Markov-Chain transition probabilities for 3 variables

Question

I am a bit confused as I need to calculate the Markov-Chain transition probabilites for 3 variables.

Example data, let's assume a sequence of letters at specific and progressively-constant time steps:

Q
Q
E
Q
C
C
E

What are my transitional probabilities?

My (wrong) understanding is:

P(Q|Q) = 1
P(Q|E) = 1
P(Q|C) = 0

P(E|E) = 0
P(E|C) = 1
P(E|Q) = 1

P(C|C) = 1
P(C|E) = 0
P(C|Q) = 1

And therefore my (wrong) transition matrix will be:

   Q   E   C
Q  1   1   1  
E  1   0   0
C  0   1   1

note row sums are not = 1

What am I missing? The same approach works with 2 variables and here it seems that I need to divide each row by the number of probabilities > 0 to make the row sums =1.

Thanks

Answer 1

I'll give one example and hopefully it will be evident how it can be applied to the rest:

$P(X(t) = E | X(t-1) = Q)$ can be estimated empirically as "the percentage of instances of Q that are followed by E". So in this case, there are 3 instances of Q, one of which is followed by an instance of E, meaning that $P(X(t) = E | X(t-1) = Q)$ is equal to 1/3.

Side note: I think when you write P(E|Q), you mean the expression I wrote above, but it's probably better to write it how I did just to be clear that you mean "the probability of E coming directly after Q".

Answer 2

In any Transition Probability Matrix, the row sum must be equal to 1. Your Probabilities are incorrect. It should be

P(Q|Q) = 1/3
p(Q|E) = 1
P(Q|C) = 0

P(E|E) = 0
P(E|C) = 1/2
P(E|Q) = 1/3

P(C|C) = 1/2
P(C|E) = 0
P(C|Q) = 1/3

Hence the Transition Probability Matrix becomes :

   Q    E    C
Q  1/3  1/3  1/3  
E  1    0    0
C  0    1/2  1/2