Background: Markov chains could be used to model a plethora of phenomena that happen in our world. The only assumption that we would have to accept is the fact that what we are trying to model depends only on the last step, and not on all previous steps $($the whole history$).$

For example, Sahin and Sen $(2001)$ model hourly wind speeds in a NW part of Turkey as a Markov chain $(x_n{)}_{\text{n}\text{i}\text{n}\text{N}}$ with $7$ states representing different wind speed levels. Since in Python arrays are indexed starting from $0,$ let us consider the states to be $S=\{0,1,2,3,4,5,6\},$ with $0$ representing the lowest wind speed level. The transition matrix is given by:

$P=\{\begin{array}{ccc}\text{:}[,& 0,1,2,3,4,5,& 6]\\ 0,& 0\mathrm{.}756,0\mathrm{.}113,0\mathrm{.}129,0\mathrm{.}002,0,0,& 0\\ 1,& 0\mathrm{.}174,0\mathrm{.}821,0\mathrm{.}004,0\mathrm{.}001,0,0,& 0\\ 2,& 0\mathrm{.}141,0\mathrm{.}001,0\mathrm{.}776,0\mathrm{.}082,0,0,& 0\\ 3,& 0\mathrm{.}003,0,0\mathrm{.}192,0\mathrm{.}753,0\mathrm{.}052,0,& 0\\ 4,& 0,0,0\mathrm{.}002,0\mathrm{.}227,0\mathrm{.}735,0\mathrm{.}036,& 0\\ 5,& 0,0,0,0\mathrm{.}007,0\mathrm{.}367,0\mathrm{.}604,& 0\mathrm{.}022\\ 6,& 0,0,0,0,0\mathrm{.}053,0\mathrm{.}158,& 0\mathrm{.}789\end{array}$

As usual, we start with loading some packages:

import numpy as np

import pandas as pd

from numpy import linalg

import scipy. linalg

import matplotlib.pyplot as plt