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:
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:
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?
10. Use the Lagrange multiplier method to find the minimum of a function x12 + x22 under the constraint x1 + x2 = 2. What if instead, we have a fuzzy constraint -- that the sum x1 + x2 is approximately 2, where "approximately 2" is described by a membership function x/2 for x from 0 to 2 and 1 - x/2 for x from 2 to 4. For the fuzzy case, just describe the formulas, no need to find the actual location of the minimum.