Inhomogeneous Noise Research Articles

Limits on primordial non-Gaussianity from Minkowski Functionals of theWMAPtemperature anisotropies

We present an analysis of the Minkowski Functionals (MFs) describing the Wilkinson Microwave Anisotropy Probe (WMAP) 3-yr temperature maps to place limits on possible levels of primordial non-Gaussianity. In particular, we apply perturbative formulae for the MFs to give constraints on the usual non-linear coupling constant fNL. The theoretical predictions are found to agree with the MFs of simulated cosmic microwave background (CMB) maps including the full effects of radiative transfer. The agreement is also very good even when the simulation maps include various observational artefacts, including the pixel window function, beam smearing, inhomogeneous noise and the survey mask. We accordingly find that these analytical formulae can be applied directly to observational measurements of fNL without relying on non-Gaussian simulations. Considering the bin-to-bin covariance of the MFs in WMAP in a chi-square analysis, we find that the primordial non-Gaussianity parameter is constrained to lie in the range -70 < fNL < 91 [95 per cent confidence level (C.L.)] using the Q + V + W co-added maps.

Primordial non-Gaussianity estimator: The inhomogeneous noise effect

Since the inhomogeneous instrument noise can produce extra non-Gaussianity in the cosmic microwave background (CMB) anisotropy, its effect should be carefully subtracted in the primordial non-Gaussianity estimation. We calculate the probability distribution function of the CMB anisotropy for a local type of non-Gaussianity, from which the optimal estimator in the general case (inhomogeneous noise and cut sky) is obtained. The new estimator obtained here is different from the popular one, since the inhomogeneous noise and cut sky effects are completely accounted for. The CMB anisotropy in the new estimator is noise weighted. The noise weight is different from that used by the WMAP Collaboration in their 5-year data analysis. Although it is still difficult to calculate the new estimator rigorously, for the case of the slightly inhomogeneous noise, there exists a series expansion method to compute the new estimator. Each order in the series is suppressed by two factors, (({sigma}{sup 2}/{sigma}{sub i}{sup 2})-1) and (C{sub l}/C{sub l}{sup tot}), which make the method feasible. Through the Edgeworth expansion we can generalize our discussion to other types of non-Gaussianity.

Z-matrix formalism for quantitative noise assessment of covariance nuclear magnetic resonance spectra.

Due to the limited sensitivity of many nuclear magnetic resonance (NMR) applications, careful consideration must be given to the effect of NMR data processing on spectral noise. This work presents analytical relationships as well as simulated and experimental results characterizing the propagation of noise by unsymmetric covariance NMR processing, which concatenates two NMR spectra along a common dimension, resulting in a new spectrum showing spin correlations as cross peaks that are not directly measured in either of the two input spectra. It is shown how the unsymmetric covariance spectrum possesses an inhomogeneous noise distribution across the spectrum with the least amount of noise in regions whose rows and columns do not contain any cross or diagonal peaks and with the largest amount of noise on top of signal peaks. Therefore, methods of noise estimation commonly used in Fourier transform spectroscopy underestimate the amount of uncertainty in unsymmetric covariance spectra. Different data processing procedures, including the Z-matrix formalism, thresholding, and maxima ratio scaling, are described to assess noise contributions and to reduce noise inhomogeneity. In particular, determination of a Z score, which measures the difference in standard deviations of a statistic from its mean, for each spectral point yields a Z matrix, which indicates whether a given peak intensity above a threshold arises from the covariance of signals in the input spectra or whether it is likely to be caused by noise. Application to an unsymmetric covariance spectrum, obtained by concatenating two 2D (13)C-(1)H heteronuclear, single quantum coherence (HSQC) and (13)C-(1)H heteronuclear, multiple bond correlation (HMBC) spectra of a metabolite mixture along their common proton dimension, reveals that for sufficiently sensitive input spectra the reduction in sensitivity due to covariance processing is modest.

Fast Estimator of Primordial Non‐Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background. II. Partial Sky Coverage and Inhomogeneous Noise

In our recent paper (Yadav et al. 2007) we described a fast cubic (bispectrum) estimator of the amplitude of primordial non-Gaussianity of local type, f_{NL}, from a combined analysis of the Cosmic Microwave Background (CMB) temperature and E-polarization observations. In this paper we generalize the estimator to deal with a partial sky coverage as well as inhomogeneous noise. Our generalized estimator is still computationally efficient, scaling as O(N^3/2) compared to the O(N^5/2) scaling of the brute force bispectrum calculation for sky maps with N pixels. Upcoming CMB experiments are expected to yield high-sensitivity temperature and E-polarization data. Our generalized estimator will allow us to optimally utilize the combined CMB temperature and E-polarization information from these realistic experiments, and to constrain primordial non-Gaussianity.

Information-theory-based snake adapted to inhomogeneous intensity variations

A new snake-based segmentation technique of a single object (simply connected) in the presence of inhomogeneous Gaussian noise is proposed, in which the mean in each region is modeled as a polynomial function of the coordinates and which is thus adapted to inhomogeneous illumination. It is shown that the minimization of the stochastic complexity of the image, which can be implemented efficiently, allows one to automatically estimate not only the number and the position of the nodes of the polygonal contour used to describe the object but also the degree of the polynomials that model the variations of the mean.

General solution to theE−Bmixing problem

We derive a general ansatz for optimizing pseudo-${C}_{\ensuremath{\ell}}$ estimators used to measure cosmic microwave background anisotropy power spectra, and apply it to the recently proposed pure pseudo-${C}_{\ensuremath{\ell}}$ formalism, to obtain an estimator which achieves near-optimal $B$-mode power spectrum errors for any specified noise distribution while minimizing leakage from ambiguous modes. Our technique should be relevant for upcoming cosmic microwave background polarization experiments searching for $B$-mode polarization. We compare our technique both to the theoretical limits based on a full Fisher matrix calculation and to the standard pseudo-${C}_{\ensuremath{\ell}}$ technique. We demonstrate it by applying it to a fiducial survey with realistic inhomogeneous noise, complex boundaries, point source masking, and a noise level comparable to what is expected for next-generation experiments ($\ensuremath{\sim}5.75\text{ }\text{ }\ensuremath{\mu}\mathrm{K}\mathrm{\text{\ensuremath{-}}}\mathrm{arcmin}$). For such an experiment our technique could improve the constraints on the amplitude of a gravity wave background by over a factor of 10 compared to what could be obtained using ordinary pseudo-${C}_{\ensuremath{\ell}}$, coming quite close to saturating the theoretical limit. Constraints on the amplitude of the lensing $B$-modes are improved by about a factor of 3.

Non-additive coupling enables stable propagation of synchronous spiking in purely random networks

Precise timing of spikes and synchronization in the millisecond range has been experimentally observed in different neuronal systems. Their occurrence correlates with external stimuli and is thus considered a key feature of neural computation. The dynamical origin of precise and coordinated spike timing, however, is not well understood. Here we show in a modeling study that synchronous spiking activity of subgroups can persist and propagate in purely random networks if we take into account the non-additive nature of dendritic input integration that was recently uncovered experimentally. We find a transition to stable propagation of synchronized spiking: For additive coupling and at low strength of non-additivity, synchronous spiking dies out; above a critical strength, stable propagation of a group of synchronized spikes becomes possible. We derive a map giving the average future size of a synchronous group in terms of the current group size. For networks with homogeneous parameters the map can be obtained analytically and reveals that the discontinuous transition found is due to a tangent bifurcation at a critical strength of non-additivity. We discuss the mechanism underlying this transition and its consequences for networks with inhomogeneous parameters and additional external noise. Prominent 'synfire-chain' models for the stable propagation of synchronous activity in cortical networks require the existence of feed-forward structures that are superimposed on otherwise randomly connected local cortical circuits. It is unclear, however, whether feed-forward structures actually exist in such local circuits. Our study suggests that additional structural features of the network connectivity may not be required for the propagation of synchronous spiking activity if synaptic interactions exhibit non-additive dendritic integration.

Noise can delay and advance the collapse of spatiotemporal chaos

Spatiotemporal chaos on a regular ring network of excitable Gray-Scott dynamical elements collapses to a stable asymptotic state. We find that the addition of dynamical noise clearly influences the spatiotemporal pattern and the transient lifetime of spatiotemporal chaos. Spatially uniform noise significantly decreases the average lifetime of spatiotemporal chaos due to an enlargement of regions of local collapse. For spatially inhomogeneous noise the collapse is maximally delayed at an intermediate noise level, but drastically advanced for larger noise levels.

BISTABLE KINETIC MODEL DRIVEN BY AN INHOMOGENEOUS NOISE

Steady state properties of a Brownian particle moving in a bistable kinetic system is investigated in the presence of an inhomogeneous noise. It is found that the inhomogeneous noise makes the system change from two-state system to one-state system, namely, the inhomogeneous noise can induce a phase transition. The particle will always move from a large intensity noise region to a small intensity region.

Pseudo-Cℓestimators which do not mix E and B modes

Pseudo-$C_\ell$ quadratic estimators for CMB temperature and polarization power spectra have been used in the analysis pipelines of many CMB experiments, such as WMAP and Boomerang. In the polarization case, these estimators mix E and B modes, in the sense that the estimated B-mode power is nonzero for a noiseless CMB realization which contains only E-modes. Recently, Challinor & Chon showed that for moderately sized surveys ($f_{sky} \sim 0.01$), this mixing limits the gravity wave B-mode signal which can be detected using pseudo-$C_\ell$ estimators to $T/S \sim 0.05$. We modify the pseudo-$C_\ell$ construction, defining ``pure'' pseudo-$C_\ell$ estimators, which do not mix E and B modes in this sense. We study these estimators in detail for a survey geometry similar to that which has been proposed for the QUIET experiment, for a variety of noise levels, and both homogeneous and inhomogeneous noise. For noise levels $\simle 20$ $\mu$K-arcmin, our modification significantly improves the B-mode power spectrum errors obtained using pseudo-$C_\ell$ estimators. In the homogeneous case, we compute optimal power spectrum errors using a Fisher matrix approach, and show that our pure pseudo-$C_\ell$ estimators are roughly 80% of optimal, across a wide range of noise levels. There is no limit, imposed by the estimators alone, to the value of $T/S$ which can be detected.

Hybrid estimation of cosmic microwave background polarization power spectra

This paper generalizes the hybrid power spectrum estimation method developed in a previous paper to the estimation of polarization power spectra of the cosmic microwave background radiation. The hybrid power spectrum estimator is unbiased and we show that it is close to optimal at all multipoles, provided the pixel noise satisfies certain constraints. Furthermore, the hybrid estimator is computationally fast and can easily be incorporated in a Monte Carlo chain for Planck-sized data sets, which may be essential to account for realistic noise covariance and systematic errors. Simple formulae are given for the covariance matrices, including uncorrelated inhomogeneous instrumental noise, and these are tested extensively against numerical simulations. We compare the behaviour of simple pseudo-Cℓ (PCL) estimates with maximum-likelihood estimates at low multipoles. For realistic sky cuts, maximum-likelihood estimates reduce very significantly the mixing of E and B modes. To achieve limits on the scalar–tensor ratio of r≪ 0.1 from sky maps with realistic sky cuts, maximum-likelihood methods, or PCL estimators based on unambiguous E and B modes, will be essential.

Multi-array passive search in a littoral environment

Acoustic barrier tactics were developed during the Cold War for deep, uniform underwater environments. Oceanographic and acoustic conditions in littoral environments are extremely complex and dynamic. The spatial and temporal variability of low-frequency signal and noise fields in these complex environments destroys the basic homogeneous assumption associated with standard tactical search concepts. Genetic algorithms (GAs) have been applied to both the signal and noise parts of this problem to produce near-optimal, nonstandard search tracks that maximize probability of detection in such inhomogeneous noise fields. In the present work, a GA was used to optimize tactics of several passive searchers by constructing optimal barriers in complex, littoral environments. The dynamic ambient noise model (DANM) was used to produce realistic, low-frequency, directional noise fields, based on discrete ship tracks. The cost function to be minimized was the probability that a target crossed the barrier undetected. Both standard and GA-derived tactics were evaluated and compared. The results show the importance of nonstandard tactics and careful consideration of environmental complexity when designing optimal passive search tactics. [Work sponsored by NAVSEA under the LCS project.]

Heat equation with strongly inhomogeneous noise

We consider the heat equation in dimension one with singular drift and inhomogeneous space–time white noise. In particular, the quadratic variation measure of the white noise is not required to be absolutely continuous w.r.t. the Lebesgue measure, neither in space nor in time. Under some assumptions we give statements on strong and weak existence as well as strong and weak uniqueness of continuous solutions.

Using peak distribution of the cosmic microwave background for WMAP and Planck data analysis: Formalism and simulations

We implement and further refine the recently proposed method at Kashlinsky et al. ([CITE]) (KHA) for a time efficient extraction of the power spectrum from future cosmic microwave background (CMB) maps. The method is based on the clustering properties of peaks and troughs of the Gaussian CMB sky. The procedure takes only steps where is the fraction of pixels with standard deviations in the map of N pixels. We use the new statistic introduced in KHA, , which characterizes spatial clustering of the CMB sky peaks of progressively increasing thresholds. The tiny fraction of the remaining pixels (peaks and troughs) contains the required information on the CMB power spectrum of the entire map. The threshold ν is the only parameter that determines the accuracy of the final spectrum. We performed detailed numerical simulations for parameters of the two-year WMAP and Planck CMB sky data including cosmological signal, inhomogeneous noise and foreground residuals. In all cases we find that the method can recover the power spectrum out to the Nyquist scale of the experiment channel. We discuss how the error bars scale with ν allowing to decide between accuracy and speed. The method can determine with significant accuracy the CMB power spectrum from the upcoming CMB maps in only ~ operations.

Yet another object detection application (YODA)

We present a software package performing object detection, photometry and star-galaxy classification specially developed to cope with multi-band imaging data as are common in modern extragalactic imaging surveys, and to be modular and therefore easily adaptable to specific needs. Specifically, the package is designed to work under conditions of inhomogeneous background noise across the detection frame, and, additionally, perform accurate aperture photometry in image sets not sharing a common coordinate system or pixel scale as is often the case in present-day extragalactic survey work. The package has been used for the MUNICS survey. The package is described in detail and its performance is compared to existing software using real imaging data.

Optimal method of linear regression in laser remote sensing.

Remote lidar sensing in the photon-counting mode is now the commonly accepted method for studying atmospheric processes in the lower and free atmosphere. However, when processing signals obtained from lidar measurements, investigators necessarily face the problem of achieving accuracy in reconstructing the atmospheric parameters despite the presence of inhomogeneous noise in the measured signals. We propose an optimal method of linear regression (OMLR) of signals. The accuracy of the the method for the reconstructed signal is estimated. An example of application of the OMLR to the reconstruction of the temperature profile from the data obtained with a Raman lidar at the Siberian Lidar Station of the Institute of Atmospheric Optics (Tomsk, Russia) is given. The proposed method is distinguished by simplicity of interpretation of the criteria used, based on careful adherence to statistical principles. This method is shown to be an efficient auxiliary tool for the processing of measured data.

Dynamics of the batch minority game with inhomogeneous decision noise.

We study the dynamics of a version of the batch minority game, with random external information and with different types of inhomogeneous decision noise (additive and multiplicative), using generating functional techniques à la De Dominicis. The control parameters in this model are the ratio alpha=p/N of the number p of possible values for the external information over the number N of trading agents, and the statistical properties of the agents' decision noise parameters. The presence of decision noise is found to have the general effect of damping macroscopic oscillations, which explains why in certain parameter regions it can effectively reduce the market volatility, as observed in earlier studies. In the limit N-->infinity we (i) solve the first few time steps of the dynamics (for any alpha), (ii) calculate the location alpha(c) of the phase transition (signaling the onset of anomalous response), and (iii) solve the statics for alpha>alpha(c). We find that alpha(c) is not sensitive to additive decision noise, but we arrive at nontrivial phase diagrams in the case of multiplicative noise. Our theoretical results find excellent confirmation in numerical simulations.

Specific coil design for SENSE: a six-element cardiac array.

In sensitivity encoding (SENSE), the effects of inhomogeneous spatial sensitivity of surface coils are utilized for signal localization in addition to common Fourier encoding using magnetic field gradients. Unlike standard Fourier MRI, SENSE images exhibit an inhomogeneous noise distribution, which crucially depends on the geometrical sensitivity relations of the coils used. Thus, for optimum signal-to-noise-ratio (SNR) and noise homogeneity, specialized coil configurations are called for. In this article we study the implications of SENSE imaging for coil layout by means of simulations and imaging experiments in a phantom and in vivo. New, specific design principles are identified. For SENSE imaging, the elements of a coil array should be smaller than for common phased-array imaging. Furthermore, adjacent coil elements should not overlap. Based on the findings of initial investigations, a configuration of six coils was designed and built specifically for cardiac applications. The in vivo evaluation of this array showed a considerable SNR increase in SENSE images, as compared with a conventional array. Magn Reson Med 45:495-504, 2001.

Decadal Variability in a Simplified Wind-Driven Ocean Model

The impact of an unsteady wind forcing on oceanic low-frequency variability is conceptually studied using a reduced-gravity shallow-water model. A time-averaged wind forcing and a simple ocean‐atmosphere coupling is completed by a stochastic component (spatially coherent white noise) representing the effect of atmospheric transient eddies. To account for the observed concentration of eddy activity along the North Atlantic and North Pacific storm tracks the variance of the stochastic forcing is chosen to be spatially inhomogeneous. Low-frequency variability of the basin-averaged energetics shows a dominant spectral peak with an amplitude depending on the inhomogeneity of the stochastic forcing and the time-averaged wind stress. The period of the variability is unexpected considering baroclinic Rossby waves forced by the ocean‐atmosphere coupling only. This variability can be explained by ‘‘spatial resonance’’ of the forced baroclinic Rossby wave and the Reynolds momentum flux induced by the spatially inhomogeneous white noise.

Reheating in the presence of inhomogeneous noise

Explosive particle production due to parametric resonance is a crucial feature of reheating in an inflationary cosmology. Coherent oscillations of the inflaton field lead to a periodically varying mass in the evolution equation of matter and gravitational fluctuations and often induce a parametric resonance instability. In a previous paper (hep-ph/9709273) it was shown that homogeneous (i.e. space independent) noise leads to an increase of the generalized Floquet exponent for all modes, at least if the noise is temporally uncorrelated. Here we extend the results to the physically more realistic case of spatially inhomogeneous noise. We demonstrate - modulo some mathematical fine points which are addressed in a companion paper - that the Floquet exponent is a non- decreasing function of the amplitude of the noise. We provide numerical evidence for an even stronger statement, namely that in the presence of inhomogeneous noise, the Floquet exponent of each mode is larger than the maximal Floquet exponent of the system in the absence of noise.