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Unformatted text preview: ECE 496 HOMEWORK ASSIGNMENT III Spring 2007 1. Consider the three-state Markov chain in Diagram 1. (a) Find the transition matrix P and the stationary distribution π * for this Markov chain. (b) Find labels for the arrows in Diagram 2 so that the resulting Markov chain has the same stationary distribution as the Markov chain in (a). 2. Consider the rather trivial GA that operates on binary “strings” of length L = 1 and populations of size n = 2 with fitness function f (1) = 5 and f (0) = 1, and with p m = . 01 (note that crossover is somewhat meaningless here). Assume fitness-proportional selection. (a) List all the possible populations and give each one a label. (b) Draw the transition diagram for the Markov chain on population space corre- sponding to this GA. (c) Find the transition matrix P of the Markov chain. (d) Find (at least approximately) the stationary distribution of the Markov chain. (e) Based on your answer to (c), find the long-term average fitness of the population, that is, find ¯ f ∞ = lim T →∞ 1 T T X t =1 ¯ f ( t ) , where ¯ f ( t ) is the average fitness of the population at time t . Note that ¯ f ∞ is independent of the initial population. 3. Mitchell asserts in Chapter 4 that the number of possible populations of size n for a standard GA operating on binary strings of length L is given by 2 L + n- 1 2 L- 1 ! . This problem steps you through a proof of that fact. (a) Show that there are 2 L populations in which only one string is represented and ` 2 L n ´ populations in which exactly n strings are represented. (b) Show that there are ` 2 L 2 ´` n- 1 1 ´ populations in which exactly two strings are rep- resented. (Suggestion: first you need to pick which two strings are represented; then you have to pick a “dividing line” in the population so that one of the strings...
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- Spring '07
- Algorithms, Markov chain, Empty set, representative, binary strings