arXiv subjects

Domenico Marinucci

Publications and source records attributed to Domenico Marinucci.

At least 73 records · Page 4Linked to original sources

arXiv2013

On the Limiting Behaviour of Needlets Polyspectra

This paper provides quantitative Central Limit Theorems for nonlinear transforms of spherical random fields, in the high frequency limit. The sequences of fields that we consider are represented as smoothed averages of spherical Gaussian eigenfunctions and can be viewed as random coefficients from continuous wavelets/needlets; as such, they are of immediate interest for spherical data analysis. In particular, we focus on so-called needlets polyspectra, which are popular tools for nonGaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on Stein-Maliavin approximations for nonlinear transforms of Gaussian fields, and on an explicit derivation on the high-frequency limit of their variances, which may have some independent interest.

math.PR↗

arXiv2013

Needlet-Whittle Estimates on the Unit Sphere

We study the asymptotic behaviour of needlets-based approximate maximum likelihood estimators for the spectral parameters of Gaussian and isotropic spherical random fields. We prove consistency and asymptotic Gaussianity, in the high-frequency limit, thus generalizing earlier results by Durastanti et al. (2011) based upon standard Fourier analysis on the sphere. The asymptotic results are then illustrated by an extensive Monte Carlo study.

math.ST↗

arXiv2013

Mean-Square Continuity on Homogeneous Spaces of Compact Groups

We show that any finite-variance, isotropic random field on a compact group is necessarily mean-square continuous, under standard measurability assumptions. The result extends to isotropic random fields defined on homogeneous spaces where the group acts continuously.

math.PR↗

arXiv2012

On Nonlinear Functionals of Random Spherical Eigenfunctions

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.

math-ph↗

arXiv2012

Normal Approximations for Wavelet Coefficients on Spherical Poisson Fields

We compute explicit upper bounds on the distance between the law of a multivariate Gaussian distribution and the joint law of wavelets/needlets coefficients based on a homogeneous spherical Poisson field. In particular, we develop some results from Peccati and Zheng (2011), based on Malliavin calculus and Stein's methods, to assess the rate of convergence to Gaussianity for a triangular array of needlet coefficients with growing dimensions. Our results are motivated by astrophysical and cosmological applications, in particular related to the search for point sources in Cosmic Rays data.

math.PR↗

arXiv2012

Detection of new point sources in WMAP 7 year data using internal templates and needlets

We have developed a new needlet based method to detect point sources in cosmic microwave background (CMB) maps and have applied it to the WMAP 7 year data. We use both the individual frequency channels as well as internal templates, the difference between pairs of frequency channels, with the advantage that the CMB component is eliminated. Using the area of the sky outside the Kq85 galactic mask, we detect a total of 2102 point sources at the $5σ$ level in either the frequency maps or the internal templates. Of these, 1116 are detected either at $5σ$ directly in the frequency channels or at $5σ$ in the internal templates and $\geq3σ$ at the corresponding position in the frequency channels. Of the 1116 sources, 603 are detections which have not been reported so far in WMAP data. We have made a catalogue of these sources available with position and flux estimated in the WMAP channels where they are seen. In total, we identified 1029 of the 1116 sources with counterparts at 5GHz and 69 at other frequencies.

astro-ph.CO↗

arXiv2012

On the linear term correction for needlets/wavelets non-Gaussianity estimators

We derive the linear correction term for needlet and wavelet estimators of the bispectrum and the non-linearity parameter fNL on cosmic microwave background radiation data. We show that on masked WMAP-like data with anisotropic noise, the error bars improve by 10-20% and almost reach the optimal error bars obtained with the KSW estimator (Komatsu et al 2005). In the limit of full-sky and isotropic noise, this term vanishes. We apply needlet and wavelet estimators to the WMAP 7-year data and obtain our best estimate fNL=37.5 \pm 21.8.

astro-ph.CO↗

arXiv2011

The defect variance of random spherical harmonics

The defect of a function $f:M\rightarrow \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern Cosmological data analysis.

math-ph↗

arXiv2010

Adaptive Nonparametric Regression on Spin Fiber Bundles

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.

math.ST↗

arXiv2010

On the Excursion Sets of Spherical Gaussian Eigenfunctions

The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivations arising from Physics and Cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a Uniform Central Limit Theorem for the empirical measure, i.e. the proportion of spherical surface where spherical Gaussian eigenfunctions lie below a level $z$. Our proofs borrows some techniques from the literature on stationary long memory processes; in particular, we expand the empirical measure into Hermite polynomials, and establish a uniform weak reduction principle, entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion. As a result, we establish a functional central limit theorem; the limiting process is fully degenerate.

math.PR↗

arXiv2010

Mixed Needlets

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.

math.CA↗

arXiv2010

Directional Variations of the Non-Gaussianity Parameter f_NL

We investigate local variations of the primordial non-Gaussianity parameter f_NL in the WMAP data, looking for possible influence of foreground contamination in the full-sky estimate of f_NL. We first improve the needlet bispectrum estimate in (Rudjord et al. 2009) on the full-sky to f_NL= 73 +/- 31 using the KQ75 mask on the co-added V+W channel. We find no particular values of f_NL estimates close to the galactic plane and conclude that foregrounds are unlikely to affect the estimate of f_NL in the V and W bands even for the smaller KQ85 mask. In the Q band however, we find unexpectedly high values of f_NL in local estimates close to the galactic mask, as well as significant discrepancies between Q band estimates and V/W band estimates. We therefore conclude that the Q band is too contaminated to be used for non-Gaussianity studies even with the larger KQ75 mask. We further noted that the local f_NL estimates on the V+W channel are positive on all equatorial bands from the north to the south pole. The probability for this to happen in a universe with f_NL = 0 is less than one percent.

astro-ph.CO↗

arXiv2009

Ergodicity and Gaussianity for Spherical Random Fields

We investigate the relationship between ergodicity and asymptotic Gaussianity of isotropic spherical random fields, in the high-resolution (or high-frequency) limit. In particular, our results suggest that under a wide variety of circumstances the two conditions are equivalent, i.e. the sample angular power spectrum may converge to the population value if and only if the underlying field is asymptotically Gaussian, in the high frequency sense. These findings may shed some light on the role of Cosmic Variance in Cosmic Microwave Background (CMB) radiation data analysis.

math.PR↗

arXiv2009

Spin Needlets Spectral Estimation

We consider the statistical analysis of random sections of a spin fibre bundle over the sphere. These may be thought of as random fields that at each point p in $S^2$ take as a value a curve (e.g. an ellipse) living in the tangent plane at that point $T_{p}S^2$, rather than a number as in ordinary situations. The analysis of such fields is strongly motivated by applications, for instance polarization experiments in Cosmology. To investigate such fields, spin needlets were recently introduced by Geller and Marinucci (2008) and Geller et al. (2008). We consider the use of spin needlets for spin angular power spectrum estimation, in the presence of noise and missing observations, and we provide Central Limit Theorem results, in the high frequency sense; we discuss also tests for bias and asymmetries with an asymptotic justification.

math.ST↗

arXiv2009

An Estimate of the Primordial Non-Gaussianity Parameter f_NL Using the Needlet Bispectrum from WMAP

We use the full bispectrum of spherical needlets applied to the WMAP data of the cosmic microwave background as an estimator for the primordial non-Gaussianity parameter f_NL. We use needlet scales up to l_max=1000 and the KQ75 galactic cut and find f_NL=84 +/- 40 corrected for point source bias. We also introduce a set of consistency tests to validate our results against the possible influence of foreground residuals or systematic errors. In particular, fluctuations in the value of f_NL obtained from different frequency channels, different masks and different multipoles are tested against simulated maps. All variations in f_NL estimates are found statistically consistent with simulations.

astro-ph.CO↗

arXiv2009

Statistical challenges in the analysis of Cosmic Microwave Background radiation

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.

stat.AP↗

arXiv2009

On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields

We consider the correlation structure of the random coefficients for a wide class of wavelet systems on the sphere (Mexican needlets) which were recently introduced in the literature by Geller and Mayeli (2007). We provide necessary and sufficient conditions for these coefficients to be asymptotic uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high resolution sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data.

math.ST↗

arXiv2008

Spin Wavelets on the Sphere

In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In particular, we propose {\em needlet-type spin wavelets} as an extension of the needlet approach recently introduced by Narcowich, Petrushev and Ward, and then considered for more general manifolds by Geller and Mayeli. We discuss localization properties in the real and harmonic domains, and investigate stochastic properties for the analysis of spin random fields. Our results are strongly motivated by cosmological applications, in particular in connection to the analysis of Cosmic Microwave Background polarization data.

math.CA↗