Cosmological Parameter Estimation: Method

CMB anisotropy data could put powerful constraints on theories of the evolution of our Universe. Using the observations of the large number of CMB experiments, many studies have put constraints on cosmological parameters assuming different frameworks. Assuming for example inflationary paradigm, one can compute the confidence intervals on the different components of the energy densities, or the age of the Universe, inferred by the current set of CMB observations. The aim of this note is to present some of the available methods to derive the cosmological parameters with their confidence intervals from the CMB data, as well as some practical issues to investigate large number of parameters

Modulation parameter estimation of LFM interference for direct sequence spread spectrum communication system in alpha-stable noise

The linear frequency modulation (LFM) interference is one of the typical broadband interferences in direct sequence spread spectrum (DSSS) communication system. In this article, a novel modulation parameter estimation method of LFM interference is proposed for the DSSS communication system in alpha-stable noise. To accurately estimate the modulation parameters, the alpha-stable noise should be eliminated first. Thus, we formulate a new generalized extended linear chirplet transform to suppress the alpha-stable noise, for a robust time-frequency, transformation of LFM interference is realized. Then, using the Radon transform, the maximum value after transformation and the chirp rate according to the angle related to the maximum value are estimated. In addition, a generalized Fourier transform is introduced to estimate the initial frequency of the LFM interference. For the performance analysis, the Cramér-Rao lower bounds of the estimated chirp rate and the initial frequency of the LFM interference in the presence of alpha-stable noise are derived. Moreover, the asymptotic properties of the modulation parameter estimator are analyzed. Simulation results demonstrate that the performance of the proposed parameter estimation method significantly outperforms existing methods, especially in a low SNR regime

Parameter estimation by implicit sampling

Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use implicit sampling in parameter estimation problems, where the goal is to find parameters of a numerical model, e.g.~a partial differential equation (PDE), such that the output of the numerical model is compatible with (noisy) data. We use the Bayesian approach to parameter estimation, in which a posterior probability density describes the probability of the parameter conditioned on data and compute an empirical estimate of this posterior with implicit sampling. Our approach generates independent samples, so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples, are avoided. We describe a new implementation of implicit sampling for parameter estimation problems that makes use of multiple grids (coarse to fine) and BFGS optimization coupled to adjoint equations for the required gradient calculations. The implementation is "dimension independent", in the sense that a well-defined finite dimensional subspace is sampled as the mesh used for discretization of the PDE is refined. We illustrate the algorithm with an example where we estimate a diffusion coefficient in an elliptic equation from sparse and noisy pressure measurements. In the example, dimension\slash mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions

Error structures and parameter estimation

This article proposes a link between statistics and the theory of Dirichlet forms used to compute errors. The error calculus based on Dirichlet forms is an extension of classical Gauss' approach to error propagation. The aim of this paper is to derive error structures from measurements. The links with Fisher's information lay the foundations of a strong connection with experiment. We show that this connection behaves well towards changes of variables and is related to the theory of asymptotic statistics