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Create a discrete Markov chain processΒΆ

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

A discrete Markov chain X: \Omega \times \mathcal{D} \rightarrow E, where E = [\![ 0,...,p-1]\!] is a process where \mathcal{D}=\mathbb{R} discretized on the time grid (t_k)_{k \geq 0} such that:

\begin{aligned} \forall k > 0,\: \mathbb{P} ( X_{t_k} \> | \> X_{t_0},...X_{t_{k-1}} ) = \mathbb{P} ( X_{t_k} \> | \> X_{t_{k-1}} ) \end{aligned}

The transition matrix of the process \mathcal{M} = (m_{i,j}) can be defined such that:

\begin{aligned} \forall t_k \in \mathcal{D}, \forall i,j < p , \> m_{i+1,j+1} = \mathbb{P} (X_{t_{k+1}} = j \> | \> X_{t_{k}} = i) \end{aligned}

The library proposes to model it through the object DiscreteMarkovChain defined thanks to the origin X_{t_0} (which can be either deterministic or uncertain), the transition matrix \mathcal{M} 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)
plot discrete markov chain process
<matplotlib.image.AxesImage object at 0x7ff3e53bbb00>

Invert axes and add texts

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

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)
Discrete Markov chain

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)
Discrete Markov chain, 3 realizations

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 future 10 steps

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()
Three Markov chain futures, 10 steps