Course Introduction

1. Course Introduction Jan Jantzen Technical University of Denmark

2. Summary • Fuzzy sets, fuzzy logic • Fuzzy clustering • Neural nets • Neuro-fuzzy modelling

3. Course Objectives • To teach the fundamental concepts • To show some applications

4. General Approach • Get plenty of good data • Design a linear model • Replace it with a nonlinear model • Did the results improve? Else repeat step 3.

5. True Love Wife: Do you love me? Husband (Boolean logician): Yes. Wife: How much?

6. Spring Summer Autumn Winter 1 0.5 Membership 0 Time of the year Seasons

7. 1 0.8 fuzzy 0.6 Membership 0.4 crisp 0.2 0 150 160 170 180 190 200 Height [cm] Tall Persons

8. Zadeh’s Challenge Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh, 1965).

9. Fuzzy (http://www.m-w.com) • Function: adjective • Inflected Form(s): fuzz·i·er; -est • Etymology: perhaps from Low German fussig loose, spongy • Date: 1713 • 1 : marked by or giving a suggestion of fuzz <a fuzzy covering of felt> • 2 : lacking in clarity or definition <moving the camera causes fuzzy photos> • - fuzz·i·ly /'f&-z&-lE/ adverb • - fuzz·i·ness /'f&-zE-n&s/ noun

10. Fuzzy (http://www.m-w.com) • Function: adjective • Synonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague

11. 1 1 Not very young 0.8 0.8 More or less old 0.6 0.6 Young Membership Membership Old 0.4 0.4 Very young 0.2 0.2 0 0 0 50 100 0 50 100 Age [years] Age [years] Age

12. Logic Wife: Do you like my girlfriend? Husband: Very much. Wife: Then you don’t love me.

13. 1 Fuzzy 0.5 Truth Crisp 0 10 15 20 25 30 Temperature [deg C] A Warm Room

14. Fuzzy Logic Control Fuzzy logic control (FLC) may be viewed as a branch of intelligent control which serves as an emulator of human decision-making behaviour that is approximate rather than exact (C.C.Lee in Singh: Systems and Control Encyclopedia, 1992).

15. Rule Format Ri: if x is Ai and y is Bi then z is Ci

16. Implication • IF room is warm • THEN set cooling at 500 watts

17. Inference • If room is warm then set cooling at 500 watts • Temperature is 21 deg C • Set cooling at 250 watts

18. Sets {Live dinosaurs in British Museum} = 

19. Fuzzy Sets {nice days} {adults}

20. A B A B A B a) b) c) A B A B A B d) e) f) Set Operations

21. Q: Why Logic? A: Math proofs, computers Example: If either the Pirates or the Cubs loose and the Giants win, then the Dodgers will be out of first place, and I will loose a bet. ((p  c)  g)  (d  b)

22. Boolean OR

23. Fuzzy OR

24. Tautologies • [p (p  q)]  q • [(p  q) (q  r)]  (p  r) • [p (p  q)]  p  q

25. A: Tolerant of imprecision Q: Why fuzzy logic?

26. Papers with fuzzy in title (INSPEC+Math Reviews)

28. m F 0 y Example: Stopping a car

29. PD Control

30. Rule base • If distance is long and approach is fast, then brake zero • If distance is long and approach is slow, then brake zero • If distance is short and approach is fast, then brake hard • If distance is short and approach is slow, then brake zero

31. 0 -10 PID fuzzy -20 Position [m] 0 1 2 3 4 5 Time [s] 4 x 10 0 -1 Control [N] -2 0 1 2 3 4 5 Time [s] Response

33. Fuzzy Clustering • Find clusters in data • Extract rules from data • E.g., bank customer segmentation, diagnosing cancer cells

36. Cluster analysis (www.m-w.com) A statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics.

37. Vehicle Example

38. 3500 Lorries 3000 2500 Sports cars 2000 Weight [kg] 1500 Medium market cars 1000 500 100 150 200 250 300 Top speed [km/h] Vehicle Clusters

39. Example: Diagnose Cancer Cells Normal smear Severely dysplastic smear Using a small brush, cotton stick, or wooden stick, a specimen is taken from the uterin cervix and smeared onto a thin, rectangular glass plate, a slide. The purpose of the smear screening is to diagnose pre-malignant cell changes before they progress to cancer. The smear is stained using the Papanicolau method, hence the name Pap smear. Different characteristics have different colours, easy to distinguish in a microscope. A cyto-technician performs the screening in a microscope. It is time consuming and prone to error, as each slide may contain up to 300.000 cells. Dysplastic cells have undergone precancerous changes. They generally have longer and darker nuclei, and they have a tendency to cling together in large clusters. Mildly dysplastic cels have enlarged and bright nuclei. Moderately dysplastic cells have larger and darker nuclei. Severely dysplastic cells have large, dark, and often oddly shaped nuclei. The cytoplasm is dark, and it is relatively small.

40. The Perceptron • Classification • Learning

41. d + - Compare e Modifier M y u Neural Network How Use A Neural Network? • Classification or approximation ? • Training data • Examples and epoch • Pattern or batch mode ? • Test data

42. Hard limiter 1 1 0 f(x) w0 -1 w1 f(x) + w2 (a) (b) -2 0 2 x Perceptron

43. y1 3 1 u1 y2 4 2 u2 y3 5 Single Layer Perceptron y1 = sgn(w1Tu), y2 = sgn(w2Tu), y3 = sgn(w3Tu)

44. Input layer Hidden layer Output layer Multilayer Perceptron

45. Case: Sunspot Cycles

46. 100 Pos W1 u + e 0 + / + Zero W2 + + + -100 Neg W3 Input layer Hidden layer Output layer Fuzzy Rules As A NN

47. Initial MFs Final MFs 1 1 0.9 0.8 0.8 0.6 0.7 0.6 0.4 0.5 0.2 0.4 0 0.3 0 5 10 15 20 0 5 10 15 20 MFs Before And After Learning

48. A1 u1 u1,u2 AND N A2 + y B1 AND N u2 u1,u2 B2 Layer 1 2 3 4 5 ANFIS net

49. Summary • Fuzzy sets, fuzzy logic • Fuzzy clustering • Neural nets • Neuro-fuzzy modelling

50. Problems Attacked • Nonlinear • Multivariable • Operator’s rules • Learning