Frequency-Domain Stochastic Modeling of Stationary Bivariate or Complex-Valued Signals

There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. In this paper we provide a joint framework for all three representations in the context of frequency-domain stochastic modeling. This framework allows us to extend many established statistical procedures for bivariate vector time series to complex-valued and rotary representations. These include procedures for parametrically modeling signal coherence, estimating model parameters using the Whittle likelihood, performing semi-parametric modeling, and choosing between classes of nested models using model choice. We also provide a new method of testing for impropriety in complex-valued signals, which tests for noncircular or anisotropic second-order statistical structure when the signal is represented in the complex plane. Finally, we demonstrate the usefulness of our methodology in capturing the anisotropic structure of signals observed from fluid dynamic simulations of turbulence.Comment: To appear in IEEE Transactions on Signal Processin

Atmospheric Retrieval for Super-Earths: Uniquely Constraining the Atmospheric Composition with Transmission Spectroscopy

We present a retrieval method based on Bayesian analysis to infer the atmospheric compositions and surface or cloud-top pressures from transmission spectra of exoplanets with general compositions. In this study, we identify what can unambiguously be determined about the atmospheres of exoplanets from their transmission spectra by applying the retrieval method to synthetic observations of the super-Earth GJ 1214b. Our approach to infer constraints on atmospheric parameters is to compute their joint and marginal posterior probability distributions using the MCMC technique in a parallel tempering scheme. A new atmospheric parameterization is introduced that is applicable to general atmospheres in which the main constituent is not known a priori and clouds may be present. Our main finding is that a unique constraint of the mixing ratios of the absorbers and up to two spectrally inactive gases (such as N2 and primordial H2+He) is possible if the observations are sufficient to quantify both (1) the broadband transit depths in at least one absorption feature for each absorber and (2) the slope and strength of the molecular Rayleigh scattering signature. The surface or cloud-top pressure can be quantified if a surface or cloud deck is present. The mean molecular mass can be constrained from the Rayleigh slope or the shapes of absorption features, thus enabling to distinguish between cloudy hydrogen-rich atmospheres and high mean molecular mass atmospheres. We conclude, however, that without the signature of Rayleigh scattering--even with robustly detected infrared absorption features--there is no reliable way to tell if the absorber is the main constituent of the atmosphere or just a minor species with a mixing ratio of <0.1%. The retrieval method leads us to a conceptual picture of which details in transmission spectra are essential for unique characterizations of well-mixed atmospheres.Comment: 23 pages, 13 figures, accepted at ApJ, submitted to ApJ on Nov 4, 201

Single-trial multiwavelet coherence in application to neurophysiological time series

A method of single-trial coherence analysis is presented, through the application of continuous muldwavelets. Multiwavelets allow the construction of spectra and bivariate statistics such as coherence within single trials. Spectral estimates are made consistent through optimal time-frequency localization and smoothing. The use of multiwavelets is considered along with an alternative single-trial method prevalent in the literature, with the focus being on statistical, interpretive and computational aspects. The multiwavelet approach is shown to possess many desirable properties, including optimal conditioning, statistical descriptions and computational efficiency. The methods. are then applied to bivariate surrogate and neurophysiological data for calibration and comparative study. Neurophysiological data were recorded intracellularly from two spinal motoneurones innervating the posterior,biceps muscle during fictive locomotion in the decerebrated cat

The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches

We describe the details of the binned bispectrum estimator as used for the official 2013 and 2015 analyses of the temperature and polarization CMB maps from the ESA Planck satellite. The defining aspect of this estimator is the determination of a map bispectrum (3-point correlator) that has been binned in harmonic space. For a parametric determination of the non-Gaussianity in the map (the so-called fNL parameters), one takes the inner product of this binned bispectrum with theoretically motivated templates. However, as a complementary approach one can also smooth the binned bispectrum using a variable smoothing scale in order to suppress noise and make coherent features stand out above the noise. This allows one to look in a model-independent way for any statistically significant bispectral signal. This approach is useful for characterizing the bispectral shape of the galactic foreground emission, for which a theoretical prediction of the bispectral anisotropy is lacking, and for detecting a serendipitous primordial signal, for which a theoretical template has not yet been put forth. Both the template-based and the non-parametric approaches are described in this paper.Comment: Latex 42 pages with 10 figures and JCAP macros. v2: corrected small mistake in section 5.3, changed colour scale of slice figures, other minor changes and additions, matches published versio