Confidence Bounds for Spherical SamplingSchemes in Stereology

Emil Dare

doi:10.5566/ias.4116

Authors

Emil Dare Aarhus University https://orcid.org/0009-0007-5696-1272

DOI:

https://doi.org/10.5566/ias.4116

Keywords:

Nucleator, Reproducing kernel Hilbert spaces, Spherical quadrature, Stereology

Abstract

This paper considers quadrature methods for integration over the unit sphere using reproducing kernel Hilbert spaces. Bayesian quadrature weights, which minimize the worst-case integration error, and their normalized counterparts, which additionally integrate constants exactly, are studied. Deterministic error bounds are established in terms of the Sobolev regularity of the integrand and geometric properties of the sampling design. For designs obtained by applying a uniformly random rotation to a fixed point configuration, exact mean and bias identities are derived, and the variance is bounded by the squared worst-case error. The normalized estimator is unbiased, whereas the bias of the unconstrained Bayesian estimator is proportional to its squared worst-case error. To construct data-dependent bounds, an ordinary least-squares estimator of the spherical harmonic coefficients is introduced, yielding a conservative 95\% confidence interval under an assumption on the Sobolev norm of the harmonic tail. The framework is applied to the Nucleator in stereology, for which explicit error bounds are obtained for convex bodies in dimensions two and three.

References

Atkinson K, Han W (2012). Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Lecture Notes in Mathematics 2044. Berlin, Heidelberg: Springer.

Bondarenko A, Radchenko D, Viazovska M (2013). Optimal asymptotic bounds for spherical designs. Ann. of Math. 178(2):443–52.

Brauchart JS, Grabner PJ (2015). Distributing many points on spheres: Minimal energy and designs. J. Complexity 31(3):293–326.

Brauchart JS, Grabner PJ, Kusner W (2019). Hyperuniform point sets on the sphere: Deterministic aspects. Constr. Approx. 50:45–61.

Brauchart JS, Grabner PJ, Kusner W, Ziefle J (2020). Hyperuniform point sets on the sphere: Probabilistic aspects. Monatsh. Math. 192:763–81.

Brauchart JS, Saff EB, Sloan IH, Womersley RS (2014). QMC designs: Optimal order quasi Monte Carlo integration schemes on the sphere. Math. Comput. 83(290):2821–51. doi:10.1090/S0025-5718-2014-02839-1.

Christensen AS (2020). Asymptotic Results of the Cavalieri Estimator. Master's thesis in Statistics, Aarhus Universitet, Institut for Matematik, Aarhus, Denmark.

Delsarte P, Goethals JM, Seidel JJ (1977). Spherical codes and designs. Geom. Dedicata 6(3):363–88. doi:10.1007/BF03187604.

Fuselier E, Hangelbroek T, Narcowich FJ, Ward JD, Wright GB (2014). Kernel based quadrature on spheres and other homogeneous spaces. Numer. Math. 127(1):57–92.

García-Fiñana M (2006). Confidence intervals in Cavalieri sampling. J. Microsc. 222(3):146–57.

Gomez-Perez D, González-Villa J, Pausinger F (2019). Estimation of volume using the nucleator and lattice points. Image Anal. Stereol. 38(2):141–50.

González-Villa J, Cruz M, Cruz-Orive LM (2017). On the precision of the nucleator. Image Anal. Stereol. 36(2):121–32.

Gundersen HJG (1988). The nucleator. J. Microsc. 151(1):3–21.

Hatcher A (2002). Algebraic Topology. Cambridge: Cambridge University Press.

Hubbert S, Morton TM (2004). Lp-error estimates for radial basis function interpolation on the sphere. J. Approx. Theory 129(1):58–77.

Kanagawa M, Sriperumbudur BK, Fukumizu K (2020). Convergence analysis of deterministic kernel-based quadrature rules in misspecified settings. Found. Comput. Math. 20(1):155–94.

Karvonen T, Kanagawa M, Särkkä S (2019). On the positivity and magnitudes of Bayesian quadrature weights. Stat. Comput. 29:1317–33.

Kiderlen M (2008). Stability results for convex bodies in geometric tomography. Indiana Univ. Math. J. 57:1999–2038.

Leobacher G, Pillichshammer F (2014). Introduction to Quasi-Monte Carlo Integration and Applications. Compact Textbooks in Mathematics. Cham: Birkhäuser.

Lošdorfer Božič A, Čopar S (2019). Spherical structure factor and classification of hyperuniformity on the sphere. Phys. Rev. E 99(3):032601.

Lotz L, Klatt MA (2026). Persistence of asymptotic variance under transport: from hyperfluctuation to stealthy hyperuniformity. arXiv:2605.22803. doi:10.48550/arXiv.2605.22803.

Meyra AG, Zarragoicoechea GJ, Maltz AL, Lomba E, Torquato S (2019). Hyperuniformity on spherical surfaces. Phys. Rev. E 100(2):022107.

Müller C (1966). Spherical Harmonics. Heidelberg: Springer.

Narcowich FJ, Sun X, Ward JD, Wendland H (2007). Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions. Found. Comput. Math. 7:369–90.

O'Hagan A (1991). Bayes–Hermite quadrature. J. Stat. Plan. Infer. 29(3):245–60.

Saff EB, Kuijlaars ABJ (1997). Distributing many points on a sphere. Math. Intell. 19(1):5–11.

Schneider R, Weil W (2008). Stochastic and Integral Geometry. Probability and Its Applications. Berlin, Heidelberg: Springer.

Soto FJ (2025). Randomized quadrature with periodic kernels: Applications to Cavalieri volume estimation. Image Anal. Stereol. 44(3):197–208.

Stehr M, Kiderlen M (2020a). Asymptotic variance of Newton–Cotes quadratures based on randomized sampling points. Adv. Appl. Probab. 52(4):1284–1307.

Stehr M, Kiderlen M (2020b). Improving the Cavalieri estimator under non-equidistant sampling and dropouts. Image Anal. Stereol. 39(3):197–212.

Stehr M, Kiderlen M (2025). Confidence intervals for Newton–Cotes quadratures based on stationary point processes. Stat. Methods Appl. 34(1):39–67.

Torquato S, Stillinger FH (2003). Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 68(4):041113.

Vedel Jensen EB (1991). Recent developments in the stereological analysis of particles. Ann. Inst. Stat. Math. 43:455–68.

Womersley RS (2018). Efficient spherical designs with good geometric properties. In: Dick J, Kuo FY, Woźniakowski H, eds., Contemporary Computational Mathematics – A Celebration of the 80th Birthday of Ian Sloan. Cham: Springer, 1243–85.