28 research outputs found

    Adaptive Nonparametric Regression on Spin Fiber Bundles

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    The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning for instance the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008,2010) and Geller et al. (2008,2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.Comment: 40 page

    Mixed Needlets

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    The construction of needlet-type wavelets on sections of the spin line bundles over the sphere has been recently addressed in Geller and Marinucci (2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal for needlets on this spin line bundle, in which needlet coefficients arise from the usual, rather than the spin, spherical harmonics, as in the previous constructions. We label this system mixed needlets and investigate in full their properties, including localization, the exact tight frame characterization, reconstruction formula, decomposition of functional spaces, and asymptotic uncorrelation in the stochastic case. We outline astrophysical applications.Comment: 26 page

    Statistical challenges in the analysis of Cosmic Microwave Background radiation

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    An enormous amount of observations on Cosmic Microwave Background radiation has been collected in the last decade, and much more data are expected in the near future from planned or operating satellite missions. These datasets are a goldmine of information for Cosmology and Theoretical Physics; their efficient exploitation posits several intriguing challenges from the statistical point of view. In this paper we review a number of open problems in CMB data analysis and we present applications to observations from the WMAP mission.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS190 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Fractional Stochastic Partial Differential Equation for Random Tangent Fields on the Sphere

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    This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L\'{e}vy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Lo\`{e}ve expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. The variance of the increments and approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.Comment: 20 page

    The integrated angular bispectrum of weak lensing

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    We investigate three-point statistics in weak lensing convergence, through the integrated angular bispectrum. It involves measuring the three-point function of harmonic multipoles of the lensing field by estimating how the angular power spectrum (two-point function) in patches is modulated by large scale modes. This approach avoids the complexity of estimating the very large number of possible bispectrum configurations. The integrated bispectrum mainly probes the squeezed limit of the bispectrum. Previous studies have compared measurements of the integrated bispectrum on the sphere against theoretical prediction, finding overall good consistency. It is crucial, to apply this statistic to data analysis and parameter estimation, to extract the covariance matrix. This requires applying the integrated bispectrum estimator to many thousands simulations, however the current implementation is too slow. Therefore, in this thesis, we investigate a new implementation, based on the so-called flat-sky approximation, in which the power spectrum in small patches of the spherical domain is computed via a tangent-plane projection.We investigate three-point statistics in weak lensing convergence, through the integrated angular bispectrum. It involves measuring the three-point function of harmonic multipoles of the lensing field by estimating how the angular power spectrum (two-point function) in patches is modulated by large scale modes. This approach avoids the complexity of estimating the very large number of possible bispectrum configurations. The integrated bispectrum mainly probes the squeezed limit of the bispectrum. Previous studies have compared measurements of the integrated bispectrum on the sphere against theoretical prediction, finding overall good consistency. It is crucial, to apply this statistic to data analysis and parameter estimation, to extract the covariance matrix. This requires applying the integrated bispectrum estimator to many thousands simulations, however the current implementation is too slow. Therefore, in this thesis, we investigate a new implementation, based on the so-called flat-sky approximation, in which the power spectrum in small patches of the spherical domain is computed via a tangent-plane projection

    Geometric Methods for Spherical Data, with Applications to Cosmology

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    This survey is devoted to recent developments in the statistical analysis of spherical data, with a view to applications in Cosmology. We will start from a brief discussion of Cosmological questions and motivations, arguing that most Cosmological observables are spherical random fields. Then, we will introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames. We will then focus on some specific issues, including tools and algorithms for map reconstruction (\textit{i.e.}, separating the different physical components which contribute to the observed field), geometric tools for testing the assumptions of Gaussianity and isotropy, and multiple testing methods to detect contamination in the field due to point sources. Although these tools are introduced in the Cosmological context, they can be applied to other situations dealing with spherical data. Finally, we will discuss more recent and challenging issues such as the analysis of polarization data, which can be viewed as realizations of random fields taking values in spin fiber bundles.Comment: 25 pages, 6 figure
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