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Rate-optimal and computationally efficient nonparametric estimation on the circle and the sphere

Athanasios G. Georgiadis, Andrew P. Percival

TL;DR

This work tackles nonparametric density estimation on the circle $S^1$ and the sphere $S^2$ by developing rate-optimal yet computationally efficient estimators. The authors replace the standard infinite spectral expansions with finite-order (band-limited) KDEs obtained via a spectral cutoff $N_s$ and provide explicit guidance for choosing the cutoff and kernel symbol $g$. They further derive closed-form probability estimators for region probabilities on $S^1$ and $S^2$, enabling fast evaluations through incomplete Beta functions and related harmonic-analytic quantities. The approach is validated through simulations under uniform and von Mises-Fisher models and demonstrated on four case studies spanning zoology, climatology, geophysics, and astronomy, illustrating both accuracy and practical usefulness. Collectively, the methods offer a scalable, interpretable toolkit for analyzing directional and periodic phenomena with wide applicability across scientific domains.

Abstract

We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for direct implementation. After establishing these estimators, we derive closed-form expressions for probability estimates over regions of the circle and the sphere. Then, the proposed theories are supported by extensive simulation studies. The considered settings naturally arise when analyzing phenomena on the Earth's surface or in the sky (sphere), as well as directional or periodic phenomena (circle). The proposed approaches are broadly applicable, and we illustrate their usefulness through case studies in zoology, climatology, geophysics, and astronomy, which may be of independent interest. The methodologies developed here can be readily applied across a wide range of scientific domains.

Rate-optimal and computationally efficient nonparametric estimation on the circle and the sphere

TL;DR

This work tackles nonparametric density estimation on the circle and the sphere by developing rate-optimal yet computationally efficient estimators. The authors replace the standard infinite spectral expansions with finite-order (band-limited) KDEs obtained via a spectral cutoff and provide explicit guidance for choosing the cutoff and kernel symbol . They further derive closed-form probability estimators for region probabilities on and , enabling fast evaluations through incomplete Beta functions and related harmonic-analytic quantities. The approach is validated through simulations under uniform and von Mises-Fisher models and demonstrated on four case studies spanning zoology, climatology, geophysics, and astronomy, illustrating both accuracy and practical usefulness. Collectively, the methods offer a scalable, interpretable toolkit for analyzing directional and periodic phenomena with wide applicability across scientific domains.

Abstract

We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for direct implementation. After establishing these estimators, we derive closed-form expressions for probability estimates over regions of the circle and the sphere. Then, the proposed theories are supported by extensive simulation studies. The considered settings naturally arise when analyzing phenomena on the Earth's surface or in the sky (sphere), as well as directional or periodic phenomena (circle). The proposed approaches are broadly applicable, and we illustrate their usefulness through case studies in zoology, climatology, geophysics, and astronomy, which may be of independent interest. The methodologies developed here can be readily applied across a wide range of scientific domains.

Paper Structure

This paper contains 20 sections, 6 theorems, 76 equations, 11 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. The underlying geometries
  4. Legendre polynomials and spherical harmonics
  5. Density estimation on the circle and sphere
  6. Rate-optimal and computationally efficient estimators
  7. Probability estimation
  8. Simulations
  9. The uniform distribution
  10. The von Mises-Fisher's distribution
  11. Comparison of Theorem \ref{['th: probability S2']} with numerical integration
  12. Case Studies
  13. Spatially Periodic Data: Honeybee Orientations
  14. Temporally Periodic Data: Annual Precipitation Patterns
  15. Global Data: Earthquake Locations
  16. ...and 5 more sections

Key Result

Theorem 2.1

Consider a random variable $X$ distributed on $\mathbb{S}^d$, $d\in\{1,2\}$, with unknown density $f$ and $X_1,\dots,X_n$, $n\in\mathbb{N}$ iid copies of $X$. Assume that $f\in\mathcal{L}^\infty(\mathbb{S}^d)\cap\dot\mathcal{H}^s(\mathbb{S}^d)$ for some $s>0$. Let also $h=h_n=n^{-1/(2s+d)}$ and $g:\ Then the corresponding KDEs $\widehat{f}_{n,h}$ as in eq: kde on S1, for $d=1$, and eq: kde S2 for

Figures (11)

  • Figure 1: (Left to right) The plot of the uniform distribution over $\mathbb{S}^2$; a sample of $n=1000$ points from such a distribution; KDEs constructed using said data for $s=0.5$, $s=1$ and $s=2$, respectively.
  • Figure 2: (Left to right) The plot of the uniform distribution over $\mathbb{S}^1$; a sample of $n=1000$ points from such a distribution; KDEs constructed using said data for $s=0.5$, $s=1$, and $s=2$, respectively.
  • Figure 3: (Left to right) The plot of the vMF distribution over $\mathbb{S}^2$ with $\kappa=1$ and $\mu=(0,0,1)$; a sample of $n=1000$ points from such a distribution; KDEs constructed using said data for $s=0.5$, $s=1$ and $s=2$, respectively.
  • Figure 4: The estimated MISE for vMF simulated points with $\kappa=1$ and $s\in\{0.1,0.2,\dots,10\}$, showing a clear minimum around $s=2$.
  • Figure 5: (Left to right) The plot of a mixture density over $\mathbb{S}^2$; a sample of $n=1000$ points from such a distribution; KDEs constructed using said data for $s=0.5$, $s=1$ and $s=2$, respectively.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • proof
  • proof
  • ...and 2 more