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Introduction To Fuzzy Logic Dr. Emad A. El-Sebakhy Room: 22-316, Phone: 860-4263 Email: dodi05@ccse@kfupm.edu.sa What Is Fuzzy Logic ? Fuzzy logic is a powerful problem-solving methodology with a lot of applications in embedded control and information processing . Fuzzy provides a remarkably simple way to draw definite conclusions from vague, ambiguous or imprecise information. In a sense, fuzzy logic resembles human decision making with its ability to work from approximate data and find precise solutions. What Is Fuzzy Logic ? Unlike classical logic which requires a deep understanding of a system, exact equations, and precise numeric values. Fuzzy logic incorporates an alternative way of thinking, which allows modeling complex systems using a higher level of abstraction originating from our knowledge and experience. Fuzzy Logic allows expressing this knowledge with subjective concepts such as very hot, bright red, and a long time which are mapped into exact numeric ranges. What Is Fuzzy Logic ? Fuzzy Logic has been gaining increasing acceptance during the past few years. There are over two thousand commercially available products using Fuzzy Logic, ranging from washing machines to high speed trains. Nearly every application can potentially realize some of the benefits of Fuzzy Logic, such as performance, simplicity, lower cost, and productivity. Fuzziness vs. Probability 1. Fuzziness is deterministic uncertainty – probability is nondeterministic. 2. Fuzziness describes event ambiguity – probability describes event occurrence. Whether an event occurs is random. The degree to which it occurs is fuzzy. 3. Probabilistic uncertainty dissipates with increasing number of occurrences fuzziness does not. Why Use Fuzzy Logic ? An Alternative Design Methodology Which Is Simpler, And Faster Fuzzy Logic reduces the design development cycle Fuzzy Logic simplifies design complexity Fuzzy Logic improves time to market A Better Alternative Solution To Non-Linear Control Fuzzy Logic improves control performance Fuzzy Logic simplifies implementation Fuzzy Logic reduces hardware costs Some Historical Developments Fuzzy systems have been around since the 1920s, when they where first proposed by Lukaciewicz. He proposed to modify traditional false (0) true (1) reasoning to include some truth such as 0.5. Since then the approach has been further develop and systems have incorporated the methodologies: 1965 - Fuzzy Set Theory (Prof. Lofti A. Zadeh, U. of Cal. at Berkely) 1966 - Fuzzy Logic (Dr. Peter N. Morinos, Bell Labs.) 1972 - Fuzzy measure (Prof. Michio Sugeno, TIT) 1974 - Fuzzy logic controller (Prof. E.H. Mamdani, London Univ.) 1980 - Control of cement0kiln with monitor capability (Denmark) 1987 - Automatic train operation for Sendai Subway (Hitachi, Japan) 1988 - Stock Trading Expert System (Yamaichi Security, Japan). What is Fuzziness? Fuzziness is deterministic uncertainty. It is concerned with the degree to which events occurrence rather than the likelihood of their occurrence. For example, the degree to which a person is tall is a fuzzy event rather than a random event. Fuzzy Logic vs Boolean Logic Two fundamental assumptions of traditional logic: An element belongs to a set or its complement An element cannot belong to both a set and its complement -- law of excluded middle Traditional logic is crisp. Fuzzy logic violates both the above assumptions. Fuzzy Logic generalizes Boolean Logic It is very useful that the Boolean Logic is included in the Fuzzy Logic If we think of x and y as “crisp” values 0 and 1, Fuzzy logic gets back to Boolean logic. Truth values Boolean logic Fuzzy logic X Y X AND Y X OR Y MIN(X, Y) MAX(X, Y) 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 0.2 0.7 - - 0.2 0.7 1 0.4 - - 0.4 1 Fuzzy Vs. Crisp Values A crisp value is a precise number that represents the exact status of the associated phenomenon. Example: The fastest land mammal, cheetahs can accelerate from 0 to 70 mph in 3 seconds. A Lamborghini Diablo sports car accelerates from 0 to 62 mph in 4 seconds! Of course, you can use the car to go on an appointment. A fuzzy value is an ambiguous term that characterizes an imprecise or not very well understood phenomenon. Example: Cheetahs can run very fast. Fuzzy Set Theory provides the means for representing uncertainty using set theory. Example: Any velocity between 45 and 70 mph is a very fast velocity. Any velocity between 25 and 57 mph is a fast velocity. Fuzzy Sets Sets with fuzzy boundaries A = Set of tall people Crisp set A Fuzzy set A 1.0 1.0 .9 .5 Membership function 5’10’’ Heights 5’10’’ 6’2’’ Heights Fuzzy Sets Fuzzy sets and set membership is the key to decision making when faced with uncertainty (Zadeh, 1965). Classical sets contain objects that satisfy precise properties. Fuzzy sets contain objects that satisfy imprecise properties of membership (membership of an object can be approximate). Example: The set of heights from 5 to 7 feet is crisp (classical set); The set of heights in the region around 6 feet is fuzzy. A fuzzy set can be defined as a set of crisp values that can be group together with an associated fuzzy term. Example: A person between 5 ft and 6 ft belongs to the set of tall people. NOTE: Because a crisp value can belong to a fuzzy set in one context and to another set in a different context a crisp value can be associated to more than one fuzzy set. For example, the set of tall people can overlap with the set of non-tall people (an impossibility in the world of binary logic). Fuzzy Sets Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: A {( x, A ( x ))| x X } Membership Universe or Fuzzy set function universe of discourse (MF) A fuzzy set is totally characterized by a membership function (MF). Types of fuzzy sets There are basically two types of fuzzy sets: normal and subnormal. (x) (x) A 1 1 Height of the A Height of the fuzzy set fuzzy set 0 x 0 x Normal fuzzy set Subnormal fuzzy set A normal fuzzy set is one whose A subnormal fuzzy set is one whose membership function has at least membership function doesn’t have one element x in the universe an element x in the universe whose membership is unity. whose membership is unity. Fuzzy Sets with Discrete Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = “sensible number of children” X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)} Fuzzy Sets with Cont. Universes Fuzzy set B = “about 50 years old” X = Set of positive real numbers (continuous) B = {(x, B(x)) | x in X} 1 B(x) x 50 2 1 10 How to represent fuzzy sets There are two common ways to represent fuzzy sets depending if the set is discrete or continuous. Discrete fuzzy sets: A notation convention for fuzzy sets when the universe of disclosure, X, is discrete and finite, is as follows for a fuzzy set A: The is not for algebraic summation but rather A = {A(x1)/x1 + A(x2)/x2 + … } = {i [ A(xi)/xi ]} denotes the collection or aggregation of each element; hence the “+” signs are not algebraic add but are a Continuous fuzzy sets: function-theoretic union. When the universe, X, is continuous and finite, the fuzzy set A is denoted by: In both notations: The sign is not an •Not a quotient bar but a delimiter algebraic integral but a •The numerator in each term is the continuous function- A theoretic union notation = { [ A(xi)/xi } membership value in set A associated with the element of the universe for continuous variables. indicated in the denominator. Alternative Notation A fuzzy set A can be alternatively denoted as follows: X is discrete A xi X A ( xi ) / xi X is continuous A A( x) / x X Note that and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division. Fuzzy Partition Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”: lingmf.m Fuzzy Set Operations Standard Complement of a fuzzy set A: A(x) = 1- A(x) Elements for which A(x) = A(x) are called equilibrium points. Standard intersection AB: = min[A(x),B(x)] Standard union AB: (A B)(x) = max[A(x),B(x)] 1. Complement (or negation) = 1- 2. Union (logical or): Largest (maximum) membership functions. Union of A and B : A B Set A Set B Note that the membership degree is not the same as a probability although the values that it may 3. Intersection (and): Smallest (minimum) membership is not 50 / 50 take are the same (0 to 1). The chance of a 25 year old being young or not young functions. - rather the degree to which a 25 year old exemplifies young is about half (0.50). Set A Intersection of A and B : A B Set operators can be defined on fuzzy sets similarly to those on crisp sets. Set B Set A Complement of Set A : Not A Operations on Fuzzy sets If we define 3 fuzzy sets, A, B and C on the universe X, for a given element x of the universe the following are examples of operations defined on A, B and C. Some operations on fuzzy sets: Union AB(x) = A(x) \/ B(x) Intersection AB(x) = A(x) /\ B(x) Complement Ā(x) = 1 - A(x) etc. A 1 1 A 1 0.7 A B B 0.3 0.3 0 14 x 0 10 14 15 x 0 x 5 10 15 17 Union Intersection Complement The fuzzy sets A and B may be represented as: The resulting fuzzy set may be represented as: A = { 0/5 + 1/10 + 0.3/14 + 0/15 } AB = { 0/10 + 0.3/14 + 0/15 } B = { 0/10 + 1/14 + 0.7/15 + 0/17 } How would AB be represented? Membership functions All information contained in a fuzzy set is described by its membership function. core 1 Membership functions can be symmetrical or asymmetrical. 0 x boundary boundary support The Core of a membership function is defined as the region of the universe that is characterized by complete and full membership in the set. The core comprises those elements for which, (x) = 1 The support is defined as that region of the universe that is characterized by nonzero membership in the set: (x) > 0 The boundaries are defined as that region of the universe containing elements that have nonzero membership but not complete membership: 0 < (x) < 1 Membership when using fuzzy sets For crisp sets an element x in the universe X is either a member of some crisp set, say A on the universe. This binary issue of membership can be represented mathematically as: 1, A x A(x) ={ 0, A where the symbol x A(x) gives the indication of an unambiguous membership of element x in set A. Fuzzy membership extends the notion of binary membership to accommodate various “degrees of membership” on the real continuous interval [0, 1], where the endpoints conform to no membership and full membership, respectively. The sets on the universe X that can accommodate “degrees of membership” are referred as “fuzzy sets”. More Definitions MF Terminology Support MF Core Normality 1 Crossover points .5 Fuzzy singleton a a-cut, strong a-cut 0 Core X Convexity Crossover points Fuzzy numbers a - cut Bandwidth Support Symmetricity Open left or right, closed The Common Membership Functions Each fuzzy set has a membership function. These are normally trapezoidal, triangular or Gaussian (normal). These are usually normalized, that is, have a maximum value of 1. In the picture above, the fuzzy set young is described with a trapezoidal membership function. All ages less than 20 have full membership (=1) and all ages greater than 30 have no membership (=0). In between is a linear relationship between age and membership. Age 25 has =0.5. Set-Theoretic Operations fuzsetop.m subset.m Common MF Formulation x a c x trimf ( x ;a,b ,c ) max min , , 0 b a c b Triangular MF: x a d x trapmf ( x ;a,b ,c ,d ) max min ,1, , 0 Trapezoidal MF: ba d c 2 1 x c 2 Gaussian MF: gaussmf ( x ;a,b ,c ) e 1 gbellmf ( x ;a ,b ,c ) 2b Generalized bell MF: x c 1 b MF Formulation disp_mf.m 1 Sigmoidal MF: sigmf ( x ;a ,b ,c ) Extensions: 1 e a ( x c ) Absolute difference of two MF Product of two MF Membership Functions (MFs) Characteristics of MFs: Subjectivemeasures Not probability functions MFs “tall” in Asia .8 .5 “tall” in the US “tall” in NBA .1 5’10’’ Heights Fuzzy Rules and Inference Fuzzy rules are simply rules where the premises and conclusions are fuzzy. But crisp values can be incorporated as well. When using rules you have to select a fuzzy inference methodology and applied it to the conclusion. The two most popular fuzzy inference methodologies are: Rule IF A THEN B } Max-Product Inference Input (crisp) Rule Max-Min Inference { IF A THEN B Input (crisp) Fuzzy rules and their results Rule Distance (D) Velocity (V) Acceleration (A) Slow Medium IF D = Medium Medium OR V = Slow THEN A = Medium Inputs Medium Slow Medium IF D = Medium OR V = Medium THEN A = Slow Final acceleration = Centroid Fuzzy Logic/Sets: Example Let Y be the set of all flowers that are yellow. Let X, the universe of discourse, be the set of flowers in my backyard. In standard set theory, every flower x in X is either an element of Y or not. In fuzzy set theory, every flower x has a degree of yellowness mY(x). Let F be the set of all flowers that are perfumed . Let x be a flower in my backyard. If mY(x) = 0.8 and mF(x) = 0.9 then mY&F(x) = min{0.8,0.9} = 0.8. Fuzzification Fuzzification is the process of making a crisp quantity fuzzy We do this by simply recognizing that many of the quantities that we consider to be crisp and deterministic are actually Example: In the real world, hardware nondeterministic at all: They carry such as a digital scale generates crisp data, but these are subject considerable uncertainty. to experimental error. The information shown in the figure If the form of uncertainty happens to below shows one possible range of arise because of imprecision, errors for a typical weight measure and ambiguity, noise or vagueness, then the associated membership function the variable is probably fuzzy and can that might represent such imprecision. be represented by a membership Reading 1 function. 0 x -1% +1% Fuzzification The representation of imprecise data as fuzzy sets is a useful but not mandatory step when those data are used in fuzzy systems. This idea is shown in the following figure where we can consider the data as a crisp or as a fuzzy reading (Figures “a” and “b”). 1 1 Reading (crisp) Reading (fuzzy) Low voltage Medium voltage 0.4 Membership 0.3 Membership 0 voltage 0 -1% +1% x a) b) In Fig “a” we might want to compare In Fig “b” the intersection of the fuzzy set A crisp voltage reading to a fuzzy “Medium voltage” and a fuzzified voltage set, say “Low voltage”. In the figure reading occurs at a membership of we see that the crisp reading intersects 0.4. We can see that the intersection the fuzzy set at a membership of 0.3, of the two fuzzy sets is a small triangle, i.e., the fuzzy set and the reading can whose largest membership occurs at 0.4. be said to agree at a membership value of 0.3. Defuzzification methods (x) x dx Weighted average x* = (x) x (x) x* = Centroid (x) dx Max-membership or Height method ( x* ) >= ( ( x )) n Center of sums X x k=1 k (x) dx Mean-max membership x* = n x * = (a + b) / 2 X k=1 k (x) dx Fuzzy logic operations Summary Fuzzy math involves in general three operations: 1. Fuzzyfication – membership function 2. Rule evaluation 3. Defuzzyfication Fuzzyfication It makes the translation from real world values to Fuzzy world values using membership functions. The membership functions in Fig.1, translate a speed= 55 into fuzzy values (Degree of membership) SLOW=0.25, MEDIUM=0.75 and FAST=0. Rule Evaluation Rule1: If Speed=Slow and Home=Far then Gas=Increase Supose SLOW=0.25 and FAR=0.82. The rule strength will be 0.25 (The minimum value of the antecedents) and the fuzzy variable INCREASE would be also 0.25. Rule2: If Speed=Medium and Higher=Secure then Gas=Increase Suppose in this case, MEDIUM=0.75 and SECURE=0.5. Now the rule strength will be 0.5 and the fuzzy variable INCREASE would be also 0.5. So, we have two rules involving fuzzy variable INCREASE. The "Fuzzy OR" of the two rules will be 0.5 (The maximum value between the two proposed values). INCREASE=0.5 Defuzzyfication After computeing the fuzzy rules and evaluating the fuzzy variables, we will need to translate these results back to the real world. We need now a membership function for each output variable like in Fig. 2. Let the fuzzy variables be: DECREASE=0.2, SUSTAIN=0.8, and INCREASE=0.5 Defuzzyfication … Each membership function will be clipped to the value of the correspondent fuzzy variable as shown in fig.3. Defuzzyfication … Defuzzification is the process of making a fuzzy quantity crisp. There are different ways to do this and the deffuzification process to be used greatly depends on the degree of uncertainty within the fuzzy set A new output membership function is built, taking for each point in the horizontal axis, the maximum value between the three membership values. Then take the centroid. Here, Engine=+2.6 2.6 Steps Needed for Building a Fuzzy System Step 1.- Determine the values of the input and output variables. Step 2.- Fuzzify the variables: create fuzzy sets to represent the different values of the input variables. Fuzzification is the process of making a crisp quantity fuzzy. Step 3.- Create fuzzy sets for the output variables of the system. Step 4.- Generate a set of fuzzy rules based on the input and output fuzzy sets. Step 5.- Choose a deffuzification method and apply it to the results obtained from the rules that are satisfied. Step 6.- The crisp value obtained from Step 5 is the answer to your problem. Example • The inverted pendulum • Inputs: the angle and d/dt input values The fuzzy regions for the input values (a) and d/dt (b). The fuzzy regions of the output value u, indicating the movement of the pendulum base. The fuzzification of the input measures x1=1, x2 = -4. The Fuzzy Associative Matrix (FAM) for the pendulum problem. The input values are on the left and top. The fuzzy consequents (a) And their union (b). The centroid of the union (-2) is the crisp output. Characteristics and Comparison of Four AI Techniques Characteristics ES or KBS Fuzzy Logic ANNs GAs Knowledge representation Explicit Explicit Implicit Explicit Numerical computation External Inherent Inherent Inherent Training & development Depends on Depends on experts Depends on data Depends on experts & experts & data data Development time Slow to moderate Slow to moderate Moderate to fast Moderate to fast on-line processing May be slow May be slow to Fast Slow to moderate moderate Initial development Define variables Define variables Define input output Define fitness & input-output and membership parameters and function, coding and relations functions initial network chromosomes architecture Refinements Add new Add new relations, Adjust network Adjust mutations, relations, rules, tune membership weights and maybe crossover probability correct logical functions, adjust network & fitness function inconsistencies beliefs architecture Testing Expert Data & Expert Data Data Comments Used in complex Used in high 20%-25% of data Used mainly in domains & technology devices goes to training. optimization professions problems Fuzzy Logic Applications Area Application areas Fuzzy Control Subway trains Cement kilns Washing Machines Fridges Video cameras Electric shavers Fuzzy Sets Review Extension of Classical Sets Not just a membership value of in the set and out the set, 1 and 0 but partial membership value, between 1 and 0 Example: Height Tall people: say taller than or equal to 1.8m 1.8m , 2m, 3m etc member of this set 1.0 m, 1.5m or even 1.79999m not a member Real systems have measurement uncertainty sonear the border lines, many misclassifications Member Functions Membership function better than listing membership values e.g. Tall(x) = {1 if x >= 1.9m ,0 if x <= 1.7m, else ( x - 1.7 ) / 0.2 } Example: Fuzzy Short Short(x) = {0 if x >= 1.9m , 1 if x <= 1.7m else ( 1.9 - x ) / 0.2 } Fuzzy Set Operators Again Fuzzy Set: Union Intersection Complement Many possible definitions we introduce one possibility Fuzzy Set Union Union ( fA(x) and fB(x) ) = max (fA(x) , fB(x) ) Union ( Tall(x) and Short(x) ) Fuzzy Set Intersection Intersection ( fA(x) and fB(x) ) = min (fA(x) , fB(x) ) Intersection ( Tall(x) and Short(x) ) Fuzzy Set Complement Complement( fA(x) ) = 1 - fA(x) Not ( Tall(x) ) Fuzzy Logic Operators Summary Fuzzy Logic: NOT (A) = 1 - A A AND B = min( A, B) A OR B = max( A, B) Fuzzy Logic NOT Fuzzy Logic AND Fuzzy Logic OR Fuzzy Controllers Used to control a physical system Structure of a Fuzzy Controller Fuzzification Conversion of real input to fuzzy set values e.g. Medium ( x ) = { 0 if x >= 1.90 or x < 1.70, (1.90 - x)/0.1 if x >= 1.80 and x < 1.90, (x- 1.70)/0.1 if x >= 1.70 and x < 1.80 } Inference Engine Fuzzy rules based on fuzzy premises and fuzzy consequences e.g. Ifheight is Short and weight is Light then feet are Small Short( height) AND Light(weight) => Small(feet) Fuzzification & Inference Example If height is 1.7m and weight is 55kg what is the value of Size(feet) Defuzzification Rule base has many rules sosome of the output fuzzy sets will have membership value > 0 Defuzzify to get a real value from the fuzzy outputs One approach is to use a centre of gravity method Defuzzification Example Imagine we have output fuzzy set values Small membership value = 0.5 Medium membership value = 0.25 Large membership value = 0.0 What is the deffuzzified value Fuzzy Control Example Input Fuzzy Sets Angle:- -30 to 30 degrees Output Fuzzy Sets Car velocity:- -2.0 to 2.0 meters per second Fuzzy Rules If Angle is Zero then output ? If Angle is SP then output ? If Angle is SN then output ? If Angle is LP then output ? If Angle is LN then output ? Fuzzy Rule Table Extended System Make use of additional information angular velocity: -5.0 to 5.0 degrees/ second Gives better control New Fuzzy Rules Make use of old Fuzzy rules for angular velocity Zero If Angle is Zero and Angular velocity is Zero then output Zero velocity If Angle is SP and Angular velocity is Zero then output SN velocity If Angle is SN and Angular velocity is Zero then output SP velocity Table format Complete Table When angular velocity is opposite to the angle do nothing System can correct itself If Angle is SP and Angular velocity is SN then output ZE velocity etc Example Inputs:10 degrees, -3.5 degrees/sec Fuzzified Values Inference Rules Output Fuzzy Sets Defuzzified Values Example of a Fuzzy Controller A cart on a 4-meter long track. The goal is to return the cart to the center of the track with 0 velocity. The available control is to push or pull on the cart. -2m 0m 2m Cart Position m(x) left middle right 1.0 notice that sets overlap 0.0 x -2m -1m 0m 1m 2m Cart Velocity m(x) moving standing moving left still right 1.0 0.0 x -1m/s -0.5m/s 0m/s 1m/s 2m/s Cart Force m(F) pull push none 1.0 0.0 F -1N -0.5N 0N 0.5N 1N Simple Control Rules If left then push If right then pull If middle then none If moving left then push If standing still then none If moving right then pull If left and moving left then push If right and moving right then pull Fuzzy Control Algorithms Find the sensor values For example, the position might be x = -0.5 meters and v = 0. Calculate the fuzzy membership For example, mmiddle(x = -0.5) = 0.5 and mleft(x = -0.5) = 0.5. Calculate the membership of the rule antecedents for all control rules. Apply the rules Aggregate the results from all control rules “De-fuzzify” to arrive at a single-valued action recommendation. Dempster-Shafer Theory Dempster-Shafer considers sets of propositions and provides an interval within which the belief must lie. interval = [Belief, Plausibility] Belief brings together all the evidence that would lead us to believe in the proposition with some certainty. Plausibility brings together the evidence that is compatible with the proposition and is not inconsistent with it. pl(p) = 1 – bel(p) So..the interval is a measure of our belief in the proposition and the amount of information we have to support this belief. Coin Toss Example Let’s say Bart goes up to Homer and bets him $20 that the coin he has in his hand will be heads on a coin toss. Homer is like, “Yeah right, you trying to trick me boy?! It’s a two headed coin isn’t it?” Coin Toss Example Belief(Heads) = 0 Belief(not Heads) = 0 Should Homer take the bet? All of a sudden Lisa comes in and tells Bart to give her back her quarter. Homer, knowing Lisa to be honest, now thinks that maybe the coin isn’t a two-headed coin. Homer is 80% sure about this. This now increases the belief functions. Coin Toss Example Belief(Heads) = (Homer’s deduced certainty * probability of it coming up heads) = 0.8 * 0.5 = 0.4 Belief(not Heads) = (Homer’s deduced certainty * probability of it not coming up heads) = 0.8 * 0.5 = 0.4 Coin Toss Example The probability interval when being ignorant would be [0,1] for the probability of heads coming up on a coin toss. After Lisa comes into the scene, Homer deduces the coin’s probability, increasing the uncertainty, so the interval becomes [0.4, 0.6]. Smaller intervals allows the reasoning system to make decisions, based from new information. Example: Melissa is 90% reliable. She said, the computer is broken into Belief in computer being broken into = 0.9 Belief in computer not being broken into = 0 Pl(broken) = 1-0 = 1 [belief,plausibility](broken) = [0.9,1] Bill is 80% reliable. He said, the computer is broken into Belief in computer being broken into = 0.8 Belief in computer not being broken into = 0 Pl(broken) = 1-0 = 1 [belief,plausibility](broken) = [0.9,1] Probability that both of them are unreliable is 0.02 Combined [belief,plausibility](broken) = [0.98,1] Dempster-Shafer Theory … Let represent our ‘frame of discernment’, which is the set of all hypothesis. We want to attach a measure of belief to each of these hypothesis after we have been presented with some evidence. But the evidence may support subsets of . Also evidence supporting one hypothesis may alter our belief in other hypothesis. Dempster-Shafer allows us to handle these interactions. Dempster-Shafer Theory … If contains n elements then there are 2n subsets of (including the empty set ). m(p) is the current belief for each of the subsets of . Dempster-Shafer allows us to combine m’s that arise from multiple sources of evidence. Example: A patient may be suffering from Cold, Flue, migraine Headache or Meningitis Call this set of hypothesis Q = {C,F,H,M} Patient has fever, which supports {C,F,M} at 0.6 That’s, m1({C,F,M}) = 0.6, m1(Q)=0.4 Patient has extreme nausea, which supports m2({C,F,H}) = 0.7, m2(Q)=0.3 We can combine these two belief distributions m1({C,F,M}) = 0.6 m2({C,F,H}) = 0.7 m3({C,F}) = 0.42 m1(Q)=0.4 m2({C,F,H}) = 0.7 m3({C,F,H}) = 0.28 m1({C,F,M}) = 0.6 m2(Q)=0.3 m3(C,F,M)=0.18 m1(Q)=0.4 m2(Q)=0.3 m3(Q)=0.12 All the sets in m3 are non-empty and unique Example … Third evidence, lab culture supports m4({M}) = 0.8 and m4(Q) = 0.2 We can combine this with m3 m3({C,F}) = 0.42 m4({M}) = 0.8 m5’({}) = 0.336 m3({C,F,H}) = 0.28 m4({M}) = 0.8 m5’({}) = 0.224 m3(C,F,M)=0.18 m4({M}) = 0.8 m5’({M}) = 0.144 m3(Q)=0.12 m4({M}) = 0.8 m5’({M}) = 0.096 m3({C,F}) = 0.42 m4(Q)=0.2 m5’({C,F})=0.084 m3({C,F,H}) = 0.28 m4(Q)=0.2 m5’({C,F,H}) =0.056 m3(C,F,M)=0.18 m4(Q)=0.2 m5’ ({C,F,M}) = 0.036 m3(Q)=0.12 m4(Q)=0.2 m5’(Q)=0.024 The denominator is = 1- (0.336 + 0.224) = 0.44 m5({M}) = (0.144 + 0.096)/0.44 = 0.545, m5({C,F})=0.191 m5({C,F,H}) =0.127, m5’({C,F,M}) = 0.082, m5(Q)=0.055 m5({}) = 0.56 Summary on Dempster-Shafer A large belief assigned to empty set (as 0.56 in the previous example) indicates that there is conflicting evidence in belief sets. When there are large hypothesis sets and complex sets of evidence, calculations can get cumbersome. But complexity is still less than Bayesian approach. Very useful tool when stronger Bayesian conclusions may not be justified. A distinction is made between probability of a proposition given uncertain evidence, and probability of proposition given no evidence. Default Reasoning A gentle introduction Simulates human nature of qualitative reasoning. All birds fly. Emu is a bird. Therefore, Emu flies. Went to Australia and saw it does nor fly. Update your belief ! “Jumping to conclusions” “Making assumptions” To believe one thing until a reason is found to believe otherwise. Default Reasoning Example Homer is at work at the Springfield nuclear power plant. Now he’s having some coffee and doughnuts and accidentally spills it over some controls. All of a sudden there are explosions and alarms are going off in the plant. Homer is frantic and is saying, “Oh my god, oh my god, Mr. Burns is going to fire me. Oh no, what do I do, what do I do?” Then Smithers comes and tells Homer that Mr. Burns wants to see him. Default Reasoning Homer’s assumption that he is going to be fired by Mr. Burns is an example of default reasoning. Later Homer finds out that he wasn’t even at his own workstation, he was at someone else’s workstation, so Homer doesn’t have to worry about being fired now. He retracts his initial assumption, and gives a big sigh and relaxes. Default Reasoning Problems What are good default rules to have? What to do in the case where some evidence matches two default rules with different conclusions? What conclusions should be kept and which ones should be retracted? How can beliefs that have default status be used to make decisions? Fuzzy Logic (FL) vs CF We use both FL and CF to handle incomplete knowledge In FL, Precision/vagueness is expressed by membership function to a set young adult pensioner mF(20,adult)=0.6, mF(20,young)=0.4, mF(20,old)=0 Fuzzy Logic is not concerned how these distribution are created but how they are manipulated. There are many interpretations, similar to Certainty Algebra Exercises Given the fuzzy sets:- Tall(X) = { 0 if X < 1.6m (X - 1.6m) / 0.2, if 1.6m <= X < 1.8m 1, if X >= 1.8m } Short(X) = { 1 if X < 1.6m (1.8m - X) / 0.2, if 1.6m <= X < 1.8m 0, if X >= 1.8m } a). Sketch the graphs of Tall(X) and Short(X). b). i. Calculate the Union of the fuzzy sets Tall(X) and Short(X). ii. Calculate the Intersection of the fuzzy sets Tall(X) and Short(X). c). Show that the complement of Tall(X) is Short(X). Given additional fuzzy sets:- Strong(Y) = { 0 if Y < 30kg (Y - 30kg) / 20, if 30kg <= Y < 50kg 1, if Y >= 50kg } Weak(Y) = { 1 if Y < 30kg (50kg - Y) / 20, if 30kg <= Y < 50kg 0, if Y >= 50kg } and the fuzzy rules:- If Tall(X) OR Strong(Y) then Heavy(Z) If Short(X) AND Weak(Y) then Light(Z) Calculate the membership values of Heavy(Z) and Light(Z) where i. X = 1.65m, Y = 30kg ii. X = 1.70m, Y = 45kg Complexity of the system Vs. precision in its model Precision in the model Mathematical equations Model-free Methods (e.g., ANNs) Fuzzy Systems Complexity (uncertainty) of the system For systems with little complexity, hence little uncertainty, closed-form mathematical expressions provide precise description of the system. For systems that are a little more complex, but for which significant data exists, model free methods such as artificial NNs, provide a powerful and robust means to reduce uncertainty through learning, based on patterns in the available data. For most complex systems where few numerical data exists and where only ambiguous or imprecise information may be available, fuzzy reasoning provides a way to understand system behavior by allowing us to interpolate approximately between observed input and output situations.