Generate alpha-sub-Gaussian (aSG) random numbers.

The implementation is based on https://github.com/ahmd-mahm/alpha-SGNm/blob/master/asgn.m

Reference: A. Mahmood and M. Chitre, "Generating random variates for stable sub-Gaussian processes with memory", Signal Processing, Volume 131, Pages 271-279, 2017. (https://doi.org/10.1016/j.sigpro.2016.08.016.)

Arguments

• α: characteristic exponent associated with the aSGN(m) process. This is

a scalar input and should lie within collect(1.1:0.01:1.98).

• R: covariance matrix of any adjacent m+1 samples in an aSGN(m) process.

The dimension of R is equal to m+1. It should be a symmetric toeplitz matrix. The maximum acceptable size of R is 10x10

• n: number of samples required

Examples

julia> x = rand(AlphaSubGaussian(n=1000))

Generates the conditional probability f(X2|X1) if [X1, X2] is a sub-Gaussian stable random vector such that X1(i)~X2~S(alpha,delta) and rho is the correlation coefficient of the underlying Gaussian vector. We assume the joint-probabiluty is given by f(X1,X2).

Generate independent stable random numbers.

:param α: characteristic exponent (0.1 to 2.0) :param β: skew (-1 to +1) :param scale: scale parameter :param loc: location parameter (mean for α > 1, median/mode when β=0)

This implementation is based on the method in J.M. Chambers, C.L. Mallows and B.W. Stuck, "A Method for Simulating Stable Random Variables," JASA 71 (1976): 340-4. McCulloch's MATLAB implementation (1996) served as a reference in developing this code.

fit(d::Type{<:AlphaStable}, x; alg=QuickSort)

Fit an α stable distribution to data.

returns AlphaStable

α∈[0.6,2.0], β∈[-1,1] , c∈[0,∞] and δ∈[-∞,∞] are the characteristic exponent, skewness parameter, scale parameter (dispersion^1/α) and location parameter respectively.

α, β, c and δ are computed based on McCulloch (1986) fractile.

fit(d::Type{<:AlphaSubGaussian}, x, m; p=1.0)

Fit an aSGN(m) model to data via the covariation method.

The covariation method requires an additional parameter p. Ideally, 1 < p < α. In most practical impulsive scenarios p=1.0 is sufficient. m is the number of lags in the covariance matrix.

The implementation is based on https://github.com/ahmd-mahm/alpha-SGNm/blob/master/param_est/asgnfit.m

Reference: A. Mahmood and M. Chitre, "Generating random variates for stable sub-Gaussian processes with memory", Signal Processing, Volume 131, Pages 271-279, 2017. (https://doi.org/10.1016/j.sigpro.2016.08.016.)

fit(d::Type{<:SymmetricAlphaStable}, x; alg=QuickSort)

Fit a symmetric α stable distribution to data.

returns SymmetricAlphaStable

α∈[1,2], c∈[0,∞] and δ∈[-∞,∞] are the characteristic exponent, scale parameter (dispersion^1/α) and location parameter respectively.

α is computed based on McCulloch (1986) fractile. scale is computed based on Fama & Roll (1971) fractile. location is the 50% trimmed mean of the sample.