The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on the d-sphere

Abstract

Random fields on the sphere play a fundamental role in the natural sciences. This paper presents a simulation algorithm parenthetical to the spectral turning bands method used in Euclidean spaces, for simulating scalar- or vector-valued Gaussian random fields on the d-dimensional unit sphere. The simulated random field is obtained by a sum of Gegenbauer waves, each of which is variable along a randomly oriented arc and constant along the parallels orthogonal to the arc. Convergence criteria based on the Berry-Esséen inequality are proposed to choose suitable parameters for the implementation of the algorithm, which is illustrated through numerical experiments. A by-product of this work is a closed-form expression of the Schoenberg coefficients associated with the Chentsov and exponential covariance models on spheres of dimensions greater than or equal to 2.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)

    MATH  Google Scholar 

  2. Alegria, A., Cuevas, F., Diggle, P., Porcu, E.: A family of covariance functions for random fields on spheres. CSGB Research Reports, Department of Mathematics, Aarhus University (2018)

  3. Arafat, M., Gregori, P., Porcu, E.: Schoenberg coefficients and curvature at the origin of continuous isotropic definite kernels on the sphere (2018). arXiv:1807.02363v1

  4. Berry, A.C.: The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49(1), 122–136 (1941)

    MathSciNet  MATH  Article  Google Scholar 

  5. Chilès, J.-P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. John Wiley and Sons, New York (2012)

    MATH  Book  Google Scholar 

  6. Clarke, J., Alegría, A., Porcu, E.: Regularity properties and simulations of Gaussian random fields on the sphere cross time. Electron. J. Stat. 12(1), 399–426 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  7. Cuevas, F., Allard, D., Porcu, E.: Fast and exact simulation of Gaussian random fields defined on the sphere cross time. Statistics and Computing (2019). in press

  8. Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986)

    MATH  Book  Google Scholar 

  9. Dryden, I.: Statistical analysis on high-dimensional spheres and shape spaces. Ann. Stat. 33(4), 1643–1665 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  10. Emery, X., Arroyo, D., Porcu, E.: An improved spectral turning-bands algorithm for simulating stationary vector Gaussian random fields. Stoch. Environ. Res. Risk Assess. 30(7), 1863–1873 (2016)

    Article  Google Scholar 

  11. Emery, X., Furrer, R., Porcu, E.: A turning bands method for simulating isotropic Gaussian random fields on the sphere. Stat. Probab. Lett. 144, 9–15 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  12. Emery, X., Lantuéjoul, C.: TBSIM: a computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Comput. Geosci. 32(10), 1615–1628 (2006)

    Article  Google Scholar 

  13. Emery, X., Lantuéjoul, C.: A spectral approach to simulating intrinsic random fields with power and spline generalized covariances. Comput. Geosci. 12(1), 121–132 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  14. Emery, X., Porcu, E.: Simulating isotropic vector-valued Gaussian random fields on the sphere through finite harmonics approximations. Stoch. Environ. Res. Risk Assess. 33(8–9), 1659–1667 (2019)

    Article  Google Scholar 

  15. Esséen, C.: On the Liapunoff limit of error in the theory of probability. Arkiv Mat Astron. och Fysik A28, 1–19 (1942)

    MathSciNet  MATH  Google Scholar 

  16. Esséen, C.: A moment inequality with an application to the central limit theorem. Scand. Actuar. J. 39(2), 160–170 (1956)

    MathSciNet  MATH  Article  Google Scholar 

  17. Gneiting, T.: Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4), 1327–1349 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  18. Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series, and Products. Academic Press, Amsterdam (2007)

    MATH  Google Scholar 

  19. Guella, J., Menegatto, V.: Unitarily invariant strictly positive definite kernels on spheres. Positivity 22(1), 91–103 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  20. Guinness, J., Fuentes, M.: Isotropic covariance functions on spheres: some properties and modeling considerations. J. Multivar. Anal. 143, 143–152 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  21. Hannan, E.: Multiple Time Series. Wiley Series in Probability and Statistics. Wiley (2009)

  22. Hansen, L.V., Thorarinsdottir, T.L., Ovcharov, E., Gneiting, T., Richards, D.: Gaussian random particles with flexible Hausdorff dimension. Adv. Appl. Probab. 47(2), 307–327 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  23. Huang, C., Zhang, H., Robeson, S.: On the validity of commonly used covariance and variogram functions on the sphere. Math. Geosci. 43, 721–733 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  24. Jensen, J.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30, 175–193 (1906)

    MathSciNet  MATH  Article  Google Scholar 

  25. Jeong, J., Jun, M., Genton, M.G.: Spherical process models for global spatial statistics. Stat. Sci. 32(4), 501–513 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  26. Kim, D., Kim, T., Rim, S.: Some identities involving Gegenbauer polynomials. Adv. Diff. Equ. 2012, 219 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  27. Korolev, V.Y., Shevtsova, I.: On the upper bound for the absolute constant in the Berry-Esseen inequality. Theory Probab. Appl. 54(4), 638–658 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  28. Lang, A., Schwab, C.: Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann. Appl. Probab. 25(6), 3047–3094 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  29. Lantuéjoul, C.: Geostatistical Simulation: Models and Algorithms. Springer, Berlin (2002)

    MATH  Book  Google Scholar 

  30. Lantuéjoul, C., Freulon, X., Renard, D.: Spectral simulation of isotropic Gaussian random fields on a sphere. Mathematical Geosciences (2019). in press

  31. Mantoglou, A., Wilson, J.L.: The turning bands method for simulation of random fields using line generation by a spectral method. Water Resour. Res. 18(5), 1379–1394 (1982)

    Article  Google Scholar 

  32. Mardia, K., Patrangenaru, V.: Directions and projective shapes. Ann. Stat. 33(4), 1666–1699 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  33. Marinucci, D., Peccati, G.: Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. Cambridge University Press, Cambridge (2011)

    MATH  Book  Google Scholar 

  34. Matheron, G.: The intrinsic random functions and their applications. Adv. Appl. Probab. 5(3), 439–468 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  35. Moller, J., Nielsen, M., Porcu, E., Rubak, E.: Determinantal point process models on the sphere. Bernoulli 24(2), 1171–1201 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  36. Porcu, E., Alegria, A., Furrer, R.: Modeling temporally evolving and spatially globally dependent data. Int. Stat. Rev. 86(2), 344–377 (2018)

    MathSciNet  Article  Google Scholar 

  37. Rainville, E.: Special function. Chelsea Publishing Company, New York (1960)

    MATH  Google Scholar 

  38. Reimer, M.: Uniform inequalities for Gegenbauer polynomials. Acta Math. Hung. 70(1–2), 13–26 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  39. Ripley, B.: Stoch. Simul. John Wiley & Sons, New York (1987)

    Google Scholar 

  40. Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9(1), 96–108 (1942)

    MathSciNet  MATH  Article  Google Scholar 

  41. Shevtsova, I.: On the absolute constants in the Berry Esséen type inequalities for identically distributed summands (2011). arXiv:1111.6554

  42. Tompson, A., Ababou, R., Gelhar, L.: Implementation of the three-dimensional turning bands random field generator. Water Resour. Res. 25(8), 2227–2243 (1989)

    Article  Google Scholar 

  43. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results. Springer, New York (1987)

    MATH  Book  Google Scholar 

  44. Ziegel, J.: Convolution roots and differentiability of isotropic positive definite functions on spheres. Proc. Am. Math. Soc. 142(6), 2063–2077 (2014)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the funding of the National Agency for Research and Development of Chile, through grants CONICYT/FONDECYT/INICIACIÓN/ No. 11190686 (A. Alegría), CONICYT PIA AFB180004 (X. Emery) and CONICYT/FONDECYT/REGULAR/No. 1170290 (X. Emery).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alfredo Alegría.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (zip 3 KB)

Appendices

Appendices

Proof of Proposition 1

Before stating the proof of Proposition 1, we must introduce some properties of Gegenbauer polynomials. A classical duplication equation (see, e.g., Ziegel 2014, Equation 2.4) establishes that, for \(d\ge 2\), \(n,k\in {\mathbb {N}}\) and \(\varvec{x}_1,\varvec{x}_2\in {\mathbb {S}}^d\),

$$\begin{aligned}&\int _{{\mathbb {S}}^d} {G}_n^{(d-1)/2}(\varvec{\omega }^\top \varvec{x}_1) \, {G}_k^{(d-1)/2}(\varvec{\omega }^\top \varvec{x}_2) \, U(\mathrm{d}\varvec{\omega }) \nonumber \\&\quad = \frac{\delta _{n,k} (d-1)}{2n + d - 1} \, {G}_n^{(d-1)/2}(\varvec{x}_1^\top \varvec{x}_2), \end{aligned}$$
(A.1)

where U is the uniform probability measure on \({\mathbb {S}}^d\) and \(\delta _{n,k}\) denotes the Kronecker delta. For \(d=1\), one has a similar identity. Let \(n, k \in {\mathbb {N}}\), then

$$\begin{aligned}&\int _{{\mathbb {S}}^1} \cos (n \vartheta (\varvec{\omega }, \varvec{x}_1)) \cos (k \vartheta (\varvec{\omega }, \varvec{x}_2)) \, U(\mathrm{d}\varvec{\omega }) \nonumber \\&\quad = \frac{\delta _{n,k}}{2} \cos (n \vartheta (\varvec{x}_1, \varvec{x}_2)), \qquad \varvec{x}_1,\varvec{x}_2\in {\mathbb {S}}^1. \end{aligned}$$

Proof of proposition 1

We only prove the result for \(d\ge 2\), since the case \(d=1\) is completely analogous. Let Z be the random field defined in (3.2). Because \(\varepsilon \) is independent of \(\kappa \) and \(\varvec{\omega }\) and has a zero mean, it is straightforward to prove that \({\mathbb {E}}\{{Z}(\varvec{x})\} = 0\) for any \(\varvec{x}\in {\mathbb {S}}^d\). On the other hand, the covariance between any two variables \({Z}(\varvec{x}_1)\) and \({Z}(\varvec{x}_2)\), with \(\varvec{x}_1,\varvec{x}_2\in {\mathbb {S}}^d\), is:

$$\begin{aligned}&{\mathbb {E}}\{{Z}(\varvec{x}_1) {Z}(\varvec{x}_2)\} \\&\quad = {\mathbb {E}}\{\varepsilon ^2\} \sum _{n=0}^\infty \frac{b_{n,d} (2n+d-1)}{d-1} \\&\qquad \int _{{\mathbb {S}}^d} {G}_n^{(d-1)/2}(\varvec{\omega }^\top \varvec{x}_1) \, {G}_n^{(d-1)/2}(\varvec{\omega }^\top \varvec{x}_2) \, U(\text {d}{\varvec{\omega }}). \end{aligned}$$

Using (A.1) and the fact that \({\mathbb {E}}\{\varepsilon ^2\}=1\), the announced covariance function is obtained. \(\square \)

Proof of proposition 2

Again, we only prove the result for \(d\ge 2\), the one-dimensional case being similar. Let \(\varvec{Z}\) be the vector-valued random field defined in (3.6). Its mean vector is zero, insofar as \(\varepsilon \) has zero mean and is independent of \(\varvec{\omega }\), \(\iota \) and \(\kappa \).

The variance-covariance matrix between any two vectors \(\varvec{Z}(\varvec{x}_1)\) and \(\varvec{Z}(\varvec{x}_2)\), with \(\varvec{x}_1,\varvec{x}_2\in {\mathbb {S}}^d\), is:

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \{\varvec{Z}(\varvec{x}_1) \varvec{Z}(\varvec{x}_2)^\top \} \\&\quad = {\mathbb {E}} \{\varepsilon ^2 \} \sum _{n=0}^\infty \frac{2n + d -1}{d-1} \sum _{i=1}^p \varvec{\gamma }^{(i)}_{n,d}\, [\varvec{\gamma }^{(i)}_{n,d}]^\top \\&\qquad \int _{{\mathbb {S}}^d} {G}_{n}^{(d-1)/2}(\varvec{\omega }^\top \varvec{x}_1) \, {G}_{n}^{(d-1)/2}(\varvec{\omega }^\top \varvec{x}_2) \, U(\text {d}{\varvec{\omega }}). \end{aligned} \end{aligned}$$

Using property (A.1) and the fact that \(\varepsilon \) is an independent random variable with zero mean and unit variance, one obtains

$$\begin{aligned}&{\mathbb {E}}\{\varvec{Z}(\varvec{x}_1) \varvec{Z}(\varvec{x}_2)^\top \} \\&\quad = \sum _{n=0}^\infty \left\{ \sum _{i=1}^p \varvec{\gamma }^{(i)}_{n,d}[\varvec{\gamma }^{(i)}_{n,d}]^\top \right\} G_n^{(d-1)/2}(\varvec{x}_1^\top \varvec{x}_2). \end{aligned}$$

The covariance function is obtained by using (3.4).

Upper bound for the third-order absolute moment of a Gegenbauer wave

Let \(d, n \in {\mathbb {N}}\), \(d \ge 2\), \(\lambda = \frac{d-1}{2}\), \(\varvec{x} \in {\mathbb {S}}^d\) (fixed) and \(\varvec{\omega }\) uniformly distributed on \({\mathbb {S}}^d\). It is of interest to find an upper bound for the following third-order absolute moment:

$$\begin{aligned} \mu _{n,d}^3 = {\mathbb {E}} \{ |G_n^\lambda (\varvec{\omega }^T \, \varvec{x}) |^3 \}. \end{aligned}$$

By introducing spherical coordinates such that:

$$\begin{aligned} \left\{ \begin{aligned} \varvec{x}&= (1,0,\cdots ,0) \\ \varvec{\omega }&= (\cos \varphi _1,\sin \varphi _1 \cos \varphi _2, \cdots ,\sin \varphi _1 \cdots \sin \varphi _{d-1} \cos \varphi _d,\\&\quad \sin \varphi _1 \cdots \sin \varphi _{d-1} \sin \varphi _d), \end{aligned} \right. \end{aligned}$$

with \(\varphi _1, \cdots , \varphi _{d-1} \in [0,\pi ]\) and \(\varphi _d \in [0,2\pi [\), one obtains:

$$\begin{aligned} \begin{aligned} \mu _{n,d}^3&= \int _{{\mathbb {S}}^d} |G_n^\lambda (\varvec{\omega }^T \, \varvec{x}) |^3 U(\mathrm{d}\varvec{\omega }) \\&= \frac{{\varGamma }\left( \frac{d+1}{2}\right) }{2\pi ^{\frac{d+1}{2}}} \int _0^{2\pi } \mathrm{d}\varphi _d \int _0^{\pi } \sin \varphi _{d-1} \mathrm{d}\varphi _{d-1} \cdots \\&\quad \times \int _0^{\pi } \sin ^{d-2} \varphi _{2} \mathrm{d}\varphi _{2} \int _0^{\pi } |G_n^\lambda (\cos \varphi _{1}) |^3 \sin ^{d-1} \varphi _{1} \mathrm{d}\varphi _{1}. \end{aligned} \end{aligned}$$

Since \(\int _0^{\pi } \sin ^{m-1} \varphi = \frac{\sqrt{\pi } {\varGamma }(\frac{m}{2})}{{\varGamma }(\frac{m+1}{2})}\) (Gradshteyn and Ryzhik (2007), formula 3.621.5), one has:

$$\begin{aligned} \mu _{n,d}^3 = \frac{2{\varGamma }\left( \frac{d+1}{2}\right) }{\sqrt{\pi }{\varGamma }\left( \frac{d}{2}\right) } \int _0^{\frac{\pi }{2}} |G_n^\lambda (\cos \varphi _{1}) |^3 \sin ^{d-1} \varphi _{1} \mathrm{d}\varphi _{1}. \end{aligned}$$
(C.1)

To find an upper bound for such a moment, we distinguish the cases \(d=2\), \(d=3\) and \(d \ge 4\).

  • Case \(d=2\). Using inequality 22.14.3 of (Abramowitz and Stegun 1972):

    $$\begin{aligned} |G_n^{\frac{1}{2}}(\cos \varphi _{1}) |\le \sqrt{\frac{2}{n \pi \sin \varphi _1}}, \qquad \varphi _1 \in ]0,\pi [, \end{aligned}$$

    one finds

    $$\begin{aligned} \mu _{n,2}^3 \le \frac{2{\varGamma }\left( \frac{d+1}{2}\right) }{\sqrt{\pi }{\varGamma }\left( \frac{d}{2}\right) } \left( \frac{2}{n \pi }\right) ^{\frac{3}{2}} \int _0^{\frac{\pi }{2}} \frac{1}{\sqrt{\sin (\varphi _{1})}} \mathrm{d}\varphi _{1}, \end{aligned}$$

    i.e.,

    $$\begin{aligned} \mu _{n,2}^3 \le \left( \frac{2}{n \pi }\right) ^{\frac{3}{2}} \frac{{\varGamma }\left( \frac{1}{4}\right) }{\pi {\varGamma }\left( \frac{3}{4}\right) } = {\mathcal {O}}\left( n^{-\frac{3}{2}}\right) . \end{aligned}$$
    (C.2)
  • Case \(d=3\). In this case, the Gegenbauer polynomials of order \(\lambda = 1\) coincide with the Chebyshev polynomials of the second kind (Abramowitz and Stegun (1972), formulae 22.5.34 and 22.3.16):

    $$\begin{aligned} G_n^1(\cos \varphi _1) = \frac{\sin \left( (n+1)\varphi _1 \right) }{\sin (\varphi _1)}, \qquad n \in {\mathbb {N}}, \varphi _1 \in ]0,\pi [. \end{aligned}$$

    We use the following inequalities:

    $$\begin{aligned} \left\{ \begin{aligned} |\sin \left( (n+1)\varphi _1\right) |&\le (n+1)\varphi _1 \text { for } 0\le \varphi _1 \le \frac{\pi }{2(n+1)} \\ |\sin \left( (n+1)\varphi _1\right) |&\le 1 \text { for } \frac{\pi }{2(n+1)} \le \varphi _1 \le \frac{\pi }{2} \\ |\sin (\varphi _1) |&\ge \frac{2}{\pi }\varphi _1 \text { for } 0\le \varphi _1 \le \frac{\pi }{2}, \end{aligned} \right. \end{aligned}$$

    which yield:

    $$\begin{aligned} \mu _{n,3}^3\le & {} \frac{2{\varGamma }\left( \frac{d+1}{2}\right) }{\sqrt{\pi }{\varGamma }\left( \frac{d}{2}\right) } \\&\left( \frac{\pi ^3 (n+1)^3}{2^3} \int _0^{\frac{\pi }{2(n+1)}} \varphi _{1}^2 \mathrm{d}\varphi _{1} + \frac{\pi ^3}{2^3} \int _{\frac{\pi }{2(n+1)}}^{\frac{\pi }{2}} \frac{\mathrm{d}\varphi _{1}}{\varphi _{1}} \right) , \end{aligned}$$

    that is:

    $$\begin{aligned} \mu _{n,3}^3 \le \frac{\pi ^5}{48} + \frac{\pi ^2 \ln (n+1)}{2} = {\mathcal {O}}\left( \ln n\right) . \end{aligned}$$
    (C.3)
  • Case \(d \ge 4\). Let us pose \(\lambda = \frac{d-1}{2}\). For any integer \(\nu \in [1,\lambda [\), Reimer (1996) showed that there exists a constant \(\varpi _{\nu ,n}^{\lambda }\) depending on \(\nu \), n and \(\lambda \) such that

    $$\begin{aligned} |G_n^{\lambda }(\cos \varphi _{1}) |\le \varpi _{\nu ,n}^{\lambda } G_n^{\lambda }(1) |n \sin (\varphi _1) |^{-\nu }, \qquad \varphi _1 \in ]0,\pi [. \end{aligned}$$

    Plugging this inequality into (C.1), one obtains:

    $$\begin{aligned} \mu _{n,d}^3\le & {} \frac{2{\varGamma }\left( \frac{d+1}{2}\right) }{\sqrt{\pi }{\varGamma }\left( \frac{d}{2}\right) } \left( \varpi _{\nu ,n}^{\lambda } \, G_n^{\lambda }(1)\right) ^3 \, n^{-3\nu } \\&\int _0^{\frac{\pi }{2}} \sin ^{d-1-3\nu }(\varphi _1) \mathrm{d}\varphi _{1}, \end{aligned}$$

    with \(G_n^{\lambda }(1) = \frac{{\varGamma }(n+2\lambda )}{{\varGamma }(2\lambda ){\varGamma }(n+1)}\) (Abramowitz and Stegun (1972), formula 22.2.3). The above integral converges when \(d-1-3\nu \) is greater than \(-1\) (Gradshteyn and Ryzhik (2007), formula 3.621.5), in which case one has:

    $$\begin{aligned} \mu _{n,d}^3\le & {} \frac{2{\varGamma }\left( \frac{d+1}{2}\right) }{\sqrt{\pi }{\varGamma }\left( \frac{d}{2}\right) } \left( \varpi _{\nu ,n}^{\lambda } \, \frac{{\varGamma }(n+d-1)}{{\varGamma }(d-1){\varGamma }(n+1)}\right) ^3 \\&n^{-3\nu } \frac{\sqrt{\pi }{\varGamma }\left( \frac{d-3\nu }{2} \right) }{2{\varGamma }\left( \frac{d-3\nu +1}{2} \right) }. \end{aligned}$$

    Reimer (1996) showed that \(\varpi _{\nu ,n}^{\lambda } = {\mathcal {O}}(1)\) as n becomes infinitely large. Furthermore, Stirling’s approximation to the factorial implies that \(\frac{{\varGamma }(n+d-1)}{{\varGamma }(n+1)} = {\mathcal {O}}\left( n^{d-2}\right) \) (Abramowitz and Stegun (1972), formula 6.1.46). The lowest asymptotic bound is obtained by choosing \(\nu =\lfloor \frac{d}{2} \rfloor - 1\):

    $$\begin{aligned} \mu _{n,d}^3 \le {\mathcal {O}}\left( n^{3d-6-3\nu }\right) = {\mathcal {O}}\left( n^{3 \lfloor \frac{d-1}{2} \rfloor }\right) . \end{aligned}$$
    (C.4)

Calculation of Schoenberg coefficients

Let C be an isotropic covariance on \({\mathbb {S}}^d\), \(d \ge 2\), K its isotropic part, and \(\{b_{n,d}: n\in {\mathbb {N}}\}\) the associated Schoenberg sequence, as defined in (2.2). The change of variable \(t = \cos \vartheta \) in (2.4) gives

$$\begin{aligned} b_{n,d} = \frac{1}{\parallel G_n^\lambda \parallel ^2} \int _{-1}^{+1} K (\arccos t) \, G_n^\lambda (t) \, \bigl (1 - t^2 \bigl )^{\lambda - 1/2} \, d t, \end{aligned}$$

with \(\lambda = \frac{d - 1}{2} > 0\). Suppose now that \(t \, \mapsto \, K (\arccos t)\) can be expanded into a power series

$$\begin{aligned} K (\arccos t) = \sum _{k=0}^\infty \alpha _k \, t^k. \end{aligned}$$

Then, using the expansion of the monomials into Gegenbauer polynomials (Rainville 1960; Kim et al. 2012)

$$\begin{aligned} t^k = \frac{k!}{2^k} \, \sum _{\ell =0}^{\lfloor k/2 \rfloor } \frac{\lambda + k - 2 \ell }{\ell !} \, \frac{{\varGamma }(\lambda )}{{\varGamma }( \lambda + k + 1 -\ell )} \, G_{k - 2 \ell }^\lambda (t) , \end{aligned}$$

where \(\lfloor \cdot \rfloor \) is the floor function, it follows

$$\begin{aligned} b_{n,d}= & {} \frac{1}{\parallel G_n^\lambda \parallel ^2} \sum _{k=0}^\infty \alpha _k \, \frac{k!}{2^k} \sum _{\ell =0}^{\lfloor k/2 \rfloor } \frac{(\lambda + k - 2 \ell ) \, {\varGamma }(\lambda )}{\ell ! \, {\varGamma }( \lambda + k + 1 -\ell )} \\&\int _{-1}^{+1} G_{k - 2 \ell }^\lambda (t) \, G_n^\lambda (t) \, \bigl (1 - t^2 \bigl )^{\lambda - 1/2} \, d t. \end{aligned}$$

The latter integral vanishes unless \(k - 2 \ell = n \), in which case it is equal to \(\parallel G_n^\lambda \parallel ^2\). We thus obtain the generic formula

$$\begin{aligned} b_{n,d} = \sum _{\ell =0}^\infty \alpha _{n + 2 \ell } \, \frac{(n + 2 \ell )!}{2^{n + 2 \ell }} \, \frac{\lambda + n}{\ell !} \, \frac{{\varGamma }(\lambda )}{{\varGamma }( \lambda + n + \ell + 1)} \cdot \nonumber \\ \end{aligned}$$
(D.1)

The rest of the calculation must be done on a case-by-case basis. Two examples are given below.

Chentsov covariance

As a first example, consider \(K (\vartheta ) = 1 - 2 \vartheta / \pi \). The power series of \( K (\arccos t) = \frac{2}{\pi } \, \arcsin t \) is given by formula 4.4.40 of Abramowitz and Stegun (1972):

$$\begin{aligned} K ( \arccos t) = \frac{2}{\pi \sqrt{\pi }} \, \sum _{k=0}^\infty \frac{{\varGamma }(k + 1/2)}{(2 k +1) \, k!} \, t^{2k+1} , \end{aligned}$$

from which we derive \(\alpha _{2k} = 0\) and

$$\begin{aligned} \alpha _{2k+1} = \frac{2}{\pi \sqrt{\pi }} \, \frac{{\varGamma }(k + 1/2)}{(2 k +1) \, k!} \cdot \end{aligned}$$

Plugging these coefficients into (D.1), we obtain that \(b_{2n,d}= 0\) and

$$\begin{aligned} b_{2n+1,d}= & {} \frac{2}{\pi \sqrt{\pi }} \sum _{\ell =0}^\infty \frac{{\varGamma }( n + \ell + 1/2)}{(2n+2\ell +1) \, ( n + \ell )!} \\&\frac{(2n + 2 \ell +1)!}{2^{2n + 2 \ell +1}} \, \frac{\lambda + 2n+1}{\ell !} \, \frac{{\varGamma }(\lambda )}{{\varGamma }( \lambda + 2n + \ell + 2)}. \end{aligned}$$

Using the duplication formula of the gamma function (formula 6.1.18 of Abramowitz and Stegun (1972)), it comes

$$\begin{aligned} \begin{aligned} b_{2n+1,d}&= \frac{1}{\pi ^2} \sum _{\ell =0}^\infty \frac{{\varGamma }^2 ( n + \ell + 1/2) \, (\lambda + 2n+1) \, {\varGamma }(\lambda )}{\ell ! \, {\varGamma }( \lambda + 2n + \ell + 2)} \\&= \frac{(\lambda + 2n+1) \, {\varGamma }(\lambda )}{\pi ^2} \, \frac{{\varGamma }^2 ( n + 1/2)}{{\varGamma }( \lambda +2n+2)} \\&\quad {}_2F_1 \left( n+\frac{1}{2}, n+\frac{1}{2}; \lambda +2n+2; 1\right) , \end{aligned} \end{aligned}$$

where \({}_2F_1\) is the Gaussian hypergeometric function. Owing to Gauss’s theorem (formula 15.1.20 of Abramowitz and Stegun (1972)), this finally reduces to

$$\begin{aligned} b_{2n+1,d}= & {} \frac{(\lambda + 2n+1) \, {\varGamma }(\lambda ) \, {\varGamma }(\lambda +1)}{\pi ^2} \nonumber \\&\frac{{\varGamma }^2 ( n + 1/2)}{{\varGamma }^2 ( \lambda + n + 3/2)}. \end{aligned}$$
(D.2)

These coefficients can be calculated directly, or by using the recurrence relation

$$\begin{aligned} b_{2 n+1,d} = \frac{\lambda + 2 n +1}{\lambda + 2 n -1} \, \frac{(n-1/2)^2}{(\lambda + n + 1/2)^2} \, b_{2 n -1,d}, \qquad n \ge 1, \end{aligned}$$

starting from

$$\begin{aligned} b_{1,d} = \frac{{\varGamma }(\lambda ) \, {\varGamma }(\lambda +2)}{\pi \, {\varGamma }^2 (\lambda +3/2)}. \end{aligned}$$

Equation (D.2) generalizes the expressions provided by Huang et al. (2011) and Lantuéjoul et al. (2019) for the specific case when \(d=2\).

Exponential covariance

Put \(K ( \vartheta ) = \exp ( - \nu \vartheta )\) with \(\nu >0\). The power series of \( F(t) = \exp ( - \nu \arccos t )\) is required. A first derivation gives \( \sqrt{1 - t^2} F^\prime (x) - \nu F (t) = 0\). A second derivation followed by a multiplication by \(\sqrt{ 1 - t^2}\) yields \( (1-t^2) F^{\prime \prime } (t) - t F^\prime (t) - \nu \sqrt{1 - t^2} F^\prime (t) = 0\). Replacing the third term by its expression in the first derivation, we finally obtain

$$\begin{aligned} ( 1 - t^2) F^{\prime \prime } (t) - t F^\prime (t) - \nu ^2 F (t) = 0. \end{aligned}$$
(D.3)

Let us now expand F into a power series:

$$\begin{aligned} F (t) = \sum _{k=0}^\infty \alpha _k t^k. \end{aligned}$$

Owing to the expression of F and to the first derivation formula, the first two coefficients are \(\alpha _0 = \exp (- \nu \pi /2)\) and \( \alpha _1 = \nu \exp ( - \nu \pi / 2)\). More generally, if the the power series of \(F^\prime \) and \(F^{\prime \prime }\) are plugged into (D.3), then the following recurrence relation is obtained:

$$\begin{aligned} \alpha _{k+2}= & {} \alpha _k \, \frac{k^2 + \nu ^2}{(k+1)(k+2)} \\= & {} \alpha _k \, \frac{4}{(k+1)(k+2)} \left( \frac{k+i\nu }{2}\right) \left( \frac{k-i\nu }{2}\right) , \end{aligned}$$

where i is the imaginary unit. If k is even, we have

$$\begin{aligned} \alpha _{k}= & {} \alpha _0 \, \frac{2^{k}}{k!} \, \frac{{\varGamma }\left( \frac{k + i \nu }{2} \right) }{{\varGamma }\left( \frac{i \nu }{2} \right) } \, \frac{{\varGamma }\left( \frac{k - i \nu }{2} \right) }{{\varGamma }\left( - \frac{i \nu }{ 2} \right) } \\= & {} \alpha _0 \, \frac{2^{k}}{k!} \, \left|{\varGamma }\left( \frac{k + i \nu }{2} \right) \right|^2 \, \frac{\nu \sinh \left( \frac{\pi \nu }{2}\right) }{ 2 \pi }, \end{aligned}$$

the last equality being obtained by using formula 6.1.29 of Abramowitz and Stegun (1972). Likewise, if k is odd, we have

$$\begin{aligned} \alpha _{k}= & {} \alpha _1 \, \frac{2^{k-1}}{k!} \, \frac{{\varGamma }\left( \frac{k + i \nu }{2}\right) }{{\varGamma }\left( \frac{1 + i \nu }{2}\right) } \, \frac{{\varGamma }\left( \frac{k - i \nu }{2}\right) }{{\varGamma }\left( \frac{1 - i \nu }{2}\right) } \\= & {} \alpha _1 \, \frac{2^{k}}{k!} \, \left|{\varGamma }\left( \frac{k + i \nu }{2} \right) \right|^2 \, \frac{\cosh \left( \frac{\pi \nu }{2}\right) }{2 \pi }, \end{aligned}$$

the last equality being obtained by using formula 6.1.30 of Abramowitz and Stegun (1972).

Accordingly, accounting for the above expressions of \(\alpha _0\) and \(\alpha _1\), in all cases we have

$$\begin{aligned} \alpha _k = C (\nu , k) \, \frac{2^k}{k!} \, \left|{\varGamma }\left( \frac{ k + i \nu }{2} \right) \right|^2, \end{aligned}$$
(D.4)

where

$$\begin{aligned} C(\nu ,k) = {\left\{ \begin{array}{ll} \displaystyle \frac{\nu \, \exp ( - \pi \nu / 2) \, \sinh ( \pi \nu / 2)}{2 \pi } &{} \text {if} \, k \, \text {is even} \\ \displaystyle \frac{\nu \, \exp ( - \pi \nu / 2) \, \cosh ( \pi \nu / 2)}{2 \pi } &{} \text {if} \, k \, \text {is odd.} \end{array}\right. } \end{aligned}$$

Plugging this expression into formula (D.1), we obtain

$$\begin{aligned} b_{n,d}&= C (\nu ,n) \, \sum _{\ell =0}^\infty {\varGamma }\left( \ell + \frac{ n + i \nu }{2} \right) \, {\varGamma }\left( \ell + \frac{ n - i \nu }{2} \right) \\&\quad \frac{\lambda + n}{\ell !} \, \frac{{\varGamma }(\lambda )}{{\varGamma }( \lambda + n + \ell + 1)} \\&= (\lambda + n) \, {\varGamma }(\lambda ) \, C ( \nu ,n) \, \frac{{\varGamma }\bigl ( \frac{n + i \nu }{2} \bigr ) \, {\varGamma }\bigl ( \frac{n - i \nu }{2} \bigr )}{ {\varGamma }( \lambda + n + 1)} \\&\quad {}_2F_1 \left( \frac{n + i \nu }{2} , \frac{n - i \nu }{2} ; \lambda + n + 1 ; 1 \right) . \end{aligned}$$

By Gauss’s theorem, it comes

$$\begin{aligned} b_{n,d}= & {} C (\nu ,n) \, (\lambda + n) \, {\varGamma }(\lambda ) \, {\varGamma }(\lambda +1) \nonumber \\&\frac{ \displaystyle \left|{\varGamma }\left( \frac{n + i \nu }{2} \right) \right|^2}{ \displaystyle \left|{\varGamma }\left( \lambda +1 + \frac{n + i \nu }{2} \right) \right|^2}. \end{aligned}$$
(D.5)

Calculating the squared modulus of the complex-valued gamma function in the numerator of (D.5) can be done by applying the recurrence relation

$$\begin{aligned}&\left|{\varGamma }\left( \frac{n + i \nu }{2} \right) \right|^2\\&\quad = \frac{ (n-2)^2 + \nu ^2}{4} \, \left|{\varGamma }\left( \frac{n-2 + i \nu }{2} \right) \right|^2, \qquad n \ge 2, \end{aligned}$$

along with the initial values (Abramowitz and Stegun (1972), formulae 6.1.29 and 6.1.30)

$$\begin{aligned}&\left|{\varGamma }\left( \frac{i \nu }{2} \right) \right|^2 = \frac{ 2 \pi }{\nu \, \sinh (\pi \nu / 2)} \\&\left|{\varGamma }\left( \frac{1 + i \nu }{2} \right) \right|^2 = \frac{\pi }{\cosh ( \pi \nu / 2)}. \end{aligned}$$

The same procedure applies for the calculation of the denominator in (D.5). Other expressions of \(b_{n,d}\) have been provided by Arafat et al. (2018) and Lantuéjoul et al. (2019), but they are valid only when \(d=2\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alegría, A., Emery, X. & Lantuéjoul, C. The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on the d-sphere. Stat Comput 30, 1403–1418 (2020). https://doi.org/10.1007/s11222-020-09952-8

Download citation

Keywords

  • Schoenberg sequence
  • Turning Bands
  • Gegenbauer polynomials
  • Central limit approximation
  • Berry-Esséen inequality