DiscreteMarkovChain¶
(Source code, png, hires.png, pdf)
- class DiscreteMarkovChain(*args)¶
Discrete Markov chain process.
- Parameters
- origin
Distribution
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.
- transitionMatrix
SquareMatrix
, 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].
- timeGrid
TimeSeries
, optional The time grid of the process. By default, the time grid is reduced to one time stamp equal to 0.
- origin
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
- distribution
UserDefined
The stationary probability distribution of the Markov chain: its probability table is the left eigenvector of the transition matrix associated to the eigenvalue .
- distribution
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
- realization
Function
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).
- realization
- getCovarianceModel()¶
Accessor to the covariance model.
- Returns
- cov_model
CovarianceModel
Covariance model, if any.
- cov_model
- getDescription()¶
Get the description of the process.
- Returns
- description
Description
Description of the process.
- description
- 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
ProcessSample
orTimeSeries
future iterations of the process. If , prediction is a
TimeSeries
. Otherwise, it is aProcessSample
.
- prediction
- 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
- marginals
Process
Process defined with marginal(s) of the random process.
- marginals
- getName()¶
Accessor to the object’s name.
- Returns
- namestr
The name of the object.
- getOrigin()¶
Accessor to the origin.
- Returns
- origin
Distribution
The probability distribution of the origin of the Markov chain.
- origin
- getOutputDimension()¶
Get the dimension of the domain .
- Returns
- dint
Dimension of the domain .
- getRealization()¶
Get a realization of the process.
- Returns
- realization
Field
Contains a mesh over which the process is discretized and the values of the process at the vertices of the mesh.
- realization
- getSample(size)¶
Get realizations of the process.
- Parameters
- nint,
Number of realizations of the process needed.
- Returns
- processSample
ProcessSample
realizations of the random process. A process sample is a collection of fields which share the same mesh .
- processSample
- getShadowedId()¶
Accessor to the object’s shadowed id.
- Returns
- idint
Internal unique identifier.
- getTimeGrid()¶
Get the time grid of observation of the process.
- Returns
- timeGrid
RegularGrid
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).
- timeGrid
- getTransitionMatrix()¶
Accessor to the transition matrix.
- Returns
- matrix
SquareMatrix
The transition matrix of the process, of dimension .
- matrix
- getTrend()¶
Accessor to the trend.
- Returns
- trend
TrendTransform
Trend, if any.
- trend
- 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.
- setName(name)¶
Accessor to the object’s name.
- Parameters
- namestr
The name of the object.
- setOrigin(*args)¶
Accessor to the origin.
- Parameters
- origin
Distribution
or The probability distribution of the origin of the Markov chain.
- origin
- setShadowedId(id)¶
Accessor to the object’s shadowed id.
- Parameters
- idint
Internal unique identifier.
- setTimeGrid(timeGrid)¶
Set the time grid of observation of the process.
- Returns
- timeGrid
RegularGrid
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).
- timeGrid
- setTransitionMatrix(transitionMatrix)¶
Accessor to the transition matrix.
- Parameters
- matrix
SquareMatrix
The transition matrix of the process, of dimension .
- matrix
- setVisibility(visible)¶
Accessor to the object’s visibility state.
- Parameters
- visiblebool
Visibility flag.