Guide to Intelligent Systems; Second Edition; Michael Negnevitsky
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- Navigate this page:
- 5.1 Fundamentals of Fuzzy Logic
- 5.2 Set definitions
- 5.3 Defining membership functions
- 5.4 Linguistic Variables and Hedges
- Putting it all together
- 5.5 Mamdani-style inference
Fuzzy SystemsChapter 4 in the book – page 87. Statements and Definitions:
Considering the examples above, using Boolean logic we would need to define a point at which a car is really quick, but where is that. 
A Camaro is considered really quick to many people. If it runs a 14.1 second quarter mile are all cars that are slower than Camaros really slow? 
How many hairs does someone need to have on their back for it to be quite hairy? If Tom’s back has 2500 hairs, and Mary’s has 2200, is Mary lucky because she does not have a hairy back? If we define really hot as 245 degree’s or higher, is it normal for my car to run at 243 degrees? As can be seen from the example above, a car is either quick, or slow. 5.1 Fundamentals of Fuzzy LogicThe basic idea of fuzzy set theory is that an element of a fuzzy set belongs with a certain degree of membership. Thus a proposition may be partly true or partly false. This degree is a real number between 0 to 1. The universe of discourse is the range of all possible values applicable to a chosen variable. The classic example is the set of tall men. The elements of the set are all men, but they have varying degrees of membership based on their heights. The universe of discourse is their heights.
Below we graph the degree of membership versus height for both Crisp and Fuzzy sets. 
5.2 Set definitionsCrisp
fA(x) : X → 0, 1 where fA(x) = 1, if x is a member of A, 0 if x is not a member of A. Fuzzy
µA(x) : X → [ 0, 1] µA(x) = 1 if x is totally in A; µA(x) = 1 if x is not in A; 0 < µA(x) < 1 if x is partly in A.
The function µA is the function that determines an elements degree of membership in the set. Typical functions are Gaussian, Sigmoid, etc. Linear functions are used commonly, they take less computation time, thus the linear fit function. A fit vector can be used. Below are the fit vectors for the ‘short, average, tall men’ graph. For the above example the fit vectors are:
THEN sailing is good IF John is tall THEN John runs fast We use hedges (very, somewhat, quite, more or less, etc) to modify our linguistic variables.
To calculate these variable values use the line equation to find the linear function: y = m*x + b m = (y1 – y0)/(x1 – x0) b = y0 – m*x0 Then do the operation that the hedge calls for. 5.45Operations on Fuzzy sets – see page 97
Fuzzy Rules: IF x is A THEN y is B
THEN ticket is very expensive 
Let speed over speed limit = 30mph That gives a degree of membership of .77 in the somewhat fast set. Then using the .77 degree of membership in the somewhat fast set maps to a ticket costing between 80 and 100 dollars in the ticket very expensive set.
How does it all work? Rule 1: IF project_funding is adequate OR project_staffing is small THEN risk is low Rule 2: IF project_funding is marginal AND project_staffing is large THEN risk is normal Rule 3: IF project_funding is inadequate THEN risk is high Rule 1:
IF x is A3 OR y is B1 THEN z is C1
IF x is A2 AND y is B2 THEN z is C2 Rule 3 IF x is A1 THEN z is C3
5.5 Mamdani-style inference
Example: Step : Fuzzify inputs Let project funding = 35% µ(inadequate_funding) = µ(A1) = .5; µ(marginal_funding) = µ(A2) = .2; µ(adequate_funding) = µ(A3) = 0.0; Let project staffing = 60% µ(staffing_small) = µ(B1) = .1; µ(staffing_large) = µ(B2) = .7; Step 2: Evaluate the rules When we evaluate the rules, we must evaluate each all if parts (antecedents) and find values for the then parts (consequents). For conjunctions like AND and OR we do operations like min and max. Rule 1:
(A3 OR B1) ≈ max(0, .1) = .1 Mapping .1 to the risk_low set we get: C1 = .1 Rule 2: (A2 AND B2) ≈ min(.2, .7) = .2 Mapping .2 to the risk_normal set we get: C2 = .2 Rule 3: A1 = .5
Step 3: Aggregation of consequents In this step we unify the outputs of all the rules. In our example we clipped the top off each of our 3 risk sets, so now we sum these areas into one resultant area. See the bottom of figure 4.10 on page 108. Step 4: Defuzzification In order to turn our resultant fuzzy sum (we summed the clipped fuzzy sets from each of our rules in 1 big fuzzy resultant set) we find the center of gravity (COG) of the set. What we mean by center of gravity is that we want to locate the point at where we can slice the set into two equal pieces. We can use the following formula to get an estimate of the COG. COG ≈ ∑ab µA(x)x / ∑ab µA(x) µA(x) the consequent of each of our rules x the x axis value of the set b the upper range of the x domain of the set a the lower range of the x domain of the set
For our example problem: COG = (0 + 10 + 20)*.1 + (30 + 40 + 50 + 60)*.2 + (70 + 80 + 90 + 100)*.5 .1 + .1 + .1 + .2 + .2 + .2 + .2 + .5 + .5 + .5 +.5 = 3 + 36 + 170 3.1
So the aggregate risk is 67.4% Assignment questions – page 126 1, 2, 3, 4, 5, 6, 7, 8, 11 Directory: Academics -> Faculty -> pmcdowell Faculty -> Chapter 1 “Preliminaries” Reasons for Studying Concepts of Programming Languages Faculty -> Chapter 7 “Expressions and Assignment Statements” Faculty -> Deleuze and Analytic Philosophy by Faculty -> Curriculum Proposal Title of Proposal: Course Status Change Faculty -> Chapter 6: Arrays Faculty -> Chapter 5 Introduction Faculty -> Carmen Sara Cañete Quesada Faculty -> Economic Conditions for Regional Development and for First Nations pmcdowell -> Patrick McDowell Download 281.84 Kb. Share with your friends:
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