Fast and reliable Markov chain Monte Carlo technique for cosmological parameter estimation

Markov chain Monte Carlo (MCMC) techniques are now widely used for cosmological parameter estimation. Chains are generated to sample the posterior probability distribution obtained following the Bayesian approach. An important issue is how to optimize the efficiency of such sampling and how to diagnose whether a finite-length chain has adequately sampled the underlying posterior probability distribution. We show how the power spectrum of a single such finite chain may be used as a convergence diagnostic by means of a fitting function, and discuss strategies for optimizing the distribution for the proposed steps. The methods developed are applied to current cosmic microwave background and large-scale structure data interpreted using both a pure adiabatic cosmological model and a mixed adiabatic/isocurvature cosmological model including possible correlations between modes. For the latter application, because of the increased dimensionality and the presence of degeneracies, the need for tuning MCMC methods for maximum efficiency becomes particularly acute.     

On the choice of parameters in solar-structure inversion

The observed solar p-mode frequencies provide a powerful diagnostic of the internal structure of the Sun and permit us to test in considerable detail the physics used in the theory of stellar structure. Among the most commonly used techniques for inverting such helioseismic data are two implementations of the optimally localized averages (OLA) method, namely the subtractive optimally localized averages (SOLA) and multiplicative optimally localized averages (MOLA). Both are controlled by a number of parameters, the proper choice of which is very important for a reliable inference of the solar internal structure. Here we make a detailed analysis of the influence of each parameter on the solution and indicate how to arrive at an optimal set of parameters for a given data set.     

Resolving the 180° Ambiguity in Solar Vector Magnetic Field Data: Evaluating the Effects of Noise,Spatial Resolution,and Method Assumptions

The objective testing of algorithms for performing ambiguity resolution in vector magnetic field data is continued, with an examination of the effects of noise in the data. Through the use of analytic magnetic field models, two types of noise are “added” prior to resolving: noise to simulate Poisson photon noise in the observed polarization spectra, and a spatial binning to simulate the effects of unresolved structure. The results are compared through the use of quantitative metrics and performance maps. We find that while no algorithm severely propagates the effects of Poisson noise beyond very local influences, some algorithms are more robust against high photon-noise levels than others. In the case of limited spatial resolution, loss of information regarding fine-scale structure can easily result in erroneous solutions. Our tests imply that photon noise and limited spatial resolution can act so as to make assumptions used in some ambiguity resolution algorithms no longer consistent with the observed magnetogram. We confirm a finding of the earlier comparison study that results can be very sensitive to the details of the treatment of the observed boundary and the assumptions governing that treatment. We discuss the implications of these findings, given the relative sensitivities of the algorithms to the two sources of noise tested here. We also touch on further implications for interpreting observational vector magnetic field data for general solar physics research.     

Statistical Feature Recognition for Multidimensional Solar Imagery

A maximum a posteriori (MAP) technique is developed to identify solar features in cotemporal and cospatial images of line-of-sight magnetic flux, continuum intensity, and equivalent width observed with the NASA/National Solar Observatory (NSO) Spectromagnetograph (SPM). The technique facilitates human understanding of patterns in large data sets and enables systematic studies of feature characteristics for comparison with models and observations of long-term solar activity and variability. The method uses Bayes’ rule to compute the posterior probability of any feature segmentation of a trio of observed images from per-pixel, class-conditional probabilities derived from independently-segmented training images. Simulated annealing is used to find the most likely segmentation. New algorithms for computing class-conditional probabilities from three-dimensional Gaussian mixture models and interpolated histogram densities are described and compared. A new extension to the spatial smoothing in the Bayesian prior model is introduced, which can incorporate a spatial dependence such as center-to-limb variation. How the spatial scale of training segmentations affects the results is discussed, and a new method for statistical separation of quiet Sun and quiet network is presented.     

A quantitative comparison of vector magnetic field measurement and analysis techniques

We make a quantitative comparison between spectral vs filter measurement and analysis techniques for extraction of solar vector magnetic fields from polarimetric data using as a basis the accurately calibrated, high angular resolution Stokes profile data from the Advanced Stokes Polarimeter. It is shown that filter-based measurements deliver qualitative images of the field alignment for sunspots that are visually similar to images derived from the more detailed analysis of the Stokes profiles. However, quantitative comparison with least-squares fits to the full Stokes profiles show that both the strength of the field predicted by the filter-based analysis and its orientation contain substantial errors. These errors are largest for plage regions outside of sunspots, where the field strengths are inferred to be only a fraction of their true values, and errors in the orientation of 40–50° are common. Within sunspots, errors of 20° are commonplace. The greatest source of these errors is the inability of the filter-based measurements to account for the small fill fraction of magnetic fields or, equivalently, scattered light in the instrument, which reduce the degree of polarization. The uncertainties of the full profile fitting methods are also discussed, along with the errors introduced by coarser wavelength sampling of the observed Stokes profiles. The least-squares fitting procedure operates best when the profiles are sampled at least as frequently as one Doppler width of the line.On leave from the Instituto de Astrofísica de Canarias, La Laguna, Tenerife, Spain.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A full-disk image standardisation of the synoptic solar observations at the Meudon Observatory

Robust techniques are developed to put the H and Ca K line full-disk images taken at the Meudon Observatory into a standardised form of a `virtual solar image'. The techniques include limb fitting, removal of geometrical distortion, centre position and size standardisation and intensity normalisation. The limb fitting starts with an initial estimate of the solar centre using raw 12-bit image data and then applies a Canny edge-detection routine. Candidate edge points for the limb are selected using a histogram based method and the chosen points fitted to a quadratic function by minimising the algebraic distance using SVD. The five parameters of the ellipse fitting the limb are extracted from the quadratic function. These parameters are used to define an affine transformation that transforms the image shape into a circle. Transformed images are generated using the nearest neighbour, bilinear or bicubic interpolation. Intensity renormalisation is also required because of a limb darkening and other non-radial intensity variations. It is achieved by fitting a background function in polar coordinates to a set of sample points having the median intensities and by standardising the average brightness. Representative examples of intermediate and final processed results are presented in addition to the algorithms developed. The research was done for the European Grid of Solar Observations (EGSO) project.     

A fast Bayesian approach to discrete object detection in astronomical data sets – PowellSnakes I

A new fast Bayesian approach is introduced for the detection of discrete objects immersed in a diffuse background. This new method, called PowellSnakes, speeds up traditional Bayesian techniques by (i) replacing the standard form of the likelihood for the parameters characterizing the discrete objects by an alternative exact form that is much quicker to evaluate; (ii) using a simultaneous multiple minimization code based on Powell's direction set algorithm to locate rapidly the local maxima in the posterior and (iii) deciding whether each located posterior peak corresponds to a real object by performing a Bayesian model selection using an approximate evidence value based on a local Gaussian approximation to the peak. The construction of this Gaussian approximation also provides the covariance matrix of the uncertainties in the derived parameter values for the object in question. This new approach provides a speed up in performance by a factor of '100' as compared to existing Bayesian source extraction methods that use Monte Carlo Markov chain to explore the parameter space, such as that presented by Hobson & McLachlan. The method can be implemented in either real or Fourier space. In the case of objects embedded in a homogeneous random field, working in Fourier space provides a further speed up that takes advantage of the fact that the correlation matrix of the background is circulant. We illustrate the capabilities of the method by applying to some simplified toy models. Furthermore, PowellSnakes has the advantage of consistently defining the threshold for acceptance/rejection based on priors which cannot be said of the frequentist methods. We present here the first implementation of this technique (version I). Further improvements to this implementation are currently under investigation and will be published shortly. The application of the method to realistic simulated Planck observations will be presented in a forthcoming publication.     

Improving Bayesian analysis for LISA Pathfinder using an efficient Markov Chain Monte Carlo method

We present a parameter estimation procedure based on a Bayesian framework by applying a Markov Chain Monte Carlo algorithm to the calibration of the dynamical parameters of the LISA Pathfinder satellite. The method is based on the Metropolis-Hastings algorithm and a two-stage annealing treatment in order to ensure an effective exploration of the parameter space at the beginning of the chain. We compare two versions of the algorithm with an application to a LISA Pathfinder data analysis problem. The two algorithms share the same heating strategy but with one moving in coordinate directions using proposals from a multivariate Gaussian distribution, while the other uses the natural logarithm of some parameters and proposes jumps in the eigen-space of the Fisher Information matrix. The algorithm proposing jumps in the eigen-space of the Fisher Information matrix demonstrates a higher acceptance rate and a slightly better convergence towards the equilibrium parameter distributions in the application to LISA Pathfinder data. For this experiment, we return parameter values that are all within ～1σ of the injected values. When we analyse the accuracy of our parameter estimation in terms of the effect they have on the force-per-unit of mass noise, we find that the induced errors are three orders of magnitude less than the expected experimental uncertainty in the power spectral density.     

Certain problems of solar and heliospheric physics in the light of novel satellite data

Recent satellite data have given a better insight into the possible nature of extremely strong disturbances on the Sun and in the heliosphere by relating them to processes in the solar interior. The energy, momentum, and mass transfer on various spatiotemporal scales are organized in the Sun into a hierarchy of coupled nonlinear processes. Confirmation has been given to the fact that coronal mass ejections and solar flares are not linked causally but merely reflect the existence of two channels of free-energy dissipation in the solar atmosphere in the form of plasma motion and plasma emission; their relative role can be described by a corresponding nondimensional parameter. Information on the global asymmetry of the solar emission and active processes has been gained. A great diversity in the geometry of eruptive events (not necessarily associated with magnetic reconnection) has been revealed. In our opinion, the basic unresolved problems in the investigation of solar activity dictate the necessity of carrying out more accurate, absolutely calibrated measurements of the whitelight solar emission at appropriately high spatiotemporal resolutions. The development of direct and indirect techniques of measuring the electric fields and currents with the aim of reconstructing the solar and heliospheric current system remains a challenging task.     

Limitations of Bayesian Evidence applied to cosmology

There has been increasing interest by cosmologists in applying Bayesian techniques, such as Bayesian Evidence, for model selection. A typical example is in assessing whether observational data favour a cosmological constant over evolving dark energy. In this paper, the example of dark energy is used to illustrate limitations in the application of Bayesian Evidence associated with subjective judgements concerning the choice of model and priors. An analysis of recent cosmological data shows a statistically insignificant preference for a cosmological constant over simple dynamical models of dark energy. It is argued that for nested problems, as considered here, Bayesian parameter estimation can be more informative than computing Bayesian Evidence for poorly motivated physical models.     

Interpretation of observations by inversion

The most recent developments in inversion techniques of the radiative transfer equation are critically reviewed and some of their findings are summarized to illustrating their achievements. Two significantly different approaches are currently being used that deserve consideration, each characterized by whether or not the model solar atmospheres are changed iteratively by the algorithm. The comparison between the two may help in finding future inversion techniques that can solve many challenging problems of solar physics that still need to be properly settled. These problems themselves suggest strategies that look more suitable than others.     

Density and temperature diagnostics of solar emission lines from Navii and Alix

Line intensity ratios of EUV emission lines from Navii and Alix have been considered for electron density and temperature determinations within the chromosphere-corona transition region and the corona. The electron pressure within the emission region has been assumed to be a constant parameter. Theoretical line intensities for these ions have been computed using a model solar atmosphere and compared with the values as observed by ATM ultraviolet spectrometer. The observed intensities correspond to the average quiet-Sun conditions near solar minimum.     

East-west inclination of large-scale photospheric magnetic fields

Sixteen years of WSO magnetogram data have been studied to determine the solar cycle variation and latitude dependence of the east-west inclination of photospheric magnetic field lines. East-west inclination is here defined as the angle between a field line and its local radial vector, as projected onto the plane of the latitude and line of sight. Inclination is determined by a least-squares fit of observed magnetic fields to a simple projection model, and is found to depend on polarity and to change with the solar cycle. Leading and following polarities are tipped towards each by about 9° and have an overall net tilt in the direction of rotation (to the west) of 0.6°. New cycles are seen to begin at high latitudes and to grow through the lower latitudes over approximately 5 years, providing evidence for an extended cycle length of 16–18 years.     

MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics

We present further development and the first public release of our multimodal nested sampling algorithm, called M ulti N est . This Bayesian inference tool calculates the evidence, with an associated error estimate, and produces posterior samples from distributions that may contain multiple modes and pronounced (curving) degeneracies in high dimensions. The developments presented here lead to further substantial improvements in sampling efficiency and robustness, as compared to the original algorithm presented in Feroz & Hobson, which itself significantly outperformed existing Markov chain Monte Carlo techniques in a wide range of astrophysical inference problems. The accuracy and economy of the M ulti N est algorithm are demonstrated by application to two toy problems and to a cosmological inference problem focusing on the extension of the vanilla Λ cold dark matter model to include spatial curvature and a varying equation of state for dark energy. The M ulti N est software, which is fully parallelized using MPI and includes an interface to C osmo MC, is available at http://www.mrao.cam.ac.uk/software/multinest/ . It will also be released as part of the SuperBayeS package, for the analysis of supersymmetric theories of particle physics, at http://www.superbayes.org .     

Corotational Tomography of Heliospheric Features Using Global Thomson Scattering Data

The Air Force/NASA Solar Mass Ejection Imager (SMEI) will provide two-dimensional images of the sky in visible light with high (0.1%) photometric precision, and unprecedented sky coverage and cadence. To optimize the information available from these images they must be interpreted in three dimensions. We have developed a Computer Assisted Tomography (CAT) technique that fits a three-dimensional kinematic heliospheric model to remotely-sensed Thomson scattering observations. This technique is designed specifically to determine the corotating background solar wind component from data provided by instruments like SMEI. Here, we present results from this technique applied to the Helios spacecraft photometer observations. The tomography program iterates to a least-squares solution of observed brightnesses using solar rotation, spacecraft motion and solar wind outflow to provide perspective views of each point in space covered by the observations. The corotational tomography described here is essentially the same as used by Jackson et al. (1998) for the analysis of interplanetary scintillation (IPS) observations. While IPS observations are related indirectly to the solar wind density through an assumed (and uncertain) relationship between small-scale density fluctuations and density, Thomson scattering physics is more straightforward, i.e., the observed brightness depends linearly on the solar wind density everywhere in the heliosphere. Consequently, Thomson scattering tomography can use a more direct density-convergence criterion to match observed Helios photometer brightness to brightness calculated from the model density. The general similarities between results based on IPS and Thomson scattering tomography validate both techniques and confirm that both observe the same type of solar wind structures. We show results for Carrington rotation 1653 near solar minimum. We find that longitudinally segmented dense structures corotate with the Sun and emanate from near the solar equator. We discuss the locations of these dense structures with respect to the heliospheric current sheet and regions of activity on the solar surface.     

The statistical significance of the low cosmic microwave background mulitipoles

The Wilkinson Microwave Anisotropy Probe ( WMAP ) has measured lower amplitudes for the temperature quadrupole and octopole anisotropies than expected in the best fitting (concordance) Λ-dominated cold dark matter (ΛCDM) cosmology. Some authors have argued that this discrepancy may require new physics. However, the statistical significance of this result is not clear. Some authors have applied frequentist arguments and claim that the discrepancy would occur by chance about 1 time in 700, if the concordance model is correct. Other authors have used Bayesian arguments to claim that the data show marginal evidence for new physics. I investigate these confusing and apparently conflicting claims in this Letter using a frequentist analysis and a simplified Bayesian analysis. On either analysis, I conclude that the WMAP results are consistent with the concordance ΛCDM model.     

On Re-sampling of Solar Images

Digital image data are now commonly used throughout the field of solar physics. Many steps of image data analysis, including image co-alignment, perspective reprojection of the solar surface, and compensation for solar rotation, require re-sampling original telescope image data under a distorting coordinate transformation. The most common image re-sampling methods introduce significant, unnecessary flaws into the data. More correct techniques have been known in the computer graphics community for some time but remain little known within the solar community and hence deserve further presentation. Furthermore, image distortion under specialized coordinate transformations is a powerful analysis technique with applications well beyond image resizing and perspective compensation. Here I give a brief overview of the mathematics of data re-sampling under arbitrary distortions, present a simple algorithm for optimized re-sampling, give some examples of distortion as an analysis tool, and introduce scientific image distortion software that is freely available over the Internet. ``First get your facts straight. Then you can distort them as you please.' – Mark Twain     

He I 1083 nm OSCILLATIONS AND DOWNFLOWS NEAR THE NORTH SOLAR POLE

Imaging spectroscopic data of the Sii 1082.7 nm (photospheric) and Hei 1083.0 nm (chromospheric) spectral lines were taken starting 22:05 UT on 23 July, 1996 with the NASA/NSO Spectromagnetograph at the NSO/Kitt Peak Vacuum Telescope. Observations were made near the north solar pole, with a field of view of 100 by 400 arc sec and with a temporal cadence of 53 s for 2 hr. Simple fitting to the line profiles measured the line position, depth, and spectral full-width at half-maximum. Power spectra of the velocity oscillations in each line were computed, and the oscillation power in the 2 to 6 mHz frequency band versus view angle was measured to search for horizontal oscillations. Horizontal waves are not detected to limiting amplitudes (1) of 22 m s^-1 in the chromosphere and 9 m s^-1 in the photosphere. These values are used to estimate limits for the energy flux into the corona. The amplitude of radial oscillations in the chromosphere is twice that of the photosphere. No statistically meaningful oscillation power is measured in the spectral parameters of the Hei line in the emission shell seen above the continuum limb. Finally, rapidly evolving red-shift events are observed in the Hei 1083 nm line on the disk; these events are some sort of coronal rain, and there are about 40 of these events on the solar disk at any moment.     

On model selection forecasting, dark energy and modified gravity

The Fisher matrix approach allows one to calculate in advance how well a given experiment will be able to estimate model parameters, and has been an invaluable tool in experimental design. In the same spirit, we present here a method to predict how well a given experiment can distinguish between different models, regardless of their parameters. From a Bayesian viewpoint, this involves computation of the Bayesian evidence. In this paper, we generalize the Fisher matrix approach from the context of parameter fitting to that of model testing, and show how the expected evidence can be computed under the same simplifying assumption of a Gaussian likelihood as the Fisher matrix approach for parameter estimation. With this 'Laplace approximation' all that is needed to compute the expected evidence is the Fisher matrix itself. We illustrate the method with a study of how well upcoming and planned experiments should perform at distinguishing between dark energy models and modified gravity theories. In particular, we consider the combination of 3D weak lensing, for which planned and proposed wide-field multiband imaging surveys will provide suitable data, and probes of the expansion history of the Universe, such as proposed supernova and baryonic acoustic oscillations surveys. We find that proposed large-scale weak-lensing surveys from space should be able readily to distinguish General Relativity from modified gravity models.