Properties of the Cosmological Density Distribution Function

The properties of the probability distribution function of the cosmological continuous density field are studied. We present further developments and compare dynamically motivated methods to derive the PDF. One of them is based on the Zel'dovich approximation (ZA). We extend this method for arbitrary initial conditions, regardless of whether they are Gaussian or not. The other approach is based on perturbation theory with Gaussian initial fluctuations. We include the smoothing effects in the PDFs. We examine the relationships between the shapes of the PDFs and the moments. It is found that formally there are no moments in the ZA, but a way to resolve this issue is proposed, based on the regularization of integrals. A closed form for the generating function of the moments in the ZA is also presented, including the smoothing effects. We suggest the methods to build PDFs out of the whole series of the moments, or out of a limited number of moments -- the Edgeworth expansion. The last approach gives us an alternative method to evaluate the skewness and kurtosis by measuring the PDF around its peak. We note a general connection between the generating function of moments for small r.m.s $\sigma$ and the non-linear evolution of the overdense spherical fluctuation in the dynamical models. All these approaches have been applied in 1D case where the ZA is exact, and simple analytical results are obtained. The 3D case is analyzed in the same manner and we found a mutual agreement in the PDFs derived by different methods in the the quasi-linear regime. Numerical CDM simulation was used to validate the accuracy of considered approximations. We explain the successful log-normal fit of the PDF from that simulation at moderate $\sigma$ as mere fortune, but not as a universal form of density PDF in general.Comment: 30 pages in Plain Tex, 1 table and 11 figures available as postscript files by anonymous ftp from ftp.cita.utoronto.ca in directory /cita/francis/lev, IFA-94-1

Edgeworth Expansion Based Model for the Convolutional Noise pdf

Recently, the Edgeworth expansion up to order 4 was used to represent the convolutional noise probability density function (pdf) in the conditional expectation calculations where the source pdf was modeled with the maximum entropy density approximation technique. However, the applied Lagrange multipliers were not the appropriate ones for the chosen model for the convolutional noise pdf. In this paper we use the Edgeworth expansion up to order 4 and up to order 6 to model the convolutional noise pdf. We derive the appropriate Lagrange multipliers, thus obtaining new closed-form approximated expressions for the conditional expectation and mean square error (MSE) as a byproduct. Simulation results indicate hardly any equalization improvement with Edgeworth expansion up to order 4 when using optimal Lagrange multipliers over a nonoptimal set. In addition, there is no justification for using the Edgeworth expansion up to order 6 over the Edgeworth expansion up to order 4 for the 16QAM and easy channel case. However, Edgeworth expansion up to order 6 leads to improved equalization performance compared to the Edgeworth expansion up to order 4 for the 16QAM and hard channel case as well as for the case where the 64QAM is sent via an easy channel

Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation

International audienceThis review discusses recent advances in geophysical data assimilation beyond Gaussian statistical modeling, in the fields of meteorology, oceanography, as well as atmospheric chemistry. The non-Gaussian features are stressed rather than the nonlinearity of the dynamical models, although both aspects are entangled. Ideas recently proposed to deal with these non-Gaussian issues, in order to improve the state or parameter estimation, are emphasized. The general Bayesian solution to the estimation problem and the techniques to solve it are first presented, as well as the obstacles that hinder their use in high-dimensional and complex systems. Approximations to the Bayesian solution relying on Gaussian, or on second-order moment closure, have been wholly adopted in geophysical data assimilation (e.g., Kalman filters and quadratic variational solutions). Yet, nonlinear and non-Gaussian effects remain. They essentially originate in the nonlinear models and in the non-Gaussian priors. How these effects are handled within algorithms based on Gaussian assumptions is then described. Statistical tools that can diagnose them and measure deviations from Gaussianity are recalled. The following advanced techniques that seek to handle the estimation problem beyond Gaussianity are reviewed: maximum entropy filter, Gaussian anamorphosis, non-Gaussian priors, particle filter with an ensemble Kalman filter as a proposal distribution, maximum entropy on the mean, or strictly Bayesian inferences for large linear models, etc. Several ideas are illustrated with recent or original examples that possess some features of high-dimensional systems. Many of the new approaches are well understood only in special cases and have difficulties that remain to be circumvented. Some of the suggested approaches are quite promising, and sometimes already successful for moderately large though specific geophysical applications. Hints are given as to where progress might come from

Gaussian Filter based on Deterministic Sampling for High Quality Nonlinear Estimation

In this paper, a Gaussian filter for nonlinear Bayesian estimation is introduced that is based on a deterministic sample selection scheme. For an effective sample selection, a parametric density function representation of the sample points is employed, which allows approximating the cumulative distribution function of the prior Gaussian density. The computationally demanding parts of the optimization problem formulated for approximation are carried out off-line for obtaining an efficient filter, whose estimation quality can be altered by adjusting the number of used sample points. The improved performance of the proposed Gaussian filter compared to the well-known unscented Kalman fiter is demonstrated by means of two examples

Indirect likelihood inference

Given a sample from a fully specified parametric model, let Zn be a given finite-dimensional statistic - for example, an initial estimator or a set of sample moments. We propose to (re-)estimate the parameters of the model by maximizing the likelihood of Zn. We call this the maximum indirect likelihood (MIL) estimator. We also propose a computationally tractable Bayesian version of the estimator which we refer to as a Bayesian Indirect Likelihood (BIL) estimator. In most cases, the density of the statistic will be of unknown form, and we develop simulated versions of the MIL and BIL estimators. We show that the indirect likelihood estimators are consistent and asymptotically normally distributed, with the same asymptotic variance as that of the corresponding efficient two-step GMM estimator based on the same statistic. However, our likelihood-based estimators, by taking into account the full finite-sample distribution of the statistic, are higher order efficient relative to GMM-type estimators. Furthermore, in many cases they enjoy a bias reduction property similar to that of the indirect inference estimator. Monte Carlo results for a number of applications including dynamic and nonlinear panel data models, a structural auction model and two DSGE models show that the proposed estimators indeed have attractive finite sample properties

Parameter estimation and the statistics of nonlinear cosmic fields

The large scale distribution of matter in the universe contains valuable information about fundamental cosmological parameters, the properties of dark matter and the formation processes of galaxies. The best hope of retrieving this information lies in providing a statistical description of the matter distribution that may be used for comparing models with observation. Unfortunately much of the important information lies on scales below which nonlinear gravitational effects have taken hold, complicating both models and statistics considerably. This thesis deals with the distribution of matter - mass and galaxies - on such scales. The aim is to develop new statistical tools that make use of the nonlinear evolution for the purposes of constraining cosmological models.A new derivation for the 1 -point probability distribution function (PDF) for density inhomogeneities is presented first. The calculation is based upon an exact statistical treatment, using the Chapman -Kolmogorov equation and second order Eulerian perturbation theory to propagate the initial density field into the nonlinear regime. The analysis yields the generating function for moments, allowing for a straightforward derivation of the skewness. A new dependance upon the perturbation spectrum is found for the skewness at second order. The results of the analysis are compared against other methods for deriving the 1 -point PDF and against data from numerical N -body simulations. Good agreement is found in both cases.The 1 -point PDF for galaxies is derived next, taking into account nonlinear biasing of the density field and the distorting effects associated with working in redshift space. Once again perturbation theory is used to evolve the density field into the nonlinear regime and the Chapman -Kolmogorov equation to propagate the initial probabilities. Transformation of the dark matter density to a biased galaxy distribution is done through an Eulerian biasing prescription, expanding the nonlinear bias function to second order. An advantage of the Chapman- Kolmogorov approach is the natural way that different initial conditions and biasing models may be incorporated. It is shown that the method is general enough to allow a non -deterministic (hidden variable) bias. The dependance on cosmological parameters of the evolution of the galaxy 1 -point PDF is demonstrated and a method for differentiating between degenerate models in linear theory is presented. A new derivation of the skewness for a biased density field in red - shift space is also given and shown to depend significantly on the density and bias parameters. The results are compared favourably with those of numerical simulations.Finally a new, general formalism for analysing parameter information from non - Gaussian cosmic fields is developed. The method is general enough for application to a range of problems including the measurement of parameters from galaxy redshift surveys, weak lensing surveys and velocity field surveys. It may also be used to test for non -Gaussianity in the Cosmic Microwave Background. Generalising maximum likelihood analysis to second order, the non -Gaussian Fisher information matrix is derived and the detailed shapes of likelihood surfaces in parameter space are explored via a parameter entropy function. Concentrating on non -Gaussianity due to nonlinear evolution under gravity, the generalised Fisher analysis is applied to a model of a Galaxy redshift survey, including the effects of biasing, redshift space distortions and shot noise. Incorporating second order moments into the parameter estimation is found to have a large effect, breaking all of the degeneracies between parameters. The results indicate that using nonlinear likelihood analysis may yield parameter uncertainties around the few percent level from forthcoming large galaxy redshift surveys

Moment bounds for non-linear functionals of the periodogram

In this paper, we prove the validity of the Edgeworth expansion of the Discrete Fourier transforms of some linear time series. This result is applied to approach moments of non linear functionals of the periodogram. As an illustration, we give an expression of the mean square error of the Geweke and Porter-Hudak estimator of the long memory parameter. We prove that this estimator is rate optimal, extending the result of Giraitis, Robinson, Samarov (1997) from Gaussian to linear processes

Efficient and Robust Signal Detection Algorithms for the Communication Applications

Signal detection and estimation has been prevalent in signal processing and communications for many years. The relevant studies deal with the processing of information-bearing signals for the purpose of information extraction. Nevertheless, new robust and efficient signal detection and estimation techniques are still in demand since there emerge more and more practical applications which rely on them. In this dissertation work, we proposed several novel signal detection schemes for wireless communications applications, such as source localization algorithm, spectrum sensing method, and normality test. The associated theories and practice in robustness, computational complexity, and overall system performance evaluation are also provided

Generation of Vorticity and Velocity Dispersion by Orbit Crossing

We study the generation of vorticity and velocity dispersion by orbit crossing using cosmological numerical simulations, and calculate the backreaction of these effects on the evolution of large-scale density and velocity divergence power spectra. We use Delaunay tessellations to define the velocity field, showing that the power spectra of velocity divergence and vorticity measured in this way are unbiased and have better noise properties than for standard interpolation methods that deal with mass weighted velocities. We show that high resolution simulations are required to recover the correct large-scale vorticity power spectrum, while poor resolution can spuriously amplify its amplitude by more than one order of magnitude. We measure the scalar and vector modes of the stress tensor induced by orbit crossing using an adaptive technique, showing that its vector modes lead, when input into the vorticity evolution equation, to the same vorticity power spectrum obtained from the Delaunay method. We incorporate orbit crossing corrections to the evolution of large scale density and velocity fields in perturbation theory by using the measured stress tensor modes. We find that at large scales (k~0.1 h/Mpc) vector modes have very little effect in the density power spectrum, while scalar modes (velocity dispersion) can induce percent level corrections at z=0, particularly in the velocity divergence power spectrum. In addition, we show that the velocity power spectrum is smaller than predicted by linear theory until well into the nonlinear regime, with little contribution from virial velocities.Comment: 27 pages, 14 figures. v2: reorganization of the material, new appendix. Accepted by PR