# Create a discrete Markov chain processΒΆ

This example details first how to create and manipulate a discrete Markov chain.

A discrete Markov chain , where is a process where discretized on the time grid such that:

The transition matrix of the process can be defined such that:

The library proposes to model it through the object DiscreteMarkovChain defined thanks to the origin (which can be either deterministic or uncertain), the transition matrix and the time grid.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)


Define a generic function to plot matrices

def plotMatrix(matrix, texts=False, origin=None, colorbar=False, extent=None, **kwargs):
"""Generic procedure for displaying a matrix with or without text overlay and color bar"""
res = plt.matshow(matrix, origin=origin, extent=extent, **kwargs)
if texts:
if extent is None:
extent = (-0.5, matrix.getNbColumns() - 0.5, -0.5, matrix.getNbRows() - 0.5)
x_step = (extent[1] - extent[0]) / matrix.getNbColumns()
y_step = (extent[3] - extent[2]) / matrix.getNbRows()
for i in range(matrix.getNbColumns()):
for j in range(matrix.getNbRows()):
c = round(
matrix[j if origin == "lower" else (matrix.getNbRows() - j - 1), i],
2,
)
plt.text(
i * x_step + extent[0] + x_step / 2,
j * y_step + extent[2] + y_step / 2,
str(c),
va="center",
ha="center",
)
if colorbar:
plt.colorbar(res)


Define the origin

origin = ot.Dirac(0.0)


Define the transition matrix

transition = ot.SquareMatrix(
[
[0.1, 0.3, 0.5, 0.1],
[0.6, 0.1, 0.2, 0.1],
[0.4, 0.3, 0.1, 0.2],
[0.2, 0.0, 0.1, 0.7],
]
)


Visualize the transition matrix

plt.matshow(transition)

<matplotlib.image.AxesImage object at 0x7ff3e53bbb00>


plotMatrix(
transition,
cmap="gray",
texts=True,
origin="lower",
colorbar=True,
alpha=0.5,
vmin=0,
vmax=1,
)


Define an 1-d mesh

tgrid = ot.RegularGrid(0.0, 1.0, 50)


Markov chain definition and realization

process = ot.DiscreteMarkovChain(origin, transition, tgrid)
real = process.getRealization()
graph = real.drawMarginal(0)
graph.setTitle("Discrete Markov chain")
view = viewer.View(graph)


Get several realizations

process.setTimeGrid(ot.RegularGrid(0.0, 1.0, 20))
reals = process.getSample(3)
graph = reals.drawMarginal(0)
graph.setTitle("Discrete Markov chain, 3 realizations")
view = viewer.View(graph)


Markov chain future 10 steps

future = process.getFuture(10)
graph = future.drawMarginal(0)
graph.setTitle("Markov chain future 10 steps")
view = viewer.View(graph)


Markov chain 3 different futures

futures = process.getFuture(10, 3)
graph = futures.drawMarginal(0)
graph.setTitle("Three Markov chain futures, 10 steps")
view = viewer.View(graph)
plt.show()