On the boundary properties of Bernstein estimators on the simplex

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature. Our results generalize those found in Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (2020) in the interior of the simplex. Different parts of the boundary having different dimensions makes the analysis more difficult.Comment: 11 pages, 0 figure

Asymptotic properties of Bernstein estimators on the simplex

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when $d = 2$. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the $d$-dimensional simplex. Our results generalize the ones in Leblanc (2012) and Babu et al. (2002), who treated the case $d = 1$, and significantly extend those found in Tenbusch (1994). In particular, our rates of convergence for the MSE and MISE are optimal.Comment: 22 pages, 1 figur

Models beyond the Dirichlet process

Bayesian nonparametric inference is a relatively young area of research and it has recently undergone a strong development. Most of its success can be explained by the considerable degree of exibility it ensures in statistical modelling, if compared to parametric alternatives, and by the emergence of new and ecient simulation techniques that make nonparametric models amenable to concrete use in a number of applied statistical problems. Since its introduction in 1973 by T.S. Ferguson, the Dirichlet process has emerged as a cornerstone in Bayesian nonparametrics. Nonetheless, in some cases of interest for statistical applications the Dirichlet process is not an adequate prior choice and alternative nonparametric models need to be devised. In this paper we provide a review of Bayesian nonparametric models that go beyond the Dirichlet process.

Models beyond the Dirichlet process

Bayesian nonparametric inference is a relatively young area of research and it has recently undergone a strong development. Most of its success can be explained by the considerable degree of flexibility it ensures in statistical modelling, if compared to parametric alternatives, and by the emergence of new and efficient simulation techniques that make nonparametric models amenable to concrete use in a number of applied statistical problems. Since its introduction in 1973 by T.S. Ferguson, the Dirichlet process has emerged as a cornerstone in Bayesian nonparametrics. Nonetheless, in some cases of interest for statistical applications the Dirichlet process is not an adequate prior choice and alternative nonparametric models need to be devised. In this paper we provide a review of Bayesian nonparametric models that go beyond the Dirichlet process.

Multi-resolution mapping and planning for UAV navigation in attitude constrained environments

In this thesis we aim to bridge the gap between high quality map reconstruction and Unmanned Aerial Vehicles (UAVs) SE(3) motion planning in challenging environments with narrow openings, such as disaster areas, which requires attitude to be considered. We propose an efficient system that leverages the concept of adaptive-resolution volumetric mapping, which naturally integrates with the hierarchical decomposition of space in an octree data structure. Instead of a Truncated Signed Distance Function (TSDF), we adopt mapping of occupancy probabilities in log-odds representation, which allows representation of both surfaces, as well as the entire free, i.e.\ observed space, as opposed to unobserved space. We introduce a method for choosing resolution -on the fly- in real-time by means of a multi-scale max-min pooling of the input depth image. The notion of explicit free space mapping paired with the spatial hierarchy in the data structure, as well as map resolution, allows for collision queries, as needed for robot motion planning, at unprecedented speed. Our mapping strategy supports pinhole cameras as well as spherical sensor models. Additionally, we introduce a first-of-a-kind global minimum cost path search method based on A* that considers attitude along the path. State-of-the-art methods incorporate attitude only in the refinement stage. To make the problem tractable, our method exploits an adaptive and coarse-to-fine approach using global and local A* runs, plus an efficient method to introduce the UAV attitude in the process. We integrate our method with an SE(3) trajectory optimisation method based on a safe-flight-corridor, yielding a complete path planning pipeline. We quantitatively evaluate our mapping strategy in terms of mapping accuracy, memory, runtime performance, and planning performance showing improvements over the state-of-the-art, particularly in cases requiring high resolution maps. Furthermore, extensive evaluation is undertaken using the AirSim flight simulator under closed loop control in a set of randomised maps, allowing us to quantitatively assess our path initialisation method. We show that it achieves significantly higher success rates than the baselines, at a reduced computational burden.Open Acces

Measuring Planck beams with planets

Aims. Accurate measurement of the cosmic microwave background (CMB) anisotropy requires precise knowledge of the instrument beam. We explore how well the Planck beams will be determined from observations of planets, developing techniques that are also appropriate for other experiments. Methods. We simulate planet observations with a Planck-like scanning strategy, telescope beams, noise, and detector properties. Then we employ both parametric and non-parametric techniques, reconstructing beams directly from the time-ordered data. With a faithful parameterization of the beam shape, we can constrain certain detector properties, such as the time constants of the detectors, to high precision. Alternatively, we decompose the beam using an orthogonal basis. For both techniques, we characterize the errors in the beam reconstruction with Monte Carlo realizations. For a simplified scanning strategy, we study the impact on estimation of the CMB power spectrum. Finally, we explore the consequences for measuring cosmological parameters, focusing on the spectral index of primordial scalar perturbations, n_s. Results. The quality of the power spectrum measurement will be significantly influenced by the optical modeling of the telescope. In our most conservative case, using no information about the optics except the measurement of planets, we find that a single transit of Jupiter across the focal plane will measure the beam window functions to better than 0.3% for the channels at 100–217 GHz that are the most sensitive to the CMB. Constraining the beam with optical modeling can lead to much higher quality reconstruction. Conclusions. Depending on the optical modeling, the beam errors may be a significant contribution to the measurement systematics for n_s