Medium)

THEN species is setosa;

and seek to establish membership functions for our input linguistic terms, perhaps from a training set, that will maximize the number of correct classifications predicted.

It is likely that the data would permit omitting a number of rules of this type, perhaps half, so that the actual number of rules might be around 15 or 20.

There is, however, another approach that will reduce the number of rules to three.

We achieve this by revising the linguistic terms for our input variables. We now define these linguistic terms (members) for input discrete fuzzy setsPL,PW,SL, andSWas setosa,versicolor, andvirginica. (Ofcourse, the membership function for setosain linguistic variablePLwould be different from the membership function forsetosain linguistic variablePW, e.g.) Our three rules now become:

IF (PL is setosa AND PW is setosa AND SL is setosa AND SW is setosa)

THEN species is setosa;

Determination of membership functions from test data becomes easier. We can simply determine the distribution of the input variables for each of the three species, and use these data to derive our membership functions.

In general, our method for reducing the number of rules to a manageable minimum amounts to the top-down method of defining the problem by first defining our outputs, then defining our inputs, and their discretization in a way relevant to the desired outputs. Combinations of the discretized inputs that do not relate to a desired output are simply omitted from our rule set as irrelevant to our goal.

7.11 SUMMARY

Inference in an expert system is the process of drawing conclusions from data that is deriving new data or truth values from input data and truth values. The new data may

be the final conclusions, or in multistep reasoning, may be intermediate conclusions that constitute input to the next step.

7.11.1 Data Types and Their Truth Values

Available attribute data types and their truth values may be summarized as:

. Integers, floats, and strings. A single truth value is attached to any value these attributes may have. This truth value is itself an attribute, accessed by appending.cfto the attribute name.

. Fuzzy numbers. Values are any number from the real line. Truth values are defined by a parameterized membership function that maps any real number value onto its truth value (grade of membership). Fuzzy numbers are used primarily in approximate comparisons in a rule antecedent. A full range of approximate comparison operators is available corresponding to the conventional Boolean numerical comparison operators furnished by most computer languages.

. Discrete fuzzy sets. Values are defined as the names of the members of the fuzzy set. Each member has a single truth value, its grade of membership of that member in the fuzzy set.

. [wec4]Membership functions. If a discrete fuzzy set is a linguistic variable, whose members are linguistic terms describing a numeric quantity, member-ship functions are attached, one to each linguistic term. These functions map any real or fuzzy number onto the grade of membership of the corresponding linguistic term. The grade of membership of the linguistic term is the truth value of its membership function.

7.11.2 Types of Fuzzy Reasoning

We assume that the truth value of an antecedent has been determined and combined with the truth value of the rule itself to furnish the antecedent confidenceP. We will denote the truth values of data to be modified asB’, the modified truth value, andB, the existing truth value.

In data modification by a rule, we consider three types of inference and define these types quantitatively, in terms of the antecedent confidence and the truth value of data to be modified.

. Monotonic reasoning. Here truth values in the consequent are nondecreasing.

B’ = P OR B

Monotonic reasoning is useful when modifying values of scalar data, or grades of membership of discrete fuzzy sets.

. Non-monotonic reasoning. Here truth values in the consequent may increase, decrease or stay the same.

7.11 SUMMARY 137

B’ = P

Non-monotonic reasoning is useful when modifying truth values directly, especially when invalidating data previously believed to be true.

. Monotonic downward. Here truth values in the consequent are nonincreasing.

B’ = P AND B

This type of reasoning is useful when combining the grade of membership of a linguistic term with its membership function prior to defuzzification.

. Approximate reasoning, defined as B’ = A⁰ o [A IMPLIES B]

in which o denotes fuzzy composition and IMPLIES denotes any fuzzy implication operator that reduces to the classical implication for crisp operands.

We define desirable properties for inference in an expert system for these inference types. We then show that the definitions of monotonic, non-monotonic, and monotonic downward reasoning all satisfy these desirable properties for their reasoning types, but the approximate reasoning method fails to satisfy all desirable properties for any type of reasoning. We note that Mamdani inference, which uses a fuzzy AND operator in place of an implication operator, is precisely the same as our monotonic downward method.

The question of inference in a fuzzy expert systems boils down to the defining of the way in which values and truth values of consequent data are inferred when a rule is fired, knowing the prior truth values of the antecedent and of consequent data.

7.12 QUESTIONS

7.2 What are the advantages and disadvantages of using separate rules for fuzzi-fication and defuzzifuzzi-fication of discrete fuzzy sets?

7.3 We wish to classify regions of an image from several numeric measurements.

We set up linguistic variables for the input data, and a non-numeric discrete fuzzy set if possible region classifications. The input variables have been fuz-zified. We have several rules whose consequent would set the grade of mem-bership of classification Artifact. Two of these rules are concurrently fireable;

of these, one would set the truth value of Artifact to 0.7, and one would set it to 0.4,

a. What is the default inference method?

b. What will the truth value of Artifact be after the rules have fired?

c. What rationale can you give for using the default inference method?

7.4 We wish to use the information from a physiological monitor to evaluate the condition of a patient in an intensive care unit. The data include heart rate, sys-tolic and diassys-tolic blood pressures, percent of oxygen saturation in both arter-ial and venous blood, and temperature. We set up linguistic variables for each of these measurements, with five linguistic terms in each linguistic vari-able, and fuzzify the input data. The data are collected nearly continuously (1 sample every 2 s), so we can calculate rates of change for each input vari-able, and set up corresponding linguistic variables for the rates of change.

Our output consists of two discrete fuzzy sets, one for present condition (good, fair, poor, bad) and one for changes (improving. stable, deteriorating). We write rules whose consequents are present condition (condition is good), and other rules whose consequents are rate of change (state is deteriorating).

We have several rules that have the same consequent. Rule A would set the grade of membership of “deteriorating” to 0.1; rule B would set its grade of membership to 0.2; and rule C would set its grade of membership to 0.5.

Since our rules are fired in parallel, all three rules are fired concurrently. To what value should we set the grade of membership of “deteriorating”?

7.5 We have five input variables with four possible values for each, and six output variable values.

a. How many rules will be required using the conventional IRC method?

b. How many rules will be required by the Combs Union Rule Configuration method?

7.12 QUESTIONS 139