arXiv subjects

M. P. Hobson

Publications and source records attributed to M. P. Hobson.

At least 109 records · Page 6Linked to original sources

arXiv2004

Simulation of non-Gaussian CMB maps

A simple method is presented for the rapid simulation of statistically-isotropic non-Gaussian maps of CMB temperature fluctuations with a given power spectrum and analytically-calculable bispectrum and higher-order polyspectra. The $n$th-order correlators of the pixel values may also be calculated analytically. The cumulants of the simulated map may be used to obtain an expression for the probability density function of the pixel temperatures. The statistical properties of the simulated map are determined by the univariate non-Gaussian distribution from which pixel values are drawn independently in the first stage of the simulation process. We illustrate the method using a non-Gaussian distribution derived from the wavefunctions of the harmonic oscillator. The basic simulation method is easily extended to produce non-Gaussian maps with a given power spectrum and diagonal bispectrum.

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arXiv2004

A fast directional continuous spherical wavelet transform

A fast algorithm for Antoine and Vandergheynst's (1998) directional continuous spherical wavelet transform (CSWT) is presented. Computational requirements are reduced by a factor of O(\sqrt{N}), when N is the number of pixels on the sphere. The spherical Mexican hat wavelet Gaussianity analysis of the WMAP 1-year data performed by Vielva et al. (2003) is reproduced and confirmed using the fast CSWT. The proposed extension to directional analysis is inherently afforded by the fast CSWT algorithm.

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arXiv2004

All-sky component separation in the presence of anisotropic noise and dust temperature variations

We present an extension of the harmonic-space maximum-entropy component separation method (MEM) for multi-frequency CMB observations that allows one to perform the separation with more plausible assumptions about the receiver noise and foreground astrophysical components. Component separation is considered in the presence of spatially-varying noise variance and spectral properties of the foreground components. It is shown that, if not taken properly into account, the presence of spatially-varying foreground spectra, in particular, can severely reduce the accuracy of the component separation. Nevertheless, by extending the basic method to accommodate such behaviour and the presence of anisotropic noise, we find that the accuracy of the component separation can be improved to a level comparable with previous investigations in which these effects were not present.

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arXiv2004

Foreground separation using a flexible maximum-entropy algorithm: an application to COBE data

A flexible maximum-entropy component separation algorithm is presented that accommodates anisotropic noise, incomplete sky-coverage and uncertainties in the spectral parameters of foregrounds. The capabilities of the method are determined by first applying it to simulated spherical microwave data sets emulating the COBE-DMR, COBE-DIRBE and Haslam surveys. Using these simulations we find that is very difficult to determine unambiguously the spectral parameters of the galactic components for this data set due to their high level of noise. Nevertheless, we show that is possible to find a robust CMB reconstruction, especially at the high galactic latitude. The method is then applied to these real data sets to obtain reconstructions of the CMB component and galactic foreground emission over the whole sky. The best reconstructions are found for values of the spectral parameters: T_d=19 K, alpha_d=2, beta_ff=-0.19 and beta_syn=-0.8. The CMB map has been recovered with an estimated statistical error of \sim 22 muK on an angular scale of 7 degrees outside the galactic cut whereas the low galactic latitude region presents contamination from the foreground emissions.

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arXiv2004

Modal decomposition of astronomical images with application to shapelets

The decomposition of an image into a linear combination of digitised basis functions is an everyday task in astronomy. A general method is presented for performing such a decomposition optimally into an arbitrary set of digitised basis functions, which may be linearly dependent, non-orthogonal and incomplete. It is shown that such circumstances may result even from the digitisation of continuous basis functions that are orthogonal and complete. In particular, digitised shapelet basis functions are investigated and are shown to suffer from such difficulties. As a result the standard method of performing shapelet analysis produces unnecessarily inaccurate decompositions. The optimal method presented here is shown to yield more accurate decompositions in all cases.

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arXiv2003

Bayesian joint analysis of cluster weak lensing and Sunyaev-Zel'dovich effect data

As the quality of the available galaxy cluster data improves, the models fitted to these data might be expected to become increasingly complex. Here we present the Bayesian approach to the problem of cluster data modelling: starting from simple, physically motivated parameterised functions to describe the cluster's gas density, gravitational potential and temperature, we explore the high-dimensional parameter spaces with a Markov-Chain Monte-Carlo sampler, and compute the Bayesian evidence in order to make probabilistic statements about the models tested. In this way sufficiently good data will enable the models to be distinguished, enhancing our astrophysical understanding; in any case the models may be marginalised over in the correct way when estimating global, perhaps cosmological, parameters. In this work we apply this methodology to two sets of simulated interferometric Sunyaev-Zel'dovich effect and gravitational weak lensing data, corresponding to current and next-generation telescopes. We calculate the expected precision on the measurement of the cluster gas fraction from such experiments, and investigate the effect of the primordial CMB fluctuations on their accuracy. We find that data from instruments such as AMI, when combined with wide-field ground-based weak lensing data, should allow both cluster model selection and estimation of gas fractions to a precision of better than 30 percent for a given cluster.

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arXiv2003

Maximum-entropy image reconstruction using wavelets

Wavelet functions allow the sparse and efficient representation of a signal at different scales. Recently the application of wavelets to the denoising of maps of cosmic microwave background (CMB) fluctuations has been proposed. The maximum-entropy method (MEM) is also often used for enhancing astronomical images and has been applied to CMB data. In this paper, we give a systematic discussion of combining these two approaches by the use of the MEM in wavelet bases for the denoising and deconvolution of CMB maps and more general images. Certain types of wavelet transforms, such as the a trous transform, can be viewed as a multi-channel intrinsic correlation function (ICF). We find that the wavelet MEM has lower reconstruction residuals than conventional pixel-basis MEM in the case when the signal-to-noise ratio is low and the point spread function narrow. Furthermore, the Bayesian evidence for the wavelet MEM reconstructions is generally higher for a wide range of images. From a Bayesian point of view, the wavelet basis thus provides a better model of the image.

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arXiv2003

Cosmology from Cluster SZ and Weak Lensing Data

Weak gravitational lensing and the Sunyaev-Zel'dovich effect provide complementary information on the composition of clusters of galaxies. Preliminary results from cluster SZ observations with the Very Small Array are presented. A Bayesian approach to combining this data with wide field lensing data is then outlined; this allows the relative probabilities of cluster models of varying complexity to be computed. A simple simulation is used to demonstrate the importance of cluster model selection in cosmological parameter determination.

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arXiv2002

Maximum-likelihood astrometric geometry calibration of interferometric telescopes: application to the Very Small Array

Interferometers require accurate determination of the array configuration in order to produce reliable observations. A method is presented for finding the maximum-likelihood estimate of the telescope geometry, and of other instrumental parameters, astrometrically from the visibility timelines obtained from observations of celestial calibrator sources. The method copes systematically with complicated and unconventional antenna and array geometries, with electronic bandpasses that are different for each antenna radiometer, and with low signal-to-noise ratios for the calibrators. The technique automatically focusses on the geometry errors that are most significant for astronomical observation. We apply this method to observations made with the Very Small Array and constrain some 450 telescope parameters, such as the antenna positions, effective observing frequencies and correlator amplitudes and phase shifts; this requires only ~ 1 h of CPU time on a typical workstation.

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arXiv2002

Analytic marginalization over CMB calibration and beam uncertainty

With the increased accuracy and angular scale coverage of the recent CMB experiments it has become important to include calibration and beam uncertainties when estimating cosmological parameters. This requires an integration over possible values of the calibration and beam size, which can be done numerically but increases computation times. We present a fast and simple algorithm for marginalization over beam and calibration errors by analytical integration. We also illustrate the effect of incorporating these uncertainties by calculating the constraints on various cosmological and inflationary parameters including the spectral index n_s and the physical baryon density Omega_b h^2, using the latest CMB data. We find that parameter constraints are significantly changed when calibration/beam uncertainties are taken into account. Typically the best fit parameters are shifted and the errors bars are increased by up to fifty per cent for e.g. n_s and Omega_b h^2, although as expected there is no change for Omega_K, because it is constrained by the positions of the peaks.

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arXiv2002

A Bayesian approach to discrete object detection in astronomical datasets

A Bayesian approach is presented for detecting and characterising the signal from discrete objects embedded in a diffuse background. The approach centres around the evaluation of the posterior distribution for the parameters of the discrete objects, given the observed data, and defines the theoretically-optimal procedure for parametrised object detection. Two alternative strategies are investigated: the simultaneous detection of all the discrete objects in the dataset, and the iterative detection of objects. In both cases, the parameter space characterising the object(s) is explored using Markov-Chain Monte-Carlo sampling. For the iterative detection of objects, another approach is to locate the global maximum of the posterior at each iteration using a simulated annealing downhill simplex algorithm. The techniques are applied to a two-dimensional toy problem consisting of Gaussian objects embedded in uncorrelated pixel noise. A cosmological illustration of the iterative approach is also presented, in which the thermal and kinetic Sunyaev-Zel'dovich effects from clusters of galaxies are detected in microwave maps dominated by emission from primordial cosmic microwave background anisotropies.

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arXiv2002

All-sky component separation for the Planck mission

An harmonic-space maximum-entropy method (MEM) is presented for separating the emission from different physical components in all-sky observations by the forthcoming Planck satellite. The analysis is performed at full Planck resolution, with a pixel size of 1.7 arcmin, which corresponds to a maximum multipole of around 6000. The simulated Planck data include emission from the CMB, the kinetic and thermal Sunyaev-Zel'dovich (SZ) effects from galaxy clusters, as well as Galactic dust, free-free and synchrotron emission. Our simulations also assume homogeneous, uncorrelated pixel noise, although this is not a requirement of the method. We find that the MEM technique produces faithful reconstructions of the main input components over the whole sky, without the need to perform a Galactic cut. The CMB power spectrum is accurately recovered up to a multipole of around 2000. The algorithm is parallelised so that the entire reconstruction can be performed in around 6 hr using 30 R10000 processors on an SGI Origin 2000 supercomputer and requires 14 Gb of memory.

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arXiv2002

Maximum-entropy weak lens reconstruction: improved methods and application to data

We develop the maximum-entropy weak shear mass reconstruction method presented in earlier papers by taking each background galaxy image shape as an independent estimator of the reduced shear field and incorporating an intrinsic smoothness into the reconstruction. The characteristic length scale of this smoothing is determined by Bayesian methods. Within this algorithm the uncertainties due to both the intrinsic distribution of galaxy shapes and galaxy shape estimation are carried through to the final mass reconstruction, and the mass within arbitrarily shaped apertures can be calculated with corresponding uncertainties. We apply this method to two clusters taken from n-body simulations using mock observations corresponding to Keck LRIS and mosaiced HST WFPC2 fields. We demonstrate that the Bayesian choice of smoothing length is sensible and that masses within apertures (including one on a filamentary structure) are reliable, provided the field of view is not too small. We apply the method to data taken on the cluster MS1054-03 using the Keck LRIS (Clowe et al. 2000) and HST (Hoekstra et al. 2000), finding results in agreement with this previous work; we also present reconstructions with optimal smoothing lengths, and mass estimates which do not rely on any assumptions of circular symmetry. The code used in this work (LensEnt2) is available from the web.

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arXiv2002

Filtering techniques for the detection of Sunyaev-Zel'dovich clusters in multifrequency CMB maps

The problem of detecting Sunyaev-Zel'dovich (SZ) clusters in multifrequency CMB observations is investigated using a number of filtering techniques. A multifilter approach is introduced, which optimizes the detection of SZ clusters on microwave maps. An alternative method is also investigated, in which maps at different frequencies are combined in an optimal manner so that existing filtering techniques can be applied to the single combined map. The SZ profiles are approximated by the circularly-symmetric template $τ(x) = [1 +(x/r_c)^2]^{-λ}$, with $λ\simeq \tfrac{1}{2}$ and $x\equiv |\vec{x}|$, where the core radius $r_c$ and the overall amplitude of the effect are not fixed a priori, but are determined from the data. The background emission is modelled by a homogeneous and isotropic random field, characterized by a cross-power spectrum $P_{ν_1 ν_2}(q)$ with $q\equiv |\vec{q}|$. The filtering methods are illustrated by application to simulated Planck observations of a $12.8^\circ \times 12.8^\circ$ patch of sky in 10 frequency channels. Our simulations suggest that the Planck instrument should detect $\approx 10000$ SZ clusters in 2/3 of the sky. Moreover, we find the catalogue to be complete for fluxes $S > 170$ mJy at 300 GHz.

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arXiv2002

Combining cosmological datasets: hyperparameters and Bayesian evidence

A method is presented for performing joint analyses of cosmological datasets, in which the weight assigned to each dataset is determined directly by it own statistical properties. The weights are considered in a Bayesian context as a set of hyperparameters, which are then marginalised over in order to recover the posterior distribution as a function only of the cosmological parameters of interest. In the case of a Gaussian likelihood function, this marginalisation may be performed analytically. Calculation of the Bayesian evidence for the data, with and without the introduction of hyperparameters, enables a direct determination of whether the data warrant the introduction of weights into the analysis; this generalises the standard likelihood ratio approach to model comparison. The method is illustrated by application to the classic toy problem of fitting a straight line to a set of data. A cosmological illustration of the technique is also presented, in which the latest measurements of the cosmic microwave background power spectrum are used to infer constraints on cosmological parameters.

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arXiv2002

Maximum-likelihood estimation of the CMB power spectrum from interferometer observations

A maximum-likelihood method is presented for estimating the power spectrum of anisotropies in the cosmic microwave background (CMB) from interferometer observations. The method calculates flat band-power estimates in separate bins in l-space, together with confidence intervals on the power in each bin. For multifrequency data, the power spectrum of the other foreground components may also be recovered. Advantage is taken of several characteristic properties of interferometer data, which together allow the fast calculation of the likelihood function. The method may be applied to single-field or mosaiced observations, and proper account can be taken of non-coplanar baselines. The method is illustrated by application to simulated data from the Very Small Array.

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arXiv2001

Combining maximum-entropy and the mexican hat wavelet to reconstruct the microwave sky

We present a maximum-entropy method (MEM) and `Mexican Hat' wavelet (MHW) joint analysis to recover the different components of the microwave sky from simulated observations by the ESA Planck satellite in a small patch of the sky. This combined method allows one to improve the CMB, Sunyaev-Zel'dovich and Galactic foregrounds separation achieved by the MEM technique alone. In particular, the reconstructed CMB map is free from any bright point source contamination. The joint analysis also produces point source catalogues at each Planck frequency which are more complete and accurate than those obtained by each method on its own. The results are especially improved at high frequencies where infrared galaxies dominate the point source contribution. Although this joint technique has been performed on simulated Planck data, it could be easily applied to other multifrequency CMB experiments, such as the forthcoming NASA MAP satellite or the recently performed Boomerang and MAXIMA experiments.

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arXiv2001

Regularization and Inverse Problems

An overview is given of Bayesian inversion and regularization procedures. In particular, the conceptual basis of the maximum entropy method (MEM) is discussed, and extensions to positive/negative and complex data are highlighted. Other deconvolution methods are also discussed within the Bayesian context, focusing mainly on the comparison of Wiener filtering, Massive Inference and the Pixon method, using examples from both astronomical and non-astronomical applications.

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