Computational Intelligence (CS-702 (A)) Unit 2 Notes

How unit 2 is examined

Fuzzy sets, operations, relations, rules, inference and defuzzification; the marks sit in fuzzy set operations (numericals, alpha cuts) and defuzzification methods.

Fuzzy sets and operations

<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">High weight</span>

Definition. <mark>A fuzzy set $A$ on a universe $X$ is a set of pairs $(x, \mu_A(x))$ where the membership function $\mu_A : X \to [0,1]$ gives the degree to which $x$ belongs to $A$.</mark>

Key points.

A crisp set allows membership 0 or 1 only, whereas a fuzzy set allows any value in $[0,1]$, so it models vague concepts such as "tall" or "warm".
A finite fuzzy set is written $A = \mu_1/x_1 + \mu_2/x_2 + \dots$, where the slash pairs a membership with its element and the plus is only a separator.
Union takes the larger membership, $\mu_{A\cup B}(x)=\max(\mu_A,\mu_B)$, and intersection takes the smaller, $\mu_{A\cap B}(x)=\min(\mu_A,\mu_B)$.
Complement is $\mu_{\bar A}(x)=1-\mu_A(x)$, and difference is $A-B = A\cap\bar B$, that is $\min(\mu_A, 1-\mu_B)$.
Bounded sum is $\min(1,\mu_A+\mu_B)$; algebraic sum is $\mu_A+\mu_B-\mu_A\mu_B$; algebraic product is $\mu_A\mu_B$.
Cardinality is $|A|=\sum_x \mu_A(x)$; subsethood is $S(A,B)=\dfrac{|A\cap B|}{|A|}$ and equality is $E(A,B)=\dfrac{|A\cap B|}{|A\cup B|}$.
Alpha cut: $A_\alpha=\{x\mid \mu_A(x)\ge\alpha\}$; strong alpha cut: $A_{\alpha+}=\{x\mid \mu_A(x)>\alpha\}$; level set: $\Lambda_A=\{\alpha\mid \mu_A(x)=\alpha \text{ for some } x\}$. Each cut is a crisp set.

Example (alpha cuts). $A=0.2/a+0.5/b+0.7/c+1/d$. Then $A_{0.2}=\{a,b,c,d\}$, $A_{0.5}=\{b,c,d\}$, $A_{0.7}=\{c,d\}$, $A_1=\{d\}$. Strong cut $A_{0.5+}=\{c,d\}$ and $A_{0.7+}=\{d\}$. Level set $\Lambda_A=\{0.2,0.5,0.7,1\}$.

Example (Dec 2020). $X=\{0.4,0.7,0.8,1\}$, $Y=\{0.3,0.6,0.5,0.9\}$ on $x_1..x_4$; $1-Y=\{0.7,0.4,0.5,0.1\}$.

Operation	x1	x2	x3	x4
$X\cup Y$ (max)	0.4	0.7	0.8	1
$X\cap Y$ (min)	0.3	0.6	0.5	0.9
$X-Y=\min(X,1-Y)$	0.4	0.4	0.5	0.1
$X+Y$ bounded sum	0.7	1	1	1
$X+Y$ algebraic sum	0.58	0.88	0.90	0.99

Example (Nov 2023). $A=\{0.5,1,0.6,0.8\}$, $B=\{1,0.5,0.1,1\}$ on $\{3,5,7,8\}$.

Measure	3	5	7	8
$A+B=a+b-ab$	1	1	0.64	1
$A\cdot B=ab$	0.5	0.5	0.06	0.8
$\min(A,B)$	0.5	0.5	0.1	0.8
$\max(A,B)$	1	1	0.6	1

$|A|=2.9$, $|A\cap B|=1.9$, $|A\cup B|=3.6$. $S(A,B)=1.9/2.9=$ 0.655; $E(A,B)=1.9/3.6=$ 0.528.

Answer frame. Open with the definition of a fuzzy set; write the formula for each operation before substituting; tabulate element by element; close with the final sets in $\mu/x$ form. For the alpha-cut question, define all three with notation, then use your own 4-element set and list the crisp set for each $\alpha$.

Pitfall: Do not use max for the sum; "$X+Y$" is bounded sum (capped at 1), and difference is $\min(\mu_X,1-\mu_Y)$, not $\mu_X-\mu_Y$.

Asked: [14 marks] (Dec 2020) Consider fuzzy sets X and Y as given; calculate $X\cup Y$, $X\cap Y$, $X-Y$, $X+Y$. Asked: [7 marks] (Nov 2023) For fuzzy sets A and B on $\{3,5,7,8\}$ obtain algebraic sum, algebraic product, subsethood measure $S(A,B)$ and equality measure $E(A,B)$. Asked: [7 marks] (Nov 2023) Define alpha cut, strong alpha cut sets and level sets of a given fuzzy set with your data set.

Membership Functions

Definition. A membership function maps each element of the universe to a degree in $[0,1]$; it defines the shape of the fuzzy set.

Key points.

Triangular function with corners $a,b,c$: $\mu=\dfrac{x-a}{b-a}$ for $a\le x\le b$, $\dfrac{c-x}{c-b}$ for $b\le x\le c$, else 0.
Trapezoidal has four corners $a,b,c,d$ with a flat top of 1 between $b$ and $c$.
Gaussian $e^{-(x-m)^2/2\sigma^2}$ is smooth; sigmoid suits open-ended sets like "high".
Support is where $\mu>0$, core where $\mu=1$, and height is the maximum $\mu$.

Concept of Fuzzy relations and their composition

Definition. <mark>A fuzzy relation $R$ from $X$ to $Y$ is a fuzzy set on $X\times Y$, with $\mu_R(x,y)\in[0,1]$ giving the strength of relation between $x$ and $y$.</mark>

Key points.

It is represented as a matrix whose rows are elements of $X$ and columns elements of $Y$.
Max-min composition of $R$ (X to Y) and $S$ (Y to Z) is $\mu_{R\circ S}(x,z)=\max_y\min(\mu_R(x,y),\mu_S(y,z))$.
Max-product composition uses $\max_y\big(\mu_R(x,y)\cdot\mu_S(y,z)\big)$ instead.
Composition works like matrix multiplication with min (or product) for multiply and max for add.

Example. $R=\begin{bmatrix}0.3&0.7\\0.6&0.2\end{bmatrix}$, $S=\begin{bmatrix}0.5&0.9\\0.8&0.4\end{bmatrix}$. Max-min: $T_{11}=\max(0.3,0.7)=0.7$, $T_{12}=\max(0.3,0.4)=0.4$, $T_{21}=\max(0.5,0.2)=0.5$, $T_{22}=\max(0.6,0.2)=0.6$, so $T=\begin{bmatrix}0.7&0.4\\0.5&0.6\end{bmatrix}$. Max-product gives $\begin{bmatrix}0.56&0.28\\0.30&0.54\end{bmatrix}$.

Asked: [7 marks] (Nov 2023) Explain the concept of fuzzy relations and their composition with the help of examples.

Concept of Fuzzy Measures

Definition. A fuzzy measure $g$ assigns to each crisp subset of $X$ a value in $[0,1]$ showing the evidence that an element belongs to that subset.

Key points.

It satisfies $g(\emptyset)=0$, $g(X)=1$ and monotonicity: if $A\subseteq B$ then $g(A)\le g(B)$.
Belief, plausibility, possibility and necessity measures are special cases.
It measures uncertainty about which crisp set holds, while a membership function measures vagueness of the set itself.

Fuzzy Rules

Definition. A fuzzy rule is an IF-THEN statement, "IF $x$ is $A$ THEN $y$ is $B$", where $A$ and $B$ are linguistic values.

Key points.

The IF part is the antecedent and the THEN part the consequent.
Antecedents combine with AND (min), OR (max) and NOT (complement).
A collection of rules is the rule base, which captures expert knowledge.
Example: IF temperature is high THEN fan speed is fast.

Inferencing

Definition. Fuzzy inference derives a fuzzy output from fuzzy inputs by applying the rule base.

Key points.

Mamdani inference has fuzzy sets as consequents; it clips the consequent at the rule's firing strength (min), aggregates rules by max and then defuzzifies.
Sugeno inference has a crisp function as consequent, $z=ax+by+c$, and outputs a weighted average, so it needs no defuzzifier.
Firing strength of a rule is the min of its antecedent memberships.
Generalised modus ponens: from "$x$ is $A'$" and "IF $x$ is $A$ THEN $y$ is $B$" infer "$y$ is $B'$".

Fuzzy Control - Selection of Membership Functions

Definition. A fuzzy controller controls a plant using linguistic IF-THEN rules instead of a mathematical model; its performance depends on the chosen membership functions.

Key points.

Triangular and trapezoidal functions are preferred because they are simple and fast to compute.
Three to seven sets per variable (such as NB, NS, ZE, PS, PB) are typical.
Adjacent sets should overlap by about 25 to 50 percent so that every input fires at least one rule.
Narrow sets near the set point give fine control; wide sets far away give quick response.

Fuzzyfication

Definition. Fuzzification converts a crisp input into degrees of membership in the input linguistic sets.

Key points.

The fuzzifier reads a sensor value $x_0$ and evaluates $\mu_A(x_0)$ for each set.
A singleton fuzzifier treats the input as a single point with membership 1.
Example: temperature 30 C may be 0.3 "warm" and 0.7 "hot".
It is the first stage of the fuzzy system.

Rule Based Design & Inferencing

Definition. A rule-based fuzzy system is built from four blocks: fuzzifier, rule base, inference engine and defuzzifier.

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Key points.

Design steps: choose input and output variables, define membership functions, write the rule base, choose the inference method, choose the defuzzifier and tune.
The rule base can hold at most (sets per input)$^{\text{inputs}}$ rules; two inputs with 5 sets each need up to 25 rules.
The inference engine finds each rule's firing strength and combines all rule outputs.

Defuzzyfication

Definition. <mark>Defuzzification is the conversion of a fuzzy output set into a single crisp value that can drive the real system.</mark>

Key points.

Centroid (centre of gravity) picks the point that balances the area: $z^*=\dfrac{\int \mu(z)\,z\,dz}{\int \mu(z)\,dz}$, or $\dfrac{\sum \mu(z_i)z_i}{\sum\mu(z_i)}$ for discrete sets; it is the most accurate but the costliest.
Bisector (centre of area) is the $z$ that splits the area under $\mu$ into two equal halves.
Mean of maxima (MOM) is the average of all $z$ where $\mu$ is at its maximum.
Smallest of maxima (SOM) is the least $z$ at maximum membership; largest of maxima (LOM) is the greatest such $z$.
Weighted average is $z^*=\dfrac{\sum \mu_i \bar z_i}{\sum \mu_i}$, where $\bar z_i$ is the peak of each clipped output set; it is quick but valid only for symmetric sets.
Maxima methods are fast but ignore the shape of the set, while centroid uses the full shape.

Steps (centroid).

Step 1: Aggregate all rule outputs into one output fuzzy set.
Step 2: Multiply each z by its membership and add: numerator.
Step 3: Add all memberships: denominator.
Step 4: Divide numerator by denominator to get crisp z*.

Example. Points $z=2,4,6,8$ with $\mu=0.2,0.6,1,0.4$: numerator $0.4+2.4+6+3.2=12$, denominator $2.2$, so $z^*=12/2.2=5.45$. MOM $=6$, SOM $=$ LOM $=6$ (single peak).

Answer frame. Open with the definition (fuzzy to crisp); draw a triangle-like output set marking the centroid, MOM, SOM and LOM positions; develop centroid first with its formula and example, then bisector, MOM/SOM/LOM, weighted average; close by stating that centroid is the most widely used. For the "write a short note on any two" form, pick Defuzzification and Fuzzy Rules.

Asked: [14 marks] (Dec 2020, Nov 2023) Define defuzzification and explain the different defuzzification methods. Write a short note on any two: i) Defuzzification ii) Genetic Operators iii) Bee Colony Optimization iv) Fuzzy Rules.

Last-minute revision

Fuzzy set: membership $\mu\in[0,1]$; union max, intersection min, complement $1-\mu$.
Difference $X-Y=\min(\mu_X,1-\mu_Y)$; bounded sum $=\min(1,a+b)$.
Algebraic sum $a+b-ab$; algebraic product $ab$.
$S(A,B)=|A\cap B|/|A|$; $E(A,B)=|A\cap B|/|A\cup B|$; $|A|=\sum\mu$.
Dec 2020 numerical: union 0.4,0.7,0.8,1; intersection 0.3,0.6,0.5,0.9; difference 0.4,0.4,0.5,0.1.
Nov 2023 numerical: $S=0.655$, $E=0.528$.
Alpha cut uses $\ge\alpha$; strong cut uses $>\alpha$; level set is the set of distinct membership values.
Max-min composition: $\max_y\min(\mu_R,\mu_S)$.
Mamdani has fuzzy consequents; Sugeno has crisp linear consequents.
Centroid $=\sum\mu z/\sum\mu$; methods: centroid, bisector, MOM, SOM, LOM, weighted average.

Memory hooks

Union = Upper (max), Intersection = Inferior (min).
"Strong" cut is strictly greater, so it loses the boundary element.
Fuzzy system flow: Fuzzify, Rules, Infer, Defuzzify (FRID).
SOM smallest, LOM largest, MOM middle of the maxima.

Coverage checklist

Fuzzy sets and operations: Dec 2020 union/intersection/difference/sum numerical; Nov 2023 algebraic sum/product, subsethood, equality; Nov 2023 alpha cut, strong alpha cut, level set.
Membership Functions: no past questions; triangular, trapezoidal, Gaussian.
Concept of Fuzzy relations and their composition: Nov 2023 fuzzy relations and composition.
Concept of Fuzzy Measures: no past questions.
Fuzzy Rules: covered; also appears as a short-note option in the Dec 2020 / Nov 2023 question.
Inferencing: no past questions.
Fuzzy Control - Selection of Membership Functions: no past questions.
Fuzzyfication: no past questions.
Rule Based Design & Inferencing: no past questions.
Defuzzyfication: Dec 2020, Nov 2023 defuzzification methods and short note.

Account

Computational Intelligence (CS-702 (A)) - Unit 2 Short Notes

How unit 2 is examined

Fuzzy sets and operations

Membership Functions

Concept of Fuzzy relations and their composition

Concept of Fuzzy Measures

Fuzzy Rules

Inferencing

Fuzzy Control - Selection of Membership Functions

Fuzzyfication

Rule Based Design & Inferencing

Defuzzyfication

Last-minute revision

Memory hooks

Coverage checklist