inference {frbs}

The process of fuzzy reasoning

Description

Inference refers to the process of fuzzy reasoning.

Usage

inference(MF, rule, names.varinput, type.tnorm, type.snorm)

Arguments

MF

a matrix of the degrees of membership functions which is a result of the fuzzifier.

rule

a matrix or list of fuzzy IF-THEN rules. See rulebase.

names.varinput

a list of names of the input variables.

type.tnorm

a value which represents the type of t-norm to be used:

• 1 or MIN means standard t-norm: min(x1, x2).

• 2 or HAMACHER means Hamacher product: (x1 * x2)/(x1 + x2 - x1 * x2).

• 3 or YAGER means Yager class: 1- min(1, ((1 - x1) + (1 - x2))).

• 4 or PRODUCT means product: (x1 * x2).

• 5 or BOUNDED means bounded product: max(0, x1 + x2 - 1).

type.snorm

a value which represents the type of s-norm to be used:

• 1 or MAX means standard s-norm: max(x1, x2).

• 2 or HAMACHER means Hamacher sum: (x1 + x2 - 2x1 * x2) / 1 - x1 * x2.

• 3 or YAGER means Yager class: min(1, (x1 + x2)).

• 4 or SUM means sum: (x1 + x2 - x1 * x2).

• 5 or BOUNDED means bounded sum: min(1, x1 + x2).

Details

In this function, fuzzy reasoning is conducted based on Mamdani and Takagi Sugeno Kang model. Furthermore, there are some formula for conjunction and disjunction operators.

The Mamdani model: A fuzzy system with, e.g., two inputs x1 and x2 (antecedents) and a single output y (consequent) is described by the following fuzzy IF-THEN rule:

IF x1 is A1 and x2 is A2 THEN y is B

where A1 and A2 are the fuzzy sets representing the antecent pairs and B is the fuzzy set representing the consequent.

The Takagi Sugeno Kang model: Suppose we have two inputs x1 and x2 and output y, then the fuzzy IF-THEN rule is as follows:

IF x1 is A1 and x2 is A2 THEN y is y = f(x1, x2)

where y = f(x1, x2) is a crisp function in the consequent part which is usually a polynomial function, and A1 and A2 are the fuzzy sets representing the antecent pairs.

Futhermore, this function has the following capabilities:

• It supports unary operators (not) and binary operators (AND and OR).

• It provides linguistic hedge (extremely, very, somewhat, and slightly).

• there are several methods for the t-norm and s-norm.

Value

a matrix of the degrees of the rules.

See Also

defuzzifier, rulebase, and fuzzifier.