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We describe an innovative statistical approach for the ab initio simultaneous analysis of the formation history and morphology of the large-scale structure of the inhomogeneous Universe. Our algorithm explores the joint posterior distribution of the many millions of parameters involved via efficient Hamiltonian Markov Chain Monte Carlo sampling. We describe its application to the Sloan Digital Sky Survey data release 7 and an additional non-linear filtering step. We illustrate the use of our findings for cosmic web analysis: identification of structures via tidal shear analysis and inference of dark matter voids.
The standard Bayesian model formalism comparison cannot be applied to most cosmological models as they lack well-motivated parameter priors. However, if the data-set being used is separable, then it is possible to use some of the data to obtain the necessary parameter distributions, the rest of the data being retained for model comparison. While such methods are not fully prescriptive, they provide a route to applying Bayesian model comparison in cosmological situations where it could not otherwise be used.
In astronomical and cosmological studies one often wishes to infer some properties of an infinite-dimensional field indexed within a finite-dimensional metric space given only a finite collection of noisy observational data. Bayesian inference offers an increasingly-popular strategy to overcome the inherent ill-posedness of this signal reconstruction challenge. However, there remains a great deal of confusion within the astronomical community regarding the appropriate mathematical devices for framing such analyses and the diversity of available computational procedures for recovering posterior functionals. In this brief research note I will attempt to clarify both these issues from an “applied statistics” perpective, with insights garnered from my post-astronomy experiences as a computational Bayesian / epidemiological geostatistician.
We present a method to map multivariate non-Gaussian posterior probability densities into Gaussian ones via nonlinear Box-Cox transformations, and generalizations thereof. This is analogous to the search for normal parameters in the CMB, but can in principle be applied to any probability density that is continuous and unimodal. The search for the optimally Gaussianizing transformation amongst the Box-Cox family is performed via a maximum likelihood formalism. We can judge the quality of the found transformation a posteriori: qualitatively via statistical tests of Gaussianity, and more illustratively by how well it reproduces the credible regions. The method permits an analytical reconstruction of the posterior from a sample, e.g. a Markov chain, and simplifies the subsequent joint analysis with other experiments. Furthermore, it permits the characterization of a non-Gaussian posterior in a compact and efficient way. The expression for the non-Gaussian posterior can be employed to find analytic formulae for the Bayesian evidence, and consequently be used for model comparison.
The extraction of foreground and CMB maps from multi-frequency observations relies mostly on the different frequency behavior of the different components. Existing Bayesian methods additionally make use of a Gaussian prior for the CMB whose correlation structure is described by an unknown angular power spectrum. We argue for the natural extension of this by using non-trivial priors also for the foreground components. Focusing on diffuse Galactic foregrounds, we propose a log-normal model including unknown spatial correlations within each component and cross-correlations between the different foreground components. We present case studies at low resolution that demonstrate the superior performance of this model when compared to an analysis with flat priors for all components.
A range of Bayesian tools has become widely used in cosmological data treatment and parameter inference (see Kunz et al.2007, Trotta 2008, Amendola et al.2013). With increasingly big datasets and higher precision, tools that enable us to further enhance the accuracy of our measurements gain importance. Here we present an approach based on internal robustness, introduced in Amendola et al. (2013) and adopted in Heneka et al. (2014), to identify biased subsets of data and hidden correlation in a model independent way.
We propose a Bayesian method to measure the total Galactic extinction parameters, RV and AV. Validation tests based on the simulated data indicate that the method can achieve the accuracy of around 0.01 mag. We apply this method to the SDSS BHB stars in the northern Galactic cap and find that the derived extinctions are highly consistent with those from Schlegel et al. (1998). It suggests that the Bayesian method is promising for the extinction estimation, even the reddening values are close to the observational errors.
A new non-parametric method based on Gaussian Processes (GP) was proposed recently to measure the Hubble constant H0. The freedom in this approach comes in the chosen covariance function, which determines how smooth the process is and how nearby points are correlated. We perform coverage tests with a thousand mock samples within the ΛCDM model in order to determine what covariance function provides the least biased results. The function Matérn(5/2) is the best with sligthly higher errors than other covariance functions, although much more stable when compared to standard parametric analyses.
We consider the problem of estimating an unknown density or regression curve from data. In the parametric setting, the curve to estimate is modelled by a function which is known up to the value of a finite number of parameters. We consider the nonparametric setting, where the curve is not modelled a priori. We focus on kernel methods, which are popular nonparametric techniques that can be used for both density and regression estimation. While these methods are appropriate when the data are observed accurately, they cannot be directly applied to astronomical data, which are often measured with a certain degree of error. It is well known in the statistics literature that when the observations are measured with errors, nonparametric procedures become biased, and need to be adjusted for the errors. Correction techniques have been developed, and are often referred to as deconvolution methods. We introduce those methods, in both the homoscedastic and heteroscedastic error cases, and discuss their practical implementation.
We describe modifications to the joint stepwise maximum likelihood method of Cole (2011) in order to simultaneously fit the GAMA-II galaxy luminosity function (LF), corrected for radial density variations, and its evolution with redshift. The whole sample is reasonably well-fit with luminosity (Q) and density (P) evolution parameters Q, P ≈ 0.8, 1.7. Red galaxies show larger luminosity but smaller density evolution than blue galaxies, as expected.
The main feature of the spatial large-scale galaxy distribution is its intricate network of galaxy filaments. This network is spanned by the galaxy locations that can be interpreted as a three-dimensional point distribution. The global properties of the point process can be measured by different statistical methods, which, however, do not describe directly the structure elements. The morphology of the large-scale structure, on the other hand, is an important property of the galaxy distribution. Here, we apply an object point process with interactions (the Bisous model) to trace and extract the filamentary network in the presently largest galaxy redshift survey, the Sloan Digital Sky Survey (SDSS data release 10). We search for multi-scale filaments in the galaxy distribution that have a radius of about 0.5, 1.0, 2.0, and 4.0 h−1 Mpc. We extract the spines of the filamentary network and divide the detected network into single filaments.
We propose a computationally feasible estimator for the needlet trispectrum, which develops earlier work on the bispectrum by Donzelli et al. (2012). Our proposal seems to enjoy a number of useful properties, in particular a) the construction exploits the localization properties of the needlet system, and hence it automatically handles masked regions; b) the procedure incorporates a quadratic correction term to correct for the presence of instrumental noise and sky-cuts; c) it is possible to provide analytic results on its statistical properties, which can serve as a guidance for simulations. The needlet trispectrum we present here provides the natural building blocks for the efficient estimation of nonlinearity parameters on CMB data, and in particular for the third order constants gNL and τNL.
The presence of multiple fields during inflation might seed a detectable amount of non-Gaussianity in the curvature perturbations, which in turn becomes observable in present data sets like the cosmic microwave background (CMB) or the large scale structure (LSS). Within this proceeding we present a fully analytic method to infer inflationary parameters from observations by exploiting higher-order statistics of the curvature perturbations. To keep this analyticity, and thereby to dispense with numerically expensive sampling techniques, a saddle-point approximation is introduced whose precision has been validated for a numerical toy example. Applied to real data, this approach might enable to discriminate among the still viable models of inflation.
The integrated Sachs-Wolfe effect was recently detected at a level of 4.4σ by [Granett et al. (2008)], by stacking compensated CMB temperature patches corresponding to superstructures in the universe. We test the reported signal using realistic gaussian random realizations of the CMB sky, based on the temperature power spectrum predicted by the concordance ΛCDM model. Such simulations provide a complementary approach to the largely used N-body simulations and allow to include the contaminant effects due to small-scale temperature fluctuations. We also apply our pipeline to foreground-cleaned CMB sky maps using the [Granett et al. (2008)] voids/clusters catalog. We confirm the detection of a signal, which depart from the null hypothesis by 3.5σ, and we report a tension with our theoretical estimates at a significance of about 2.5σ.
A transformed auto-correlation method is presented here, where a received signal is transformed based on a priori reflecting model, and then the transformed signal is cross-correlated to its original one. If the model is correct, after transformation, the reflected signal will be coherent to the transmitted signal, with zero delay. A map of transformed auto-correlation function with zero delay can be generated in a given parametric space. The significant peaks in the map may indicate the possible reflectors nearby the central transmitter. The true values of the parameters of reflectors can be estimated at the same time.
The primordial power spectrum is an indirect probe of inflation or other structure-formation mechanisms. We introduce a new method, named PRISM, to estimate this spectrum from the empirical cosmic microwave background (CMB) power spectrum. This is a sparsity-based inversion method, which leverages a sparsity prior on features in the primordial spectrum in a wavelet dictionary to regularise the inverse problem. This non-parametric approach is able to reconstruct the global shape as well as localised features of the primordial spectrum accurately and proves to be robust for detecting deviations from the currently favoured scale-invariant spectrum. We investigate the strength of this method on a set of WMAP nine-year simulated data for three types of primordial spectra and then process the WMAP nine-year data as well as the Planck PR1 data. We find no significant departures from a near scale-invariant spectrum.
A new spin wavelet transform on the sphere is proposed to analyse the polarisation of the cosmic microwave background (CMB), a spin ± 2 signal observed on the celestial sphere. The scalar directional scale-discretised wavelet transform on the sphere is extended to analyse signals of arbitrary spin. The resulting spin scale-discretised wavelet transform probes the directional intensity of spin signals. A procedure is presented using this new spin wavelet transform to recover E- and B-mode signals from partial-sky observations of CMB polarisation.
The incredible variety of galaxy shapes cannot be summarized by human defined discrete classes of shapes without causing a possibly large loss of information. Dictionary learning and sparse coding allow us to reduce the high dimensional space of shapes into a manageable low dimensional continuous vector space. Statistical inference can be done in the reduced space via probability distribution estimation and manifold estimation.
Spectroscopic redshift surveys are an incredibly valuable tool in cosmology, allowing us to trace the distribution of galaxies as a function of distance and, thus, trace the evolution of structure formation in the Universe. However, estimating the redshifts from spectra with low signal-to-noise is difficult, and such data are often either discarded or require human classification of spectral lines to obtain the galaxy redshift. Darth Fader offers an automated method for estimating the redshifts of galaxies in the low signal-to-noise regime. Using a sophisticated, wavelet-based technique, galaxy spectra can be separated into continuum, line and noise components, and the lines can then be cross-correlated with template spectra in order to estimate the redshifts. Cross-matching of the identified lines then allows for a cleaning of the resulting catalogue, effectively removing the vast majority of erroneous redshift estimates and resulting in a highly pure, highly accurate redshift catalogue. Darth Fader allows us to effectively use low signal-to-noise galaxy spectra, and dramatically reduces the number of human hours required to do this, allowing spectroscopic surveys to probe deeper into the formation history of the Universe.
We present a simple construction of spherical wavelets for the unit ball, which we label Radial 3D Needlets. We envisage an experimental framework where data are collected on concentric spheres with the same pixelization at different radial distances from the origin. The unit ball is hence viewed as a tensor product of the unit interval with the unit sphere: a set of eigenfunctions is therefore defined on the corresponding Laplacian operator. Wavelets are then constructed by a smooth convolution of the projectors defined by these eigenfunctions. Localization properties may be rigorously shown to hold in the real and harmonic domain, and an exact reconstruction formula holds; the system allows a very convenient computational implementation.