11.10. Markov Matrix Homework¶
City planners use stochastic Markov matrices to analyze trends in land use. Researchers collect data on how each parcel of land is used and track changes. A city conducted such a study and published a Markov matrix showing changes in land use between the years 2015 and 2020.
The categories are:
Residential.
Office.
Commercial.
Parking.
Vacant.
The columns represent the transition probabilities for the initial usage of the land in 2015. The rows represent the probabilities for the land transitioning to a particular usage in 2020. For example, land that was in residential use in 2015 (column 1) had a 0.85 probability of still being in residential use in 2020 (row 1) and a 0.05 probability of being vacant in 2020 (row 5).
Let the ratios of land use in 2015 be as follows.
M = np.array([[0.85, 0.05, 0.05, 0.05, 0.10],
[0.02, 0.55, 0.20, 0.10, 0.10],
[0.03, 0.15, 0.65, 0.25, 0.20],
[0.05, 0.20, 0.05, 0.55, 0.20],
[0.05, 0.05, 0.05, 0.05, 0.40]]); print(M)
start = np.array([[0.25, 0.3, 0.2, 0.1, 0.15]]).T; print(start)
We will assume that the conditions driving the changes are consistent. Use Python to determine the following. See Application: Markov Matrices as needed.
Calculate the vectors showing the expected land use in years 2025, 2030, and 2035. Find powers of \(\bf{M}\) to determine this.
Find the eigenvectors and eigenvalues of the Markov matrix.
Calculate the change of basis coefficients needed to express \(\mathbf{M}^k \bm{start}\) as a sum of products of the eigenvectors and eigenvalues.
Using the eigenvectors, eigenvalues, and coefficients, calculate (matrix multiplication) the expected land usage in year 2040 (5 sample periods from 2015).
Using the eigenvectors, eigenvalues, and coefficients, find the steady state land usage.
