Lecture 3: Fuzzy If-then & Fuzzy Reasoning

Fuzzy If-then rule

Fuzzy If-then rules basically encodes fuzzy relations.

If $x$ is $A$, then $y$ is $B$.

In $A$ and $B$ are linguistic values defined by fuzzy sets on universes of discourse $X$ and $Y$. $x$ is $A$ => Antecedent / premise ; $y$ is $B$ => Consequence / conclusion

Fuzzy Reasoning (aka) Generalised Modus Ponens

Existing information are encoded as rules using the [[#Fuzzy If-then rule]] and is used to make real-time inference. But the question is how to decode the encoded information?

There are various methods for decoding, a few popular ones are shown below

1. Mamdani’s implication $$R=A\to B=A\times B=\underset{X\times Y}{\int }{\mu }_A(x)\wedge {\mu }_B(y)/(x,y)$$

2. Zadeh’s max-min implication $$R=A\to B=\underset{X\times Y}{\int }(1-{\mu }_A(x))\vee {\mu }_A(x)\wedge {\mu }_B(y)/(x,y)$$

But for the course, only Mamdani’s interpretation is used for decoding. However, a more generalised form this decoding processes (inference process) is called compositional rule of inference. This is explained by the below image.

Single Rule With Single Antecedent

A rule signifies the existing encoded information (If $x$ is $A$, then $y$ is $B$). On the other hand, antecedent is the fact that is provided to be inferred ($x$ is $A^{‘}$).

In such cases,

$${\mu }_{B^{{}^{\prime }}}(y)={\mu }_{A^{{}^{\prime }}}(x)\circ (A\to B)$$

$${\mu }_{B^{{}^{\prime }}}(y)={\vee }_x[{\mu }_{A^{{}^{\prime }}}(x)\wedge {\mu }_A(x)\wedge {\mu }_B(y)]$$

$${\mu }_{B^{{}^{\prime }}}(y)=[{\vee }_x({\mu }_{A^{{}^{\prime }}}(x)\wedge {\mu }_A(x))]\wedge {\mu }_B(y)$$

Single Rule With Multiple Antecedent

Rules: If $x$ is $A$ and $y$ is $B$, then $z$ is $C$

Antecedent: $x$ is $A^{{}^{\prime }}$ and $y$ is $B^{{}^{\prime }}$

${\mu }_{C^{{}^{\prime }}}(z)=[{\vee }_x({\mu }_{A^{{}^{\prime }}}(x)\wedge {\mu }_A(x))]\wedge [{\vee }_y({\mu }_{B^{{}^{\prime }}}(y)\wedge {\mu }_B(y))]\wedge {\mu }_C(z)$

$$W_1={\vee }_x({\mu }_{A^{{}^{\prime }}}(x)\wedge {\mu }_A(x))$$

$$W_2={\vee }_y(\mu \ast B^{{}^{\prime }}(y)\wedge {\mu }_B(y))$$

Multiple Rules With Multiple Antecedent

Rules:

1. $A_1\times B_1\to C_1$
2. $A_2\times B_2\to C_2$

Antecedent:

1. $x$ is $A^{{}^{\prime }}$ and $y$ is $B^{{}^{\prime }}$

$${\mu }_{C^{{}^{\prime }}}(z)=(A^{{}^{\prime }}\times B^{{}^{\prime }})\circ (R_1\vee R_2)$$

$${\mu }_{C^{{}^{\prime }}}(z)=((A^{{}^{\prime }}\times B^{{}^{\prime }})\circ R_1)\vee ((A^{{}^{\prime }}\times B^{{}^{\prime }})\circ R_2)$$

Defuzzification

Centroid of Area (COA) or Centre of Gravity (COG)

Largest of Maximum (LOM)