Transition matrices to describe the probability of moving between distinct states or bins. We look at how to use a transition matrix to make short-term and long-term predictions. We start by creating a diagram and a matrix to represent the probabilities of moving between three bins (A, B, and C), including the probability of staying in the same bin. We can then compute the probabilities of an item being in a particular bin after a certain number of moves using powers of the transition matrix.
Happily these computations involve stochastic matrices, where columns sum to one. We observe that as the matrix is raised to higher powers, it stabilizes, indicating long-term probabilities. I point out the amazing result that initial conditions become irrelevant in long-term predictions.
Towards the end, a special case of a two-state system is examined using eigenvalues and eigenvectors, demonstrating how to calculate the long-term behavior in such a system. The proof at the end is optional--it requires knowledge of eigenvalues and eigenvectors (and mentions matrix diagonalization). However, do check out the form of the vector u⃗ for the two-state case.
(Mathematical Modeling Lecture 4.3)
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21 июл 2021
