Question

In particular, our process has a finite number of different possible states, namely: anger, disgust, fear, joy, sadness, analytical, confident and tentative. If we assume that this process is stationary and Markovian, there must exists a 9×9 transition matrix A encoding the probabilities in Eq. (1) by:

a_ij=P(X_n = s_i | X_n-1 = s_j)

where si, sj are any two of the moods defined above and where n is any point in time.

Given dataset for a Markov chain model. Find a reasonable way to estimate the transition matrix A and compute it. The full dataset to be used can be found at https://storage.googleapis.com/matrices-su18/markov.csv.

Answer 1

Since the dataset is not available i have tried to explain how to construct a transition matrix in a general way.

In a markov chain the transition probability of moving from state sj at time n-1 tp state i at time n (say from anger to fear) is obtained as follows :

• Pji = ( Number of transitions from state sj to state si / Number of transitions out of state sj)

The state spaces given are anger, disgust, fear, sadness, joy, analytical, confident and tentative. Given, a dataset find the probabilities of moving from one state to another.

Say we represent the transition matrix it as follows:

   Anger Disgust Fear Sadness Joy Analytical Confident Tentative

Anger x    x x x    x x x x

Disgust y    y y y    y y y y

Fear z z z z    z z z z

Sadness and so on......

Joy

Analytical

Confident

Tentative

• Each of the values ( x or y or z and so on) in the above matrix is the transition probability from state in the column to the state in the row. They cannot be negative, and the sum of each row is equal to 1.