In this article we will discuss about:- 1. Introduction to Fuzzy Inference System 2. Architecture of a Fuzzy Inference System.
Introduction to Fuzzy Inference System:
Let us review inference in logic. It is based on principle of syllogism (deductive reasoning).
Consider the statements:
If A implies B
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If B implies C
Infer- A implies C
Now take a look at the formalization made by Prof. Zedah (which might look like mathematic), with help of an example.
If a woman is beautiful then she has a short life; written in fuzzy set as:
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If X is F
then Y is G
where, F and G are linguistic values (labels) defined by fuzzy sets on the ranges (universe of discourse) X and Y respectively. X is a variable which means some one who belongs to a particular set of women. Let U be a universal set of all women.
For example:
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U = {Arisha, Pariza, Krishna}
X variable can be anyone of them.
X is a membership function which assigns a degree of beauty, to each member of U.
Also F is a fuzzy set over U.
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We can give Arisha any degree of beauty, mµF (Arisha), by our subjectivity. The membership function is providing semantics of the fuzzy set to mean beautiful women.
The variable Y shows the age when she (beautiful woman) dies. Let V be another universal set of age –
V = {1, 2,….. 100}
Then G is the fuzzy set over V to mean a short life. The membership function of G, mµF (Y), shows the degree of short life, given age Y in V. This degree (membership function of short life) can be given by us.
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Now let us consider a given fact:
She is pretty.
X is F’
Where X is a variable on U and F’ is a new fuzzy set over U to mean ‘pretty’; the fuzzy set F’ need not necessarily be equal to fuzzy set F of the premise. This is one of the characteristics of fuzzy sets; binary-based symbolic logic cannot derive any conclusion from incomplete premise.
Formalization of Fuzzy Inference System:
Using the fuzzy sets, the fuzzy inference can be formalised as:
If X is F
Then Y is G
where F and F’ are fuzzy sets over U
What can we say about the span of life of pretty women?
We wish to have a consequence
Y is G’
G’ is a fuzzy set of set of ages, V.
Many methods have been proposed for this purpose, but we will use method proposed by Prof. Zadeh. It is required to identify from F and G a fuzzy relation R, over U and V which has a consequence over G’ on V’ from the given F. This method is similar as that used in guessing the diseases from symptoms. The other inference methods use the same inference scheme but the definitions of R are different.
There is no unique manipulation in defining fuzzy relations. Whatever fuzzy relation is defined, we expect that the conclusion, G’ would be closer to “short life” as fuzzy set ‘pretty’ more closely matches ‘beautiful’. If the two fuzzy sets were independent then we would know nothing about a woman’s life span, meaning thereby that there would be an unknown conclusion.
The consequence G’ is given by a membership function other than language. If we want to have human readable outputs, we need to find the optimal words to approximate the membership function. This requirement leads us to an interesting problem of linguistic approximation of fuzzy sets.
Now let us turn to the problem: what if a premise such as “a beautiful woman has a shorter life” is not completely true. Let us say that it is “almost true”.
That is:
If X is Fs
then Y is G
is almost true.
The ‘true’ of ‘almost true’ can be assigned to linguistic truth value defined with a fuzzy set, T, on the set of truth values from 0 to 1.
The result can be seen as follows:
1. Compute the fuzzy relation R over U and V from F and G.
2. Modify relation R to R’ for the fuzzy set T to mean “almost true”.
3. With the relation R’, have the fuzzy set G’ of V from the R’ and fact F.
Thus:
4. Uncertain knowledge is described in our spoken language (natural, language), which gives a label specifying a fuzzy set.
5. Knowledge is translated into membership functions, which provide semantics to our knowledge. The process is called fuzzification.
6. The membership functions are appropriate representations in computers and are retranslated into language before they show output to us. The process is called defuzzification or resolved simple number.
Architecture of a Fuzzy Inference System:
A fuzzy inference system is a rule-based system which uses fuzzy logic, rather than Boolean logic, to reason about data. It uses fuzzy set theory to map inputs (features) to outputs (classes) in the case of fuzzy classification.
The Basic Structure of Fuzzy information System is shown in Fig. (13.9)
1. A fuzzifier, which translates crisp (real-valued) inputs into fuzzy values.
2. An inference engine, which applies a fuzzy reasoning mechanism to obtain a fuzzy output using the rules contained in the knowledge base. These fuzzy rules, which define the connection between input and output fuzzy variables have: IF antecedent THEN consequent format.
3. A defuzzifier, which translates this latter output into a crisp value; and
4. A knowledge base, which contains both an ensemble of fuzzy rules, known as the rule base, and an ensemble of membership functions known as the database.
The inference engine using the rules contained in the rule base performs the decision-making process. These fuzzy rules, which define the connection between input and output fuzzy variables have the form: If antecedent then consequent.
Fuzzification:
Establishes the fact-base of the fuzzy system. First, identifies the input and output of the system. Fuzzification then defines appropriate IF THEN rules and uses raw data to derive a membership function.
For example, consider an air conditioning system which samples the current temperature and moisture levels to determine the optimal circulation level. In this case, the inputs consist of the current temperature and moisture level. The fuzzy system outputs the optimal air circulation level: ‘none’, ‘low’, or ‘high’.
The following fuzzy rules are used:
1. R1:
If the room is hot, circulate the air a lot.
2. R2:
If the room is cool, do not circulate the air.
2. R3:
It the room is cool and moist, circulate the air slightly.
Finally, a knowledge engineer must determine two membership functions: one which maps temperatures to fuzzy values and the other which maps moisture measurements to fuzzy values.
Inference Engine:
As inputs are received by the system, inference engine evaluates all IF THEN rules and determines their truth values. If a given input does not precisely correspond to an IF THEN rule, then partial matching of the input data is used to interpolate an answer.
For example, suppose that the air conditioning system has measured temperature and moisture levels and mapped them to the fuzzy values of 0.7 and 0.1 respectively. The system now needs to infer the truth of each fuzzy rule presented above. In this example we use the simplest method, MAX-MIN. Basically, this method sets the fuzzy value of the THEN clause (or conclusion) to the fuzzy value of the IF clause. Thus, the method infers fuzzy values of 0.7, 0.1 and 0.1 for rules 1, 2 and 3 respectively.
Composition:
Combines all fuzzy conclusions obtained by inference into a single conclusion. Different fuzzy rules might have different conclusions, so it is necessary to consider all rules.
For example, each inference conclusion about the air conditioning system suggests a different action; rule 1 suggests a ‘high’ circulation level, rule 2 suggests turning off air circulation, and rule 3 suggests a ‘low’ circulation level. The max-min method uses the maximum fuzzy value of the inference conclusions as the final conclusion (In particular, composition selects a fuzzy value of 0.7 since this was the highest fuzzy value associated with the inference conclusions). Do you agree?
Defuzzification:
Converts the fuzzy value obtained from composition into a ‘crisp’ value; this process is often complex since the resulting fuzzy set might not translate directly into a crisp value. Defuzzification is necessary since controllers of physical systems require discrete signals.
Defuzzification is a process, which maps from a space defined over an fuzzy input universe of discourse into a space of non-fuzzy (crisp) number., it Is intuitive that fuzzification and defuzzification should be reversible. If we are going to use fuzzy sets to make decisions, then ultimately getting fuzzy answer is no help to us. We need to convert from a fuzzy answer to an actual number.
Many methods have been used for defuzzification by various processes but most of them are problem dependent. There has been no rule, which guides how to select a method suitable to solve given problem.
There are five steps of the fuzzy inference process:
1. Fuzzification of the input variables,
2. Application of the fuzzy operator (AND or OR) in the antecedent,
3. Implication from the antecedent to the consequent,
4. Aggregation of the consequent across the rules, and
5. Defuzzification.
1. Fuzzify Inputs:
The first step is to take the inputs and determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions.
2. Apply Fuzzy Operator:
Once the inputs have been fuzzified the degree to which each part of the antecedent has been satisfied for each rule is known. If the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one number which represents the result of the antecedent for that rule. This number, then be applied to the output function. The input to the fuzzy operator is two or more membership values from fuzzified input variables. The output is a single truth value.
3. Apply Implication Method:
Before applying the implication method, the care of the rules’ weights has to be taken. Every rule has a weight (a number between 0 and 1), which is applied to the number given by the antecedent. Once proper weighting has been assigned to each rule, the implication method is deployed.
A consequent is a fuzzy set represented by a membership function, which weights appropriately the linguistic characteristics which attributed to it. The consequent is reshaped using a function associated with the antecedent (a single number). The input for the implication process is a single number given by the antecedent, and the output is a fuzzy set.
4. Aggregate All Outputs:
Since decisions are based on the testing of all of the rules in Fuzzy Inference System, the rules must be combined in some manner in order to make a decision. Aggregation is the process by which the fuzzy sets which represent the outputs of each rule are combined into a single fuzzy set.
Aggregation only occurs one e for each output variable. The input of the aggregation process is the list of truncated output function returned by the implication process for each rule. The output of the aggregation process is one fuzzy set for each output variable.
5. Defuzzify:
The input for the defuzzification process is a fuzzy set (the aggregate output fuzzy set) and the output is a single number. As much as fuzziness helps the rule evaluation during the intermediate steps, the final desired output for each variable is generally a single number.
In this way, the inference involved in our daily life can be implemented in computers.