.. _MarkovHW:

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.

.. math::

   \mathbf{M} = \begin{bmatrix}
                   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
                \end{bmatrix}

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.

.. math:: \bm{start} = \begin{bmatrix}
                         0.25 \\  0.3 \\  0.2 \\  0.1 \\  0.15 
                       \end{bmatrix}

::

    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
:ref:`markov` as needed.

#. Calculate the vectors showing the expected land use in years 2025,
   2030, and 2035. Find powers of :math:`\bf{M}` to determine this.

#. Find the eigenvectors and eigenvalues of the Markov matrix.

#. Calculate the change of basis coefficients needed to express
   :math:`\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.