Stationary Noise Process Research Articles

Resonance fluorescence of noisy systems

Light scattering from resonantly or nearly resonantly excited systems, known as resonance fluorescence (RF), has been gaining importance as a versatile tool for investigating quantum states of matter and readout of quantum information, recently including also the inherently noisy solid state systems. In this work we develop a general theory of RF in the low excitation limit on systems in which the transition energy is subject to noise for two important classes of noise processes: white noise fluctuations that lead to phase diffusion and an arbitrary stationary Markovian noise process on a finite set of states. We apply the latter to the case of random telegraph noise (TN) and a sum of an arbitrary number of identical random TN contributions. We show that different classes of noise influence the RF spectrum in a characteristic way. Hence, the spectrum carries information on the characteristics of noise present in the physical system.

Joint non-parametric estimation of mean and auto-covariances for Gaussian processes

Gaussian processes that can be decomposed into a smooth mean function and a stationary autocorrelated noise process are considered and a fully automatic nonparametric method to simultaneous estimation of mean and auto-covariance functions of such processes is developed. The proposed empirical Bayes approach is data-driven, numerically efficient, and allows for the construction of confidence sets for the mean function. Performance is demonstrated in simulations and real data analysis. The method is implemented in the R package eBsc.1

A search for planetary companions around 800 pulsars from the Jodrell Bank pulsar timing programme

ABSTRACT We have searched for planetary companions around 800 pulsars monitored at the Jodrell Bank Observatory, with both circular and eccentric orbits of periods between 20 d and 17 yr and inclination-dependent planetary masses from 10−4 to $100\, \mathrm{M}_{\oplus }$. Using a Bayesian framework, we simultaneously model pulsar timing parameters and a stationary noise process with a power-law power spectral density. We put limits on the projected masses of any planetary companions, which reach as low as 1/100th of the mass of the Moon ($\sim 10^{-4}\, \mathrm{M}_{\oplus }$). We find that two-thirds of our pulsars are highly unlikely to host any companions above $2-8\, \mathrm{M}_{\oplus }$. Our results imply that fewer than $0.5{{\ \rm per\ cent}}$ of pulsars could host terrestrial planets as large as those known to orbit PSR B1257+12 ($\sim 4\, \mathrm{M}_{\oplus }$); however, the smaller planet in this system ($\sim 0.02\, \mathrm{M}_{\oplus }$) would be undetectable in $95{{\ \rm per\ cent}}$ of our sample, hidden by both instrumental and intrinsic noise processes, although it is not clear whether such tiny planets could exist in isolation. We detect significant periodicities in 15 pulsars; however, we find that intrinsic quasi-periodic magnetospheric effects can mimic the influence of a planet, and for the majority of these cases we believe this to be the origin of the detected periodicity. Notably, we find that the highly periodic oscillations in PSR B0144+59 are correlated with changes in the pulse profile and therefore can be attributed to magnetospheric effects. We believe the most plausible candidate for planetary companions in our sample is PSR J2007+3120.

Time-frequency analysis of gravitational wave data

Data from gravitational wave detectors are recorded as time series that include contributions from myriad noise sources in addition to any gravitational wave signals. When regularly sampled data are available, such as for ground based and future space based interferometers, analyses are typically performed in the frequency domain, where stationary (time invariant) noise processes can be modeled very efficiently. In reality, detector noise is not stationary due to a combination of short duration noise transients and longer duration drifts in the power spectrum. This non-stationarity produces correlations across samples at different frequencies, obviating the main advantage of a frequency domain analysis. Here an alternative time-frequency approach to gravitational wave data analysis is proposed that uses discrete, orthogonal wavelet wavepackets. The time domain data is mapped onto a uniform grid of time-frequency pixels. For locally stationary noise - that is, noise with an adiabatically varying spectrum - the time-frequency pixels are uncorrelated, which greatly simplifies the calculation of quantities such as the likelihood. Moreover, the gravitational wave signals from binary systems can be compactly represented as a collection of lines in time-frequency space, resulting in a computational cost for computing waveforms and likelihoods that scales as the square root of the number of time samples, as opposed to the linear scaling for time or frequency based analyses. Key to this approach is having fast methods for computing binary signals directly in the wavelet domain. Multiple fast transform methods are developed in detail.

Capacity of Two-Way Channels With Symmetry Properties

In this paper, we make use of channel symmetry properties to determine the capacity region of three types of two-way networks: 1) two-user memoryless two-way channels (TWCs); 2) two-user TWCs with memory; and 3) three-user multiaccess/degraded broadcast (MA/DB) TWCs. For each network, symmetry conditions under which a Shannon-type random coding inner bound (under independent non-adaptive inputs) is tight are given. For two-user memoryless TWCs, prior results are substantially generalized by viewing a TWC as two interacting state-dependent one-way channels. The capacity of symmetric TWCs with memory, whose outputs are functions of the inputs and independent stationary and ergodic noise processes, is also obtained. Moreover, various channel symmetry properties under which the Shannon-type inner bound is tight are identified for three-user MA/DB TWCs. The results not only enlarge the class of symmetric TWCs whose capacity region can be exactly determined but also imply that interactive adaptive coding, not improving capacity, is unnecessary for such channels.

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic processis considered.Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the processes are not known exactly while some sets of admissible spectral densities of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral densities.

Discrete Multi-Tone Digital Subscriber Loop Performance in the Face of Impulsive Noise

As an important solution to “the last mile” access, digital subscriber loops (DSLs) are still maintained in a huge plant to support low-cost but high-quality broadband network access through telephone lines. The discrete multi-tone (DMT) transmissions constitute a baseband version of the ubiquitous orthogonal frequency division multiplexing. While the DMT is ideally suited to deal with the frequency selective channel in DSL, the presence of bursty impulsive noise tends to severely degrade the transmission performance. In this paper, we analyze the statistics of impulsive noise and its effects on the received signals, with the aid of a hidden semi-Markov process. The closed-form bit error rate expression is derived for the DMT system for $Q$ -ary quadrature amplitude modulation under practical noise conditions and for measured dispersive DSL channels. Instead of relying on the simplified stationary and impulsive noise process, our noise model considers both the temporal and spectral characteristics based on the measurement results. The simulation results confirm the accuracy of the formulas derived and quantify the impact both of the impulsive noise and of the dispersive channel in DSL.

Fifteen years ofXMM–NewtonandChandramonitoring of Sgr A★: evidence for a recent increase in the bright flaring rate

We present a study of the X-ray flaring activity of Sgr A* during all the 150 XMM-Newton and Chandra observations pointed at the Milky Way center over the last 15 years. This includes the latest XMM-Newton and Chandra campaigns devoted to monitoring the closest approach of the very red Br-Gamma emitting object called G2. The entire dataset analysed extends from September 1999 through November 2014. We employed a Bayesian block analysis to investigate any possible variations in the characteristics (frequency, energetics, peak intensity, duration) of the flaring events that Sgr A* has exhibited since their discovery in 2001. We observe that the total bright-or-very bright flare luminosity of Sgr A* increased between 2013-2014 by a factor of 2-3 (~3.5 sigma significance). We also observe an increase (~99.9% significance) from 0.27+-0.04 to 2.5+-1.0 day^-1 of the bright-or-very bright flaring rate of Sgr A*, starting in late summer 2014, which happens to be about six months after G2's peri-center passage. This might indicate that clustering is a general property of bright flares and that it is associated with a stationary noise process producing flares not uniformly distributed in time (similar to what is observed in other quiescent black holes). If so, the variation in flaring properties would be revealed only now because of the increased monitoring frequency. Alternatively, this may be the first sign of an excess accretion activity induced by the close passage of G2. More observations are necessary to distinguish between these two hypotheses.

Analyzing long-term correlated stochastic processes by means of recurrence networks: potentials and pitfalls.

Long-range correlated processes are ubiquitous, ranging from climate variables to financial time series. One paradigmatic example for such processes is fractional Brownian motion (fBm). In this work, we highlight the potentials and conceptual as well as practical limitations when applying the recently proposed recurrence network (RN) approach to fBm and related stochastic processes. In particular, we demonstrate that the results of a previous application of RN analysis to fBm [Liu et al. Phys. Rev. E 89, 032814 (2014)] are mainly due to an inappropriate treatment disregarding the intrinsic nonstationarity of such processes. Complementarily, we analyze some RN properties of the closely related stationary fractional Gaussian noise (fGn) processes and find that the resulting network properties are well-defined and behave as one would expect from basic conceptual considerations. Our results demonstrate that RN analysis can indeed provide meaningful results for stationary stochastic processes, given a proper selection of its intrinsic methodological parameters, whereas it is prone to fail to uniquely retrieve RN properties for nonstationary stochastic processes like fBm.

Low-rank approximations for large stationary covariance matrices, as used in the Bayesian and generalized-least-squares analysis of pulsar-timing data

Many data-analysis problems involve large dense matrices that describe the covariance of stationary noise processes; the computational cost of inverting these matrices, or equivalently of solving linear systems that contain them, is often a practical limit for the analysis. We describe two general, practical, and accurate methods to approximate stationary covariance matrices as low-rank matrix products featuring carefully chosen spectral components. These methods can be used to greatly accelerate data-analysis methods in many contexts, such as the Bayesian and generalized-least-squares analysis of pulsar-timing residuals.

Comments on: Extensions of some classical methods in change point analysis

First of all, I would like to compliment the authors for writing this very fine survey paper on some recent developments in change point analysis. There is a lot of current research activity devoted to change point analysis, and many articles have appeared since the publication of the seminal book by Csorgő and Horvath (1997). Among the topics covered in the present paper are empirical process techniques, Darling-Erdős laws, changes in correlations, changes in regression parameters, sequential testing, panel models, and functional data. Horvath and Rice provide an excellent survey of some of the recent results, which will be helpful to all researchers in this area. In my discussion, I will complement the paper by Horvath and Rice by presenting some recent results on U-statistics based robust change-point tests for time series. In a series of papers, see Dehling and Fried (2012), Dehling, Fried, Garcia and Wendler (2013), Dehling, Rooch and Taqqu (2013a, 2013b), and Betken (2014), we have investigated such tests, and derived their asymptotic distribution, both in the case of short range as well as long range dependent time series. We consider a model where the data are generated by Xi = μi + i, where μi is an unknown signal, and where i is a stationary ergodic noise process with E( i) = 0. Given the observations X1, . . . , Xn, we wish to test the hypothesis of no change, i.e. H : μ1 = . . . = μn, against the alternative

Error-Covariance Analysis of the Total Least-Squares Problem

This paper derives and analyzes the estimate error covariances associated with both the nonstationary and stationary noise process cases with uncorrelated elementwise components for the total least-squares problem. The nonstationary case is derived directly from the associated unconstrained total least-squares loss function. The stationary case is derived by using a linear expansion of the total least-squares estimate equation, which involves a first-order expansion of the associated singular value decomposition matrices. The actual solution for the error covariance is evaluated at the true variables, which are unknown in practice. Two common approaches to overcome this difficulty are used; the first involves using the measurements directly and the second involves using the estimates, which are more accurate than the measurements. This paper shows that using the latter greatly simplifies the error-covariance solution for the stationary case. Simulation results using bearings-only point estimation are shown to quantify the theoretical derivations.

Properties of Digital Switching Currents in Fully CMOS Combinational Logic

In this paper, we present a model to derive statistical properties of digital noise due to logic transitions of gates in a fully CMOS combinational circuit. Switching activity of logic gates in a digital system is a deterministic process, depending on both circuit parameters and input signals. However, the huge number of logic blocks in a complex IC makes digital switching a cognitively stochastic process. For a combinational logic network, we can model digital switching currents as stationary shot noise processes, deriving both their amplitude distributions and their power spectral densities. From the spectra of digital currents, we can also calculate the spectral components and the rms value of disturbances injected into the on-chip power supply lines. The stochastic model for switching currents has been validated by comparing theoretical results with circuit simulations.

A Pure-Jump Transaction-Level Price Model Yielding Cointegration

We propose a new transaction-level bivariate log-price model that yields fractional or standard cointegration. The model provides a link between market microstructure and lower-frequency observations. The two ingredients of our model are a long-memory stochastic duration process for the waiting times, {τk}, between trades and a pair of stationary noise processes, ({ek} and {ηk}), which determine the jump sizes in the pure-jump log-price process. Our model includes feedback between the disturbances of the two log-price series at the transaction level, which induces standard or fractional cointegration for any fixed sampling interval Δt. We prove that the cointegrating parameter can be consistently estimated by the ordinary least squares estimator, and we obtain a lower bound on the rate of convergence. We propose transaction-level method-of-moments estimators of the other parameters in our model and discuss the consistency of these estimators.

Exploring the qualitative behavior of uncertain dynamical systems

The global behavior of uncertain nonlinear systems by means of the cell mapping method is investigated. The systems under consideration are governed by ordinary differential equations where uncertainty is characterized by a bounded stationary noise process. In order to make cell mapping methods applicable to this particular situation generalizations of the theory become necessary. The numerical approach is based on a reformulation of the dynamics in terms of a Markov process whose qualitative behavior is analyzed. We demonstrate the usefulness of the presented scheme in the engineering field while addressing an example problem from technical mechanics.

DOA Estimation of Quasi-Stationary Signals With Less Sensors Than Sources and Unknown Spatial Noise Covariance: A Khatri–Rao Subspace Approach

In real-world applications such as those for speech and audio, there are signals that are nonstationary but can be modeled as being stationary within local time frames. Such signals are generally called quasi-stationary or locally stationary signals. This paper considers the problem of direction-of-arrival (DOA) estimation of quasi-stationary signals. Specifically, in our problem formulation we assume: i) sensor array of uniform linear structure; ii) mutually uncorrelated wide-sense quasi-stationary source signals; and iii) wide-sense stationary noise process with unknown, possibly nonwhite, spatial covariance. Under the assumptions above and by judiciously examining the structures of local second-order statistics (SOSs), we develop a Khatri-Rao (KR) subspace approach that has two notable advantages. First, through an identifiability analysis, it is proven that this KR subspace approach can operate even when the number of sensors is about half of the number of sources. The idea behind is to make use of a ¿virtual¿ array structure provided inherently in the local SOS model, of which the degree of freedom is about twice of that of the physical array. Second, the KR formulation naturally provides a simple yet effective way of eliminating the unknown spatial noise covariance from the signal SOSs. Extensive simulation results are provided to demonstrate the effectiveness of the KR subspace approach under various situations.

Coherent network analysis technique for discriminating gravitational-wave bursts from instrumental noise

The sensitivity of current searches for gravitational-wave bursts is limited by non-Gaussian, nonstationary noise transients which are common in real detectors. Existing techniques for detecting gravitational-wave bursts assume the output of the detector network to be the sum of a stationary Gaussian noise process and a gravitational-wave signal. These techniques often fail in the presence of noise nonstationarities by incorrectly identifying such transients as possible gravitational-wave bursts. Furthermore, consistency tests currently used to try to eliminate these noise transients are not applicable to general networks of detectors with different orientations and noise spectra. In order to address this problem we introduce a fully coherent consistency test that is robust against noise nonstationarities and allows one to distinguish between gravitational-wave bursts and noise transients in general detector networks. This technique does not require any a priori knowledge of the putative burst waveform.

Use of prewhitening in climate regime shift detection

Time series of observations generated by a stationary red noise process are characterized by long intervals when the observations remain above or below the overall mean value. These intervals can be easily misinterpreted as “climatic regimes” with different statistics. A “prewhitening” procedure that removes the red noise component from the time series prior to an application of a regime shift detection technique is discussed. The key elements of this procedure are subsampling and bias correction of the least squares estimate of the serial correlation. A new technique to obtain a bias‐corrected estimate of the autoregressive parameter is proposed. It is shown that the Pacific Decadal Oscillation (PDO) appears to be more than just a manifestation of a red noise process.

Assessment of long-range correlation in time series: How to avoid pitfalls

Due to the ubiquity of time series with long-range correlation in many areas of science and engineering, analysis and modeling of such data is an important problem. While the field seems to be mature, three major issues have not been satisfactorily resolved. (i) Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? (ii) The mathematical theory of long-range correlation concerns the behavior of the correlation of the time series for very large times. A measured time series is finite, however. How can we relate the fractal scaling break at a specific time scale to important parameters of the data? (iii) An important technique in assessing long-range correlation in a time series is to construct a random walk process from the data, under the assumption that the data are like a stationary noise process. Due to the difficulty in determining whether a time series is stationary or not, however, one cannot be 100% sure whether the data should be treated as a noise or a random walk process. Is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process, and vice versa? In this paper, we seek to gain important insights into these issues by examining three model systems, the autoregressive process of order 1, on-off intermittency, and Lévy motions, and considering an important engineering problem, target detection within sea-clutter radar returns. We also provide a few rules of thumb to safeguard against misinterpretations of long-range correlation in a time series, and discuss relevance of this study to pattern recognition.

On the nonlinear modeling of shot noise

We introduced a nonlinear shot-noise model, a natural generalization of the "classic" shot-noise model, which differs markedly from the existing linear shot-noise models. This model produces a wide spectrum of stationary noise processes. Because of its intrinsic nonlinearity, the model's resulting noise processes are capable of displaying a rich variety of both amplitudal and temporal statistical behaviors. Surprisingly, the nonlinear model is amenable to mathematical analysis and yields closed-form formulae for the characterizing statistics of its resulting noise processes.