Goodness-of-fit test for noisy directional data

TL;DR

This paper develops a nonparametric goodness-of-fit test for noisy spherical data, capable of detecting deviations from uniformity under various noise conditions, with theoretical optimality and practical applications demonstrated.

Contribution

It introduces an adaptive nonparametric test for spherical data with noise, achieving optimal separation rates without prior knowledge of smoothness, and extends to super smooth noise cases.

Findings

The test achieves the optimal separation rate for detecting non-uniformity.

The procedure is adaptable to unknown smoothness levels.

Applications demonstrate effectiveness on real astrophysics and paleomagnetism data.

Abstract

We consider spherical data $X_i$ noised by a random rotation $\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \C\psi_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $\nu$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $\psi_N^{ad}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ and we provide a procedure which reaches this rate. We also deal with the case of super smooth noise. We illustrate the theory by implementing our test procedure for various kinds of noise on SO(3) and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.