1 Brief Overview of Markov Processes To begin with, we will investigate systems of subjects in various states under cons

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1 Brief Overview Of Markov Processes To Begin With We Will Investigate Systems Of Subjects In Various States Under Cons 1 (160.92 KiB) Viewed 21 times

1 Brief Overview of Markov Processes To begin with, we will investigate systems of subjects in various states under consideration. There are a fixed number n of states, a fixed number of subjects and we use a vector x to indicate the number of subjects in each state. Each component of x denotes the number of subjects in each state. That is, the first component of x is the number of subjects in the first state, etc. Sometimes, we indicate the fraction of subjects in each state instead of the total number. In this fraction case, the sum of the components of x must sum to one. At a fixed interval of time, the subjects change their state (or remain in the same state). Each subject must be in a state after this transition. We model this transition by considering the probability that a subject originally in the j-th state ends up in the i-th state after the transition, denoted Pij. You can see from your textbook or the video that if we take Px, we get a vector with the distribution of subjects into the states after the transition. If we do m transitions, we can find the distribution of subjects into the states by taking pix, where xo is the initial distribution. We assume these transition probabilities are independent of time. We call such a process a Markov chain or Markov process. In the Dropbox link, you will find a .m file with the transition matrix P for this project and an initial distribution vector. You will note that we will be using the form of the state vector x of having the components add to one. This means each component will represent the ratio of the whole that shares that characteristic. Xo = 2 Description of the Application In this project, you will investigate the land use of a small city over time. We will look at whether land is used for residential, commerical, civic, or industrial purposes. The state vector x starts in 2021 as [0.357 0.20 0.05 [0.40 where the first component indicates the ratio of land in the city used for residential purposes, the second component is the ratio of land in the city used for commercial purposes, the third component is the ratio of land used for civic purposes, and the four compenent is the ratio of land used for industrial purposes. The transition matrix for how the properties change EVERY FIVE YEARS (so the fixed interval of time is five years) is [0.70 0.15 0.00 0.00 P= 0.12 0.70 0.00 0.06 0.10 0.05 0.95 0.01 0.08 0.10 0.05 0.93 (these are defined in the provided .m file on D2L). 1. (8 points) Use MATLAB investigate what happens to the state vector after undergoing twenty transi- tions (or 100 years). What do you notice happening to the state vector? A for loop in MATLAB may be helpful. 2. (10 points) Use the eig command in MATLAB to calculate the eigenvalues and corresponding eigenvec- tors of P. Make sure to give eig the correct number of outputs (see online help with MATLAB for use of the eig command). Make sure to use [V,D]=eig(...) to get a matrix V of eigenvectors and diagonal
matrix D of eigenvalues of P (think of diagonalization). MATLAB will give you eigenvectors that have norm one. 3. (15 points) You should see that one is an eigenvalue. What do the eigenvectors corresponding to the eigenvalue one look like think about the eigenspace E.)? In particular, find an eigenvector in E, whose components add up to one (Hint: Find an appropriate t). How does this relate to what you observe in question 1? 4. (10 points) Use MATLAB to explore what happens in transitions if we start with an eigenvector of all non-negative entries corresponding to the eigenvalue one. Why does your answer make sense in light of the definitions of eigenvalues and eigenvectors?