Fuzzy Logic

Write 3-5 fuzzy rules that determine heart attack risk, using:

Three ‘universes of discourse’ (UoD): diet, exercise, and risk

2 or 3 fuzzy classes per UoD, and their membership functions (represent graphically)

Show fuzzy inference for one set of sample data

Fuzzy Rules

Diet is low AND Exercise is high → Balanced
Diet is high OR Exercise is low → Unbalanced
Balanced → Risk is low
Unbalanced → Risk is high

Membership Functions

Diet is high: $\200dpi \inline \tiny f_{diet-high} (x) = \frac{1}{5000} \cdot x$

diet high membership chart

Diet is low: $\200dpi \inline \tiny f_{diet-low} (x) = 1 - \frac{1}{5000} \cdot x$

diet low membership chart

Exercise is high: $\200dpi \inline \tiny f_{exercise-high} (x) = \frac{1}{2000} \cdot x$

exercise high membership chart

Exercise is low: $\200dpi \inline \tiny f_{exercise-high} (x) = 1 - \frac{1}{2000} \cdot x$

exercise low membership chart

Balanced: $\200dpi \inline \tiny f_{balanced} (x) = 0.8 - \frac{1}{125} \cdot x$

balanced membership chart

Unbalanced: $\200dpi \inline \tiny f_{unbalanced} (x) = \frac{1}{125} \cdot x$

unbalanced membership chart

Risk is high: $\200dpi \inline \tiny f_{risk-high} (x) = \frac{1}{125} \cdot x$

risk high membership chart

Risk is low: $\200dpi \inline \tiny f_{risk-low} (x) = 0.8 - \frac{1}{125} \cdot x$

risk low membership chart

Sample Data

Diet: consume 3000 calories per day.

Exercise: burn 1000 calories per day.

What is the risk of heart disease?

Fuzzification

Membership for Diet High:

$\200dpi \inline \tiny \begin{align*} f_{diet-high} (3000) & = \frac {1} {5000} \cdot 3000 \\ & = 0.6 \end{align}$

Membership for Diet Low:

$\200dpi \inline \tiny \begin{align*} f_{diet-low} (3000) & = 1 - \frac {1} {5000} \cdot 3000 \\ & = 0.4 \end{align}$

Membership for Exercise High:

$\200dpi \inline \tiny \begin{align*} f_{exercise-high} (1000) & = \frac {1} {2000} \cdot 3000 \\ & = 0.5 \end{align}$

Membership for Exercise Low:

$\200dpi \inline \tiny \begin{align*} f_{exercise-low} (1000) & = 1 - \frac {1} {2000} \cdot 3000 \\ & = 0.5 \end{align}$

Rule Evaluation

Diet is low AND Exercise is high → Balanced

Membership for Balanced:

$\200dpi \inline \tiny \begin{align*} f_{balanced} & = min [f_{diet-low}, f_{exercise-high}] \\ & = min [0.4, 0.5] \\ & = 0.4 \end{align}$
Diet is high OR Exercise is low → Unbalanced

Membership for Unbalanced:

$\200dpi \inline \tiny \begin{align*} f_{unbalanced} & = max [f_{diet-high}, f_{exercise-low}] \\ & = max [0.6, 0.5] \\ & = 0.6 \end{align}$
Balanced → Risk is low

Membership for Risk Low:

$\200dpi \inline \tiny \begin{align*} f_{risk-low} & = f_{balanced} \\ & = 0.4 \end{align}$

Likelihood of heart disease cut off:

$\200dpi \inline \tiny \begin{align*} f_{risk-low} (x) & = 0.8 - \frac {1} {125} \cdot x \\ & = 0.4 \\ & \Rightarrow x = 50 \end{align}$

Membership function for Risk Low:
Unbalanced → Risk is high

Membership for Risk High:

$\200dpi \inline \tiny \begin{align*} f_{risk-high} & = f_{unbalanced} \\ & = 0.6 \end{align}$

Likelihood of heart disease cut off:

$\200dpi \inline \tiny \begin{align*} f_{risk-high} (x) & = \frac {1} {125} \cdot x \\ & = 0.6 \\ & \Rightarrow x = 75 \end{align}$

Membership function for Risk High:

Aggregation

The aggregated membership for Risk High and Risk Low:

aggregated risk membership chart

The aggregated Risk function:

$\200dpi \inline \tiny \begin{align*} f_{aggreged-risk}(x) = \begin{cases} 0.4, & \mbox{if }x \in [0, 50) \\ 3x+1, & \mbox{if }x \in [50, 75) \\ 3x+1, & \mbox{if }x \in [75, 100] \end{cases} \end{align}$

Defuzzification

The Center of Gravity (COG) method is used for the defuzzification.

$\200dpi \inline \tiny \begin{align*} COG = \frac {\displaystyle \sum_{x=a}^b f_A(x) \cdot x} {\displaystyle \sum_{x=a}^b f_A(x)} \end{align}$

The integration of the aggregated Risk membership function returns a likelihood of 47.5% of heart disease.

$\200dpi \inline \tiny \begin{align*} & \int_{0}^{100} f_{aggregated-risk} \cdot \, dx \\ & = \int_{0}^{50} 0.4 \cdot \, dx + \int_{50}^{75} \frac {1} {125} \cdot x \cdot \, dx + \int_{75}^{100} 0.6 \cdot \, dx \\ & = 50 \cdot 0.4 + \frac {1} {125} \cdot \begin{bmatrix} \frac {x^2} {2} \end{bmatrix}_{50}^{75} + 25 \cdot 0.6 \\ & = 20 + \frac {1} {250} \cdot \begin{pmatrix} 75^2 - 50^2 \end{pmatrix} + 15 \\ & = 47.5 \end{align}$