# FractionalGaussianNoiseProcess

FractionalGaussianNoiseProcess[μ,σ,h]

represents a fractional Gaussian noise process with drift μ, volatility σ, and Hurst index h.

FractionalGaussianNoiseProcess[h]

represents a fractional Gaussian noise process with drift 0, volatility 1, and Hurst index h.

# Details

• FractionalGaussianNoiseProcess is a continuous-time and continuous-state random process.
• FractionalGaussianNoiseProcess is a Gaussian process with mean function and covariance function .
• FractionalGaussianNoiseProcess[μ,σ,h] is equivalent to TransformedProcess[x[t+1]-x[t],xFractionalBrownianMotionProcess[μ,σ,h],t].
• FractionalGaussianNoiseProcess allows μ to be any real number, σ to be any positive real number, and h to be a real number between 0 and 1.
• FractionalGaussianNoiseProcess can be used with such functions as RandomFunction and CovarianceFunction.

# Examples

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## Basic Examples(3)

Simulate a fractional Gaussian noise process:

Plot the path:

Mean and variance functions are constant:

Covariance function:

## Scope(11)

### Basic Uses(6)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for various Hurst indices:

Process parameter estimation:

Correlation function as a function of lags:

Absolute correlation function:

### Process Slice Properties(5)

Univariate SliceDistribution:

First-order probability density function does not depend on time:

Compare with the density function of a normal distribution:

Multivariate slice distribution:

Slice distribution of higher order will not autoevaluate:

Second-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis are constant:

Generating functions:

CentralMoment and its generating function:

Cumulant and its generating function:

## Generalizations & Extensions(1)

Useful shortcut evaluates to its full-form counterpart:

## Applications(1)

Consider the time series of yearly minimal water levels of the Nile River for the years 6221281:

Fit a fractional Gaussian noise process:

Compare means of the data and the model:

Compare covariance functions:

Simulate the values for the following 100 years:

## Properties & Relations(7)

FractionalGaussianNoiseProcess is weakly stationary:

Fractional Gaussian noise process is mean ergodic:

The process is weakly stationary:

Calculate absolute correlation function:

Find value of the strip integral:

Check if the limit of the integral is 0 to conclude mean ergodicity:

Fractional Gaussian noise does not have independent increments for :

Compare to the product of expectations:

Fractional Gaussian noise has a long memory for a Hurst parameter greater than 1/2:

Covariance function for a long memory process is not summable:

For a short memory, process is summable:

Conditional cumulative probability distribution:

Fractional Gaussian noise is self-similar:

Calculate scaled sums of each path:

Fit a normal distribution:

Compare to the slice distribution of the process:

Probability density histogram and functions:

Compare to non-weakly stationary FractionalBrownianMotionProcess:

## Neat Examples(3)

Simulate a fractional Gaussian noise process in two dimensions:

Study the behavior of fractional Gaussian noise depending on the Hurst parameter:

Simulate paths from a fractional Gaussian noise process:

Take a slice at 1 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1:

Wolfram Research (2014), FractionalGaussianNoiseProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalGaussianNoiseProcess.html.

#### Text

Wolfram Research (2014), FractionalGaussianNoiseProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalGaussianNoiseProcess.html.

#### CMS

Wolfram Language. 2014. "FractionalGaussianNoiseProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalGaussianNoiseProcess.html.

#### APA

Wolfram Language. (2014). FractionalGaussianNoiseProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalGaussianNoiseProcess.html

#### BibTeX

@misc{reference.wolfram_2024_fractionalgaussiannoiseprocess, author="Wolfram Research", title="{FractionalGaussianNoiseProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FractionalGaussianNoiseProcess.html}", note=[Accessed: 29-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_fractionalgaussiannoiseprocess, organization={Wolfram Research}, title={FractionalGaussianNoiseProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/FractionalGaussianNoiseProcess.html}, note=[Accessed: 29-May-2024 ]}