Taking the American election as an example, the historical data of the past ten elections are obtained first, and then the transfer matrix of voters' intentions is obtained according to the historical data. We assume that we get the following transfer matrix (obviously this data is not true):
In this way, a difference equation group is formed.
r n+ 1 = 0.75 r n+0.20d n+0.40 I n
d n+ 1 = 0.05 r n+0.60d n+0.20 I n
I n+ 1 = 0.20 r n+0.20d n+0.40 I n
According to the content of our previous difference equation, we can infer the long-term trend of voters' voting intention.
Finally, the long-term trend is: 56% of the candidates are * * * and the party, 19% are Democrats, and 25% are independent candidates.
This problem can also be solved directly by matrix.
There is another theorem about the transfer matrix properties of Markov chains called Chapman-Kolmogorov equation:
That is to say, P (m) = (P ij (m)) is an m-step transition matrix from state I to state J, which should be easy for friends who are familiar with matrix operation to prove.
We have got the one-step transition matrix, and we only need to do one iteration: