Power-law Noise Research Articles

Investigation of noises in the EPN weekly time series

The constantly growing needs of permanent stations’ velocities users cause their stability level to increase. To this research we included more than 150 stations located across Europe operating within the EUREF Permanent Network (EPN) with weekly changes in the ITRF2005 reference frame. The obvious long-range dependencies in the stochastic part of GPS time series were proven by Ljung-Box test. Moreover, the results of Detrended Fluctuation Analysis (DFA) comprised in 90 % in the long-time range dependency, indicated an autocorrelation of the signal with a possible coloured noise, similar to the flicker one. The noise in the GPS time series was analysed here with the Maximum Likelihood Estimation (MLE). The amplitudes of white noise for the white, flicker and random-walk combination vary between 0.5 to 3 mm for horizontal components and between 1.5 up to 9 mm for the vertical component. The amplitudes of white and flicker noise show a clear geographical latitude dependence with a correlation with its minimum centred at approximately 50°N. The assumption of white plus flicker plus randomwalk noise led to stations’ velocity errors from 0 to 3 mm/year for the flicker noise and from 0 up to 7 mm/year for the random-walk. The spectral indices for an approach assuming the occurrence of the white and power-law noise model in weekly time series varied between –1.7 to –0.2 and –1.4 to –0.2 for horizontal and vertical changes. The performed spatial filtering (weighted stacking method) resulted in decreasing the power-law amplitudes by averagely 1-2 mm/year -κ/4 and moving the spectral indices closer to zero, with amplitudes and indices median decreased by 5 %, compared to the time series before stacking. ARTICLE INFO

Power colours: simple X-ray binary variability comparison

We demonstrate a new method of variability classification using observations of black hole X-ray binaries. Using `power colours' -- ratios of integrated power in different Fourier frequency bands -- we can clearly differentiate different canonical black hole states as the objects evolve during outburst. We analyse (~ 2400) Rossi X-ray Timing Explorer observations of 12 transient low mass black hole X-ray binaries and find that the path taken around the power colour-colour diagram as the sources evolve is highly consistent from object to object. We discuss how the consistency observed in the power colour-colour diagram between different objects allows for easy state classification based on only a few observations, and show how the power-spectral shapes can be simply classified using a single parameter, the power-spectral `hue'. To illustrate the benefits of our simple model-independent approach, we show that the persistent high mass X-ray binary Cyg X-1 shows very similar power-spectral evolution to the transient black hole sources, with the main difference being caused by a combination of a lack of quasi-periodic oscillations and an excess of low-frequency power-law noise in the Cyg X-1 power spectra during the transitional state. We also compare the transient objects to the neutron star atoll source Aquila X-1, demonstrating that it traces a different path in the power colour-colour plot. Thus, power-colours could be an effective method to classify newly discovered X-ray binaries.

Power-Law Noises over General Spatial Domains and on Nonstandard Meshes

Power-law noises abound in nature and have been observed extensively in both time series and spatially varying environmental parameters. Although recent years have seen the extension of traditional stochastic partial differential equations to include systems driven by fractional Brownian motion, spatially distributed scale-invariance has received comparatively little attention, especially for parameters defined over nonstandard spatial domains. This paper discusses the extension of power-law noises to general spatial domains by outlining their theoretical underpinnings as well as addressing their numerical simulation on arbitrary meshes. Three computational algorithms are presented for efficiently generating their sample paths, accompanied by numerous numerical illustrations.

Probability distributions and confidence intervals for simulated power law noise.

A method for simulating power law noise in clocks and oscillators is presented based on modification of the spectrum of white phase noise, then Fourier transforming to the time domain. Symmetric real matrices are introduced whose traces-the sums of their eigenvalues-are equal to the Allan variances, in overlapping or non-overlapping forms, as well as for the corresponding forms of the modified Allan variance. We show that the standard expressions for spectral densities, and their relations to Allan variance, are obtained with this method. The matrix eigenvalues determine probability distributions for observing a variance at an arbitrary value of the sampling interval τ, and hence for estimating confidence in the measurements. Examples are presented for the common power-law noises. Extension to other variances such as the Hadamard variance, and variances with dead time, are discussed.

Effect of the span of Australian GPS coordinate time series in establishing an optimal noise model

The span of coordinate time series affects the determination of an optimal noise model. We analyzed position data recorded for 10 continuous Global Positioning System (GPS) sites from 1998.0 to mid-2009 on the Australian Plate to estimate the best noise model and thereafter obtain the true uncertainties of the velocity, employing the maximum likelihood estimation (MLE) method. MLE was employed to analyze the data in four ways. In the first two analyses, the noise was assumed to be a combination of flicker noise and white noise for the raw time series and spatially filtered time series. In the final two analyses, the spectral indices and amplitudes were simultaneously estimated for a power law noise plus white noise model for the raw time series and spatially filtered time series. We conclude that the noise model of GPS time series in Australia can be best described as the combination of flicker noise and white noise. Velocity uncertainties fall below ∼0.2 mm/yr when the time span exceeds ∼9.5 years. A comparison of noise amplitudes and maximum likelihood estimation values between the raw and spatially filtered time series suggests that traditional spatial filtering to remove common-mode errors might not be applicable to the raw time series of this region.

Irregular variations in GPS time series by probability and noise analysis

The character of the topocentric components in ETRF2000(R08) from the Polish ASG-EUPOS system was analysed using skewness and kurtosis derived from the data probability density function. The data from 115 permanent GPS stations with a time span of more than 5 years were used. The main goal of this research was to show that any unmodelled systematics can disrupt the results. The obtained median values of skewness and kurtosis clearly indicate the discrepancies between the assumed normality of the GPS time series distribution and the reality, mainly due to stochastic and/or deterministic parts that are still present in the data. The quadratic relationship between skewness and kurtosis was developed with the empirically determined constants. The noise analysis with the Maximum Likelihood Estimation was also performed with the assumptions of white and white plus power-law noise. The estimated spectral indices for power law noises are close to −1 (flicker noise). The uncertainties of the intraplate velocities with white and white plus power law noise assumptions were calculated. It was shown that these uncertainties can be underestimated up to 5 mm/year. Additionally we made 1000 simulation which were aimed at showing how the values of skewness and kurtosis are changed when some mismodelled part of the data remains in the time series. We assumed different values of trend (1 and 6 mm/year), seasonal amplitudes (1 and 6 mm) and offsets (3 and 10 mm) with flicker noise of 1 mm amplitude to see how the metrics we use are biased.

Velocity autocorrelation of a free particle driven by a Mittag-Leffler noise: fractional dynamics and temporal behaviors.

We investigate the dynamical phase diagram of the generalized Langevin equation of the free particle driven by a Mittag-Leffler noise and show critical curves and a critical value of the exponent parameter of the Mittag-Leffler function that mark different dynamical regimes. By considering that the modeling of a Mittag-Leffer memory kernel corresponds to a power-law second-order memory kernel, we show that the generalized Langevin equation of the velocity autocorrelation function (VACF) is transformed in a fractional Langevin equation. In the superdiffusive case our results exhibit oscillations and negative correlations of the VACF that are not provided by the usual power-law noise model.

Optimizing passive quantum clocks

We describe protocols for passive atomic clocks based on quantum interrogation of the atoms. Unlike previous techniques, our protocols are adaptive and take advantage of prior information about the clock's state. To reduce deviations from an ideal clock, each interrogation is optimized by means of a semidefinite program for atomic state preparation and measurement whose objective function depends on the prior information. Our knowledge of the clock's state is maintained according to a Bayesian model that accounts for noise and measurement results. We implement a full simulation of a running clock with power-law noise models and find significant improvements by applying our techniques.

Computational assessment of stereoscopic viewing a sequence of stereo pairs of breast tomosynthesis projection images.

Digital breast tomosynthesis (DBT) is a 3D imaging technology in which an x-ray fan beam rotates around the breast, producing a series of projection images. The imaging geometry of DBT lends itself naturally to stereo viewing because a stereo pair can be easily formed by two projection images with a reasonable separation angle. Stereo viewing may reveal the 3D structures of breasts thus has the potential to increase the sensitivity and specificity of breast imaging. In this study, we conduct a simulation study that mimics the detection of breast lesions on stereoscopic viewing of DBT projections. The presentation approach we investigate here is one in which the reader is presented with a sequence of stereo pairs from a rotating point of view. We render voxel datasets that contain random 3D power-law noise to model normal breast tissues with different breast densities. A 3D Gaussian signal is inserted to some of the datasets to model the presence of a breast lesion. Sequences of stereo pairs of projection images are generated for each voxel dataset by varying the projection angles of the two views. The diagnostic performance, in terms of the accuracy of binary decisions on the presence of the simulated lesions, is evaluated with a numerical model observer.

Uncertainties of geodetic velocities from permanent GPS observations:The Sudeten case study

The assumption that the GPS-derived coordinate time series behave like the white noise leads to the underestimation of the velocity uncertainties. The permanent stations velocities are required nowadays with the stability at the level of 0.1 mm/year in case of using them to the kinematic reference frame realization or numerous geodynamic interpretations. Therefore, it is required to know which kind of noise models describes the time series of geodetic coordinates better. These noises can be the effect of e.g. mismodelled satellite orbits or improperly mounted antennae. In the following paper, three combinations of noise models were assumed: the white noise model only, to show how the omitting of correlations in the stochastic part of the GPS time series underestimates the velocity uncertainties; the combination of the white plus flicker plus randomwalk noise to estimate which of the power-law noises with the known spectral index prevails in the GPS time series; and the combination of the white plus power-law noise to estimate which coloured noise fits the stochastic part the best. We analyzed more than 40 stations (belonging to the ASG-EUPOS and EPN networks) with 5 years of observations from the area of Sudeten. The results show that the amplitudes of the white noise for the assumption of white noise only model range from 0.6 to 2 mm for the horizontal (North and East) components and from 2 to 8 mm for the vertical (Up) component, while the velocity uncertainties range from 0.01 to 0.13 mm/year for the horizontal velocities and from 0.04 to 0.24 for the vertical one. For the combination of the white, flicker and random-walk noise, the white noise is at the same level in the entire analyzed area, while the flicker noise prevails over other kinds of noise for all components. For the North and East its amplitude ranges from 2 to 5 mm*yr –0.25 , while for the U component it is 2-3 times larger. Here, the uncertainty of permanent stations velocity ranges from 0.1 mm/year for the horizontal components to even 1.1 mm/year for the vertical one, which is over 10 times larger than the assumed level of stability of permanent stations velocity. For the last combination, the obtained spectral indices range from –1.6 to –0.4 for the horizontal components and from –1.0 to –0.4 for the vertical component, which confirms the prevalence of fractional white and Brownian motion quite close to flicker noise with amplitudes of power-law noise of 25 mm*yr κ/4 and 4-12 mm*yr

Detection of distinct power spectra in soft and hard X-ray bands in the hard state of GRS 1915+105★

The well-known black hole X-ray binary GRS 1915+105 is a unique source in the sense that it cannot be classified within the standard picture of black hole binary states. In this work we study archival XMM-Newton observations taken between 2003 and 2004 of the \c{hi} variability class of GRS 1915+105, which corresponds to the hard state in the standard black hole X-ray binary state classification. The crucial point of our study is that by using XMM-Newton data we can access the variability below 3 keV, an energy range that is not covered with RXTE. We focus on the study of the power spectral shape in the soft and hard X-ray band, in light of our work done with Swift on MAXI J1659-152. In the hard band (above 2.5 keV) power density spectra consist of band-limited noise and quasi-periodic oscillations, corresponding to the power spectral shape seen in the hard or intermediate state, while in the soft band the averaged power density spectrum is consistent with a power-law noise, corresponding to the power spectral shape usually seen in the soft state. The coexisting of two different power spectral shapes in the soft and hard band, where the soft band power spectrum is dominated by a power-law noise, is consistent with MAXI J1659-152, and confirms the energy dependence of power spectral states. Our additional spectral analysis shows that the disc component does contribute to the soft band flux. These findings support that the observed black hole power spectral state depends on which spectral component we are looking at, which implies that power spectral analysis is probably a more sensitive method than spectral modeling to trace the emergence of the disc component in the hard or intermediate state.

Observer efficiency in free-localization tasks with correlated noise.

The efficiency of visual tasks involving localization has traditionally been evaluated using forced choice experiments that capitalize on independence across locations to simplify the performance of the ideal observer. However, developments in ideal observer analysis have shown how an ideal observer can be defined for free-localization tasks, where a target can appear anywhere in a defined search region and subjects respond by localizing the target. Since these tasks are representative of many real-world search tasks, it is of interest to evaluate the efficiency of observer performance in them. The central question of this work is whether humans are able to effectively use the information in a free-localization task relative to a similar task where target location is fixed. We use a yes-no detection task at a cued location as the reference for this comparison. Each of the tasks is evaluated using a Gaussian target profile embedded in four different Gaussian noise backgrounds having power-law noise power spectra with exponents ranging from 0 to 3. The free localization task had a square 6.7° search region. We report on two follow-up studies investigating efficiency in a detect-and-localize task, and the effect of processing the white-noise backgrounds. In the fixed-location detection task, we find average observer efficiency ranges from 35 to 59% for the different noise backgrounds. Observer efficiency improves dramatically in the tasks involving localization, ranging from 63 to 82% in the forced localization tasks and from 78 to 92% in the detect-and- localize tasks. Performance in white noise, the lowest efficiency condition, was improved by filtering to give them a power-law exponent of 2. Classification images, used to examine spatial frequency weights for the tasks, show better tuning to ideal weights in the free-localization tasks. The high absolute levels of efficiency suggest that observers are well-adapted to free-localization tasks.

RADIO VARIABILITY AND RANDOM WALK NOISE PROPERTIES OF FOUR BLAZARS

We present the results of a time series analysis of the long-term radio lightcurves of four blazars: 3C 279, 3C 345, 3C 446, and BL Lacertae. We exploit the data base of the University of Michigan Radio Astronomy Observatory (UMRAO) monitoring program which provides densely sampled lightcurves spanning 32 years in time in three frequency bands located at 4.8, 8, and 14.5 GHz. Our sources show mostly flat or inverted (spectral indices -0.5 < alpha < 0) spectra, in agreement with optically thick emission. All lightcurves show strong variability on all time scales. Analyzing the time lags between the lightcurves from different frequency bands, we find that we can distinguish high-peaking flares and low-peaking flares in accord with the classification of Valtaoja et al. (1992). The periodograms (temporal power spectra) of the observed lightcurves are consistent with random-walk powerlaw noise without any indication of (quasi-)periodic variability. The fact that all four sources studied are in agreement with being random-walk noise emitters at radio wavelengths suggests that such behavior is a general property of blazars.

Langevin equation for a free particle driven by power law type of noises

We study a generalized Langevin equation for a free particle driven by N internal noises. The mean square displacement and velocity autocorrelation function are derived in case of a mixture of Dirac delta, power law and Mittag-Leffler noises. Additionally, a frictional memory kernel of distributed order is considered. The long time limit and short time limit are analyzed, and the dominant contributions of noises on particle dynamics is discussed. Various different diffusive behaviors (subdiffusion, superdiffusion, normal diffusion, ultraslow diffusion) are obtained. The considered problem may be used in the theory of anomalous diffusion in complex environment.

The biphase explained: understanding the asymmetries in coupled Fourier components of astronomical time series

We make the first attempt to estimate and interpret the biphase data for astronomical time series. The biphase is the phase of the bispectrum, which is the Fourier domain equivalent of the three-point correlation function. The bispectrum measures two key nonlinear properties of a time series -- its reversability in time, and the symmetry about the mean of its flux distribution -- for triplets of frequencies. Like other Fourier methods, it is especially valuable for working with time series which contain large numbers of cycles at the period of interest, but in which the signal-to-noise at a given frequency is small in any individual cycle, either because of measurement errors, or because of the contributions from signals at other frequencies. This has long been the case for studies of X-ray binaries, but is increasingly becoming true for stellar variability (both intrinsic and due to planetary transits) in the Kepler era. We attempt in this paper also to present some simple examples to give a more intuitive understanding of the meaning of the bispectrum to readers, in order to help to understand where it may be applicable in astronomy. In particular, we give illustrative examples of what biphases may be shown by common astrophysical time series such as pulsars, eclipsers, stars in the instability strip, and solar flares. We then discuss applications to the biphase data for understanding the shapes of the quasi-periodic oscillations of GRS 1915+105 and the coupling of the quasi-periodic oscillations to the power-law noise in that system.

Extracting Colored Noise Statistics in Time Series via Negentropy

In the analysis of some specific time series (e.g., Global Positioning System coordinate time series, chaotic time series, human brain imaging), the noise is generally modeled as a sum of a power-law noise and white noise. Some existing softwares estimate the amplitude of the noise components using convex optimization (e.g., Levenberg-Marquadt) applied to a log-likelihood cost function. This work studies a novel cost function based on an approximation of the negentropy. Restricting the study to simulated time series with flicker noise plus white noise, we demonstrate that this cost function is convex. Then, we show thanks to numerical approximations that it is possible to obtain an accurate estimate of the amplitude of the colored noise for various lengths of the time series as long as the ratio between the colored noise amplitude and the white noise is smaller than 0.6. The results demonstrate that with our proposed cost function we can improve the accuracy by around 5% when compared with the log-likelihood ones with simulated time series shorter than 1400 samples.

Validation of a power-law noise model for simulating small-scale breast tissue

We have validated a small-scale breast tissue model based on power-law noise. A set of 110 patient images served as truth. The statistical model parameters were determined by matching the radially averaged power-spectrum of the projected simulated tissue with that of the central tomosynthesis patient breast projections. Observer performance in a signal-known exactly detection task in simulated and actual breast backgrounds was compared. Observers included human readers, a pre-whitening observer model and a channelized Hotelling observer model. For all observers, good agreement between performance in the simulated and actual backgrounds was found, both in the tomosynthesis central projections and the reconstructed images. This tissue model can be used for breast x-ray imaging system optimization. The complete statistical description of the model is provided.

ENERGY-DEPENDENT POWER SPECTRAL STATES AND ORIGIN OF APERIODIC VARIABILITY IN BLACK HOLE BINARIES

We found the black hole candidate MAXI J1659-152 showed distinct power spectra, i.e., a power-law noise (PLN) vs. band-limited noise (BLN) plus quasi-periodic oscillations (QPOs), below and above about 2 keV respectively, in the observations with the Swift and the RXTE during the 2010 outburst, indicating a high energy cut-off of the PLN and a low energy cut-off of the BLN and the QPOs around 2 keV. The emergence of the PLN and the fading of the BLN and the QPOs initially took place from below 2 keV when the source entered the hard intermediate state and finally settled in the soft state three weeks later. The evolution was accompanied by the emergence of the disk spectral component and decreases in the amplitudes of variability in the soft X-ray and the hard X-ray bands. Our results indicate that the PLN is associated with the optically thick disk in both hard and intermediate states, and power spectral state is independent of the X-ray energy spectral state in a broadband view. We suggest that in the hard and the intermediate state, the BLN and the QPOs emerge from the innermost hot flow subjected to Comptonization, while the PLN originates from the optically thick disk further out. The energy cut-offs of the PLN and the BLN or QPOs then follow the temperature of the seed photons from the inner edge of the optically thick disk, while the high frequency cut-off of the PLN follows the orbital frequency at the inner edge of the optically thick disk as well.

Extracting White Noise Statistics in GPS Coordinate Time Series

The noise in GPS coordinate time series is known to follow a power-law noise model with different components (white noise, flicker noise, and random walk). This work proposes an algorithm to estimate the white noise statistics, through the decomposition of the GPS coordinate time series into a sequence of sub time series using the empirical mode decomposition algorithm. The proposed algorithm estimates the Hurst parameter for each sub time series and then selects the sub time series related to the white noise based on the Hurst parameter criterion. Both simulated GPS coordinate time series and real data are employed to test this new method; the results are compared to those of the standard (CATS software) maximum-likelihood (ML) estimator approach. The results demonstrate that this proposed algorithm has very low computational complexity and can be more than 100 times faster than the CATS ML method, at the cost of a moderate increase of the uncertainty (~5%) of the white noise amplitude. Reliable white noise statistics are useful for a range of applications including improving the filtering of GPS time series, checking the validity of estimated coseismic offsets, and estimating unbiased uncertainties of site velocities. The low complexity and computational efficiency of the algorithm can greatly speed up the processing of geodetic time series.

Intermittency in relation with 1/f noise and stochastic differential equations

One of the models of intermittency is on-off intermittency, arising due to time-dependent forcing of a bifurcation parameter through a bifurcation point. For on-off intermittency, the power spectral density (PSD) of the time-dependent deviation from the invariant subspace in a low frequency region exhibits 1/f power-law noise. Here, we investigate a mechanism of intermittency, similar to the on-off intermittency, occurring in nonlinear dynamical systems with invariant subspace. In contrast to the on-off intermittency, we consider the case where the transverse Lyapunov exponent is zero. We show that for such nonlinear dynamical systems, the power spectral density of the deviation from the invariant subspace can have 1/fβ form in a wide range of frequencies. That is, such nonlinear systems exhibit 1/f noise. The connection with the stochastic differential equations generating 1/fβ noise is established and analyzed, as well.