A spectral algorithm to simulate nonstationary random fields on spheres and multifractal star-shaped random sets

Abstract

An extension of the turning arcs algorithm is proposed for simulating a random field on the two-dimensional sphere with a second-order dependency structure associated with a locally varying Schoenberg sequence. In particular, the correlation range as well as the fractal index of the simulated random field, obtained as a weighted sum of Legendre waves with random degrees, may vary from place to place on the spherical surface. The proposed algorithm is illustrated with numerical examples, a by-product of which is a closed-form expression for two new correlation functions (exponential-Bessel and hypergeometric models) on the sphere, together with their respective Schoenberg sequences. The applicability of our findings is also described via the emulation of three-dimensional multifractal star-shaped random sets.

Appendices

Appendices

A. Proof of Proposition 1

The positive semidefiniteness of (7) is a direct consequence of the addition theorem for spherical harmonic functions, which is described below. The set of spherical harmonic functions, $\{{Y}_{nm}: n\in {\mathbb {N}}, m = -n, \ldots , n\}$, is an orthogonal basis of the Hilbert space of complex-valued square integrable functions on ${\mathbb {S}}^2$. Explicit expressions for these functions can be found in Olver et al. (2010, formula 14.30.1) and Marinucci and Peccati (2011). The addition theorem for spherical harmonic functions (Olver et al. 2010, formula 14.30.9) establishes that

$$\begin{aligned} {P}_n(\varvec{x}_1^\top \varvec{x}_2)& = \frac{4\pi }{2n+1} \sum _{m=0}^n\bigg [ \hbox {Re} \left\{ {Y}_{nm}(\varvec{x}_1) \right\} \hbox {Re} \left\{ {Y}_{nm}(\varvec{x}_2)\right\} \\&+\hbox {Im} \left\{ {Y}_{nm}(\varvec{x}_1)\right\} \hbox {Im} \left\{ {Y}_{nm}(\varvec{x}_2)\right\} \bigg ],\\&\quad \varvec{x}_1, \varvec{x}_2\in {\mathbb {S}}^2, \end{aligned}$$

where $\hbox {Re}$ and $\hbox {Im}$ represent real and imaginary parts, respectively. A straightforward calculation shows that, for any $k\in {\mathbb {N}}^{*}$, and for any system of points $\varvec{x}_1,\ldots , \varvec{x}_k \in {\mathbb {S}}^2$ and constants $a_1,\ldots ,a_k \in {\mathbb {R}}$,

$$\begin{aligned} &\sum _{i,j=1}^k a_i a_j C(\varvec{x}_i,\varvec{x}_j)\\& = \sum _{i,j=1}^k a_i a_j \sum _{n=0}^\infty \beta _n(\varvec{x}_i,\varvec{x}_j) {P}_n(\varvec{x}_i^\top \varvec{x}_j)\\& = \sum _{n=0}^\infty \frac{4\pi }{2n+1} \sum _{m=0}^n \left[ \sum _{i,j=1}^k \left\{ {c}_{nm,i} c_{nm,j} + {d}_{nm,i} d_{nm,j} \right\} \beta _n(\varvec{x}_i,\varvec{x}_j)\right] , \end{aligned}$$

where $c_{nm,i} = a_i \hbox {Re}\left\{ {Y}_{nm}(\varvec{x}_i)\right\}$ and $d_{nm,i} = a_i \hbox {Im}\left\{ {Y}_{nm}(\varvec{x}_i)\right\}$. The last expression is clearly nonnegative due to the positive semidefiniteness of the functions $\beta _n$, and the exchange order of summations is well justified by dominated convergence.

B. Proof of Proposition 2

The basic random field defined in (24) clearly has a zero expectation. On the other hand, in order to obtain its covariance function, we use the same arguments as in Alegría et al. (2020). Indeed, note that

$$\begin{aligned} {\mathbb {E}}\{{Z}(\varvec{x}_1) {Z}(\varvec{x}_2)\}& = {\mathbb {E}}(\varepsilon ^2) \sum _{n=0}^\infty \{b_n(\varvec{x}_1) b_n(\varvec{x}_2)\}^{1/2} (2n+1)\\&\times \int _{{\mathbb {S}}^2} {P}_n(\varvec{\omega }^\top \varvec{x}_1) {P}_n(\varvec{\omega }^\top \varvec{x}_2) U(\hbox {d}{\varvec{\omega }}), \\&\quad \varvec{x}_1,\varvec{x}_2\in {\mathbb {S}}^2, \end{aligned}$$

where U is the uniform probability measure on ${\mathbb {S}}^2$. The result follows from the duplication equation for Legendre polynomials (see, e.g., (Ziegel 2014, equation 2.4)): for any $n,k\in {\mathbb {N}}$,

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

where $\delta _{n,k}$ denotes the Kronecker delta.

C. Assessment of the central limit approximation

Starting with the well-known Berry–Esséen inequality, Alegría et al. (2020) showed that the Kolmogorov–Smirnov distance between the marginal distribution of $\widetilde{Z}(\varvec{x})$ as defined in (25) and a Gaussian distribution is upper bounded as follows:

$$\begin{aligned}&\underset{z\in {\mathbb {R}}}{\sup } \Bigg |{\mathbb {P}} \left( \frac{\widetilde{Z}(\varvec{x})}{C(\varvec{x},\varvec{x})^{1/2}} < z \right) - G(z) \Bigg |\le \frac{\xi \, {\mathbb {E}}( |\varepsilon |^3)}{C(\varvec{x},\varvec{x})^{3/2} \, L^{1/2}} \\&\times \sum _{n}\frac{b_{n}(\varvec{x})^{3/2} \, (2n + 1)^{3/2} \, {\mathbb {E}} \left( |P_n(\varvec{\omega }^T \varvec{x}) |^3 \right) }{\zeta _n^{1/2}}, \end{aligned}$$

where the sum is extended over all the integers n such that $\zeta _n$ is positive, G is the standard Gaussian cumulative distribution function, $\xi$ is a constant between 0.4097 and 0.4748, and ${\mathbb {E}} \left( |P_n(\varvec{\omega }^T \varvec{x}) |^3 \right)$ does not depend on $\varvec{x}$ and behaves as ${\mathcal {O}}(n^{-3/2})$ at large n. Now, in the four examples presented in Sect. 4.2, the simulated random field $\widetilde{Z}(\varvec{x})$ has a unit variance, $\varepsilon$ has a Rademacher distribution and $\{\zeta _{n}: n \in {\mathbb {N}}\}$ is the probability mass sequence of a shifted zeta distribution with parameter 2, so that $C(\varvec{x},\varvec{x})=1$, ${\mathbb {E}}( |\varepsilon |^3)=1$ and $\zeta _n^{-1/2} = \pi \, (n+1) / \sqrt{6}$. Accordingly:

$$\begin{aligned} \underset{z\in {\mathbb {R}}}{\sup } \Bigg |{\mathbb {P}} \left( \widetilde{Z}(\varvec{x}) < z \right) - G(z) \Bigg |\le \frac{\xi }{L^{1/2}} \sum _{n} b_{n}(\varvec{x})^{3/2} \, \tau _n, \end{aligned}$$

with $\tau _n = (2n + 1)^{3/2} \, {\mathbb {E}} \left( |P_n(\varvec{\omega }^T \varvec{x}) |^3 \right) \, \pi \, (n+1) / \sqrt{6} = {\mathcal {O}}(n)$ as $n \rightarrow +\infty$. Furthermore, one has:

• $b_{n}(\varvec{x}) = {\mathcal {O}}(n^{-2\nu (\varvec{x})-1})$ with $\nu (\varvec{x}) \in [0.2,1.8]$ (Legendre-Matérn model);

• $b_{n}(\varvec{x}) = {\mathcal {O}}(a(\varvec{x})^n)$ with $a(\varvec{x}) \in [0.1,0.9]$ (multiquadric model);

• $b_{n}(\varvec{x}) = {\mathcal {O}}\left(\frac{a(\varvec{x})^n}{n!}\right)$ with $a(\varvec{x}) \in [0.1,8.0]$ (exponential-Bessel model);

• $b_{n}(\varvec{x}) = {\mathcal {O}}(a(\varvec{x})^n \, n^{\nu (\varvec{x})-1})$, with $a(\varvec{x}) \in [0.1,0.9]$ and $\nu (\varvec{x}) \in [1,19]$ (hypergeometric model).

As a result, in all the four cases, the sequence $\{b_{n}(\varvec{x})^{3/2} \, \tau _n: n \in {\mathbb {N}}\}$ is summable, hence, the Berry–Esséen upper bound is finite and proportional to $L^{-1/2}$. By increasing L, it is possible to ensure that the distance between the marginal distribution of the simulated random field and a standard Gaussian distribution is less that any given positive threshold.