frbs (version 3.2-0)

# rulebase: The rule checking function

## Description

This function checks the consistency of a rule definition (given by the user). The rulebase consists of several fuzzy IF-THEN rules. The rules could be in a list or matrix type. Generally, there are three types of rule structures which are rules based on Mamdani, Takagi Sugeno Kang and fuzzy rule-based classification systems (FRBCS).

For rules of the Mamdani model, there are 2 parts in each rule, the antecedent and the consequent part, which are separated by "->".

## Usage

rulebase(type.model, rule, func.tsk = NULL)

## Arguments

type.model

a value determining the type of model to use. Here, MAMDANI and TSK mean Mamdani and Takagi Sugeno Kang model, respectively.

rule

a matrix or list of rules.

func.tsk

a matrix representing the consequent parts of rules in Takagi Sugeno Kang formulation.

## Value

fuzzy IF-THEN rule base

## Details

For example: r1 <- c("a1","and","b1","->", "c1")

It means that "IF input.variable1 is a1 and input.variable2 is b1 THEN output.variable is c1"

Here, ("a1", "and", "b1") is the antecedent, with "a1" and "b1" being linguistic terms, and ("c1") is the consequent part.

A fuzzy IF-THEN rule base with several rules is defined in the following way:

r1 <- c("not a1","and","b1", "->", "c1")

r2 <- c("a2","or","b2", "->", "c2")

r3 <- c("a3","or","b2", "->", "c3")

rule <- list(r1,r2,r3)

For rules of the Takagi Sugeno Kang model, the rules are at first defined without the consequent part, e.g.:

r1 <- c("a1",1,"b1","->")

r2 <- c("a2",2,"b2", "->")

r3 <- c("a3","2","b2", "->")

rule <- list(r1,r2,r3)

The consequences are defined then as a matrix fun_tsk, which contains the linear equations of the consequences of the rules. The dimension of this matrix is [<number_of_rules>, <number_of_variables> + 1]. The matrix has one extra column for the constants. If there is no constant, a zero is put.

So, for example, if we have 3 rules and 2 linguistic variables (A, B), the matrix fun_tsk has dim(3,3), as in:

func.tsk <- matrix(c(1, 1, 5, 2, 1, 3, 1, 2, 2), nrow=3, ncol=3, byrow = TRUE)

Furthermore, we can represent linguistic hedges within the rules. The kinds of hedges used are

• "extremely" reduces the truth value. For example, "extremely a1" means membership function $$a1 = \mu(a1)^3$$.

• "very" reduces the truth value. For example, "very a1" means membership function $$a1 = \mu(a1)^2$$.

• "somewhat" increases the truth value. For example, "somewhat a1" means membership function $$a1 = \mu(a1)^0.5$$.

• "slightly" increases the truth value. For example, "slightly a1" means membership function $$a1 = \mu(a1)^0.33$$

An example of fuzzy IF-THEN rules using linguistic hedge is:

r1 <- c("very a1","and","b1","->","c1")

r2 <- c("a2",2,"b2", "->", "c2")

r3 <- c("a3","2","slightly b2", "->", "c3")

rule <- list(r1,r2,r3)

Furthermore, the following is an example in order to give names to the linguistic terms in the input and output variables.

varinput.1 <- c("a1", "a2", "a3")

varinput.2 <- c("b1", "b2")

names.varinput <- c(varinput.1, varinput.2)

names.varoutput <- c("c1", "c2", "c3")

In case of FRBCS model, the structure of rules are quite similar with Takagi Sugeno Kang model. But, instead of using linear equation in consequent part, consequent parts in FRBCS are represented by class. For example, Take into account that consequent parts expresses classes.

rule<-matrix(c("v.1_a.2","and","v.2_a.2","and","v.3_a.3","and","v.4_a.2","->","3",

"v.1_a.2","and","v.2_a.3","and","v.3_a.1","and","v.4_a.3","->","1",

"v.1_a.2","and","v.2_a.2","and","v.3_a.2","and","v.4_a.2","->","2"),

nrow=3, byrow=TRUE)

Where, "1", "2", "3" represent class 1, 2, and 3.

Noted that all rules included in rule base must have the same length as many as the number of variables. However, we can ignore some input variables by defining the "dont_care" value. For example, the rule ("not a1","and","dont_care", "->", "c1") refers to the rule ("not a1" "->", "c1"). Furthermore, if we are using the learning methods, the fuzzy IF-THEN rules will be generated automatically as the outputs of frbs.learn.

## See Also

defuzzifier, inference, and fuzzifier