I have just started learning about Markov chain and have a trouble determining appropriate transition matrix:
Suppose that whether or not it rains today depends on previous weather conditions through the last three days. Show how this system may be analyzed by using a Markov chain by determining how many states are needed and describing the general form of the P matrix. Also suppose that if it has rained for the past three days, then it well rain today with probability .8; if it did not rain for any of the past three days, then it will rain today with probability.2; and if any other case, the weather today will, with probability .6., be the same as yesterday. Determine the matrix P for this Markov chain.
This question has a very detailed explanation but I still don't get it.
Can someone please explain me at least one row? Also how are you suppose to read notations, does YYYY= P(00) means that it will rain today given that it has rained for the past three days? For instance, P(00) = Y YYY = 0.8 because it has rained for the past three days (YYY) then it will rain today.
P(10) = Y NYY = 0. Why 0? Why it isnt 0.6? but P(30) = Y YYN =0.6 I have spent so much time looking for explanations but every website just gives transition matrix without explaining me obtained values.
Let's agree first that a reasonable state space is $$S = \{YYY, YYN, YNY, NYY, YNN, NYN, NNY, NNN\}$$ where, for example, $YYY$ means it rained today and the previous two days, whereas $YYN$ means that it didn't rain today, but did rain the previous two days, etc. To be clear the transitions will be to move to the weather tomorrow, today and the day before. So, for example, $YYY \to YYN$ means the transition from the state where today and the previous two days rained into the state where it doesn't rain tomorrow (the "next step's today") and did rain the previous two days. Now $P(YYY \to YYN) = .2$ whereas $P(YYY \to YYY) = .8$ by your story. However, state changes as $P(\ast \ast Y \to \ast N \ast) = 0$ since there is no way that if it rained today, then you transition to where it doesn't rain today (I used the symbol $\ast$ means any weather).. does this make sense? If so, I imagine using $S$ as I have listed, you can create the transition matrix.