Exam January 2014, questions PDF

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FUZZY SETS AND FUZZY LOGIC SYSTEMS 2014...

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The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 3 MODULE, SPRING 2013-2014

FUZZY SETS AND FUZZY LOGIC SYSTEMS

Time allowed 1 hour

Candidates may complete the front cover of their answer book and sign their desk card but must NOT write anything else until the start of the examination period is announced. Answer Question 1, and any ONE from Questions 2 and 3. Each question carries 30 marks. Only silent, self contained calculators with a Single-Line Display or Dual-Line Display are permitted in this examination. No electronic devices capable of storing and retrieving text, including electronic dictionaries, may be used.

Marks available for sections of questions are shown in brackets in the right-hand margin.

Dictionaries are not allowed with one exception. Those whose first language is not English may use a standard translation dictionary to translate between that language and English provided that neither language is the subject of this examination. Subject specific translation dictionaries are not permitted. DO NOT turn examination paper over until instructed to do so.

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1

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(a)

Discuss, with some brief examples, the inadequacies of classical 2-valued logic in describing the real world. [8]

(b)

You are presented with two tables, A and B, on each of which are placed ten opaque bottles. You are told that the bottles on table A have each been filled with liquid that is drinkable with probability 0.9. You are then told that the bottles on table B have each been filled with liquid which belongs to the fuzzy set of drinkable liquids with membership 0.9. You are told you have to choose one bottle and drink its contents. Discuss which bottle you choose and why. [8]

(c)

Two fuzzy sets, A and B, (written in standard Zadeh notation) are given as: A= 0.0/1 + 0.3/2 + 0.6/3 + 0.9/4 + 0.8/5 + 0.5/6 + 0.3/7 + 0.2/8 B= 0.4/1 + 0.8/2 + 0.5/3 + 0.3/4 + 0.6/5 + 0.9/6 + 1.0/7 + 0.5/8 Define the ‘standard’ (Zadeh) fuzzy operators for negation (NOT), intersection (AND) and union (OR), and use them to calculate the following fuzzy sets: (i) NOT A (ii) (NOT A) AND B (iii) A OR B [6]

(d)

Give informal definitions of the properties of normality and convexity for fuzzy sets, and say which of the fuzzy sets A and B, in Q1(c) above, satisfies each property. [4]

(e)

Briefly explain what a t-norm and t-conorm is (there is no need to give the theoretical axioms), and give an example of each which is different from the ‘standard’ operators defined in Q1(c) above. [4]

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2

(a)

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Given the linguistic input variables, Age and Height, each with the given terms: Age young= 1/0 + 0.9/10 + 0.8/20 + 0.5/30 + 0.2/40 + 0.0/50 middle= 0.2/20 +0.5/30 + 0.7/40 + 1/50 + 0.7/50 + 0.5/60 + 0.2/70 old= 0.1/40 + 0.3/50 + 0.5/60 + 0.7/70 + 0.9/80 + 1/90 + 1/100 Height short= 1/1.4 + 0.75/1.5 + 0.5/1.6 + 0.25/1.7 average= 0.5/1.6 + 1/0.7 + 0.5/1.8 tall= 0.2/1.6 + 0.5/1.7 + 0.75/1.8 + 1/1.9 + 1/2.0 and the single linguistic output variable Employ, with the three terms: Employ bad= 1/0 + 1/1 + 0.7/2 + 0.5/3 + 0.3/4 fair= 0.1/3 + 0.5/4 + 1/5 + 0.5/6 + 0.1/7 good= 0.3/6 + 0.5/7 + 0.7/8 + 1/9 + 1/10 calculate the fuzzy set result of applying Mamdani inference on input values of Age=40 and Height=1.6 when using the rules: R1: IF Age is young OR Height is tall THEN Employ is good R2: IF Age is middle THEN Employ is fair R3: IF Age is old AND Height is short THEN Employ is bad Note that you should not defuzzify the output set at this stage. [10]

(b)

Describe and give basic formulae for two alternative methods of numeric defuzzification. Provide an example of two obviously different output sets for which one of these defuzzification methods gives the same result. [8]

(c)

Calculate the defuzzified value of your output set from Q2(a) above, using centre-of-gravity defuzzification. If you have not been able to derive a set for Q2(a), then write down your own fuzzy set defined on a universe of discourse of the integers from 0 to 10, and calculate its defuzzified value. [6]

(d)

Describe the role of similarity measures in the process of linguistic defuzzification. Give examples of two different similarity measures. [6]

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3

(a)

In the context of fuzzy modelling, explain in detail each of the following terms: (i) the structure of a fuzzy model (ii) the parameters of a fuzzy model (iii) direct and indirect objective functions

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[2] [2] [4]

(b)

Create an example fuzzy inference system (FIS) with parameterised membership functions (m.f.s), consisting of two linguistic input variables and one linguistic output variable, with each variable having three membership functions. Ensure there are at least three tuneable parameters in your example FIS. Draw carefully labelled diagrams showing the three m.f.s in each variable, and explain carefully what the parameters control. [6]

(c)

Outline how you would go about tuning your parameterised FIS above, explaining the terms exhaustive-search, Monte-Carlo search and hill-climbing. [6]

(d)

Explain the function of each of the layers in the ANFIS model above, and hence explain how it replicates Sugeno (TSK) inference. What order is the replicated Sugeno (TSK) model? [10]

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