The setting is the same as in the Demonstration "

Three Alternatives to the Likelihood Maximization for Estimating a Centered Matérn (3/2) Process", except that the "differentiability" parameter (often denoted by

) is fixed here to 1. This value 1 for

is classically used in two dimensions following the seminal work of Whittle [1]. The corresponding autocorrelation function has the simple expression

(the process variance

being denoted by

), where

is a modified Bessel function of the second kind (see the Details section in the help page for the

BesselK function) and

is still called the inverse-range parameter,

being the "range" (or "support") of

.

One important difference with the case

is that the considered process, even regularly sampled (here observed at

in the set

), no longer coincides with any ARMA time series. Fast Fourier transforms (FFT) are then invoked for the simulation of such a series.

The same three methods as in the case

are implemented here for estimating

and

, namely: Cressie's weighted least-squares method [4], the "hybrid" Zhang–Zimmerman method [5], and the recently proposed "CGEM-EV" method [6], here in the case of no measurement error. For each of the three methods, the estimate of the so-called "microergodic coefficient", which is now

(the general expression for

being

), is naturally defined as the square of the product of the estimates of

and

.

For all the simulated time series, the true variance

is fixed to 1 (this choice is without loss of generality), and various

can be tried for the true value of the inverse-range parameter. The legends for the table of the numerical results are similar to the ones in the above-mentioned Demonstration for the case

.

Let

denote the candidate correlation matrix of

, with inverse-range

, that is, the

matrix whose element in the

row and

column is

. The only difficulty regarding implementation in the case of no measurement error, of both the hybrid method and CGEM-EV, is the computation of the quadratic form

, the so-called "Gibbs energy of

associated with a given

".

The computation of

uses a conjugate-gradient (CG) solver preconditioned by a classical factored sparse approximate inverse (FSAI) preconditioning (see [7] for a recent survey), each product by

being obtained via FFT from the standard embedding of

in a circulant

matrix.

It is important to notice that

becomes ill-conditioned for large

and

decreasing toward the lower-bound

used here (as in the Demonstration for

); so using a preconditioner is a mandatory requirement to accelerate the CG convergence.

It is observed here that this implementation is quite fast, even for

.