We provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behavior of our procedure for any $\mathbb {L}_p$ loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We also perform a numerical study which proves that the procedure shows very promising properties in practice as well.Comment: Published in at http://dx.doi.org/10.1214/10-AOS858 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider spherical data X i noised by a random rotation ε i ∈ SO(3) so that only the sample Z i = ε i X i , i = 1, . . . , 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 ≥ Cψ N and f is in a Sobolev space with smoothness s". For a noise density f ε 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 ad N (s) = (N/ 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.

ObjectivesTo define the characteristics of post-traumatic headache with cluster headache phenotype (PTH-CH) and to compare these characteristics with primary CH.MethodsA retrospective study was conducted of patients seen between 2007 and 2017 in a headache centre and diagnosed with PTH-CH that developed within 7 days of head trauma. A control cohort included 553 patients with primary CH without any history of trauma who attended the headache clinic during the same period. Data including demographics, attack characteristics and response to treatments were recorded.ResultsTwenty-six patients with PTH-CH were identified. Multivariate analysis revealed significant associations between PTH-CH and family history of CH (OR 3.32, 95% CI 1.31 to 8.63), chronic form (OR 3.29, 95% CI 1.70 to 6.49), parietal (OR 14.82, 95% CI 6.32 to 37.39) or temporal (OR 2.04, 95% CI 1.10 to 3.84) location of pain, and presence of prominent cranial autonomic features during attacks (miosis OR 11.24, 95% CI 3.21 to 41.34; eyelid oedema OR 5.79, 95% CI 2.57 to 13.82; rhinorrhoea OR 2.65, 95% CI 1.26 to 5.86; facial sweating OR 2.53, 95% CI 1.33 to 4.93). Patients with PTH-CH were at a higher risk of being intractable to acute (OR 12.34, 95% CI 2.51 to 64.73) and preventive (OR 16.98, 95% CI 6.88 to 45.52) treatments and of suffering from associated chronic migraine (OR 10.35, 95% CI 3.96 to 28.82).ConclusionThis largest series of PTH-CH defines it as a unique entity with specific evolutive profile. Patients with PTH-CH are more likely to suffer from the chronic variant, have marked autonomic features, be intractable to treatment and have associated chronic migraine compared with primary CH.