1. MAT120 Asst. Prof. Dr. Ferhat PAKDAMAR (Civil Engineer) M Blok - M106 pakdamar@gtu.edu.tr Gebze Technical University Department of Architecture Spring – 2014/2015 Week 2

2. Subjects WeekSubjectsMethods 111.02.2015Introduction 218.02.2015 Set Theory and Fuzzy Logic.Term Paper 325.02.2015 Real Numbers, Complex numbers, Coordinate Systems. 404.03.2015 Functions, Linear equations 511.03.2015 Matrices 618.03.2015Matrice operations 725.03.2015MIDTERM EXAM MT 801.04.2015 Limit. Derivatives, Basic derivative rules 908.04.2015 Term Paper presentationsDead line for TP 1015.04.2015 Integration by parts, 1122.04.2015 Area and volume Integrals 1229.04.2015 Introduction to Numeric Analysis 1306.05.2015 Introduction to Statistics. 1413.05.2015Review 15 Review 16 FINAL EXAM FINAL

3. LOGIC «the study of the principles of correct reasoning» «the study of the principles of reasoning» «the branch of philosophy that analyzes inference» «logic is a branch of philosophy» …..

4. Conditional Statements Different kinds of reasoning in Geometry? kinds of reasoning 1.intuition reasoning 2.analogy Analogy is a cognitive process of transferring information or meaning from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. 3.inductive reasoning 4.deductive reasoning 5.abstract reasoning

5. Logical Set Properties The term "Boolean algebra" honors George Boole (1815– 1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.

6. INTRODUCTION TO FUZZY LOGIC

7. What Is Fuzzy Logic?

8. Why do we use FL?

9.  Fuzzy logic is conceptually easy to understand.  Fuzzy logic is flexible.  Fuzzy logic is tolerant of imprecise data.  Fuzzy logic can model nonlinear functions of arbitrary complexity.  Fuzzy logic can be built on top of the experience of experts.  Fuzzy logic can be blended with conventional control techniques.  Fuzzy logic is based on natural language.

10. When not to use FL?  Fuzzy logic is not a cure-all. Fuzzy logic is a convenient way to map an input space to an output space. If you find it’s not convenient, try something else. If a simpler solution already exists, use it. Fuzzy logic is the codification of common sense—use common sense when you implement it and you will probably make the right decision. Many controllers, for example, do a fine job without using fuzzy logic. However, if you take the time to become familiar with fuzzy logic, you’ll see it can be a very powerful tool for dealing quickly and efficiently with imprecision and nonlinearity.

11. Fuzzy Sets  Set is a group, that could form by the elements of living or non-living creatures or concepts. Elemanları belirlenebilen canlı ve cansız varlıklar, yada kavramların oluşturduğu topluluğa denir.  To say a group is set:  Each element of set has to be defined clearly  Each element has to be different  Each element has to named by a capital letter In other words, as the description or perspectiveness of Crisp Set. Either “an apple is an element of the set 100% or not. (Primary School 5)

12. E A E Classic (Crisp) SetFuzzy Set Contradiction Excluded Middle   

13. 1.80 Membership Functions

14. Shapes of Membership Functions If the function was changed from triangle to any type, then set should been train.

16. Let us assume that, this is the Height set  As a result of using more subsets. The intuitional expressions that can not be explain in Crisp Sets, such as more, much, very, very much, extremely, a little, a few....etc. can be add and explain in Fuzzy Sets.  The subsets which will be releated to each other with AND or OR must be intersect  At least one subset must be intersect in a set tall short 1.50 1.70 1.80 1.90 2.10 Height very tall avarage 1.0 0.0 Membership Degree

17. H µ Crisp Heat Set and Subsets -Edges are crisp ! Fuzzy Heat Set and Subsets -Edges are not crisp !!! Cold 1.0 CoolWarm Hot 0.025 50 H µ ~ 1.0 Cold CoolWarm Hot ~ §If the set would be related to any set then Membership degree of at least one element, in the subsets of sets must be “1.0”.

18. Logical Operators AND, OR, NOT (min, max, 1-A)

19. Set demonstration of operators

20. c-means or k-means classification medium 1.0 0.0 R Q low high medium small very high big very big

21. If-Then Rules  Fuzzy sets and fuzzy operators are the subjects and verbs of fuzzy logic.  if x is A then y is B  The if-part of the rule “x is A” is called antecedent, while the then-part of the rule “y is B” is called consequent.  if service is good then tip is average

22.  The ancedent of a rule can have multiple parts :  IF sky is gray AND wind is strong THEN...  IF the distance between the cars is short AND speed is slow THEN hold the gas pedal steady.  The consequent of a rule can also have multiple parts :  IF temperature is cold THEN hot water valve is open AND cold water valve is shut.

24.  Fuzzy Inputs  All fuzzy statements in the antecedent are in a degree of membership between 0 and 1.  Applying Fuzzy Operator  If there are multiple parts to the antecedent, apply fuzzy logic operators and resolve the antecedent to a single number between 0 and 1.  Applying Implication Method  Use the degree of support for the entire rule to shape the output fuzzy set.

25. If I have datas but I don’t know the rules ! a lot 1.0 0.0 Car Carbonmonoksit afew some a little some a lot I VI V IV III II IX VIII VII

26. Fuzzy Inference Systems  Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. The process of fuzzy interference : Membership functions Fuzzy logic operators if-then rules

27. Fuzzy Inference System (FIS)

28. Fuzzy Inference Systems  RULE BASE, If-then rules are kept in rule base.  Rule base can be constructed using one of the following methods:  An expert’s experience  Modelling a human operator actions  A fuzzy model of the process  Learning  Copying an existing controller

29. Fuzzy Inference Systems  Issues in constructing the rule-base:  Number of rules;  number of inputs : n  number of fuzzy sets for each input : m  number of rules: m n  Completeness; A proper control action should be inferred for every input.  Consistency;

30. Fuzzy Inference Systems  FUZZIFICATION  The fuzzy sets are created for every crisp input  INFERENCE ENGINE  Output sets for each rule are aggregated into a single output fuzzy set.

31. Fuzzy Inference Systems  DEFUZZIFICATION  The resulting set is defuzzified, or resolved into a single number.  Some of the defuzzification methods:  Center of area/gravity  Center of sums  Center of Largest area  First of Maxima  Middle of maxima  Height

32. Fuzzy Inference Systems  Different Types of Fuzzy Systems  Mandani Fuzzy Inferencing System  finds the centroid of two dimensional function.  The Tsukamata Fuzzy Model  The Sugeno Model  We use weighted average of a few data points.  Used for the output membership functions are either linear or constant.

33. Fuzzy Inference Systems  MANDANI-TYPE INFERENCE  Mandani-type inference is the most commonly seen methodology.It was among the first control systems built using fuzzy theory.  Expects the output functions to be fuzzy sets.  After the aggregation process, there is a fuzzy set for each output variable that needs defuzzification.  Finds the centroid of a two dimensional function.

34. An example

35.  The first step is to take the inputs & determine the degree of membership in the qualifying lingusitic set (always the interval between 0 and 1 )  The example is built on three rules with different fuzzy linguistic sets :  service ---> poor, good, excellent  food ---> rancid, delicious  tip ---> cheap, average, generous.

36. Fuzzify Inputs

37.  Applying Fuzzy Operator  We use AND, OR operators.

39. Result of defuzzification

41. Fuzzy Inference Systems  Fuzzy inference systems have been successfully applied in fields such as:  Automatic control  Data classification  Decision analysis  Expert systems  Computer vision

42. A few Applications  Definition of the problem  General window  Rule editor  Rule viewer

43. -Shower -Tipper -Surprise sample!

44. Conclusion  We can use fuzzy to modelling all real LIFE easily  If we have data, then we do not need expert. If we have expert, then we do not need data. But if we have both of them, then better.  We can work with WORDS in Fuzzy.  Computers do not need to work a lot of time with Fuzzy.

45. References -Fuzzy Logic with Engineering Applications, Timothy J.Ross, McGraw-Hill, Inc. -Bulanık Mantık ve Modelleme İlkeleri, Zekai Şen, Bilge Kültür Sanar 2001. -A Course in Fuzzy Systems and Control, Li-Xin Wang, Prentice-Hall Int., Inc. -Mühendislikte Modern Yöntemler Sempozyumu Bildiri Kitabı, İTÜ 2001. -Fuzzy Logic with Engineering Applications Ders notları, Zekai Şen -www.fuzzytech.com -www.bumat.com.tr -www.yapayzeka.org -www.ieee.org

46. Have a nice week!