FS Part 1 : Uncertainty, Intervals and FS. André Bigand, LISIC, ULCO Université Lille-Nord de France Web:www-lisic.univ-littoral.

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1 FS Part 1 : Uncertainty, Intervals and FS André Bigand, LISIC, ULCO Université Lille-Nord de France Web:www-lisic.univ-littoral.fr\~bigand\ EILCO 18/01/2016

2 Introduction to Fuzzy Set Theory OBJECTIVES 1. To introduce fuzzy sets and how they are used 2. To define some types of uncertainty and study what methods are used to with each of the types. 3. To define fuzzy numbers, fuzzy logic and how they are used 4. To study methods of how fuzzy sets can be constructed 5. To see how fuzzy set theory is used and applied in research (LISIC) August 12, 2003 I. INTRODUCTION: Math Clinic Fall

3 Outline Uncertainty and mathematics Uncertainty and metrology What are Fss? Expert systems with FS Applications

4 Uncertainty, uncertainties? In order to deal with indeterminacy mathematically, two axiomatic systems have been founded, namely, probability theory and uncertainty theory. When no samples are available to estimate a probability distribution, we have to invite some domain experts to evaluate the belief degree that each event will happen. In order to rationally deal with personal belief degrees, uncertainty theory was founded in 2007 and subsequently studied by many researchers. Nowadays, uncertainty theory has become a branch of mathematics (Zadeh, 1965,...).

5 Uncertainty and Metrology A measure is the realization of a random variable x Type A uncertainty : evaluation method given by a statistical analysis of measures (More than 30 observations, Gaussian law, χ² law) Ex. : x µ t Type B uncertainty : evaluation method given by mean of others methods ( Less than 5 observations) Ex. : Sensor Accuracy ( Is there a link between μ and the true value of measure) Precision interval : x ± 10%

6 Intervals and Sets Crisp set : A = {x1, x2, x3,..., xn} Fa : X {0, 1} Charateristics function : X μ Y X Interval computations are particular case of Robust Statistics. Union and Intersection operators : Min and Max connectives.

7 Intervals and Fuzzy Sets (FSs)

8 Intervals and FSs Fuzzy sets are sets that have gradations of belonging EXAMPLES: Green BIG Near Classical sets, either an element belongs or it does not EXAMPLES: Set of integers a real number is an integer or not You are either in an airplane or not Your bank account is x dollars and y cents

9 Fuzzy sets, FSs Fuzzy set : A = {(x, μa(x)) x ϵ X} Fa : X [0, 1] X Membership function : μ X

10 Fss and Intervals

11 FSs and Intervals The relation between fuzzy and interval techniques is well known; e.g., due to the fact that a fuzzy number can be represented as a nested family of intervals (alpha-cuts), level-bylevel interval techniques are often used to process fuzzy data. At present, researchers in fuzzy data processing mainly used interval techniques originally designed for non-fuzzy applications, techniques which are often taken from textbooks and are, therefore, already outperformed by more recent and more efficient methods. One of the main objectives of the proposed special session is to make the fuzzy community at-large better acquainted with the latest, most efficient interval techniques, especially with techniques specifically developed for solving fuzzy-related problems. Another objective is to combine fuzzy and interval techniques, so that we will be able to use the combined techniques in (frequent) practical situations where both types of uncertainty are present: for example, when some quantities are known with interval uncertainty (e.g., coming from measurements), while other quantities are known with fuzzy uncertainty (coming from expert estimates).

12 Membership Functions MF choice?

13 FS and Uncertainty

14 FS and Uncertainty Vague or Imprecise knowledge : Imprecision with FS (Valued knowledge) Ex. : Image's pixel Linguistic variable (small, big, high, ) Incompletness

15 Basic Information : α-cuts,... FS= (α-cuts) : kernel α Support

16 Basic Information : l-levels Utilizing alpha-levels to obtain fuzzy polynomials, we have: ép ù º éê pa- ( x), pa+ ( x)ùú = ( x ) ë ûa ë û {zîr : z = p (x), d Î [ z ] } d z2+a z1+a z2-a z1-a x1 x x2 i i a

17 Fss Operators Basic Operators : Union and Intersections Other operators : t-norms (T) and t-conorms (optimistic and pessimistic decision)

18 Linguistic Information FS : fuzzy numbers,. FS : linguistic variables (small, medium,, A1, A2) : grammar Rule If x is small Then u is small A1 A2 U1 U2 μ = 1 X U

19 Inference Machine (FIS) Rules : If x is Ai then u is Ui (i є I) Representation : X F(L(X)) F(U) U Inference : μ((ui) = sup(iєi)(μ(ai) T μ(ai,u))) U1 A1 A2 x U2

20 Rules? Rules obtained form experts (examples) Rules automatically obtained (control, PI)...

21 First Order Logic Variables x, y, z,. Constants a, b, c,. Functional symbols f, g,. Logical connectives (conjonction), (disjonction), (negation), (implication) Quantifiers Ɐ, ⱻ,.

22 Example : Tipper Tipper.fis ;

23 Tipper : FIS Ex.Matlab :

24 Applications Expert systems Control Image processing Regression, classification Task planification under uncertainty Data base query...

25 Application to smart cars (security)

26 Application to smart cars (gradual) System :

27 Application to smart cars (comfort) Stereo :

28 Application to smarter homes:gradual information Why comfort?

29 Rules for PI (Why fuzzy PI?) ε έ PI u PI + saturation

30 Rules for PI ε derε -L -M -S -O +O +S +M +L +L +O +S +M +M +M +L +L +L +M -S -O +S +M +M +M +L +L +S -M -S -O +S +S +S +M +L +O -L -M -S +O +O +S +M +L -O -L -M -S -O -O +S +M +L -S -L -M -S -S -S +O +S +M -M -L -L -M -M -M -S +O +O

31 Image Processing From Image to Fss : an image with L gray levels can be associated to FSs regarding a predefined image property (brightness, homogeneity, noisiness, edginess,...) g1 g2 g3 μ1 μ2 μ3

32 Image processing Dark Gray Bright

33 Image Processing (for an image MxN) Fuzzy geometry : area (Σ Σ μ(m,n)), Measure of fuzziness :γ = (2/MN) Σ min(μ(m,n),1-μ(m,n)) Fuzzy entropy : H= (1/MNln2) Σ S(m,n) with S(μ(m,n)= -μ.ln(μ) (1-μ).ln(1 - μ)

34 FS Generation Example (figures from Cidalia Fonte & Lodwick) f(z) 100% 80% 60% 40% 20% 0% altitude z The membership function of points to the fuzzy set is given by: 1982 meters above the 20m level f z x, y x, y T 100

35 Fuzzy interpolation z x

36 Fuzzy classification (fcm) y x

37 Fuzzy grammars : similarity Similarity (between patterns) : sim(a,b) = Argmax(min(d(A,B))) d(a,b)? (distance, fuzzy integral, ) Entropy

38 Conclusion Numerous applications : Control Vision Data base query Scheduling.. Drawback : Imprecise and Vague data modelization But Noise (random data)?

39 Fuzzy Sets (FSs) First Order FS : Many Applications (..Bloch.. ) : * Imprecision : Interval * Image processing (pixel imprecision) : (10V +/- 1%) Filtering, clustering, * FS modelization : Interval valued segmentation... Randomness?

40 Type-2 FSs * Imprecise FS : * Applications : All the aplications of FS But with a second degree of freedom to modelize uncertainty (Image processing, classification,.)

41 Biblio Concernant la presentation de logique floue qq ref. biblio: * le mieux: Driankov, An introduction to fuzzy control, Springer * En francais: H. Buhler, Reglage par logique floue, Presses polytech. et univ. romandes. * L. Foulloy et coll.: Commande floue 1 et 2, Hermes Lavoisier (en francais, + pointu) * B. Bouchon-Meunier, L. Foulloy, M. Ramdani: Logique floue, exercices corriges (Cepadues Ed.) * P. Borne et coll.: Commande floue (collection automatique, etudes de cas), Ed. Technip