interpretation of Markov transition field

Question

Markov transition fields come from this article: https://www.researchgate.net/publication/282181246_Spatially_Encoding_Temporal_Correlations_to_Classify_Temporal_Data_Using_Convolutional_Neural_Networks

They are quite well explained here: https://lazyprogrammer.me/convert-a-time-series-into-an-image/

M_kl (one element of the transition field) is then just the probability that we saw a direct one-step (i.e. Markovian) transition from q_k to q_l in the time series.

But the article states:

By assigning the probability from the quantile at time step i to the quantile at time step j at each pixel M_ij , the MTF M actually encodes multi-step transition probabilities of the time series. M_i,j||i−j|=k denotes the transition probability between the points with time interval k.

and

By scattering the first-order transition probability into the temporally ordered matrix, MTFs encode the transition dynamics between different time lags k.

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I don't understand what is meant by these sentences because the transition matrix really just contains probabilities of transitions between quantiles of neighboring elements in the time series, i.e. number of steps is always considered to be 1 by definition so I don't get how it can be true that it "encodes the transition dynamics between different time lags k" - k is a difference between two timestamps in the time series. The k (number of steps to reach another quantile) is not taken into account in the definition of Markov transition field.

And actually it would make sense to me if it was taken into account and we would put into the Markov transition field the probabilities that one quantile changes into another in k steps. Why it isn't like that?

Answer 1

Imagine you have three states, A B C, and your transition probability matrix looks like this:

  A  | B  | C
A .5 | .5 |  0
B .3 | .4 | .3
C  0 | .5 |  5

(e.g. from state A there's a .5 chance of staying in A and a .5 chance of transitioning to B)

It's easy to see that this encodes the probabilities for the three possible lag-1 transitions starting at A:

• A -> A (p = .5)
• A -> B (p = .5)
• A -> C (p = 0)

The trick is that it you can calculate the lag-2 transition probabilities by multiplying together the lag-1 probabilities:

• A -> A -> A: p = .5 * .5 = .25
• A -> A -> B: p = .5 * .5 = .25
• A -> B -> A: p = .5 * .3 = .15
• A -> B -> B: p = .5 * .4 = .20
• A -> B -> C: p = .5 * .3 = .15

Finally, if you just want to know the probability of being in state B two steps after being in state A, you simply add up the probabilities of all the chains of transitions that would get you there:

P(A -> ? -> B) = P(A -> A -> B) + P(A -> B -> B) = .45

Answer 2

I did try some other interpretations. E.g. consider that the points i, j have a distance |i - j| between them and therefore calculate the probability of transition between the respective quantiles for the points in that number of steps (i.e. on the main diagonal there will be just 1s as the distance is zero).

Needless to say, so far I have had more success with the original definition (where the probability is always that one quantile changes into another in a single step) when the resulting field is employed into a simple machine learning model.