Computational Intelligence (AD-702 (C)) - Unit 2 Short Notes

How unit 2 is examined

This unit covers fuzzy sets, membership functions, relations, measures, and the fuzzy controller pipeline (fuzzification, rule base, inference, defuzzification). No topic was asked in the recent papers, so learn the definitions and formulas below.

Fuzzy sets and operations

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

Key points.

1. A crisp set allows only membership 0 or 1, whereas a fuzzy set allows any value between 0 and 1, so it models vague terms such as "tall" or "warm".
2. Union is the maximum: $\mu_{A\cup B}(x)=\max(\mu_A(x),\mu_B(x))$.
3. Intersection is the minimum: $\mu_{A\cap B}(x)=\min(\mu_A(x),\mu_B(x))$.
4. Complement is $\mu_{\bar A}(x)=1-\mu_A(x)$; unlike crisp sets, $A\cup\bar A\ne X$ and $A\cap\bar A\ne\emptyset$.

Membership Functions

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Definition. <mark>A membership function is a curve that maps each input value of the universe to a degree of membership between 0 and 1.</mark>

Key points.

1. The triangular function with feet $a,c$ and peak $b$ is $\mu(x)=\max\left(\min\left(\frac{x-a}{b-a},\frac{c-x}{c-b}\right),0\right)$.
2. The trapezoidal function has a flat top of membership 1 between two middle points, so it suits a range that is fully "true".
3. The Gaussian function $\mu(x)=e^{-(x-c)^2/2\sigma^2}$ is smooth and differentiable, and the bell and sigmoid functions are other common shapes.
4. The height of a set is its maximum membership, and its support is the set of points with membership above 0.

Concept of Fuzzy relations and their composition

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Definition. <mark>A fuzzy relation $R$ from $X$ to $Y$ is a fuzzy set on $X\times Y$ with membership $\mu_R(x,y)\in[0,1]$, usually stored as a matrix.</mark>

Key points.

1. A fuzzy relation expresses a graded connection such as "x is much greater than y", instead of a yes or no link.
2. Composition of $R$ (on $X\times Y$) and $S$ (on $Y\times Z$) gives a relation on $X\times Z$.
3. Max-min composition is $\mu_{R\circ S}(x,z)=\max_y\min\left(\mu_R(x,y),\mu_S(y,z)\right)$, worked like matrix multiplication with min for product and max for sum.
4. Max-product composition replaces min by the ordinary product.

Concept of Fuzzy Measures

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Definition. ==A fuzzy measure $g$ assigns each subset of $X$ a value in $[0,1]$ showing the degree of evidence that an element belongs to that subset, with $g(\emptyset)=0$, $g(X)=1$ and monotonicity.==

Key points.

1. Monotonicity means that if $A\subseteq B$ then $g(A)\le g(B)$.
2. A fuzzy measure needs no additivity, so it generalises a probability measure.
3. The Sugeno $\lambda$-measure satisfies $g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)$ for disjoint $A,B$, with $\lambda>-1$.
4. Possibility and necessity measures, and belief and plausibility, are special fuzzy measures.

Fuzzy Rules, Inferencing

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Definition. <mark>A fuzzy rule is an IF-THEN statement, "IF x is A THEN y is B", where A and B are fuzzy sets, and inferencing derives the output fuzzy set from the inputs using such rules.</mark>

Key points.

1. The IF part is the antecedent and the THEN part is the consequent; several antecedents are joined by AND (min) or OR (max).
2. Generalised modus ponens lets the system conclude B' from the premise "x is A'" even when A' only approximately matches A.
3. Mamdani implication clips or scales the consequent by the rule's firing strength, using min or product.
4. The outputs of all fired rules are aggregated, usually by max, into one fuzzy set.

Fuzzy Control - Selection of Membership Functions

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Definition. <mark>A fuzzy controller is a rule-based controller that turns crisp measurements into a crisp control action through fuzzification, rule base, inference engine and defuzzification.</mark>

Key points.

1. Triangular and trapezoidal functions are preferred in control because they are simple and fast to compute.
2. Each variable usually has 3 to 7 linguistic terms (Negative Big ... Zero ... Positive Big), with neighbouring sets overlapping by about 25 to 50 percent.
3. The membership functions should cover the whole input range so that at least one rule always fires.
4. Mamdani controllers give fuzzy outputs that are defuzzified, whereas Sugeno controllers give crisp linear or constant outputs.

Fuzzyfication

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Definition. <mark>Fuzzification is the conversion of a crisp input value into degrees of membership in the linguistic fuzzy sets of that variable.</mark>

Key points.

1. It is the first stage of the fuzzy system, applied to each measured input.
2. A crisp value is read against the membership curves; for example a temperature of 30 may be 0.4 "warm" and 0.6 "hot".
3. The chosen scaling and shape of the membership functions decide how sensitive the controller is.
4. The output of this stage is a set of membership degrees that feed the antecedents of the rules.

Rule Based Design & Inferencing

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Definition. <mark>Rule based design means building the rule base from expert knowledge or data, and the inference engine evaluates those rules on the fuzzified inputs to produce an output fuzzy set.</mark>

Key points.

1. Design steps are to choose inputs and outputs, define linguistic terms, write the rules, and tune the membership functions.
2. A rule base should be complete (every input combination is covered) and consistent (no contradicting rules).
3. Firing strength of a rule is the min (AND) of its antecedent memberships.
4. Mamdani inference clips each consequent at its firing strength and aggregates by max, whereas Sugeno uses a weighted average of crisp rule outputs.

Defuzzyfication

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Definition. <mark>Defuzzification converts the aggregated output fuzzy set into a single crisp value that can drive the actuator.</mark>

Key points.

1. The centroid (centre of area) method gives $z^*=\frac{\int \mu(z)\,z\,dz}{\int \mu(z)\,dz}$, or $\frac{\sum \mu_i z_i}{\sum \mu_i}$ in the discrete case.
2. The weighted average method uses only the peak of each clipped output set and is cheaper, but works for symmetric sets.
3. Mean of maxima averages the points with highest membership, and the max-membership method picks the peak value.
4. Centroid is the most used because it gives a smooth output that reflects all the fired rules.

Last-minute revision

• Fuzzy membership $\mu_A(x)\in[0,1]$; union = max, intersection = min, complement = $1-\mu$.
• Triangular MF is defined by three points $a,b,c$; trapezoidal by four.
• Max-min composition: $\mu_{R\circ S}(x,z)=\max_y\min(\mu_R,\mu_S)$.
• Fuzzy measure: $g(\emptyset)=0$, $g(X)=1$, monotone; Sugeno adds $\lambda g(A)g(B)$.
• Rule form: IF x is A THEN y is B; firing strength = min of antecedents.
• Mamdani clips the consequent and aggregates with max; Sugeno output is crisp.
• Controller pipeline: fuzzification, rule base, inference, defuzzification.
• Centroid: $z^*=\sum\mu_i z_i/\sum\mu_i$.
• Other defuzzifiers: weighted average, mean of maxima, max membership.

Memory hooks

• "Union Max, Intersection Min": U is a tall letter (max), the small n of intersection is min.
• Composition = matrix multiplication with min in place of times and max in place of plus.
• Controller order: F-R-I-D (Fuzzify, Rules, Infer, Defuzzify).
• Centroid = balance point of the shape, like a centre of mass.

Coverage checklist

• Fuzzy sets and operations: definition, union, intersection, complement (no past questions).
• Membership Functions: triangular, trapezoidal, Gaussian (no past questions).
• Concept of Fuzzy relations and their composition: max-min and max-product (no past questions).
• Concept of Fuzzy Measures: axioms, Sugeno measure (no past questions).
• Fuzzy Rules, Inferencing: IF-THEN, modus ponens, Mamdani (no past questions).
• Fuzzy Control - Selection of Membership Functions: controller structure, choice of functions (no past questions).
• Fuzzyfication: crisp to membership degrees (no past questions).
• Rule Based Design & Inferencing: rule base design, firing strength (no past questions).
• Defuzzyfication: centroid, weighted average, mean of maxima (no past questions).