ARMA¶
(Source code, png, hires.png, pdf)

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

__init__
(self, *args)¶ Initialize self. See help(type(self)) for accurate signature.

computeNThermalization
(self, epsilon)¶ Accessor to the stored state of the ARMA process.
 Parameters
 epsfloat,
 Returns
 Ntherint,
The number of iterations of the ARMA process before being stationary and independent of its initial 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
(self)¶ Accessor to the AR coefficients of the ARMA process.
 Returns
 ARCoeff
ARMACoefficients
The AR coefficients of the linear recurrence defining the process.
 ARCoeff

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getContinuousRealization
(self)¶ 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
(self)¶ Accessor to the covariance model.
 Returns
 cov_model
CovarianceModel
Covariance model, if any.
 cov_model

getDescription
(self)¶ Get the description of the process.
 Returns
 description
Description
Description of the process.
 description

getFuture
(self, *args)¶ Get possible futures from the current state of the ARMA process.
 Parameters
 Nitint,
The number of time stamps of the future.
 Nrealint,
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
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getInputDimension
(self)¶ Get the dimension of the domain .
 Returns
 nint
Dimension of the domain : .

getMACoefficients
(self)¶ Accessor to the MA coefficients of the ARMA process.
 Returns
 MACoeff
ARMACoefficients
The MA coefficients of the linear recurrence defining the process.
 MACoeff

getMarginal
(self, *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

getNThermalization
(self)¶ Accessor to the number of time stamps used to thermalize the process.
 Returns
 Ntherint,
The number of time stamps used to make the ARMA realization be independent from its actual state.
Default precision is: .

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getOutputDimension
(self)¶ Get the dimension of the domain .
 Returns
 dint
Dimension of the domain .

getRealization
(self)¶ 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
(self, 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
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getState
(self)¶ 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.
 ARMAstate

getTimeGrid
(self)¶ 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

getTrend
(self)¶ Accessor to the trend.
 Returns
 trend
TrendTransform
Trend, if any.
 trend

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

getWhiteNoise
(self)¶ Accessor to the white noise defining the ARMA process.
 Returns
 whiteNoise
WhiteNoise
The white noise used in the linear recurrence of the ARMA process.
 whiteNoise

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

isComposite
(self)¶ Test whether the process is composite or not.
 Returns
 isCompositebool
True if the process is composite (built upon a function and a process).

isNormal
(self)¶ 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
(self)¶ 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
(self, description)¶ Set the description of the process.
 Parameters
 descriptionsequence of str
Description of the process.

setNThermalization
(self, n)¶ Accessor to the number of time stamps used to thermalize the process.
 Parameters
 Ntherint,
The number of time stamps used to make the ARMA realization independent from its actual state.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setState
(self, 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.
 ARMAstate

setTimeGrid
(self, 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

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.

setWhiteNoise
(self, 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.
 whiteNoise