Using Markov Chains for weather forecast in Weatherland.

markov chainsstochastic-processes

Weatherland has only two types of days – sunny and rainy. It is known that before every sunny day there are two rainy days and every fifth day after a rainy day is also rainy. The Weatherlands weather forecast for today is sunny.

  1. Find any possible realisations of the chain for the next 11 days.
  2. Is it a Markov chain? Why? If not then how could change the given process to fit in the rules of Markov Chain? Write out the transition Matrix.
  3. What are the probabilites of sunny and rainy days if $t \rightarrow \inf$

So I tried to solve the following problem and my steps are here. I am pretty sure about the 1. and 3. step but I am full of doubts with second (I have an idea but I am unable to construct the transition Matrix for it). I hope that someone can give me feedback about the current answer and my ideas about the 2. task to ensure that I am on the right (or completely wrong) track…

  1. Chain realisations for the next 11 days (and a bit more):
    Chain realisations as table

a) We know that today is sunny day (let's mark it in the table with letter S and yellow colour)

b) Before every sunny day there are two rainy days (lets mark them in the table with letter R and blue colour)

c) Every fifth day after a rainy day is rainy, so 3th, 4th, 8th, 9th …. 5n-1 and 5n-2 days are rainy

d) The first two days can't be sunny because before every sunny day there are two rainy days

e) Followed by the fourth point we can say that every 5n+1 and 5n+2 are rainy days

f) 6.1 The weather on every 5th day can be both rainy and sunny, so P(S)=1/2 and P(R)=1/2. If it is rainy then the there will never be a sunny day. If it's sunny then we have the same situation again in the fifth day after last sunny day.

Conclusion: The first three rows show the possible realisations for the next 11 days.

  1. From the table at least it is pretty clear to see that if $t \rightarrow \inf $ then the change of rainy days $r \rightarrow 1$ and sunny days $s \rightarrow 0$.

I hope that the previous answer is correct, because now is the question that I do not know the answer to.

  1. I believe that it is not a Markov chain because even though we know that today's weather is sunny we still have to take into account that before every sunny day there are two rainy days and therefore the future is not dependent only on the present but also the past.
    So we should change the task that every fifth day is either sunny or rainy and then we get the transition matrix of:
    \begin{pmatrix}
    1/2 & 1/2\\
    0 & 1
    \end{pmatrix}
    where $a_{1,1} = 1/2$ – today is sunny and the fifth day will be sunny;

$a_{1,2} = 1/2 $ – today is sunny and the fifth day will be rainy;

$a_{2,1} = 0 $ – today is rainy and the fifth day is rainy;

$a_{2,2} = 1$ – today is rainy and the fifth day is rainy.

So… Am I on the right track or have I driven straight in to the deep dark forest?

Best Answer

The chart for $1$ is fine. $3$ is also correct (assuming that the probability of a rainy day/sunny day is $50/50$ when it is not already guaranteed).

The Markov chain you set up for $2$ could work (and would actually be the best way given the starting condition that today it's sunny), but the problem is that you could only move along days if they are equivalent mod $5$. You couldn't directly set up a relation between day $1$ and day $2$. A more exhaustive way, when the starting condition may be different, would be to use a $32$ state Markov chain. Each state would be the weather for the last $5$ days (including the current day).

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