The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

Figures & data

Table 1 Coefficients of the rational approximation for $\beta =3/4$ (exponential covariance on ${\mathbb{R}}^2$) for m = 1, 2, 3, normalized so that c_m = 1.

m	b0	c0	b1	c1	b2	c2	b3	c3	b4
1	1.69e–2	7.69e–2	8.06e–1	1	2.57e–1
2	8.08e–4	5.30e–3	1.98e–1	4.05e–1	1.07	1	1.41e–1
3	3.72e–5	3.27e–4	3.03e–2	8.57e–2	6.84e–1	1.00	1.28	1	9.17e–2

Fig. 1 The L₂- and $L_{\infty }$-errors of the covariance functions for different values of ν for the different approximation methods. When ν = 1, all methods are exact.

Table 2 Covariance errors ($\times 100$) and computing times in seconds ($\times 100$) for sampling from the rational SPDE approximation u (with $\beta =3/4$) and, in parentheses, for evaluating $\text{log}\hspace{0.17em}\left|{\mathbf{Q}}_{\mathbf{x}|\mathbf{y}}\right|$.

		Rational SPDE approximation			Standard SPDE approach
n		m = 1	m = 2	m = 3	β = 2	β = 3	β = 4
57^2	Error	1.849	1.339	1.415	2.259	2.173	2.147
	Time	1.5 (3.2)	1.8 (5.2)	2.7 (8.7)	1.7 (2.6)	1.7 (3.9)	2.2 (6.3)
85^2	Error	1.720	0.757	0.807	0.953	0.928	0.921
	Time	3.1 (8.4)	5.0 (14)	7.6 (25)	3.0 (8.2)	5.8 (13)	7.9 (22)
115^2	Error	1.559	0.526	0.501	0.509	0.498	0.494
	Time	7.6 (22)	11 (34)	18 (57)	6.3 (18)	11 (35)	18 (53)

NOTE: These values are also given for the standard SPDE approach with $\beta =2,3,4$.

Table 3 Results of the parameter estimation.

		Rational samples	Matérn samples
	Truth	Estimate	Coarse mesh	Fine mesh
κ	10	10.026 (0.5661)	10.966 (1.8060)	10.864 (0.4414)
${\varphi }^2$	1.0	1.0014 (0.0228)	1.1089 (0.6155)	0.9743 (0.0210)
${\sigma }^2$	0.1	0.1001 (0.0009)	0.3016 (0.0036)	0.2320 (0.0044)
ν	0.5	0.5011 (0.0168)	0.5554 (0.0991)	0.5462 (0.0138)

NOTE: For each parameter estimate, the mean of 100 different estimates is shown, with the corresponding standard deviation in parentheses.

Fig. 2 Average summer precipitation residuals (in cm) for 1979 and the FEM mesh.

Fig. 3 Nine basis functions modeling the parameters for the nonstationary models.

Table 4 Model-dependent results for (i) the log-likelihood, (ii) the pseudo cross-validation scores (RMSE, CRPS, LS, each $\times 100$) averaged over ten replicates, and (iii) the computational time for one evaluation of the likelihood averaged over 100 computations.

	Log-likelihood	RMSE	CRPS	LS	Time
Stationary β = 1	219773	4.206	2.295	177.2	0.125
Stationary fractional	220255	4.167	2.274	178.2	0.412
Nonstationary β = 1	225969	4.194	2.266	182.1	0.121
Nonstationary fractional	226095	4.170	2.254	182.4	0.416

Fig. 4 Estimated marginal standard deviations (top row) and contours of 0.7 correlation of the correlation function for selected locations (bottom row), for the fractional (left column) and β = 1 (right column) models.