{'fraction_non_alphanumeric': 0.024825783972125436, 'fraction_numerical': 0.005226480836236934, 'mean_word_length': 5.224932249322493, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nonparametric density estimation on the rotation group

In many applications, the collected data takes values in a manifold rather than in an Euclidian space. Numerous examples of such datasets can be found in the fields of robotics, satellite control attitude, chemistry, crystallography [1]. In statistics, many applications have lead to the study of datasets on spheres or oriented frames [2] as well as on special matrix manifolds (Grassmann and Stiefel manifolds) [3]. A common important problem in all these applications is to model the random behaviour of the underlying physical system. For this purpose, many well known distributions have been extended to manifolds [3] or new distribution have been introduced to cope with the special structure of curved spaces [2]. Such distributions are usefull to deal with some parametric estimation problems and are well suited to situations where the random phenomena at action have a known model. In this study, we will consider the case where the underlying distribution model is not known, as well as the case where mixtures can occur in the observed data. In order to perform density estimation in such situations, we will consider nonparametric techniques to obtain estimates of probability densities from samples with values on manifolds. In particular, we will focuse on the Lie group SO(3), i.e. the group of rotations in 3D space. Among the existing nonparametric methods, the proposed work will concentrate on Kernel and Wavelet methods for probability density estimation [4, 5]. The specificity of the datasets will lead to the definition of estimators involving concepts from noncommutative harmonic analysis (Fourier series on groups and manifolds) [1] and wavelets on manifolds [6]. The internship will consist in proposing estimators for probability density based on the above mentioned tools, and in studying their performances and asymptotic behaviours. A systematic comparison will be performed between Kernel and Wavelet approaches. The proposed estimators will be tested on simulated datasets, including samples from well known distributions and mixtures on SO(3). Applications to real datasets from motion capture and Xray molecules scattering will possibly be carried out as well. Programming will be made in Python langage.