What is all the Fuzz about? Engineering Fuzzy Systems CPSC 433 - PDF document

Fuzzy Systems in Knowledge What is all the Fuzz about? Engineering Fuzzy Systems CPSC 433 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Fuzzy Systems What does Fuzzy Logic mean? • Fuzzy logic was introduced by Lot � Zadeh 1. Motivation � UC Berkeley � in 1965. 2. Fuzzy Sets • Fuzzy logic is based on fuzzy set theory, an 3. Fuzzy Numbers extension of classical set theory. • Fuzzy logic attempts to formalize 4. Fuzzy Sets and Fuzzy Rules � approximate � knowledge and reasoning. 5. Extracting Fuzzy Models from Data • Fuzzy logic did not attract any attention until 6. Examples of Fuzzy Systems the 1980s � fuzzy controller applications � . Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Fuzzy is Just Human Fuzzy is Economical • Humans primarily use fuzzy terms: large, sma � , • Zadeh: � Make use of the leeway of fuzziness. � fast, slow, warm, cold, ... • Fuzziness as a principle of economics: • W e say: • Precision is expensive. � If the weather is nice and I have a little time, I will probably go for a hike along the Bow. � • Only apply as much precision to a problem as necessary. • W e don � t say: • Example: Backing into a parking space � If the temperature is above 24 degrees and the cloud cover is less than 10 � , and I have 3 hours time, I will � How long would it take if we had to park a car with a go for a hike with a probability of 0.47. � precision of ±2 mm? Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction

Fuzzy Systems Basics of Fuzzy Sets 1. Motivation • Example: the set of � young people � 2. Fuzzy Sets young = { x ∈ P | age ( x ) ≤ 20 } 3. Fuzzy Numbers • W e can de � ne a characteristic function for this set: 4. Fuzzy Sets and Fuzzy Rules � 1 : age ( x ) ≤ 20 µ young ( x ) = 5. Extracting Fuzzy Models from Data 0 : 20 < age ( x ) 6. Examples of Fuzzy Systems Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Basics of Fuzzy Sets Fuzzy Membership Function • Fuzzy set theory o � ers a variable notion of membership : • A person of age 25 could still belong to the set of young people, but only to a degree of less than one, say 0.9.  1 : age ( x ) ≤ 20  1 − age ( x ) − 20 µ young ( x ) = : 20 < age ( x ) ≤ 30 10  1 : age ( x ) ≤ 20  0 : 30 < age ( x )  1 − age ( x ) − 20 µ young ( x ) = : 20 < age ( x ) ≤ 30 10 • Now the set of young contains people with ages  0 : 30 < age ( x ) between 20 and 30, with a linearly decreasing degree of membership. Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Shapes for Membership Fcts. Parameters of FMFs • Support : s A := � x : � A � x � > 0 � • The area where the membership function is positive. Triangle: [a,b,c] Trapezoid: [a,b,c,d] • Core : c A := � x : � A � x � = 1 � • The area for which elements have a maximum degree of membership to the fuzzy set A. Singleton: [a,m] Gaussian: [a, � ] Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction

Parameters of FMFs Support: s A := � x : � A � x � > 0 � • � � Cut : � A := � x : � A � x � = � � • The cut through the membership function of A at Triangle: [a,b,c] Trapezoid: [a,b,c,d] height a. • Height : h A := max x � � A � x � � • The maximum value of the membership function of Singleton: [a,m] A. Gaussian: [a, � ] Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction � � Cut: � A := � x : � A � x � = � � Core: c A := � x : � A � x � = 1 � � � Triangle: [a,b,c] Trapezoid: [a,b,c,d] Triangle: [a,b,c] Trapezoid: [a,b,c,d] m=1 � � Singleton: [a,m] Singleton: [a,m] Gaussian: [a, � ] Gaussian: [a, � ] Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Height: h A := max x � � A � x � � Linguistic V ariables Height = 1 Height = 1 • The covering of a variable domain with several fuzzy sets, together with a corresponding semantics, de � nes a linguistic variable . Triangle: [a,b,c] Trapezoid: [a,b,c,d] • Example: linguistic variable ag � Height = 1 Height = m Singleton: [a,m] Gaussian: [a, � ] Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction

Granulation Fuzzy Granules • Granulation results in a grouping of objects • Using fuzzy sets, we can incorporate the fact into imprecise clusters of fuzzy granules . that no sharp boundaries between � groups � , such as young , middle � aged , and old , exist. • The objects forming a granule are drawn together by similarity. • The corresponding membership functions overlap in certain areas, forming non � crisp • This can be seen as a form of fuzzy data � fuzzy � boundaries. compression. • This compositional way of de � ning fuzzy sets over a domain of a variable is called granulation . Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Finding Fuzzy Granules Fuzzy Systems • If expert knowledge on a domain is not 1. Motivation available, an automatic granulation is used. 2. Fuzzy Sets 3. Fuzzy Numbers 4. Fuzzy Sets and Fuzzy Rules • Standard granulation using an odd number of membership functions: 5. Extracting Fuzzy Models from Data • NL: negative large, NM: negative medium, NS: 6. Examples of Fuzzy Systems negative small, Z: zero, ... Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction Fuzzy vs. Crisp Numbers Fuzzy Numbers as Fuzzy Sets • Fuzzy numbers are a special type of fuzzy sets • Real � world measurements are always with speci � c membership functions: imprecise. • � A must be normalized � c A � ∅ � . • Usually, such imprecise measurements are modeled through • � A must be singular . There is precisely one point which lies inside the core, modeling the typical value • a crisp number x , denoting the most typical value, � = modal value � of the fuzzy number. • together with an interval, describing the amount of • � A must be monotonically increasing left of the core imprecision. and monotonically decreasing on the right � only one • In a linguistic sense: � about x � peak! � . Christian Jacob, University of Calgary Christian Jacob, University of Calgary CPSC 433 � Arti � cial Intelligence: An Introduction CPSC 433 � Arti � cial Intelligence: An Introduction