[SOLVED] 29650 Engineering Mathematics 2 – Tutorial sheet 4

29650 Engineering Mathematics 2 – Tutorial sheet 4

Question 1

A 4 state Markov model has states s1, s2, s3 and s4, initial state probability vector P
0 and transition probability matrix A given by

(1)

Questions:

1. Calculate the probability of the sequence s = s1s1s3s2s4s4

2. Calculate the state probability vectors P
1 and P
2 at times t = 1 and t = 2 respectively.

Question 2

Two sequence generators X and Y output sequences of symbols a, b, c. They are modelled as 3 state Markov models MX and MY, respectively.

MX has parameters

(2)

MY has parameters

(3)

In both cases the symbols a, b, c correspond to states 1, 2, 3 respectively. Generator X trans-mits signals 3 times more often than generator Y.

Question:

The sequence a, b, c is received. Which generator did it most probably come from? (Don’t forget what you learnt in the first three weeks of last Semester…. Bayes Theorem!)

Question 3

I didn’t tell you how to do this in the lectures, but if you understand what a Markov process is it should be easy!

A sequence generator X transmits the following sequences of symbols a, b:

(4)

Questions:

1. Use the sequences to estimate the parameters P0 and A of a 2 state Markov model of X.

2. Calculate P1, P2 and P3

3. Calculate limt→∞P
t

4. Verify your answer by finding the eigenvalues and eigenvectors of AT.

Question 4

A 3-state Markov process has parameters P0 and A given by:

(5)

State 3 is the exit state. A simple way to generate a random initial state from P0
is:

1. Generate a random number r uniformly distributed over [0, 1]

2. If r ≤ 0.7 output a else output b

Having generated the first state the process continues as follows:

1. Choose the row of A corresponding to the current state s

2. Generate a random number r uniformly distributed over [0, 1]

3. If r ≤ as,1 output a, else if r ≤ as,1 + as,2 output b, else you have reached the exit state – don’t output anything – stop.

4. If the new state is state 3 then stop, else return to step 2.

Questions:

1. What is the probability that the model generates a sequence of length exactly 2 symbols?

2. What is the value of Pt as t → ∞?

3. Use the following sequence of random numbers to generate as many sequences of out-puts a and b as possible from the model: 0.31, 0.53, 0.17, 0.6, 0.26, 0.65, 0.69, 0.75, 0.45, 0.08, 0.23, 0.91, 0.15, 0.83, 0.54, 0.99, 0.08, 0.44, 0.11, 0.96

4. Use the sequences that you have created to estimate the parameters of the Markov model that generated them.

5. How similar is this new model to the ‘correct’ model?