噪声方向数据的拟合优度检验
Goodness-of-fit test for noisy directional data
摘要：
我们考虑随机旋转$\varepsilon_i\in$ SO（3）加噪声的球形数据$X_i$，使得仅观察到样本$Z_i=\varepsilon_iX_i$，$i=1,\dots, N$。定义了一个非参数检验过程来区分$H_0:$ "$X_i$的密度$f$是球上的均匀密度$f_0$“和$H_1:$" $\|f-f_0\|_2^2\geq \Cψ_N$和$f$是在一个光滑的Sobolev空间$s$”.对于噪声密度$f_\varepsilon$与平滑指数$ν$，我们表明，自适应程序（即$s$不假定为已知）不能有一个更快的分离速度比$ψ_N^{ad}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2ν+1)}$，我们提供了一个程序，达到这个速度。我们还处理超平滑噪声的情况。我们通过对SO（3）上的各种噪声执行我们的测试程序并将其与其他程序进行比较来说明理论。提供了天体物理学和古地磁学中实际数据的应用。
英文摘要：
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 $H_1:$ ''$\|f-f_0\|_2^2\geq \Cψ_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $ν$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $ψ_N^{ad}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2ν+1)}$ 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.
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