# FARIMAProcess

FARIMAProcess[{a1,,ap},d,{b1,,bq},v]

represents an autoregressive fractionally integrated moving-average process such that its d difference is an ARMAProcess[{a1,,ap},{b1,,bq,v].

FARIMAProcess[{a1,,ap},d,{b1,,bq},Σ]

represents a vector autoregressive fractionally integrated moving-average process (y1(t), ,yn(t)) such that its (d,,d) difference is a vector ARMAProcess.

FARIMAProcess[{a1,,ap},{d1,,dn},{b1,,bq},Σ]

represents a vector autoregressive fractionally integrated moving-average process (y1(t), ,yn(t)) such that its (d1,,dn) difference is a vector ARMAProcess.

# Details

• FARIMAProcess is also known as ARFIMA or long-memory time series.
• FARIMAProcess is a discrete-time and continuous-state random process.
• The FARIMA process is described by the difference equations , where is the state output, is the white noise input, and is the shift operator.
• The scalar FARIMA process has transfer function , where .
• The vector FARIMA process has transfer matrix , where , and where is the × identity matrix.
• A scalar FARIMA process should have real coefficients ai, bj, real integrating parameter d such that , and a positive variance v.
• An -dimensional vector FARIMA process should have real coefficient matrices ai and bj of dimensions ×, real integrating parameters di such that or real integrating parameter d such that , and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
• FARIMAProcess[p,d,q] and FARIMAProcess[p,q] represent a FARIMA process of orders p and q with known or unknown integration order d for use in EstimatedProcess and related functions.
• FARIMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

# Examples

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## Basic Examples(3)

Simulate a FARIMA process:

Covariance function:

Correlation function:

Partial correlation function:

## Scope(25)

### Basic Uses(8)

Simulate an ensemble of paths:

Simulate with given precision:

Simulate a first-order scalar process:

Sample paths for positive and negative values of the integration parameter:

Simulate a two-dimensional process:

Create a 2D sample path function from the data:

The color of the path is the function of time:

Create a 3D sample path function with time:

The color of the path is the function of time:

Simulate a three-dimensional process:

Create a sample path function from the data:

The color of the path is the function of time:

Estimate process parameters:

Compare the sample covariance function with that of the estimated process:

Estimate integration order of the fractional noise:

Compare the sample correlation function with that of the estimated process:

Forecast future values:

Find the forecast for the next 20 steps:

Show the forecast path:

Plot the data and the forecasted values:

### Covariance and Spectrum(5)

Closed-form correlation function for pure FARIMA:

For FARIMA with autoregressive and moving-average components available numerically:

Partial correlation has closed form for special case:

Correlation matrix:

Special case of pure FARIMA:

Covariance matrix:

Special case of pure FARIMA:

Power spectral density:

Vector FARIMAProcess:

### Stationarity and Invertibility(4)

Check if a time series is weakly stationary:

Find conditions for a process to be weakly stationary:

Find invertibility conditions:

Check if a time series is invertible:

Find its invertible representation:

### Estimation Methods(2)

The available methods for estimating a FARIMAProcess:

Compare log likelihoods:

The available methods for estimating fractional noise:

Spectral estimator allows you to specify windows used for PowerSpectralDensity calculation:

Spectral estimator allows the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

### Process Slice Properties(5)

Single time SliceDistribution:

Multiple time slice distributions:

First-order probability density function:

Stationary mean and variance:

Compare with the density function of a normal distribution:

Compute the expectation of an expression:

Calculate a probability:

Skewness and kurtosis:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

### Representations(1)

Approximate with an ARMAProcess:

Approximate with an MAProcess:

Approximate with an ARProcess:

Compare random samples of the process and its approximation:

## Applications(1)

Consider the time series of yearly minimal water levels of the Nile River for the years 6221284:

Centralize the data:

Find a FARIMA model:

Forecast the future values:

Adjust back to the mean level and proper time stamps:

Plot Nile minimum flow together with the hundred-year forecast:

## Properties & Relations(5)

The correlations are summable for -1/2<d<0:

For 0<d<1/2, the sum of correlations diverges:

FARIMAProcess has a long memory for positive integration orders:

FARIMAProcess is a generalization of an ARMAProcess:

FARIMAProcess is a generalization of an ARProcess:

FARIMAProcess is a generalization of an MAProcess:

## Possible Issues(2)

ToInvertibleTimeSeries does not always exist:

Method of moments is only supported for estimation of fractional noise:

Use the other solver:

## Neat Examples(2)

Simulate a three-dimensional FARIMAProcess:

Simulate paths from a FARIMA process:

Take a slice at 50 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 50:

Wolfram Research (2012), FARIMAProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FARIMAProcess.html.

#### Text

Wolfram Research (2012), FARIMAProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FARIMAProcess.html.

#### CMS

Wolfram Language. 2012. "FARIMAProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FARIMAProcess.html.

#### APA

Wolfram Language. (2012). FARIMAProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FARIMAProcess.html

#### BibTeX

@misc{reference.wolfram_2024_farimaprocess, author="Wolfram Research", title="{FARIMAProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/FARIMAProcess.html}", note=[Accessed: 26-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_farimaprocess, organization={Wolfram Research}, title={FARIMAProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/FARIMAProcess.html}, note=[Accessed: 26-May-2024 ]}