# DiscreteMarkovChain¶

class DiscreteMarkovChain(*args)

Discrete Markov chain process.

Parameters
originDistribution or , optional

Probability distribution of the Markov chain origin, i.e. state of the process at . By default, the origin is set to a Dirac distribution of value 0.0.

transitionMatrixSquareMatrix, optional

Transition matrix of the process. The matrix must be square, of dimension equal to the number of possible states of the process. By default, the transition matrix of the process is set to the 1x1 matrix [1].

timeGridTimeSeries, optional

The time grid of the process. By default, the time grid is reduced to one time stamp equal to 0.

Notes

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

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

The transition matrix of the process is square, and its dimension is equal to the number of states of the process. Besides, is a stochastic matrix, i.e.:

The origin of the process must be provided either as a deterministic value , or as a probability distribution. In this case, the distribution of must be 1D, and its support must be a part of .

Examples

Create a Markov chain:

>>> import openturns as ot
>>> timeGrid = ot.RegularGrid(0, 0.1, 10)
>>> transitionMatrix = ot.SquareMatrix([[0.9,0.05,0.05],[0.7,0.0,0.3],[0.8,0.0,0.2]])
>>> origin = 0
>>> myMarkovChain = ot.DiscreteMarkovChain(origin, transitionMatrix, timeGrid)

Get a realization:

>>> myReal = myMarkovChain.getRealization()

Methods

 Compute the stationary distribution of the Markov chain. exportToDOTFile(filename) Export to DOT graph. Accessor to the object’s name. Get a continuous realization. Accessor to the covariance model. Get the description of the process. getFuture(*args) Prediction of the future iterations of the process. getId() Accessor to the object’s id. Get the dimension of the domain . getMarginal(*args) Get the marginal of the random process. getMesh() Get the mesh. getName() Accessor to the object’s name. Accessor to the origin. Get the dimension of the domain . Get a realization of the process. getSample(size) Get realizations of the process. Accessor to the object’s shadowed id. Get the time grid of observation of the process. Accessor to the transition matrix. getTrend() Accessor to the trend. Accessor to the object’s visibility state. hasName() Test if the object is named. Test if the object has a distinguishable name. Test whether the process is composite or not. isNormal() Test whether the process is normal or not. Test whether the process is stationary or not. setDescription(description) Set the description of the process. setMesh(mesh) Set the mesh. setName(name) Accessor to the object’s name. setOrigin(*args) Accessor to the origin. setShadowedId(id) Accessor to the object’s shadowed id. setTimeGrid(timeGrid) Set the time grid of observation of the process. setTransitionMatrix(transitionMatrix) Accessor to the transition matrix. setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

computeStationaryDistribution()

Compute the stationary distribution of the Markov chain.

Returns
distributionUserDefined

The stationary probability distribution of the Markov chain: its probability table is the left eigenvector of the transition matrix associated to the eigenvalue .

Examples

Compute the stationary distribution of a Markov chain:

>>> import openturns as ot
>>> timeGrid = ot.RegularGrid(0, 0.1, 10)
>>> transitionMatrix = ot.SquareMatrix([[0.9,0.05,0.05],[0.7,0.0,0.3],[0.8,0.0,0.2]])
>>> origin = 0
>>> myMarkovChain = ot.DiscreteMarkovChain(origin, transitionMatrix, timeGrid)
>>> distribution = myMarkovChain.computeStationaryDistribution()
>>> print(distribution)
UserDefined({x = [0], p = 0.333333}, {x = [1], p = 0.333333}, {x = [2], p = 0.333333})
exportToDOTFile(filename)

Export to DOT graph.

DOT is a graph description language.

Parameters
filenamestr

The name of the file to be written.

Notes

The graph can be customized using the following ResourceMap string entries:

• DiscreteMarkovChain-DOTArcColor

• DiscreteMarkovChain-DOTLayout

• DiscreteMarkovChain-DOTNodeColor

• DiscreteMarkovChain-DOTNodeShape

getClassName()

Accessor to the object’s name.

Returns
class_namestr

The object class name (object.__class__.__name__).

getContinuousRealization()

Get a continuous realization.

Returns
realizationFunction

According to the process, the continuous realizations are built:

• either using a dedicated functional model if it exists: e.g. a functional basis process.

• or using an interpolation from a discrete realization of the process on : in dimension , a linear interpolation and in dimension , a piecewise constant function (the value at a given position is equal to the value at the nearest vertex of the mesh of the process).

getCovarianceModel()

Accessor to the covariance model.

Returns
cov_modelCovarianceModel

Covariance model, if any.

getDescription()

Get the description of the process.

Returns
descriptionDescription

Description of the process.

getFuture(*args)

Prediction of the future iterations of the process.

Parameters
stepNumberint,

Number of future steps.

sizeint, , optional

Number of futures needed. Default is 1.

Returns
prediction

future iterations of the process. If , prediction is a TimeSeries. Otherwise, it is a ProcessSample.

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getInputDimension()

Get the dimension of the domain .

Returns
nint

Dimension of the domain : .

getMarginal(*args)

Get the marginal of the random process.

Parameters
kint or list of ints

Index of the marginal(s) needed.

Returns
marginalsProcess

Process defined with marginal(s) of the random process.

getMesh()

Get the mesh.

Returns
meshMesh

Mesh over which the domain is discretized.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOrigin()

Accessor to the origin.

Returns
originDistribution

The probability distribution of the origin of the Markov chain.

getOutputDimension()

Get the dimension of the domain .

Returns
dint

Dimension of the domain .

getRealization()

Get a realization of the process.

Returns
realizationField

Contains a mesh over which the process is discretized and the values of the process at the vertices of the mesh.

getSample(size)

Get realizations of the process.

Parameters
nint,

Number of realizations of the process needed.

Returns
processSampleProcessSample

realizations of the random process. A process sample is a collection of fields which share the same mesh .

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getTimeGrid()

Get the time grid of observation of the process.

Returns
timeGridRegularGrid

Time grid of a process when the mesh associated to the process can be interpreted as a RegularGrid. We check if the vertices of the mesh are scalar and are regularly spaced in but we don’t check if the connectivity of the mesh is conform to the one of a regular grid (without any hole and composed of ordered instants).

getTransitionMatrix()

Accessor to the transition matrix.

Returns
matrixSquareMatrix

The transition matrix of the process, of dimension .

getTrend()

Accessor to the trend.

Returns
trendTrendTransform

Trend, if any.

getVisibility()

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

True if the name is not empty and not the default one.

isComposite()

Test whether the process is composite or not.

Returns
isCompositebool

True if the process is composite (built upon a function and a process).

isNormal()

Test whether the process is normal or not.

Returns
isNormalbool

True if the process is normal.

Notes

A stochastic process is normal if all its finite dimensional joint distributions are normal, which means that for all and , with , there is and such that:

where , and and is the symmetric matrix:

A Gaussian process is entirely defined by its mean function and its covariance function (or correlation function ).

isStationary()

Test whether the process is stationary or not.

Returns
isStationarybool

True if the process is stationary.

Notes

A process is stationary if its distribution is invariant by translation: , , , we have:

setDescription(description)

Set the description of the process.

Parameters
descriptionsequence of str

Description of the process.

setMesh(mesh)

Set the mesh.

Parameters
meshMesh

Mesh over which the domain is discretized.

setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setOrigin(*args)

Accessor to the origin.

Parameters
origin

The probability distribution of the origin of the Markov chain.

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setTimeGrid(timeGrid)

Set the time grid of observation of the process.

Returns
timeGridRegularGrid

Time grid of observation of the process when the mesh associated to the process can be interpreted as a RegularGrid. We check if the vertices of the mesh are scalar and are regularly spaced in but we don’t check if the connectivity of the mesh is conform to the one of a regular grid (without any hole and composed of ordered instants).

setTransitionMatrix(transitionMatrix)

Accessor to the transition matrix.

Parameters
matrixSquareMatrix

The transition matrix of the process, of dimension .

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.