14.16. Markov Matrix of Land UseΒΆ

Stochastic Markov matrices are used by city planners to analyze trends in land use. Researchers collect data of how each parcel of land is used and track changes. The near by city conducted such a study and published a Markov matrix showing changes in land use between the years 2015 and 2020.

The categories are:

  1. Residential
  2. Office
  3. Commercial
  4. Parking
  5. Vacant

\bf{M} = \mat{0.85 0.05 0.05 0.05 0.10; 0.025 0.55 0.20 0.10 0.10; 0.025 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}

The columns represents the transition probabilities for the initial usage of the land in 2015. The rows represent the probabilities for the land transitioning to a usage in 2020. For example, land that was in residential use in 2015 (column 1) had a 0.85 chance of still being in residential use in 2020 (row 1) and a 0.05 chance of being vacant in 2020 (row 5).

Let the ratios of land use in 2015 be as follows.

\mathbf{start} = \vector{0.25; 0.3; 0.2; 0.1; 0.15}

For convenience, the file LandUse.mat may be loaded to create the \bf{M} matrix and \bm{start} vector.

We will assume that the conditions driving the changes are steady. Use the MATLAB Command Window to determine the following. See Application: Markov Matrices and Change of Basis and Difference Equations as needed. Copy the contents from the Command Window into a text file to submit on Canvas.

  1. Calculate the vectors showing the expect land use in years 2025, 2030, and 2035. Just find powers of \bf{M} to determine this.
  2. Find the eigenvectors and eigenvalues of the Markov matrix.
  3. Calculate the change of basis coefficients needed to express \mathbf{M}^k \bm{start} as a sum of products of the eigenvectors and eigenvalues.
  4. Using the eigenvectors, eigenvalues, and coefficients calculate (matrix multiplication) the expected land usage in year 2040 (5 sample periods from 2015).
  5. Using the eigenvectors, eigenvalues, and coefficients find the steady state land usage.