AI with Fuzzy Logic

Lessons

Senior 6 · ICT

AI with Fuzzy Logic

5 lessons

Lesson 2 of 5

Fuzzy Sets and Membership Functions

25 min

Describing a word with a graph

A school buys a fan controller for a hot classroom in Bugesera. Before it can decide what to do, it needs to know what "cool", "comfortable" and "hot" mean, in numbers. Each word becomes a fuzzy set on the temperature scale, and each set is described by a membership function: a graph that gives, for every temperature, how fully that temperature belongs to the word. Choosing these functions is the first design step of every fuzzy system.

Definition

Linguistic variable: a quantity whose values are words. Example: classroom temperature, with the values (called terms) cool, comfortable and hot.

Universe of discourse: the full range of values the quantity can take. Example: 10 °C to 40 °C.

Membership function $\mu_A(x)$ : a function that gives, for each $x$ in the universe, the degree to which $x$ belongs to the fuzzy set $A$ , a number in $[0, 1]$ .

Fuzzification: turning a measured number into its degrees of membership in each term. Example: 21 °C becomes cool 0.2, comfortable 0.6, hot 0.

The common shapes

Triangular, with feet at $a$ and $c$ and peak at $b$ :

$\mu(x) = \max\left( \min\left( \frac{x - a}{b - a},\; \frac{c - x}{c - b} \right),\; 0 \right)$

Trapezoidal, with feet at $a$ and $d$ and a flat top from $b$ to $c$ :

$\mu(x) = \max\left( \min\left( \frac{x - a}{b - a},\; 1,\; \frac{d - x}{d - c} \right),\; 0 \right)$

A trapezoid with $a = b$ at the left end of the universe (or $c = d$ at the right end) is called a shoulder: it stays at 1 all the way to the edge, as "cool" and "hot" do below.

Gaussian (bell-shaped), centred on $c$ with width $\sigma$ :

$\mu(x) = e^{-\frac{(x - c)^2}{2\sigma^2}}$

Triangles and trapezoids are quick to compute and easy to explain, so most teaching examples and many small controllers use them. Gaussian curves are smooth everywhere, which some optimisation methods need.

Definition

Features of a fuzzy set $A$ :

Support: all $x$ with $\mu_A(x) > 0$ . For the triangle (0, 2, 4), the open interval from 0 to 4.
Core: all $x$ with $\mu_A(x) = 1$ . For the trapezoid (2, 4, 6, 8), the interval from 4 to 6.
Crossover points: where $\mu_A(x) = 0.5$ . For the triangle (0, 2, 4), at 1 and 3.
Height: the largest membership value. A set of height 1 is called normal.

Three sets for one classroom

The fan controller uses these three sets. Notice that neighbouring sets overlap: between 18 °C and 22 °C a temperature is partly cool and partly comfortable. Overlap is what makes the controller change smoothly; with no overlap it would jump from one behaviour to the next like a crisp switch.

Term	Shape and parameters	Core	Support
cool	Left shoulder (10, 10, 17, 22)	10 to 17 °C	10 to just under 22 °C
comfortable	Triangle (18, 23, 28)	23 °C only	between 18 and 28 °C
hot	Right shoulder (24, 30, 40, 40)	30 to 40 °C	above 24 °C

Worked example 1: fuzzify 21 °C and 26 °C

At 21 °C.

Cool: 21 is on the falling side between 17 and 22: $\mu = \frac{22 - 21}{22 - 17} = \frac{1}{5} = 0.2$ .
Comfortable: 21 is on the rising side between 18 and 23: $\mu = \frac{21 - 18}{23 - 18} = \frac{3}{5} = 0.6$ .
Hot: 21 is below 24, so $\mu = 0$ .

At 26 °C.

Cool: above 22, so 0.
Comfortable: on the falling side between 23 and 28: $\mu = \frac{28 - 26}{28 - 23} = \frac{2}{5} = 0.4$ .
Hot: on the rising side between 24 and 30: $\mu = \frac{26 - 24}{30 - 24} = \frac{2}{6} \approx 0.33$ .

The memberships at one temperature need not add up to 1: at 21 °C they add to 0.8, and at 26 °C to about 0.73. Fuzzy memberships are not shares of a whole.

Worked example 2: "about 25 °C" as a bell

A Gaussian set "about 25 °C" has $c = 25$ and $\sigma = 2$ . Find the membership of 27 °C and 29 °C.

$\mu(27) = e^{-\frac{(27 - 25)^2}{2 \times 2^2}} = e^{-\frac{4}{8}} = e^{-0.5} \approx 0.607$

$\mu(29) = e^{-\frac{(29 - 25)^2}{8}} = e^{-2} \approx 0.135$

Twice as far from the centre gives far less than half the membership: the bell falls off quickly in its tails.

In code

The same calculations in Python, tested with python3. The functions follow the formulas above; fuzzify returns the three memberships for a temperature.

import math

def tri(x, a, b, c):
    """Triangle: 0 at a, rises to 1 at b, falls to 0 at c."""
    return max(min((x - a) / (b - a), (c - x) / (c - b)), 0.0)

def trap(x, a, b, c, d):
    """Trapezoid: 0 at a, 1 from b to c, 0 at d. Use a == b or c == d for a shoulder."""
    rise = (x - a) / (b - a) if b > a else 1.0
    fall = (d - x) / (d - c) if d > c else 1.0
    return max(min(rise, 1.0, fall), 0.0)

def gauss(x, c, s):
    """Gaussian bell centred on c with width s."""
    return math.exp(-((x - c) ** 2) / (2 * s ** 2))

def fuzzify(t):
    return {"cool": round(trap(t, 10, 10, 17, 22), 3),
            "comfortable": round(tri(t, 18, 23, 28), 3),
            "hot": round(trap(t, 24, 30, 40, 40), 3)}

for t in (15, 21, 23, 26, 32):
    print(t, fuzzify(t))
print(round(gauss(27, 25, 2), 3), round(gauss(29, 25, 2), 3))

Output:

15 {'cool': 1.0, 'comfortable': 0.0, 'hot': 0.0}
21 {'cool': 0.2, 'comfortable': 0.6, 'hot': 0.0}
23 {'cool': 0.0, 'comfortable': 1.0, 'hot': 0.0}
26 {'cool': 0.0, 'comfortable': 0.4, 'hot': 0.333}
32 {'cool': 0.0, 'comfortable': 0.0, 'hot': 1.0}
0.607 0.135

The printed values match the worked examples: 0.2 and 0.6 at 21 °C, 0.4 and 0.333 at 26 °C, and 0.607 and 0.135 for the bell.

MATLAB: What Is Fuzzy Logic? (Fuzzy Logic, Part 1). Watch how a crisp input is fuzzified using overlapping membership functions, and notice the shapes used; they are the same triangles and trapezoids as in this lesson.

Warning

Read the right side of the triangle. The most common error is using the rising formula $\frac{x - a}{b - a}$ on the falling side. Always check first where $x$ is: left of the peak uses the rising side, right of the peak uses the falling side, and outside the feet the value is 0. A membership value above 1 or below 0 means you used the wrong side.

Tip

Sketch before you compute. Draw the sets, mark your $x$ with a vertical line, and see which sets it cuts and roughly how high. Then calculate. The sketch tells you which formula to use and whether your answer is sensible.

✦

Hands-on: design a linguistic variable

Choose a quantity from school life: the time a student takes to reach school (minutes), with terms short, moderate and long.
Decide the universe (for example 0 to 120 minutes) and the parameters of each set: a left shoulder, a triangle and a right shoulder, overlapping.
Draw the three sets on graph paper with a ruler.
Copy the Python program and change fuzzify to use your sets. Run it for five journey times, including one in each overlap.
Check two of the printed values by hand with the formulas.

Check: every value between 0 and 120 minutes belongs to at least one set with membership above 0, and your hand calculations match the program.

Practice

Define linguistic variable, universe of discourse and fuzzification.
For the triangle (18, 23, 28), find $μ \mu$ at 19.5 °C, 23 °C and 27 °C.
For the trapezoid (2, 4, 6, 8), state the support, the core and the crossover points.
Fuzzify 18.5 °C with the three classroom sets.
A Gaussian set has $c = 60$ and $\sigma = 5$ . Find $\mu(65)$ to 3 decimal places.
A student defines cool as (10, 10, 17, 20) and comfortable as (21, 24, 27). What is wrong with this design, and how would a controller behave at 20.5 °C?
Explain why overlapping sets give smoother control than sets that only touch.

Show answersHide answers

A linguistic variable is a quantity whose values are words (terms), such as temperature with cool, comfortable and hot. The universe of discourse is the full range of values it can take. Fuzzification turns a measured value into its degree of membership in each term.
$\frac{19.5 - 18}{5} = 0.3$ ; 1 (the peak); $\frac{28 - 27}{5} = 0.2$ .
Support: between 2 and 8. Core: from 4 to 6. Crossover points: 3 and 7.
Cool: $\frac{22 - 18.5}{5} = 0.7$ . Comfortable: $\frac{18.5 - 18}{5} = 0.1$ . Hot: 0.
$e^{-\frac{25}{50}} = e^{-0.5} \approx 0.607$ .
There is a gap: from 20 °C to 21 °C no set has membership above 0. At 20.5 °C every rule would have strength 0, so the controller would produce no output or fall back to a default. Neighbouring sets must overlap.
With overlap, a temperature near a boundary belongs partly to both sets, so both rules fire partly and the output blends between them as the temperature changes. Sets that only touch behave like crisp categories: the output jumps at the boundary.

Sources and credits

Membership function (mathematics), Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
Fuzzy set, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
Video: What Is Fuzzy Logic? | Fuzzy Logic, Part 1, MATLAB, embedded from YouTube.
The fan-controller scenario, graphs, program, worked examples, hands-on task and practice items were written and tested by Ganzaa.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Describing a word with a graph

2The common shapes

3Three sets for one classroom

4In code

5Hands-on: design a linguistic variable

Describing a word with a graph

The common shapes

Three sets for one classroom

In code

Hands-on: design a linguistic variable