Classic Estimator for the Hurst Effect

ClassicEstimators

This module contains functions to estimate them Hurst coefficient, closely related to the long memory parameter, of a time series using the rescaled range (R/S) statistic.

Author

J. Eduardo Vera-Valdés

autocorrelation(x::Array, k::Int; flag::Bool=false)

Computes the autocorrelation function of a time series.

Arguments

• x::Array: The time series.
• k::Int: The lag of the autocorrelation.

Output

• acf::Array: The autocorrelation function.

Optional arguments

• flag::Bool: If true, the autocorrelation function is displayed.

Examples

julia> autocorrelation(randn(100), 10)

autocorrelation_plot(x::Array, k::Int)

Computes the autocorrelation function of a time series.

Arguments

• x::Array: The time series.
• k::Int: The lag of the autocorrelation.

Output

• p1::Plots.Plot: The autocorrelation function.

Examples

julia> autocorrelation_plot(randn(100), 10)

autocovariance(x::Array, k::Int)

Computes the autocovariance of a time series.

Arguments

• x::Array: The time series.
• k::Int: The lag of the autocovariance.

Output

• acv::Array: The autocovariance.

Examples

julia> autocovariance(randn(100), 2)

log_variance_est(x::Array; m::Int=200)

Computes the long memory estimate using the log-variance.

Arguments

• x::Array: The time series.

Output

• d_var::Real: The estimated long memory parameter computed as (beta+1)/2, where beta is the slope of the linear regression of the log-variance plot.

Optional arguments

• m::Int: The number of partitions of the time series. Default is 20. A value of m ≈ length(x)/2 is recommended.

Notes

This function uses the linear regression method on the log-variance to estimate the long memory parameter.

Examples

julia> log_variance_est(randn(100,1); m = 40)

log_variance_plot(x::Array; m::Int=100, slope::Bool=false, slope2::Bool=false)

Produces the log-variance plot of a time series.

Arguments

• x::Array: The time series.

Output

• d_var::Real: The estimated long memory parameter computed as (beta+1)/2, where beta is the slope of the linear regression of the log of the variance plot.

Optional arguments

• m::Int: The number of partitions of the time series. Default is 20. A value of m ≈ length(x)/2 is recommended.
• slope::Bool: If true, the slope of the linear regression is displayed.
• slope2::Bool: If true, the theoretical slope for a short memory process is displayed.

Notes

This function uses the linear regression method on the log of the variance plot to estimate the long memory parameter.

Examples

julia> log_variance_plot(randn(100,1); m = 40)

rescaled_range(x::Array; k::Int = 20)

Computes the rescaled range (R/S) statistic of a time series.

Arguments

• x::Array: The time series.

Optional arguments

• k::Int: The number of partitions of the time series. Default is 20.

Output

• RS::Array: The rescaled range statistic.

Examples

julia> rescaled_range(randn(100))

rescaled_range_est(x::Array; k::Int = 20)

Estimates the long memory parameter of a time series using the rescaled range (R/S) statistic.

Arguments

• x::Array: The time series.

Output

• d::Real: The estimated long memory parameter.

Optional arguments

• k::Int: The number of partitions of the time series. Default is 20.

Notes

This function uses the linear regression method on the log of the rescaled range to estimate the long memory parameter. The Hurst coefficient is related to the long memory parameter d by the formula H = d + 1/2.

Examples

julia> rescaled_range_est(randn(100))

rescaled_range_plot(x::Array; k::Int = 100, slope::Bool = false, slope2::Bool = false)

Produces the rescaled range plot of a time series.

Arguments

• x::Array: The time series.

Output

• p1::Plots.Plot: The rescaled range plot.

Optional arguments

• k::Int: The number of partitions of the time series. Default is 100.
• slope::Bool: If true, the slope of the linear regression is displayed.
• slope2::Bool: If true, the theoretical slope for a short memory process is displayed.

Notes

This function uses the linear regression method on the log of the rescaled range to estimate the long memory parameter. The Hurst coefficient is related to the long memory parameter d by the formula H = d + 1/2.

Examples

julia> rescaled_range_plot(randn(300,1))

smean(x::Array)

Computes the sample mean of a time series.

Arguments

• x::Array: The time series.

Output

• mean::Real: The sample mean.

Examples

julia> smean(randn(100))

sstd(x::Array; k::Int=0)

Computes the sample standard deviation of a time series.

Arguments

• x::Array: The time series.

Output

• std::Real: The sample standard deviation.

Optional arguments

• k::Int: The bias-correction term.

Notes

This function divides by T instead of T-1 if no bias-correction term is provided.

Examples

julia> sstd(randn(100))

sstdk(x::Array, m::Int)

Computes the sample standard deviation of a time series using the m-th order sample mean.

Arguments

• x::Array: The time series.
• m::Int: The order of the sample mean.

Output

• std::Real: The sample standard deviation.

Examples

julia> sstdk(randn(100), 20)