Final exam for the course CS 4365/CS 5354, Summer 2010

Name: _____________________________________________________________

10 pages of notes allowed. Please place your solution to each problem on a separate sheet of paper, with your name on top of each sheet.

1. Describe two methods of eliciting degrees of certainty from experts: polling and marking on a scale. Give numerical examples of using both methods.

2. List requirements on "and"-operations (t-norms) and "or"-operations (t-conorms). Give two example of t-norms and two examples of t-conorms. Explain, on the example of one of your t-norms, why it is not a t-conorm: list all t-conorm requirements that are satisfied by this t-norm and those that are not satisfied.

3-4. If an iPhone overheats, it is necessary to let it cool down:

• if it overheats a little bit, we need to cool it down for a short period of time;
• if it overheats a lot, we must cool it down for a long period of time.
Use the fuzzy techniques to come up with the degree to which, for a x = 20 degrees overheating, it is OK to let it cool down for u = 5 minutes. Use linear interpolation to derive the corresponding membership functions:
• For "overheating a little bit": the value is 1 for x = 0 and 0 for x = 50.
• For "overheating a lot": the value is 0 for x = 0 and 1 for x = 50 (and for all larger x).
• For "short period of time": the value is 1 for u = 0 and 0 for u = 10.
• For "long period of time": the value is 0 for u = 10 and 1 for u = 20 (and for all larger values u).

5. Three people estimated the temperature as 90, 95, and 100. Use the Least Squares method to combine these three estimates into a single one. Explain how the Least Squares method is used to derive a defuzzification formula; write down the resulting formula. For extra credit: derive the formula for centroid defuzzification.

6. Let x1 = "long period of time" and x2 = "short period of time" (as in Problem 3-4). Find the alpha-cut for x1 - x2 for alpha = 0.6.

7. Find the range of the function y = (x1 - 1)2 + x1 * x2 when x1 is between 0 and 1, and x2 is between -2 and 1, by using two methods:

• the calculus-based method for finding the exact range, and
• the naive interval computations method for computing the enclosure for the range.

8. Use the crisp clustering algorithm to cluster the following 1-D data: objects are characterized by values 0.0, 1.0, 2.0, 6.0, and 7.0, we have two clusters, and the initial representatives are 0.5 and 7.5. Write down the formulas explaining how to use fuzzy clustering to cluster this data. What is the advantage of fuzzy clustering as compares to the crisp one?