Error In Power Spectrum Research Articles

Forecasts and Statistical Insights for Line Intensity Mapping Cross-correlations: A Case Study with 21 cm × [C ii]

Intensity mapping—the large-scale mapping of selected spectral lines without resolving individual sources—is quickly emerging as an efficient way to conduct large cosmological surveys. Multiple surveys covering a variety of lines (such as the hydrogen 21 cm hyperfine line, carbon-monoxide rotational lines, and [C ii] fine-structure lines, among others) are either observing or will soon be online, promising a panchromatic view of our Universe over a broad redshift range. With multiple lines potentially covering the same volume, cross-correlations have become an attractive prospect, both for probing the underlying astrophysics and for mitigating observational systematics. For example, cross-correlating 21 cm and [C ii] intensity maps during reionization could reveal the characteristic scale of ionized bubbles around the first galaxies, while simultaneously providing a convenient way to reduce independent foreground contaminants between the two surveys. However, many of the desirable properties of cross-correlations in principle emerge only under ideal conditions, such as infinite ensemble averages. In this paper, we construct an end-to-end pipeline for analyzing intensity mapping cross-correlations, enabling instrumental effects, foreground residuals, and analysis choices to be propagated through Monte Carlo simulations to a set of rigorous error properties, including error covariances, window functions, and full probability distributions for power-spectrum estimates. We use this framework to critically examine the applicability of simplifying assumptions such as the independence and Gaussianity of power-spectrum errors. As worked examples, we forecast the sensitivity of near-term and futuristic 21 cm × [C ii] cross-correlation measurements, providing recommendations for survey design.

Experimental Validation of Speech Improvement-Based Stratified Adaptive Finite-Time Saturation Control of Omnidirectional Service Robot

To implement the human-robot interactions in a noisy environment, the speech improvement-based (SIB) stratified adaptive finite-time saturation control (SAFTSC) for omnidirectional service robot (OSR) is developed. From the outset, the feature vectors of nine designed speech commands are extracted from their frequency signals and then trained by multiclass support vector machine. Two background noises are on-line filtered with the characteristic: ``the smaller error in power spectrum is, the larger recovery from noisy power spectrum is.'' Comparisons among without or with noise, and filtering are addressed. To achieve the zero pose error of OSR in finite time, an adaptive finite-time indirect trajectory (AFTIT) is constructed. To track the AFTIT with the zero error in finite time, the adaptive finite-time saturation control (AFTSC) is also established. Both AFTIT and AFTSC possess nonlinear switching gain increasing the high-frequency motion capability to fulfill the classified speech command. Simply put, the proposed SIB stratified AFTSC includes the speech improvement for classification, the AFTIT, and the AFTSC. Besides the stability of the closed-loop system is verified by the Lyapunov stability theory, three categories of SIB experiments are compared.

Mitigating contamination in LSS surveys: a comparison of methods

ABSTRACT Future large-scale structure surveys will measure the locations and shapes of billions of galaxies. The precision of such catalogues will require meticulous treatment of systematic contamination of the observed fields. We compare several existing methods for removing such systematics from galaxy clustering measurements. We show how all the methods, including the popular pseudo-Cℓ Mode Projection and Template Subtraction methods, can be interpreted under a common regression framework and use this to suggest improved estimators. We show how methods designed to mitigate systematics in the power spectrum can be used to produce clean maps, which are necessary for cosmological analyses beyond the power spectrum, and we extend current methods to treat the next-order multiplicative contamination in observed maps and power spectra, which reduced power spectrum errors from $\Delta \chi ^2_{\rm C_\ell }\simeq 10$ to ≃ 1 in simulated analyses. Two new mitigation methods are proposed, which incorporate desirable features of current state-of-the-art methods while being simpler to implement. Investigating the performance of all the methods on a common set of simulated measurements from Year 5 of the Dark Energy Survey, we test their robustness to various analysis cases. Our proposed methods produce improved maps and power spectra when compared to current methods, while requiring almost no user tuning. We end with recommendations for systematics mitigation in future surveys, and note that the methods presented are generally applicable beyond the galaxy distribution to any field with spatial systematics.

Characterizing Signal Loss in the 21 cm Reionization Power Spectrum: A Revised Study of PAPER-64

Abstract The Epoch of Reionization (EoR) is an uncharted era in our universe’s history during which the birth of the first stars and galaxies led to the ionization of neutral hydrogen in the intergalactic medium. There are many experiments investigating the EoR by tracing the 21 cm line of neutral hydrogen. Because this signal is very faint and difficult to isolate, it is crucial to develop analysis techniques that maximize sensitivity and suppress contaminants in data. It is also imperative to understand the trade-offs between different analysis methods and their effects on power spectrum estimates. Specifically, with a statistical power spectrum detection in HERA’s foreseeable future, it has become increasingly important to understand how certain analysis choices can lead to the loss of the EoR signal. In this paper, we focus on signal loss associated with power spectrum estimation. We describe the origin of this loss using both toy models and data taken by the 64-element configuration of the Donald C. Backer Precision Array for Probing the Epoch of Reionization (PAPER). In particular, we highlight how detailed investigations of signal loss have led to a revised, higher 21 cm power spectrum upper limit from PAPER-64. Additionally, we summarize errors associated with power spectrum error estimation that were previously unaccounted for. We focus on a subset of PAPER-64 data in this paper; revised power spectrum limits from the PAPER experiment are presented in a forthcoming paper by Kolopanis et al. and supersede results from previously published PAPER analyses.

Power Spectrum Equalized Scalar Representation of Wide-Angle Optical Field Propagation

It may be desirable to represent optical fields using scalar approximations, due to its simplicity. Since the optical field is an electromagnetic wave, in order to implement an optical setup, a mapping from such a scalar field to the vector electromagnetic field is needed. In the conventional scalar-to-vector field mapping, a large error in power spectrum arises in wide-angle fields due to the neglected large longitudinal component of the electric field. This error could be severe in wide-angle or off-axis imaging setups. In order to find another scalar-to-vector field mapping that compensates for this large magnitude error, first, a general constraint on monochromatic electromagnetic fields to appropriately represent them by a scalar wave in free space is developed. The development of the general constraint begins by formulating the computations of the components of the magnetic field as the outputs of linear-shift invariant (LSI) systems, where the inputs to the LSI systems are the transverse components of the electric field. Furthermore, if one of the transverse components of the electric field can be computed from the other one using a LSI operation, a scalar field, which is related to the transverse components through another LSI system, can be used to fully describe the electromagnetic field. Under this constraint, the required condition on the filters which relates the scalar field to the electric field is presented by taking into consideration the longitudinal component, such that the power spectra of the scalar field and the corresponding electric field are equal. The filters are specified for the electric fields with zero longitudinal component and simple polarization features, as well. Moreover, for the electric fields with simple polarization features, some discrete simulations are performed to compare the scalar field intensity pattern and the corresponding electric field intensity patterns for the conventional and proposed mapping cases. The simulation results show that the excessive amplification of the large frequency components is compensated by the proposed filters, and hence, the undesired effects of the filters used in the computation of the longitudinal component disappear. In this respect, if equality of the power spectra of the scalar field and the corresponding electric field is of concern in an application, the proposed scalar-to-vector wave field mapping should be used.

Bayesian modal identification with particle filtering for sediment property inversion

Sequential Bayesian filtering methods have been previously used in dispersion curve tracking for long range sound propagation in the ocean. Modal frequency probability density functions were extracted for sound speed inversion. Here, we calculate modal arrival time densities, instead, and employ them for inversion for sediment sound speed and thickness and water column depth. Bayesian mode identification is performed to this end. We investigate two methods for describing the statistical errors in power spectra, which we use in the arrival time density calculation using normal modes. We then link these densities to the parameters of interest. The approaches are tested with synthetic data as well as data collected in the Gulf of Mexico. [Work supported by ONR.]

MAXIMUM LIKELIHOOD ANALYSIS OF SYSTEMATIC ERRORS IN INTERFEROMETRIC OBSERVATIONS OF THE COSMIC MICROWAVE BACKGROUND

We investigate the impact of instrumental systematic errors in interferometric measurements of the cosmic microwave background (CMB) temperature and polarization power spectra. We simulate interferometric CMB observations to generate mock visibilities and estimate power spectra using the statistically optimal maximum likelihood technique. We define a quadratic error measure to determine allowable levels of systematic error that do not induce power spectrum errors beyond a given tolerance. As an example, in this study we focus on differential pointing errors. The effects of other systematics can be simulated by this pipeline in a straightforward manner. We find that, in order to accurately recover the underlying B-modes for r=0.01 at 28<l<384, Gaussian-distributed pointing errors must be controlled to 0.7^\circ rms for an interferometer with an antenna configuration similar to QUBIC, in agreement with analytical estimates. Only the statistical uncertainty for 28<l<88 would be changed at ~10% level. With the same instrumental configuration, we find the pointing errors would slightly bias the 2-\sigma upper limit of the tensor-to-scalar ratio r by ~10%. We also show that the impact of pointing errors on the TB and EB measurements is negligibly small.

Time-frequency analysis with Bayesian filtering methods for dispersion tracking and geoacoustic inversion in the ocean

Iterative and sequential Bayesian filtering methods have been previously used in dispersion tracking for long range sound propagation in oceanic environments. The methods rely on accurate modeling of (i) the acoustic field in the frequency domain as a function of time and (ii) the statistical model governing the behavior of errors in the measured data. Normal distributions are typically used for the latter but may not necessarily be the most accurate models for the description of the data. We investigate alternative methods for describing the statistical errors in power spectra, which we then use for linking the extracted time-frequency information to geoacoustic inversion. Probability density functions computed for frequencies and arrival times via filtering are propagated “backwards” through sound propagation models for quantification of the uncertainty in the estimation process. [Work supported by ONR].

Modelling the viscoelasticity and thermal fluctuations of fluids at the nanoscale

Simulation methodologies for modelling the viscoelasticity and thermal fluctuations of molecular fluids at nanoscale are presented. In particular, we bridge the distinct frameworks of fluctuating hydrodynamics (FHD) and molecular dynamics (MD) simulations using a coarse-graining procedure that properly considers the effective size of each fluid molecule in mapping a phase space vector of the all-atom model to field variables in the FHD representation. We also generalise the FHD equations to model non-Markovian rheological responses by incorporating coloured noise. To capture the increasing elastic responses that emerge at the nanoscale, a composite Newtonian–Maxwell rheological model is developed. Numerical simulations using a staggered discretisation scheme demonstrate that the thermodynamic states and dynamic properties (power spectra) of FHD equations match with those of all-atom MD simulations quantitatively. This agreement also unambiguously determines the wavenumber-dependent transport coefficients of the composite Newtonian–Maxwell model. To avoid the complexities associated with using wavenumber-dependent transport coefficients, we characterise the approximation of using a single set of transport coefficients. We find that for collective fluctuations with a wavelength longer than 25 Å, wavenumber-independent models can accurately describe the power spectra. For fluctuations with shorter wavelengths, relative errors in power spectra increase monotonically with wavenumber.

Cosmic shear systematics: software-hardware balance

Cosmic shear measurements rely on our ability to measure and correct the Point Spread Function (PSF) of the observations. This PSF is measured using stars in the field, which give a noisy measure at random points in the field. Using Wiener filtering, we show how errors in this PSF correction process propagate into shear power spectrum errors. This allows us to test future space-based missions, such as Euclid or JDEM, thereby allowing us to set clear engineering specifications on PSF variability. For ground-based surveys, where the variability of the PSF is dominated by the environment, we briefly discuss how our approach can also be used to study the potential of mitigation techniques such as correlating galaxy shapes in different exposures. To illustrate our approach we show that for a Euclid-like survey to be statistics limited, an initial pre-correction PSF ellipticity power spectrum, with a power-law slope of -3 must have an amplitude at l =1000 of less than 2 x 10^{-13}. This is 1500 times smaller than the typical lensing signal at this scale. We also find that the power spectrum of PSF size \dR^2) at this scale must be below 2 x 10^{-12}. Public code available as part of iCosmo at http://www.icosmo.org

Cosmological parameters from WMAP 5-year temperature maps

I calculate a hybrid cross-power spectrum estimator from the WMAP 5-year CMB temperature maps, discuss the goodness of fit, and then constrain cosmological parameters. The spectrum and results are generally consistent with previous results, though the power spectrum error bars are slightly smaller and there are small shifts at high ell. The small improvement in error bars is obtained at very low numerical cost but does not significantly improve parameter constraints. I discuss the accuracy of the likelihood model and how constraints on the optical depth translate into constraints on the reionization history allowing for helium reionization. In the appendices I propose a simple reionization parameterization that determines the history in terms of a mid-point reionization redshift, and suggest a new likelihood approximation for chi-squared-like distributions with varying skewness.

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.

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.

Inhomogeneous systematic signals in cosmic shear observations

We calculate the systematic errors in the weak gravitational lensing power spectrum which would be caused by spatially varying calibration (i.e. multiplicative) errors, such as might arise from uncorrected seeing or extinction variations. The systematic error is fully described by the angular two-point correlation function of the systematic in the case of the 2D lensing that we consider here. We investigate three specific cases: Gaussian, 'patchy', and exponential correlation functions. In order to keep systematic errors below statistical errors in future LSST-like surveys, the spatial variation of calibration should not exceed 3% rms. This conclusion is independently true for all forms of correlation function we consider. The relative size of the E- and B-mode power spectrum errors does, however, depend upon the form of the correlation function, indicating that one cannot repair the E-mode power spectrum systematics by means of the B-mode measurements.

CMB power spectrum estimation via hierarchical decomposition

We have developed a fast, accurate and generally applicable method for inferring the power spectrum and its uncertainties from maps of the cosmic microwave background (CMB) in the presence of inhomogeneous and correlated noise. For maps with 10 to 100 thousand pixels, we apply an exact power spectrum estimation algorithm to submaps of the data at various resolutions, and then combine the results in an optimal manner. To analyze larger maps efficiently one must resort to sub-optimal combinations in which cross-map power spectrum error correlations are only calculated approximately. We expect such approximations to work well in general, and in particular for the megapixel maps to come from the next generation of satellite missions.

107. 空間周波数領域に於ける X 線写真の画像解析と処理 : (第 5 報) 画像の周期断裂に起因するスペクトル誤差が周波数フィルター処理画像に及ぼす影響(第 46 回総会学術大会会員研究発表予稿)

107. 空間周波数領域に於ける X 線写真の画像解析と処理 : (第 5 報) 画像の周期断裂に起因するスペクトル誤差が周波数フィルター処理画像に及ぼす影響(第 46 回総会学術大会会員研究発表予稿)

107. 空間周波数領域に於ける X 線写真の画像解析と処理 : (第 5 報) 画像の周期断裂に起因するスペクトル誤差が周波数フィルター処理画像に及ぼす影響(CR-2 周波数処理)

107. 空間周波数領域に於ける X 線写真の画像解析と処理 : (第 5 報) 画像の周期断裂に起因するスペクトル誤差が周波数フィルター処理画像に及ぼす影響(CR-2 周波数処理)

Effects of binary quantisation when using the fast Fourier transform to compute power spectral estimates

The effects of severely quantising the multiplication coefficients in the fast Fourier transform have been studied. Rounding these off to the nearest power of 2 produces power-spectrum errors of as little as 0.08% (maximum) for a pure sinewave sampled at 64 points.

Errors in Power Spectra Due to Finite Sample

An estimate is made of the accuracy with which the ``spectrum'' of a finite sample of Gaussian noise approximates the spectrum of the original noise from which the sample was taken. The sample is considered to be spliced to form an endless loop, thereby making it a periodic function. The effect of this splice is also discussed.