arXiv subjects

M. P. Hobson

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

At least 91 records · Page 5Linked to original sources

arXiv2007

Fast cosmological parameter estimation using neural networks

We present a method for accelerating the calculation of CMB power spectra, matter power spectra and likelihood functions for use in cosmological parameter estimation. The algorithm, called CosmoNet, is based on training a multilayer perceptron neural network and shares all the advantages of the recently released Pico algorithm of Fendt & Wandelt, but has several additional benefits in terms of simplicity, computational speed, memory requirements and ease of training. We demonstrate the capabilities of CosmoNet by computing CMB power spectra over a box in the parameter space of flat ΛCDM models containing the 3σWMAP1 confidence region. We also use CosmoNet to compute the WMAP3 likelihood for flat ΛCDM models and show that marginalised posteriors on parameters derived are very similar to those obtained using CAMB and the WMAP3 code. We find that the average error in the power spectra is typically 2-3% of cosmic variance, and that CosmoNet is \sim 7 \times 10^4 faster than CAMB (for flat models) and \sim 6 \times 10^6 times faster than the official WMAP3 likelihood code. CosmoNet and an interface to CosmoMC are publically available at www.mrao.cam.ac.uk/software/cosmonet.

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arXiv2007

A Bayesian analysis of the primordial power spectrum

We present a Bayesian analysis of large-scale structure and cosmic microwave background data to constrain the form of the primordial power spectrum. We have extended the usual presumption of a scale invariant spectrum to include: (i) a running spectral index; (ii) a broken spectrum arising perhaps from an interruption of the potential driving inflation; (iii) a large scale cutoff in power as the first year WMAP results appear to indicate; (iv) a reconstruction of the spectrum in eight bins in wavenumber; and (v) a spectrum resulting from a cosmological model proposed by Lasenby & Doran, which naturally exhibits an exponential drop in power on very large scales. The result of our complete Bayesian analysis includes not only the posterior probability distribution from which parameter estimates are inferred but also the Bayesian evidence. This evidence value is greater for a model with fewer parameters unless a more complicated model provides a significantly better fit to the data, thus allowing a powerful method of model selection. We find that those models exhibiting any form of cutoff in power on large scales consistently produce higher evidences than either the Harrison-Zel'dovich or single spectral index spectra. In particular, within the best-fit concordance cosmology, we find the Lasenby & Doran spectrum to show significantly larger evidence as compared to the other models.

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arXiv2007

WMAP 3-year primordial power spectrum

We constrain the form of the primordial power spectrum using Wilkinson Microwave Anisotropy Probe (WMAP) 3-year cosmic microwave background (CMB) data (+ other high resolution CMB experiments) in addition to complementary large-scale structure (LSS) data: 2dF, SDSS, Ly-alpha forest and luminous red galaxy (LRG) data from the SDSS catalogue. We compute the comparative Bayesian evidence in addition to parameter estimates for a collection of seven models: (i) a scale invariant Harrison-Zel'dovich (H-Z) spectrum; (ii) a power-law; (iii) a running spectral index; (iv) a broken spectrum; (v) a power-law with an abrupt cutoff on large-scales; (vi) a reconstruction of the spectrum in eight bins in wavenumber; and (vii) a spectrum resulting from a cosmological model proposed by Lasenby & Doran (L-D). Using a basic dataset of WMAP3 + other CMB + 2dF + SDSS our analysis confirms that a scale-invariant spectrum is disfavoured by between 0.7 and 1.7 units of log evidence (depending on priors chosen) when compared with a power-law tilt. Moreover a running spectrum is now significantly preferred, but only when using the most constraining set of priors. The addition of Ly-alpha and LRG data independently both suggest much lower values of the running index than with basic dataset alone and interestingly the inclusion of Ly-alpha significantly disfavours a running parameterisation by more than a unit in log evidence. Overall the highest evidences, over all datasets, were obtained with a power law spectrum containing a cutoff with a significant log evidence difference of roughly 2 units. The natural tilt and exponential cutoff present in the L-D spectrum is found to be favoured decisively by a log evidence difference of over 5 units, but only for a limited study within the best-fit concordance cosmology.

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arXiv2007

Efficient Bayesian inference for multimodal problems in cosmology

Bayesian model selection provides the cosmologist with an exacting tool to distinguish between competing models based purely on the data, via the Bayesian evidence. Previous methods to calculate this quantity either lacked general applicability or were computationally demanding. However, nested sampling (Skilling 2004), which was recently applied successfully to cosmology by Muhkerjee et al. 2006, overcomes both of these impediments. Their implementation restricts the parameter space sampled, and thus improves the efficiency, using a decreasing ellipsoidal bound in the n-dimensional parameter space centred on the maximum likelihood point. However, if the likelihood function contains any multi-modality, then the ellipse is prevented from constraining the sampling region efficiently. In this paper we introduce a method of clustered ellipsoidal nested sampling which can form multiple ellipses around each individual peak in the likelihood. In addition we have implemented a method for determining the expectation and variance of the final evidence value without the need to use sampling error from repetitions of the algorithm ; this further reduces the computational load by at least an order of magnitude. We have applied our algorithm to a pair of toy models and one cosmological example where we demonstrate that the number of likelihood evaluations required is ~ 4% of that necessary for using previous algorithms. We have produced a fortran library containing our routines which can be called from any sampling code, in addition for convenience we have incorporated it into the popular CosmoMC code as CosmoClust. Both are available for download at http://www.mrao.cam.ac.uk/software/cosmoclust .

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arXiv2007

Multimodal nested sampling: an efficient and robust alternative to MCMC methods for astronomical data analysis

In performing a Bayesian analysis of astronomical data, two difficult problems often emerge. First, in estimating the parameters of some model for the data, the resulting posterior distribution may be multimodal or exhibit pronounced (curving) degeneracies, which can cause problems for traditional MCMC sampling methods. Second, in selecting between a set of competing models, calculation of the Bayesian evidence for each model is computationally expensive. The nested sampling method introduced by Skilling (2004), has greatly reduced the computational expense of calculating evidences and also produces posterior inferences as a by-product. This method has been applied successfully in cosmological applications by Mukherjee et al. (2006), but their implementation was efficient only for unimodal distributions without pronounced degeneracies. Shaw et al. (2007), recently introduced a clustered nested sampling method which is significantly more efficient in sampling from multimodal posteriors and also determines the expectation and variance of the final evidence from a single run of the algorithm, hence providing a further increase in efficiency. In this paper, we build on the work of Shaw et al. and present three new methods for sampling and evidence evaluation from distributions that may contain multiple modes and significant degeneracies; we also present an even more efficient technique for estimating the uncertainty on the evaluated evidence. These methods lead to a further substantial improvement in sampling efficiency and robustness, and are applied to toy problems to demonstrate the accuracy and economy of the evidence calculation and parameter estimation. Finally, we discuss the use of these methods in performing Bayesian object detection in astronomical datasets.

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arXiv2007

Theoretical Analysis of Astronomical Phased Arrays

Low-noise phased arrays are essential for the next generation of microwave and submillimetre wave astronomy. We analyze their behaviour from a functional perspective, and show that their operation is intimately related to the mathematical theory of frames. No assumptions are made about the orthogonality or linear independence of the synthesised beams. Frame theory allows an unambiguous assessment of whether the outputs of an array can be used to observe a field or brightness distribution within a given class. Image reconstruction is carried out using dual beams. We identify the natural modes of phased arrays, and carry out an analysis of noise. The scheme allows the expectation values, the mean-square fluctuations, and the correlations between fluctuations at the output ports of a phased array to be determined for a source in any state of spatial coherence. Both classical and photon-counting statistics are included. Our model is conceptually powerful, and suggests many simulation and image recovery techniques.

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arXiv2007

Cosmological applications of a wavelet analysis on the sphere

The cosmic microwave background (CMB) is a relic radiation of the Big Bang and as such it contains a wealth of cosmological information. Statistical analyses of the CMB, in conjunction with other cosmological observables, represent some of the most powerful techniques available to cosmologists for placing strong constraints on the cosmological parameters that describe the origin, content and evolution of the Universe. The last decade has witnessed the introduction of wavelet analyses in cosmology and, in particular, their application to the CMB. We review here spherical wavelet analyses of the CMB that test the standard cosmological concordance model. The assumption that the temperature anisotropies of the CMB are a realisation of a statistically isotropic Gaussian random field on the sphere is questioned. Deviations from both statistical isotropy and Gaussianity are detected in the reviewed works, suggesting more exotic cosmological models may be required to explain our Universe. We also review spherical wavelet analyses that independently provide evidence for dark energy, an exotic component of our Universe of which we know very little currently. The effectiveness of accounting correctly for the geometry of the sphere in the wavelet analysis of full-sky CMB data is demonstrated by the highly significant detections of physical processes and effects that are made in these reviewed works.

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arXiv2007

{\sc CosmoNet}: fast cosmological parameter estimation in non-flat models using neural networks

We present a further development of a method for accelerating the calculation of CMB power spectra, matter power spectra and likelihood functions for use in cosmological Bayesian inference. The algorithm, called {\sc CosmoNet}, is based on training a multilayer perceptron neural network. We compute CMB power spectra (up to $\ell=2000$) and matter transfer functions over a hypercube in parameter space encompassing the $4σ$ confidence region of a selection of CMB (WMAP + high resolution experiments) and large scale structure surveys (2dF and SDSS). We work in the framework of a generic 7 parameter non-flat cosmology. Additionally we use {\sc CosmoNet} to compute the WMAP 3-year, 2dF and SDSS likelihoods over the same region. We find that the average error in the power spectra is typically well below cosmic variance for spectra, and experimental likelihoods calculated to within a fraction of a log unit. We demonstrate that marginalised posteriors generated with {\sc CosmoNet} spectra agree to within a few percent of those generated by {\sc CAMB} parallelised over 4 CPUs, but are obtained 2-3 times faster on just a \emph{single} processor. Furthermore posteriors generated directly via {\sc CosmoNet} likelihoods can be obtained in less than 30 minutes on a single processor, corresponding to a speed up of a factor of $\sim 32$. We also demonstrate the capabilities of {\sc CosmoNet} by extending the CMB power spectra and matter transfer function training to a more generic 10 parameter cosmological model, including tensor modes, a varying equation of state of dark energy and massive neutrinos. {\sc CosmoNet} and interfaces to both {\sc CosmoMC} and {\sc Bayesys} are publically available at {\tt www.mrao.cam.ac.uk/software/cosmonet}.

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arXiv2007

Detection of the ISW effect and corresponding dark energy constraints made with directional spherical wavelets

Using a directional spherical wavelet analysis we detect the integrated Sachs-Wolfe (ISW) effect, indicated by a positive correlation between the first-year Wilkinson Microwave Anisotropy Probe (WMAP) and NRAO VLA Sky Survey (NVSS) data. Detections are made using both a directional extension of the spherical Mexican hat wavelet and the spherical butterfly wavelet. We examine the possibility of foreground contamination and systematics in the WMAP data and conclude that these factors are not responsible for the signal that we detect. The wavelet analysis inherently enables us to localise on the sky those regions that contribute most strongly to the correlation. On removing these localised regions the correlation that we detect is reduced in significance, as expected, but it is not eliminated, suggesting that these regions are not the sole source of correlation between the data. This finding is consistent with predictions made using the ISW effect, where one would expect weak correlations over the entire sky. In a flat universe the detection of the ISW effect provides direct and independent evidence for dark energy. We use our detection to constrain dark energy parameters by deriving a theoretical prediction for the directional wavelet covariance statistic for a given cosmological model. Comparing these predictions with the data we place constraints on the equation-of-state parameter $w$ and the vacuum energy density $Ω_Λ$. We also consider the case of a pure cosmological constant, i.e. $w=-1$. For this case we rule out a zero cosmological constant at greater than the 99.9% significance level. All parameter estimates that we obtain are consistent with the standand cosmological concordance model values.

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arXiv2006

Millimetric observation of the SZ effect in the Corona Borealis supercluster

We have observed the Corona Borealis Supercluster with the Millimeter and Infrared Testa grigia Observatory (MITO), located in the Italian Alps, at 143, 214, 272, and 353 GHz. We present a description of the measurements, data analysis, and results of the observations together with a comparison with observations performed at 33 GHz with the Very Small Array (VSA) interferometer situated at the Teide Observatory (Tenerife-Spain). Observations have been made in the direction of the supercluster towards one Cosmic Microwave Background (CMB) cold spot previously detected in a VSA temperature map. Observational strategy and data analysis are described in detail, explaining the procedures used to disentangle primary and secondary anisotropies in the resulting maps. From a first level of data analysis we find evidence in MITO data of primary anisotropy, however still with room for the presence of secondary anisotropy, especially when VSA results are included. With a second level of data analysis using map-making and the maximum entropy method we claim a weak detection of a faint signal compatible with a SZ effect characterized at most by a Comptonization parameter $y=(7.8^{+5.3}_{-4.4})\times10^{-6}$ 68% CL. The low level of confidence in the presence of a SZ signal invite us to study this sky region with higher sensitivity and angular resolution experiments like the already planned upgraded versions of VSA and MITO.

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arXiv2006

A high-significance detection of non-Gaussianity in the WMAP 3-year data using directional spherical wavelets

We repeat the directional spherical real Morlet wavelet analysis used to detect non-Gaussianity in the WMAP 1-year data (McEwen et al. 2005a), on the WMAP 3-year data. The non-Gaussian signal previously detected is indeed present in the 3-year data, although the significance of the detection is reduced. Using our most conservative method for constructing significance measures, we find the significance of the detection of non-Gaussianity drops from 98.3+/-0.4% to 94.9+/-0.7%; the significance drops from 99.3+/-0.3% to 97.2+/-0.5% using a method based on the $χ^2$ statistic. The wavelet analysis allows us to localise most likely sources of non-Gaussianity on the sky. We detect very similar localised regions in the WMAP 1-year and 3-year data, although the regions extracted appear more pronounced in the 3-year data. When all localised regions are excluded from the analysis the 3-year data is consistent with Gaussianity.

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arXiv2006

A directional continuous wavelet transform on the sphere

A new construction of a directional continuous wavelet analysis on the sphere is derived herein. We adopt the harmonic scaling idea for the spherical dilation operator recently proposed by Sanz et al. but extend the analysis to a more general directional framework. Directional wavelets are a powerful extension that allow one to also probe oriented structure in the analysed function. Our spherical wavelet methodology has the advantage that all functions and operators are defined directly on the sphere. The construction of wavelets in our framework is demonstrated with an example.

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arXiv2006

Non-Gaussianity detections in the Bianchi VIIh corrected WMAP 1-year data made with directional spherical wavelets

Many of the current anomalies reported in the Wilkinson Microwave Anisotropy Probe (WMAP) 1-year data disappear after `correcting' for the best-fit embedded Bianchi type VII_h component (Jaffe et al. 2005), albeit assuming no dark energy component. We investigate the effect of this Bianchi correction on the detections of non-Gaussianity in the WMAP data that we previously made using directional spherical wavelets (McEwen et al. 2005a). As previously discovered by Jaffe et al. (2005), the deviations from Gaussianity in the kurtosis of spherical Mexican hat wavelet coefficients are eliminated once the data is corrected for the Bianchi component. This is due to the reduction of the cold spot at Galactic coordinates (l,b)=(209^\circ,-57\circ), which Cruz et al. (2005) claim to be the source of non-Gaussianity introduced in the kurtosis. Our previous detections of non-Gaussianity observed in the skewness of spherical wavelet coefficients are not reduced by the Bianchi correction. Indeed, the most significant detection of non-Gaussianity made with the spherical real Morlet wavelet at a significant level of 98.4% remains (using a very conservative method to estimate the significance). We make our code to simulate Bianchi induced temperature fluctuations publicly available.

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arXiv2006

Fast directional continuous spherical wavelet transform algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.

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arXiv2006

A Bayesian analysis of regularised source inversions in gravitational lensing

Strong gravitational lens systems with extended sources are of special interest because they provide additional constraints on the models of the lens systems. To use a gravitational lens system for measuring the Hubble constant, one would need to determine the lens potential and the source intensity distribution simultaneously. A linear inversion method to reconstruct a pixellated source brightness distribution of a given lens potential model was introduced by Warren & Dye. In the inversion process, a regularisation on the source intensity is often needed to ensure a successful inversion with a faithful resulting source. In this paper, we use Bayesian analysis to determine the optimal regularisation constant (strength of regularisation) of a given form of regularisation and to objectively choose the optimal form of regularisation given a selection of regularisations. We consider and compare quantitatively three different forms of regularisation previously described in the literature for source inversions in gravitational lensing: zeroth-order, gradient and curvature. We use simulated data with the exact lens potential to demonstrate the method. We find that the preferred form of regularisation depends on the nature of the source distribution.

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arXiv2006

Effect of component separation on the temperature distribution of the CMB

We present a study of the effect of component separation on the recovered cosmic microwave background (CMB) temperature distribution, considering Gaussian and non-Gaussian input CMB maps. First, we extract the CMB component from simulated Planck data (in small patches of the sky) using the maximum entropy method (MEM), Wiener filter (WF) and a method based on the subtraction of foreground templates plus a linear combination of frequency channels (LCFC). We then apply a wavelet-based method to study the Gaussianity of the recovered CMB and compare it with the same analysis for the input map. When the original CMB map is Gaussian (and assuming that point sources have been removed), we find that none of the methods introduce non-Gaussianity (NG) in the CMB reconstruction. On the contrary, if the input CMB map is non-Gaussian, all the studied methods produce a reconstructed CMB with lower detections of NG than the original map. This effect is mainly due to the presence of instrumental noise. In this case, MEM tends to produce slightly higher non-Gaussian detections in the reconstructed map than WF whereas the detections are lower for the LCFC. We have also studied the effect of point sources in the MEM reconstruction. If no attempt to remove point sources is performed, they clearly contaminate the CMB reconstruction, introducing spurious NG. When the brightest point sources are removed from the data using the Mexican Hat Wavelet, the Gaussian character of the CMB is preserved. However, when analysing larger regions of the sky, the variance of our estimators will be appreciably reduced and, in this case, we expect the point source residuals to introduce spurious NG in the CMB. Thus, a careful subtraction (or masking) of point source emission is crucial when studying the Gaussianity of the CMB.

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arXiv2005

CMB observations from the CBI and VSA: A comparison of coincident maps and parameter estimation methods

We present coincident observations of the Cosmic Microwave Background (CMB) from the Very Small Array (VSA) and Cosmic Background Imager (CBI) telescopes. The consistency of the full datasets is tested in the map plane and the Fourier plane, prior to the usual compression of CMB data into flat bandpowers. Of the three mosaics observed by each group, two are found to be in excellent agreement. In the third mosaic, there is a 2 sigma discrepancy between the correlation of the data and the level expected from Monte Carlo simulations. This is shown to be consistent with increased phase calibration errors on VSA data during summer observations. We also consider the parameter estimation method of each group. The key difference is the use of the variance window function in place of the bandpower window function, an approximation used by the VSA group. A re-evaluation of the VSA parameter estimates, using bandpower windows, shows that the two methods yield consistent results.

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arXiv2005

A high-significance detection of non-Gaussianity in the WMAP 1-year data using directional spherical wavelets

A directional spherical wavelet analysis is performed to examine the Gaussianity of the WMAP 1-year data. Such an analysis is facilitated by the introduction of a fast directional continuous spherical wavelet transform. The directional nature of the analysis allows one to probe orientated structure in the data. Significant deviations from Gaussianity are detected in the skewness and kurtosis of spherical elliptical Mexican hat and real Morlet wavelet coefficients for both the WMAP and Tegmark et al. (2003) foreground-removed maps. The previous non-Gaussianity detection made by Vielva et al. (2003) using the spherical symmetric Mexican hat wavelet is confirmed, although their detection at the 99.9% significance level is only made at the 95.3% significance level using our most conservative statistical test. Furthermore, deviations from Gaussianity in the skewness of spherical real Morlet wavelet coefficients on a wavelet scale of 550 arcmin (corresponding to an effective global size on the sky of approximately 26 degrees and an internal size of 3 degrees) at an azimuthal orientation of 72 degrees, are made at the 98.3% significance level, using the same conservative method. The wavelet analysis inherently allows one to localise on the sky those regions that introduce skewness and those that introduce kurtosis. Preliminary noise analysis indicates that these detected deviation regions are not atypical and have average noise dispersion. Further analysis is required to ascertain whether these detected regions correspond to secondary or instrumental effects, or whether in fact the non-Gaussianity detected is due to intrinsic primordial fluctuations in the cosmic microwave background.

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