# ARMA¶

class ARMA(*args)

ARMA process.

Available constructors:

ARMA()

ARMA(ARCoeff, MACoeff, whiteNoise)

ARMA(ARCoeff, MACoeff, whiteNoise, ARMAstate)

Parameters: ARCoeff : ARMACoefficients The coefficients of the AR part of the recurrence : in dimension 1 and in dimension . Default is: in dimension 1 and the associated time grid is . MACoeff : ARMACoefficients The coefficients of the MA part of the recurrence : in dimension 1 and in dimension . Default is: in dimension 1 and the associated time grid is . whiteNoise : WhiteNoise The white noise distribution of the recurrent relation. Default is: the Normal distribution with zero mean and unit variance in dimension 1. ARMAstate : ARMAState The state of the ARMA process which will be extended to the next time stamps. The state is composed with values of the process and values of the white noise. This constructor is needed to get possible futurs from the current state.

Notes

An ARMA process in dimension is defined by the linear recurrence :

where and .

In dimension 1, an ARMA process is defined by:

where .

Examples

Create an ARMA(4,2) process:

>>> import openturns as ot
>>> myTimeGrid = ot.RegularGrid(0.0, 0.1, 10)
>>> myWhiteNoise = ot.WhiteNoise(ot.Triangular(-1.0, 0.0, 1.0), myTimeGrid)
>>> myARCoef = ot.ARMACoefficients([0.4, 0.3, 0.2, 0.1])
>>> myMACoef = ot.ARMACoefficients([0.4, 0.3])
>>> myARMAProcess = ot.ARMA(myARCoef, myMACoef, myWhiteNoise)

>>> myLastValues = ot.Sample([[0.6], [0.7], [0.3], [0.2]])
>>> myLastNoiseValues = ot.Sample([[1.2], [1.8]])
>>> myARMAState = ot.ARMAState(myLastValues, myLastNoiseValues)
>>> myARMAProcessWithState = ot.ARMA(myARCoef, myMACoef, myWhiteNoise, myARMAState)


Generate a realization:

>>> myTimeSeries = myARMAProcess.getContinuousRealization()


Methods

 computeNThermalization(epsilon) Accessor to the stored state of the ARMA process. getARCoefficients() Accessor to the AR coefficients of the ARMA process. getClassName() Accessor to the object’s name. getContinuousRealization() Get a continuous realization. getCovarianceModel() Accessor to the covariance model. getDescription() Get the description of the process. getFuture(*args) Get possible futures from the current state of the ARMA process. getId() Accessor to the object’s id. getInputDimension() Get the dimension of the domain . getMACoefficients() Accessor to the MA coefficients of the ARMA process. getMarginal(*args) Get the marginal of the random process. getMesh() Get the mesh. getNThermalization() Accessor to the number of time stamps used to thermalize the process. getName() Accessor to the object’s name. getOutputDimension() Get the dimension of the domain . getRealization() Get a realization of the process. getSample(size) Get realizations of the process. getShadowedId() Accessor to the object’s shadowed id. getState() Accessor to the stored state of the ARMA process. getTimeGrid() Get the time grid of observation of the process. getTrend() Accessor to the trend. getVisibility() Accessor to the object’s visibility state. getWhiteNoise() Accessor to the white noise defining the ARMA process. hasName() Test if the object is named. hasVisibleName() Test if the object has a distinguishable name. isComposite() Test whether the process is composite or not. isNormal() Test whether the process is normal or not. isStationary() Test whether the process is stationary or not. setDescription(description) Set the description of the process. setMesh(mesh) Set the mesh. setNThermalization(n) Accessor to the number of time stamps used to thermalize the process. setName(name) Accessor to the object’s name. setShadowedId(id) Accessor to the object’s shadowed id. setState(state) Accessor to the stored state of the ARMA process. setTimeGrid(timeGrid) Set the time grid of observation of the process. setVisibility(visible) Accessor to the object’s visibility state. setWhiteNoise(whiteNoise) Accessor to the white noise defining the ARMA process.
__init__(*args)

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

computeNThermalization(epsilon)

Accessor to the stored state of the ARMA process.

Parameters: eps : float, Nther : int, The number of iterations of the ARMA process before being stationary and independent of its intial state.

Notes

The thermalization number is defined as follows:

where is the integer part of a float and the are the roots of the polynomials (given here in dimension 1) :

getARCoefficients()

Accessor to the AR coefficients of the ARMA process.

Returns: ARCoeff : ARMACoefficients The AR coefficients of the linear recurrence defining the process.
getClassName()

Accessor to the object’s name.

Returns: class_name : str 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).
getCovarianceModel()

Accessor to the covariance model.

Returns: cov_model : CovarianceModel Covariance model, if any.
getDescription()

Get the description of the process.

Returns: description : Description Description of the process.
getFuture(*args)

Get possible futures from the current state of the ARMA process.

Parameters: Nit : int, The number of time stamps of the future. Nreal : int, The number of possible futures that are generated. Default is: .

Notes

• If :
A TimeSeries
One possible future of the ARMA process, from the current state over the next time stamps.
• If :
A ProcessSample
possible futures of the ARMA process, from the current state over the next time stamps.

Note that the time grid of each future begins at the last time stamp of the time grid associated to the time series which is extended.

getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getInputDimension()

Get the dimension of the domain .

Returns: n : int Dimension of the domain : .
getMACoefficients()

Accessor to the MA coefficients of the ARMA process.

Returns: MACoeff : ARMACoefficients The MA coefficients of the linear recurrence defining the process.
getMarginal(*args)

Get the marginal of the random process.

Parameters: k : int or list of ints Index of the marginal(s) needed. marginals : Process Process defined with marginal(s) of the random process.
getMesh()

Get the mesh.

Returns: mesh : Mesh Mesh over which the domain is discretized.
getNThermalization()

Accessor to the number of time stamps used to thermalize the process.

Returns: Nther : int, The number of time stamps used to make the ARMA realization be independent from its actual state. Default precision is: .
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getOutputDimension()

Get the dimension of the domain .

Returns: d : int 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.
getSample(size)

Get realizations of the process.

Parameters: n : int, Number of realizations of the process needed. processSample : ProcessSample realizations of the random process. A process sample is a collection of fields which share the same mesh .
getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
getState()

Accessor to the stored state of the ARMA process.

Returns: ARMAstate : ARMAState The state of the ARMA process which will be extended to the next time stamps. The state is composed with values of the process and values of the white noise.
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).
getTrend()

Accessor to the trend.

Returns: trend : TrendTransform Trend, if any.
getVisibility()

Accessor to the object’s visibility state.

Returns: visible : bool Visibility flag.
getWhiteNoise()

Accessor to the white noise defining the ARMA process.

Returns: whiteNoise : WhiteNoise The white noise used in the linear recurrence of the ARMA process.
hasName()

Test if the object is named.

Returns: hasName : bool True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleName : bool True if the name is not empty and not the default one.
isComposite()

Test whether the process is composite or not.

Returns: isComposite : bool True if the process is composite (built upon a function and a process).
isNormal()

Test whether the process is normal or not.

Returns: isNormal : bool 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: isStationary : bool 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: description : sequence of str Description of the process.
setMesh(mesh)

Set the mesh.

Parameters: mesh : Mesh Mesh over which the domain is discretized.
setNThermalization(n)

Accessor to the number of time stamps used to thermalize the process.

Parameters: Nther : int, The number of time stamps used to make the ARMA realization independent from its actual state.
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setState(state)

Accessor to the stored state of the ARMA process.

Parameters: ARMAstate : ARMAState The state of the ARMA process which will be extended to the next time stamps. The state is composed with values of the process and values of the white noise.
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).
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.
setWhiteNoise(whiteNoise)

Accessor to the white noise defining the ARMA process.

Parameters: whiteNoise : WhiteNoise The white noise used in the linear recurrence of the ARMA process.