Likelihood Analysis of Power Spectra and Generalized Moment Problems

We develop an approach to spectral estimation that has been advocated by Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance extension problem, by Enqvist and Karlsson. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Georgiou and Lindquist and can also be viewed as a generalization of the classical work by Burg and Jaynes on the maximum entropy method. In the present paper we present a new fast algorithm in the general case (i.e., for general Gaussian priors) and show that for priors with a specific structure the solution can be given in closed form.Comment: 17 pages, 4 figure

Distances between power spectral densities

We present several natural notions of distance between spectral density functions of (discrete-time) random processes. They are motivated by certain filtering problems. First we quantify the degradation of performance of a predictor which is designed for a particular spectral density function and then it is used to predict the values of a random process having a different spectral density. The logarithm of the ratio between the variance of the error, over the corresponding minimal (optimal) variance, produces a measure of distance between the two power spectra with several desirable properties. Analogous quantities based on smoothing problems produce alternative distances and suggest a class of measures based on fractions of generalized means of ratios of power spectral densities. These distance measures endow the manifold of spectral density functions with a (pseudo) Riemannian metric. We pursue one of the possible options for a distance measure, characterize the relevant geodesics, and compute corresponding distances.Comment: 16 pages, 4 figures; revision (July 29, 2006) includes two added section

Constrained probability distributions of correlation functions

Context: Two-point correlation functions are used throughout cosmology as a measure for the statistics of random fields. When used in Bayesian parameter estimation, their likelihood function is usually replaced by a Gaussian approximation. However, this has been shown to be insufficient. Aims: For the case of Gaussian random fields, we search for an exact probability distribution of correlation functions, which could improve the accuracy of future data analyses. Methods: We use a fully analytic approach, first expanding the random field in its Fourier modes, and then calculating the characteristic function. Finally, we derive the probability distribution function using integration by residues. We use a numerical implementation of the full analytic formula to discuss the behaviour of this function. Results: We derive the univariate and bivariate probability distribution function of the correlation functions of a Gaussian random field, and outline how higher joint distributions could be calculated. We give the results in the form of mode expansions, but in one special case we also find a closed-form expression. We calculate the moments of the distribution and, in the univariate case, we discuss the Edgeworth expansion approximation. We also comment on the difficulties in a fast and exact numerical implementation of our results, and on possible future applications.Comment: 13 pages, 5 figures, updated to match version published in A&A (slightly expanded Sects. 5.3 and 6

On the existence of a solution to a spectral estimation problem \emph{\`a la} Byrnes-Georgiou-Lindquist

A parametric spectral estimation problem in the style of Byrnes, Georgiou, and Lindquist was posed in \cite{FPZ-10}, but the existence of a solution was only proved in a special case. Based on their results, we show that a solution indeed exists given an arbitrary matrix-valued prior density. The main tool in our proof is the topological degree theory.Comment: 6 pages of two-column draft, accepted for publication in IEEE-TA

Long term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development

The scaling properties encompass in a simple analysis many of the volatility characteristics of financial markets. That is why we use them to probe the different degree of markets development. We empirically study the scaling properties of daily Foreign Exchange rates, Stock Market indices and fixed income instruments by using the generalized Hurst approach. We show that the scaling exponents are associated with characteristics of the specific markets and can be used to differentiate markets in their stage of development. The robustness of the results is tested by both Monte-Carlo studies and a computation of the scaling in the frequency-domain.Comment: 46 pages, 7 figures, accepted for publication in Journal of Banking & Financ

A Markov Chain Monte Carlo Algorithm for analysis of low signal-to-noise CMB data

We present a new Monte Carlo Markov Chain algorithm for CMB analysis in the low signal-to-noise regime. This method builds on and complements the previously described CMB Gibbs sampler, and effectively solves the low signal-to-noise inefficiency problem of the direct Gibbs sampler. The new algorithm is a simple Metropolis-Hastings sampler with a general proposal rule for the power spectrum, C_l, followed by a particular deterministic rescaling operation of the sky signal. The acceptance probability for this joint move depends on the sky map only through the difference of chi-squared between the original and proposed sky sample, which is close to unity in the low signal-to-noise regime. The algorithm is completed by alternating this move with a standard Gibbs move. Together, these two proposals constitute a computationally efficient algorithm for mapping out the full joint CMB posterior, both in the high and low signal-to-noise regimes.Comment: Submitted to Ap