4-2 - 4.2 Applications of Eigenvalues/Eigenvectors Markov...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
4.2 Applications of Eigenvalues/Eigenvectors – Markov Chains A Markov chain is a discrete-time stochastic process used to model certain systems in physics and statistics, for example. With a Markov Chain, a transition matrix is used to predict future states of the system. Eigenvalues and eigenvectors are used to calculate the steady state of the system after a long period of time. We will need two things to get started in a Markov Chain problem: the initial state vector, x 0 , and the transition matrix, A. Example: Suppose we gather data about a population of birds that inhabit three islands, named (simply) X, Y, and Z. We find that birds fly back and forth among the three islands (so the population of birds is constant) and suppose we find that the daily process happens as such: Birds on Island X: 20% stay at X; 50% fly to Y; 30% fly to Z Birds on Island Y: 20% stay on Y; 40% fly to Z; 40% fly to X Birds on Island Z: 20% stay on Z; 30% fly to X; 50% fly to Y This information is summarized below. From X From Y From Z To X .2 .4 .3 To Y .5 .2 .5 To Z .3 .4 .2 Because each column represents probabilities and the population of birds is constant, each column sums to 1 (100%). This forms our transition matrix, A = .2 .4 .3 .5 .2 .5 .3 .4 .2 é ë ê ê ê ù û ú ú ú . Suppose at the time that we begin collecting data, there are 400 birds on Island X, 300 on Island Y and 800 on Island Z. This gives us an initial state vector, x 0 = 400 300 800 é ë ê ê ê ù û ú ú ú . Note that this population, 1500, will be a constant throughout this process--we
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

4-2 - 4.2 Applications of Eigenvalues/Eigenvectors Markov...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online