ME290M Expert Systems in Mechanical Engineering Spring 1999, T-Th 12:30-2:00 pm 1165 Etcheverry Hall, Course Control No. 56369 http://best.me.berkeley.edu/~aagogino/me290m/s99

Due Date: Thursday, May 6, 1999

1. Short Project Presentation (10 points)

Be prepared to give a short project presentation to the class on May 6. Keep your presentation to 5 min. for a single person project and 10 min. for a team project. The idea is to share your progress to date with the class. The final project report is due by 5:00 pm on May 19. (I’d prefer to get some in earlier, if at all possible).

2. Solve Problem 4.12 in the text. (20 points)

Under conditional independence of E1 and E2, show that for a rule involving the conjunction of known evidence that:



 

3. Solve Problem 5.2 in the text. (10 points)

Prove that CF (H,E) + CF(H’,E) = 0

4. Solve Problem 5.3 in the text. (20 points)

Given rules

IF E₁ AND E₂ AND E₃

THEN H(CF₁)

 

IF E₄ OR E₅

THEN H(CF₂)

Where

CF₁ (E₁,e)=1

CF₁ (E₂,e)=0.5

CF₁ (E₃,e)=0.3

CF₂ (E₄,e)=0.7

CF₂ (E₅,e)=0.2

CF₁ (H,E)=0.5

CF₂ (H,E)=0.9

(a) Draw a tree illustrating how these rules support H.

(b) Calculate the certanty factors CF₁ (H,e) and CF₂ (H,e).

(c) Calculate CF_COMBINE (CF₁ (H,e) and CF₂ (H,e))

5. Solve Problem 5.11 in the text. (10 points)

   (a) Define five linguistic values for the linguistic variable "Uncertainty".

   (b) Draw appropriate functions for these values and explain your choices.

   (c) Draw the fuzzy sets for

   Not TRUE

   More or Less TRUE

   Sort Of TRUE

   Pretty TRUE

   Rather TRUE

   TRUE

   Assuming TRUE is an S-function. What re the limts of TRUE? Explain.

6. St. Petersburg Paradox Problem (10 points for part a, 20 points extra credit for parts b&c)

   You have an opportunity to invest $X in the following lottery: A biased coin is tossed. If it comes up heads (with probability "p") your money is doubled. If it comes up tails you get nothing. Your total wealth at the beginning of the game is $W and you can not borrow money (i.e., you can not have negative wealth). Answer the following questions.

   (a) Assume that you can only play this game once and you are an expected monetary value decision maker. How much would you pay for this lottery? How much would you pay if your utility function was of the form U(y) =ln(y+$W) where $W is your wealth at any time?

   Let the subscript "i" signify values at the beginning of lottery i. Let

   W_i = Total wealth after lottery i-1 and before lottery i. W₁=wealth before any investments in the lotteries are made.

   x_i = The total amount of money invested in lottery i. The wealth constraint restricts x_i to be less than or equal to W_i.

   R_i= The portion of W_i to be invested in lottery i. Thus x_i=R_iW_i.

   (b) (10 points extra credit) Assume that you can play this game twice and you are an expected value decision maker. How much would you invest in the first game and given the results of the first game how much would you invest in the second game? How much would you pay if our utility function was of the form in part (a)?

   (c) (10 points extra credit) Now assume that you can play this game as many times as you want given your total wealth constraint (i.e., if you have no money to play you can’t play). How much would you invest at each game given the results of the previous game? Show how to formulate this problem if you are an expected value monetary decision maker? Solve the problem in terms of $W and p if your utility function is of the logarithmic form in part (a)?

7. Oil Wildcatters Problem. (20 points)

   Use Hugin (or any other Bayes’ Net software you prefer) to solve the Oil Wildcatters Problem. Use the numbers provided in the text on pp.183-186. Correct any mistakes that are in the textbook version.