Plots for Long Memory

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)

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)

periodogram_plot(x::Array; slope::Bool=true)

Plots the log-periodogram of a time series x.

Arguments

• x::Vector: time series

Output

• p1::Plots.Plot: log-periodogram

Optional arguments

• slope::Bool: If true, the slope of the linear regression is displayed.

Examples

julia> periodogram_plot(randn(100,1))

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))

LMPlot(x::Array;lags::Int=50,name::String="Time series")

Returns a plot of the time series x.

Arguments

• x::Array: The time series.

Optional arguments

• lags::Int: The number of lags to use in the autocorrelation function.
• name::String: The name of the time series to use in the plot.

Output

• pp::Plot: Four plots are returned in a 2x2 grid:
  • The first plot is the time series of x.
  • The second plot is the autocorrelation function of x using the autocorrelation_plot function.
  • The third plot is the periodogram of x using the periodogram_plot function.
  • The fourth plot is the variance plot of x using the variance_plot function.

Examples

julia> LMPlot(randn(100))