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Fuzzy Logic and Fuzzy Inference

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- Why use fuzzy logic?
- Tipping example
- Fuzzy set theory
- Fuzzy inference

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- A super set of Boolean logic
- Builds upon fuzzy set theory
- Graded truth. Truth values between True and False. Not everything is either/or, true/false, black/white, on/off etc.
- Grades of membership. Class of tall men, class of far cities, class of expensive things, etc.
- Lotfi Zadeh, UC/Berkely 1965. Introduced FL to model uncertainty in natural language. Tall, far, nice, large, hot, …
- Reasoning using linguistic terms. Natural to express expert knowledge. If the weather is cold then wear warm clothing

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- Fuzzy logic provides a method to formalize reasoning when dealing with vague terms.
- Traditional computing requires finite precision which is not always possible in real world scenarios. Not every decision is either true or false, or as with Boolean logic either 0 or 1.
- Fuzzy logic allows for membership functions, or degrees of truthfulness and falsehoods. Or as with Boolean logic, not only 0 and 1 but all the numbers that fall in between.

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Pros:

- Conceptually easy to understand w/ “natural” maths
- Tolerant of imprecise data
- Universal approximation: can model arbitrary nonlinear functions
- Intuitive
- Based on linguistic terms
- Convenient way to express expert and common sense knowledge
Cons:

- Not a cure-all
- Crisp/precise models can be more efficient and even convenient
- Other approaches might be formally verified to work

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- The Basic Tipping Problem: Given a number between 0 and 10 that represents the quality of service at a restaurant what should the tip be?Cultural footnote: An average tip for a meal in the U.S. is 15%, which may vary depending on the quality of the service provided.

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- Tip = 15% of total bill

- What about quality of service?

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- Tip = linearly proportional to service from 5% to 25%tip = 0.20/10*service+0.05

- What about quality of the food?

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- The Extended Tipping Problem: Given a number between 0 and 10 that represents the quality of service and the quality of the food, at a restaurant, what should the tip be?How will this affect our tipping formula?

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- Tip = 0.20/20*(service+food)+0.05

- We want service to be more important than food quality. E.g., 80% for service and 20% for food.

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- Tip = servRatio*(.2/10*(service)+0.05) + servRatio = 80% (1-servRatio)*(.2/10*(food)+0.05);

- Seems too linear. Want 15% tip in general and deviation only for exceptionally good or bad service.

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if service < 3,

tip(f+1,s+1) = servRatio*(.1/3*(s)+.05) + ... (1-servRatio)*(.2/10*(f)+0.05);

elseif s < 7,

tip(f+1,s+1) = servRatio*(.15) + ...

(1-servRatio)*(.2/10*(f)+0.05);

else,

tip(f+1,s+1) = servRatio*(.1/3*(s-7)+.15) + ...

(1-servRatio)*(.2/10*(f)+0.05);

end;

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- Nice plot but
- ‘Complicated’ function
- Not easy to modify
- Not intuitive
- Many hard-coded parameters
- Not easy to understand

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What we want to express is:

- If service is poor then tip is cheap
- If service is good the tip is average
- If service is excellent then tip is generous
- If food is rancid then tip is cheap
- If food is delicious then tip is generous
or

- If service is poor or the food is rancid then tip is cheap
- If service is good then tip is average
- If service is excellent or food is delicious then tip is generous
We have just defined the rules for a fuzzy logic system.

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Decision function generated using the 3 rules.

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- Before we have a fuzzy solution we need to find out
- how to define terms such as poor, delicious, cheap, generous etc.
- how to combine terms using AND, OR and other connectives
- how to combine all the rules into one final output

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- Boolean/Crisp set A is a mapping for the elements of S to the set {0, 1}, i.e., A: S {0, 1}
- Characteristic function:
A(x) =

{

1if x is an element of set A

0if x is not an element of set A

- Fuzzy set F is a mapping for the elements of S to the interval [0, 1], i.e., F: S [0, 1]
- Characteristic function: 0 F(x) 1
- 1 means full membership, 0 means no membership and anything in between, e.g., 0.5 is called graded membership

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- fuzzy set A
- A = {(x,µA(x))| x X} where µA(x) is called the membership function for the fuzzy set A. X is referred to as the universe of discourse.
- The membership function associates each element x X with a value in the interval [0,1].

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- Fuzzy sets and concepts are commonly used in natural languageJohn is tallDan is smartAlex is happyThe class is hot
- E.g., the crisp set Tall can be defined as {x | height x > 1.8 meters}But what about a person with a height = 1.79 meters?What about 1.78 meters?…What about 1.52 meters?

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- In a fuzzy set a person with a height of 1.8 meters would be considered tall to a high degreeA person with a height of 1.7 meters would be considered tall to a lesser degree etc.
- The function can changefor basketball players,Danes, women, children etc.

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- The S-function can be used to define fuzzy sets
- S(x, a,b, c) =
- 0for xa
- 2(x-a/c-a)2for axb
- 1 – 2(x-c/c-a)2for bxc
- 1for x c

a

b

c

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- These membership functions are some of the commonly used membership functions in the fuzzy inference systems.
- Triangle(x; a, b, c) = 0 if x a;
= (x-a)/(b-a) if a x b;

= (c-x)/(c-b) if b x c;

= 0 if c x.

- Trapezoid(x; a, b, c, d) = 0 if x a;
= (x-a)/(b-a) if a x b;

= 1 if b x c;

= (d-x)/(d-c) 0 if c x d;

= 0, if d x.

- Sigmoid(x; a, c) = 1/(1 + exp[-a(x-c)]) where a controls slope at the crossover point x = c.

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- One dimensional fuzzy set can be extended to form its cylindrical extension on second dimension
- Fuzzy set A = “(x,y) is near (3,4)” is
- µA(x,y) = exp[- ((x-3)/2)2 -(y-4)2 ]
= exp[- ((x-3)/2)2 ]exp-(y-4)2 ]

=gaussian(x;3,2)gaussian(y;4,1)

- µA(x,y) = exp[- ((x-3)/2)2 -(y-4)2 ]
- This is a composite MF since it can be decomposed into two gaussian MFs

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a

b-a

b-a/2

b+a/2

b+a

a

- P(x, a, b) =
- S(x, b-a, b-a/2, b)for xb
- 1 – S(x,b, b+a/2, a+b)for x b
E.g., close (to a)

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- Piecewise linear: triangular etc.
- Easier to represent and calculate saves computation

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- A finite set of elements:F = 1/x1 + 2/x2 + … n/xn+ means (Boolean) set union
- For example:TALL = {0/1.0, 0/1.2, 0/1.4, 0.2/1.6, 0.8/1.7, 1.0/1.8}

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- Let X = {0, 1, 2, 3, 4, 5, 6} be a set of numbers of children a family may possibly have.
- fuzzy set A with “sensible number of children in a family” may be described by
- A = {(0, 0.1), (1, 0.3), (2, 0.7), (3, 1), (4, 0.7), (5, 0.3), (6, 0.1)}

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- X = R+ be the set of possible ages for human beings.
- fuzzy set B = “about 50 years old” may be expressed as
- B = {(x, µB(x)) | x ЄX}, where
- µB(x) = 1/(1 + ((x-50)/10)4)

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- A = ΣxiЄ XµA(xi)/ xi, if X is a collection of discrete objects,
- A = ∫X µA(x)/ x, if X is a continuous space.

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- EqualityA = BA (x)= B (x)for all x X
- ComplementA’A’ (x)= 1 - A(x)for all x X
- ContainmentA BA (x)B (x)for all x X
- UnionA BA B(x) = max(A (x),B (x))for all x X
- IntersectionA BA B(x) = min(A (x),B (x))for all x X

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- Crossover point of a fuzzy set A is a point x in X such that
- {(x, µA(x)) |µA(x) = 0.5 }
- α-cut of a fuzzy set A is set of all points x in X such that
- {(x, µA(x)) |µA(x) ≥ α }

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- Support(A) is set of all points x in X such that
- {(x, µA(x)) |µA(x) > 0 }

- core(A) is set of all points x in X such that
- {(x, µA(x)) |µA(x) =1 }

- Fuzzy set whose support is a single point in X with µA(x) =1 is called fuzzy singleton

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- AB = T(A(x), B(x)) where T is T-norm operator. There are some possible T-Norm operators.
- Minimum: min(a,b)=a ٨b
- Algebraic product: ab
- Bounded product: 0 ٧(a+b-1)

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- C(x) = AB = S(A(x), B(x)) where S is called S-norm operator.
- It is also called T-conorm
- Some of the T-conorm operators
- Maximum: S(a,b) = max(a,b)
- Algebraic sum: a+b-ab
- Bounded sum: = 1 ٨(a+b)

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A’

A

AB

AB

B

A

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tall

More or less tall

Very tall

- Modifying the meaning of a fuzzy set using hedges such as very, more or less, slightly, etc.
- Concentration or Con operator
- Very F = F2
- Dilation or Dil operator
- More or less F = F1/2
- more or less tall
= DIL(tall);

- extremely tall
= CON(CON(CON(tall)))

- etc.

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- A fuzzy relation for N sets is defined as an extension of the crisp relation to include the membership grade.R = {R(x1, x2, … xN)/(x1, x2, … xN) | xi X, i=1, … N}
which associates the membership grade, R , of each tuple.

- E.g. Friend = {0.9/(Manos, Nacho), 0.1/(Manos, Dan), 0.8/(Alex, Mike), 0.3/(Alex, John)}

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- A binary fuzzy relation is a fuzzy set in X × Y which maps each element in X × Y to a membership value between 0 and 1. If X and Y are two universes of discourse, then
- R = {((x,y), R(x, y)) | (x,y) ЄX × Y } is a binary fuzzy relation in X × Y.
- X × Y indicates cartesian product of X and Y

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- Fuzzy rules are useful for modeling human thinking, perception and judgment.
- A fuzzy if-then rule is of the form “If x is A then y is B” where A and B are linguistic values defined by fuzzy sets on universes of discourse X and Y, respectively.
- “x is A” is called antecedent and “y is B” is called consequent.

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- If pressure is high, then volume is small.
- If the road is slippery, then driving is dangerous.
- If the fruit is ripe, then it is soft.

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- The fuzzy rule “If x is A then y is B” may be abbreviated as A→ B and is interpreted as A × B.
- A fuzzy if then rule may be defined (Mamdani) as a binary fuzzy relation R on the product space X × Y.
- R = A→ B = A × B =∫X×YA(x) T-normB(y)/ (x,y).

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- Fuzzy logical operations
- Fuzzy rules
- Fuzzification
- Implication
- Aggregation
- Defuzzification

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- ABA and B
- 000
- 010
- 00
- 111

- ABA or B
- 000
- 011
- 01
- 111

- AND, OR, NOT, etc.
- NOT A = A’ = 1 - A(x)
- A AND B = A B = min(A (x),B (x))
- A OR B = A B = max(A (x),B (x))

From the following truth tables it is seen that fuzzy logic is a superset of Boolean logic.

min(A,B)

max(A,B)

1-A

Anot A

0 1

1 0

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- Use fuzzy sets and fuzzy operators as the subjects and verbs of fuzzy logic to form rules.
if x is A then y is B

where A and B are linguistic terms defined by fuzzy sets on the sets X and Y respectively.

This reads

if x == A then y = B

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- In Boolean logic:p qif p is true then q is true
- In fuzzy logic:p qif p is true to some degree then q is true to some degree.0.5p => 0.5q(partial premise implies partially)
- How?

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- Apply implication function to the rule
- Most common way is to use min to “chop-off” the consequent(prod can be used to scale the consequent)

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- Fuzzify inputsDetermine the degree of membership for all terms in the premise.If there is one term then this is the degree of support for the consequent.
- Apply fuzzy operatorIf there are multiple parts, apply logical operators to determine the degree of support for the rule.
- Apply implication methodUse degree of support for rule to shape output fuzzy set of the consequent.
How do we then combine several rules?

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- We aggregate the outputs into a single fuzzy set which combines their decisions.
- The input to aggregation is the list of truncated fuzzy sets and the output is a single fuzzy set for each variable.
- Aggregation rules: max, sum, etc.
- As long as it is commutative then the order of rule exec is irrelevant.

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- Given N observations Ei over X and hypothesis Hi over Y we have N rules: if E1 then H1if E2 then H2if EN then HN
- H = max[min(E1), min(E2), … min(EN)]

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Center of gravity

Center of largest area

- Take a fuzzy set and produce a single crisp number that represents the set.
- Practical when making a decision, taking an action etc.
I x

I

I=

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Tip = 16.7 %

Result of defuzzification

(centroid)

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- How to determine the membership functions? Usually requires fine-tuning of parameters
- Defuzzification can produce undesired results

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- Matlab’s Fuzzy Toolbox
- FuzzyClips
- Etc.

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- Quality Assurance
- Error Diagnostics
- Control Theory
- Pattern Recognition

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- Otis Elevators
- Vacuum Cleaners
- Hair Dryers
- Air Control in Soft Drink Production
- Noise Detection on Compact Disks
- Cranes
- Electric Razors
- Camcorders
- Television Sets
- Showers

Japan, Korea, China were the early adapters of

Fuzzy Logic into industrial applications

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- Medical Diagnosis
- Legal
- Stock Market Analysis
- Mineral Prospecting
- Weather Forecasting
- Economics
- Politics

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- Slides from CS 561 (Intro to AI) course, Sessions 20-21 of Paolo Pirjanian
- Slides from OperMgt 345 course of Mitch Pence, Boise State University
- Slides from CS623-Lec11-4Sept06, S.G.Sanjeevi of IIT Bombay.
- Fuzzy Logic. Fuzzy Logic - a powerful new technology.http://www.austinlinks.com/Fuzzy/
- FuzzyNet On-line. Automatic Transmission http://www.aptronix.com/fuzzynet/applnote/transmis.htm
- Garner, Martin. Weird Water and Fuzzy Logic: More notes of a Fringe Watcher.
- Generation 5. An Introduction to Fuzzy Logic. http://www.generation5.org/fuzzyintro.shtml
- Sowell, Thomas . FUZzy Logic For “Just Plain Folks” .
- You Fuzzin’ with me? http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/sbaa/article1.html
- Zadeh, Lotfi A. http://http.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
- Zadeh, L. (1965), "Fuzzy sets", Information and Control, 8: 338-353
- Jang J.S.R., (1997): ANFIS architecture. In: Neuro-fuzzy and Soft Computing (J.S. Jang, C.-T. Sun, E. Mizutani, Eds.), Prentice Hall, New Jersey

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The following, illustrates a basic “fuzzy” automatic transmission system. The transmission uses four fuzzy sensor inference inputs to control the best gear selection for the given conditions. The inputs are throttle, vehicle speed, engine speed and engine load.

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Labels and Membership Functions of Throttle.

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Labels and Membership Functions of vehicle speed.

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Labels and Membership Functions of engine speed.

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Labels and Membership Functions of engine load.

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Labels and Membership Functions of shift.

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Using the labels as defined in the previous slides, rules can be written for the fuzzy interface unit. The rules provide a tangible knowledge base required for the decision process and are represented as English like if-then statements.

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To create the fuzzy interface unit, rules such as the following would be developed to facilitate the automatic shifting of the vehicle.

If vehicle speed is low, variation of vehicle speed is small, slope resistance is positive large and accelerator is medium then mode is steep uphill mode.

If vehicle speed is medium, variation of vehicle speed is small, slope resistance is negative large and accelerator is small then mode is gentle downhill mode.

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If Mode is Steep uphill mode, the Shift is No. 2

If Mode is Gentle downhill mode, then Shift is No. 3

The previous slides illustrate how fuzzy logic can provide a powerful tool when addressing complex situations that are not feasible using conventional approaches. By employing fuzzy logic, we have the ability to include additional variables and rules to take into account factors that might improve the behavior of the control system.

* See reference (see notes): http://www.aptronix.com/fuzzynet/applnote/transmis.htm

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