Intelligent Systems II · TEE · read

Graph axes, intervals and the fractions inside membership functions

1. The maths these three topics use

Short prerequisites, with worked examples. Read once before Fuzzy → GA → PSO.

Read a graph one input at a time

The horizontal axis runs left to right and shows input x. The vertical axis runs bottom to top and shows the output. A point (3,0.4) means: input 3 gives output 0.4. In a membership graph, the vertical output is a degree between 0 and 1.

To read the graph at x=3, start at 3 on the bottom axis, move vertically until you reach the line, then move horizontally to the vertical axis. The height you read is the membership.

Halfway along a straight rise from height 0 to height 1 gives height 0.5.

Read this	Meaning	Concrete example
x < 5	Strictly less than 5	4.9 qualifies; 5 does not
x ≤ 5	Less than or equal to 5	Both 4.9 and 5 qualify
2 ≤ x ≤ 8	x is between 2 and 8 including both ends	The interval is [2,8]
[2,8)	Include 2; exclude 8	Useful for roulette interval boundaries
Piecewise formula	Choose one calculation according to where x lies	Rising branch, falling branch or outside

A straight membership line rises from (2,0) to (6,1). Find its height at x=3. [2 marks · Worked example]

Given

Left input=2; peak input=6; required input=3.

To find

Membership height.

Formula and why it applies

\text{height}=\frac{\text{distance from left}}{\text{total rising width}}

A straight line changes by the same proportion of its height as the proportion travelled along its width.

Solution

Step 1. Find distance travelled

3-2=1

Step 2. Find total width

6-2=4

Step 3. Divide distance by width

\mu(3)=\frac{1}{4}=0.25

Final answer. Membership=0.25.

A straight membership line rises from (2,0) to (6,1). Find its height at x=4. [2 marks · Worked example]

Given

Left input=2; peak input=6; required input=4.

To find

Membership height.

Formula and why it applies

\text{height}=\frac{\text{distance from left}}{\text{total rising width}}

A straight line changes by the same proportion of its height as the proportion travelled along its width.

Solution

Step 1. Find distance travelled

4-2=2

Step 2. Find total width

6-2=4

Step 3. Divide distance by width

\mu(4)=\frac{2}{4}=0.5

Final answer. Membership=0.5.

A straight membership line rises from (10,0) to (30,1). Find its height at x=25. [2 marks · Worked example]

Given

Left input=10; peak input=30; required input=25.

To find

Membership height.

Formula and why it applies

\text{height}=\frac{\text{distance from left}}{\text{total rising width}}

A straight line changes by the same proportion of its height as the proportion travelled along its width.

Solution

Step 1. Find distance travelled

25-10=15

Step 2. Find total width

30-10=20

Step 3. Divide distance by width

\mu(25)=\frac{15}{20}=0.75

Final answer. Membership=0.75.

A membership line falls from (5,1) to (8,0). Find its height at x=7. [2 marks · Worked example]

Given

Peak=5; right zero=8; x=7.

To find

Membership.

Formula and why it applies

\mu(x)=\frac{\text{distance still remaining to right zero}}{\text{total falling width}}

Solution

Step 1. Remaining distance

8-7=1

Step 2. Total falling width

8-5=3

Step 3. Divide

\mu(7)=1/3\approx0.333333

Final answer. Membership≈0.333333. Using distance already travelled would reverse the falling line.

Yaad rakho: Chadhti line: left se kitna aage? Utarti line: right tak kitna baaki? Dono mein poori width se divide.