Transition Probability Matrix

However, using a tree diagram also has its limitations: if we want to calculate the probability after a month or even half a year, the tree diagram method will no longer be efficient. Therefore, mathematicians adopted the calculation method using Matrix. The matrix below is called the “transition probability matrix”.

$ \left(\begin{array}{cccc}P_{11}&P_{12}&...&P_{1n}\\P_{21}&P_{22}&...&P_{2n}\\...&...&...&...\\P_{m1}&P_{m2}&...&P_{mn}\end{array}\right) $

Just as its name implies, each element inside the transition probability matrix describes a transition probability from state to another. Here, $ P_{11} $ represents the probability of event 1 occurring again on the second day after event 1 occurred on the first day; $ P_{21} $ represents the probability of event 1 occurring on the second day after event 2 occurred on the first day… and so on and so forth. Using this method, the transition probability matrix of the weather example can be written as:

Markovmatrix.jpg

The rows represent the current state, and the columns represent the future state. To read this matrix, one would notice that $ P_{11} $, $ P_{21} $, and $ P_{31} $ are all transition probabilities of the current state of a rainy day. This is also the case for column two with the current state of a sunny day, and column three with the current state of a cloudy day. Notice how the sum of each column and row add up to one, confirming that they are valid probabilities.

State Vectors A state vector is a column vector whose $ i $th component is the probability that the system is in the $ i $th state at that time. In the context of our example problem, if the current day has rainy weather, the state vector for the current day would be $ \left(\begin{array}{ccc}1&0&0\end{array}\right) $, where the value of the first row signifies that there is a one hundred percent chance for the current day to have rainy weather.


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