MATERNCOV Autocovariance of the Matern random process and variations. [TAU,R]=MATERNCOV(DT,N,SIGMA,ALPHA,LAMBDA) returns the autocovariance function R of a length N complex-valued Matern random process having variance SIGMA^2, slope parameter ALPHA, and damping parameter LAMBDA. DT is the sample interval. Note that LAMBDA is understood to have the same units as the inverse sample interval 1/DT. TAU is an array of time lags at which R is computed, and is given by TAU=DT*[0,1,...,N-1]. By definition, R is one-sided theoretical autocovariance at non-negative time lags. See below for the relationship between this and the full, length (2N-1) theoretical autocovariance function. Note that for LAMBDA=0, the case of fractional Brownian motion, R will contain only INFs because the autocovariance function is unbounded. The input parameters SIGMA, ALPHA, and LAMBDA, may all either be scalars or arrays of the same length M. If the latter, then the output autocovariance function R will be a matrix with N rows and M columns. [TAU,R]=MATERNCOV(DT,N,SIGMA,ALPHA,LAMBDA,NU,MU) returns the autocovariance function of various extensions of the Matern process. MATERNOISE(...,'composite') also works. See MATERNSPEC for details. See MATERNSPEC for a more thorough discussion of the Matern process. For details on the Matern process and its autocovariance function, see: Lilly, Sykulski, Early, and Olhede, (2017). Fractional Brownian motion, the Matern process, and stochastic modeling of turbulent dispersion. Nonlinear Processes in Geophysics, 24: 481--514. __________________________________________________________________ Relationship to full autocovariance For a time series of length N, the full autocovariance function RF is length 2N-1, defined at time lags -N+1,-N+2...,-1,0,1,...,N-2,N-1. The one-sided autocovariance R contains the full autocovariance RF at positive time lags. Negative lags are given by Hermitian symmetry. [TAUF,RF]=MATERNCOV(...,'full') returns the full (two-sided) autocovariance RF and the corresponding two-sided time array TAUF. RF is constructed from R as RF=[FLIPUD(CONJ(R(2:end,:));R]. __________________________________________________________________ Composite Matern [TAU,R]=MATERNCOV(DT,N,SIGMA,ALPHA,LAMBDA,NU,MU,'composite'), returns the autocovariance of the composite Matern process. See MATERNSPEC for more details. This autocovariance does not have an analytic form, but is approximated to high precision by inverse Fourier transforming the spectrum with 10 x oversampling over a 10 x longer time period, and then decimating. MATERNCOV(...,'composite',M,P) specifies the numerical oversampling parameters used in the numerical calculation. The spectrum is computed over a time window of M times the duration required, and P times the required sampling density, for a total of M*P time more points. These flags may set to optimize the tradeoff between speed and accuracy. The default behavior corresponds to M=10 and P=10. This computation method is expected to minimize aliasing effects and resolution errors. __________________________________________________________________ See also MATERNSPEC, MATERNIMP, MATERNOISE, MATERNFIT. 'materncov --t' runs some tests. Usage: [tau,R]=materncov(dt,N,sigma,alpha,lambda); [tau,R]=materncov(dt,N,sigma,alpha,lambda,nu,mu); [tau,R]=materncov(dt,N,sigma,alpha,lambda,nu,mu,'composite'); __________________________________________________________________ This is part of JLAB --- type 'help jlab' for more information (C) 2013--2017 J.M. Lilly --- type 'help jlab_license' for details