Power-law Noise Research Articles

Experimental evidence of multiaffinity of pinned interfaces

We report an imbibition experiment investigating the multiaffinity of the pinned rough interfaces. We observe density dependent multiaffine nature of the rough interfaces. Results are well compared with the existence of power law noise as well as the dependence on particle size of the morphology of the rough interfaces.

Dynamically strained ferroelastics: Statistical behavior in elastic and plastic regimes

The dynamic evolution in ferroelastic crystals under external shear is explored by computer simulation of a two-dimensional model. The characteristic geometrical patterns obtained during shear deformation include dynamic tweed in the elastic regime as well as interpenetrating needle domains in the plastic regime. As a result, the statistics of jerk energy differ in the elastic and plastic regimes. In the elastic regime the distributions of jerk energy are sensitive to temperature and initial configurations. However, in the plastic regime the jerk distributions are rather robust and do not depend much on the details of the configurations, although the geometrical pattern formed after yield is strongly influenced by the elastic constants of the materials and the configurations we used. Specifically, for all geometrical configurations we studied, the energy distribution of jerks shows a power-law noise pattern $P(E)\ensuremath{\sim}{E}^{\ensuremath{-}(\ensuremath{\gamma}\ensuremath{-}1)}(\ensuremath{\gamma}\ensuremath{-}1=1.3\ensuremath{-}2)$ at low temperatures and a Vogel-Fulcher distribution $P$($E$) \ensuremath{\sim} exp-($E/{E}_{0}$) at high temperatures. More complex behavior occurs at the crossover between these two regimes where our simulated jerk distributions are very well described by a generalized Poisson distributions $P(E)\ensuremath{\sim}{E}^{\ensuremath{-}(\ensuremath{\gamma}\ensuremath{-}1)}$ exp-($E/{E}_{0}$)${}^{n}$ with $n$ $=$ 0.4--0.5 and \ensuremath{\gamma}\ensuremath{-}1 \ensuremath{\approx} 0 (Kohlrausch law). The geometrical mechanisms for the evolution of the ferroelastic microstructure under strain deformation remain similar in all thermal regimes, whereas their thermodynamic behavior differs dramatically: on heating, from power-law statistics via the Kohlrausch law to a Vogel-Fulcher law. There is hence no simple way to predict the local evolution of the twin microstructure from just the observed statistical behavior of a ferroelastic crystal. It is shown that the Poisson distribution is a convenient way to describe the crossover behavior contained in all the experimental data without recourse to specific scaling functions or temperature-dependent cutoff lengths.

Fluctuations in multiplicative systems with jumps

Fluctuation properties of the Langevin equation including a multiplicative, power-law noise and a quadratic potential are discussed. The noise has the Levy stable distribution. If this distribution is truncated, the covariance can be derived in the limit of large time; it falls exponentially. Covariance in the stable case, studied for the Cauchy distribution, exhibits a weakly stretched exponential shape and can be approximated by the simple exponential. The dependence of that function on system parameters is determined. Then we consider a dynamics which involves the above process and obey the generalised Langevin equation, the same as for Gaussian case. The resulting distributions possess power-law tails - that fall similarly to those for the driving noise - whereas central parts can assume the Gaussian shape. Moreover, a process with the covariance 1/t at large time is constructed and the corresponding dynamical equation solved. Diffusion properties of systems for both covariances are discussed.

Fast error analysis of continuous GNSS observations with missing data

One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires $$\mathcal{O }(n^3)$$ operations for $$n$$ observations. Previous research by the authors has shown that this amount of operations can be reduced to $$\mathcal{O }(n^2)$$ for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.

Spectra of a MHD plasma forced by power-law noise

Scalings of a noise-driven two-dimensional homogeneous MHD plasma are considered by using numerical simulations. The noise is a Gaussian random process whose two-point correlation function is delta-correlated in time while obeying a power-law spectrum k2−2δ that determines the input rate of the energy where k is the inverse of the scale length. For the case of infrared forcing, δ > 2, the scaling of the energy spectrum is found to closely follow the phenomenological prediction of k−3/2. In the other case of ultraviolet forcing where δ < 2, the renormalization-group scale, which is k1−4δ/3, is found to approximately hold for the energy spectrum.

Scalings of forced Hasegawa–Mima equation by power-law noise

For the plasma modeled by the Hasegawa–Mima equation forced by a noise, Kolmogorov-type phenomenology (KP) predicts the power-law scaling in two inertial regimes. In the spatial scale, which is larger than the ion gyroradius, the energy spectrum scales as k−11/3 where the energy cascades to larger scales and k−5 where the enstrophy is transferred to smaller scales down to the dissipation range. The renormalization-group (RG) approach, however, provides different scaling k−1−4δ/3 that strongly depends on the power exponent δ of the noise spectrum. Through numerical simulations it is found that, for δ ⩽ 2 or δ ⩾ 3, the energy spectrum follows KP while, for 2 ⩽ δ ⩽ 3, RG scaling prevails.

A statistically defined anthropomorphic software breast phantom

Digital anthropomorphic breast phantoms have emerged in the past decade because of recent advances in 3D breast x-ray imaging techniques. Computer phantoms in the literature have incorporated power-law noise to represent glandular tissue and branching structures to represent linear components such as ducts. When power-law noise is added to those phantoms in one piece, the simulated fibroglandular tissue is distributed randomly throughout the breast, resulting in dense tissue placement that may not be observed in a real breast. The authors describe a method for enhancing an existing digital anthropomorphic breast phantom by adding binarized power-law noise to a limited area of the breast. Phantoms with (0.5 mm)(3) voxel size were generated using software developed by Bakic et al. Between 0% and 40% of adipose compartments in each phantom were replaced with binarized power-law noise (β = 3.0) ranging from 0.1 to 0.6 volumetric glandular fraction. The phantoms were compressed to 7.5 cm thickness, then blurred using a 3 × 3 boxcar kernel and up-sampled to (0.1 mm)(3) voxel size using trilinear interpolation. Following interpolation, the phantoms were adjusted for volumetric glandular fraction using global thresholding. Monoenergetic phantom projections were created, including quantum noise and simulated detector blur. Texture was quantified in the simulated projections using power-spectrum analysis to estimate the power-law exponent β from 25.6 × 25.6 mm(2) regions of interest. Phantoms were generated with total volumetric glandular fraction ranging from 3% to 24%. Values for β (averaged per projection view) were found to be between 2.67 and 3.73. Thus, the range of textures of the simulated breasts covers the textures observed in clinical images. Using these new techniques, digital anthropomorphic breast phantoms can be generated with a variety of glandular fractions and patterns. β values for this new phantom are comparable with published values for breast tissue in x-ray projection modalities. The combination of conspicuous linear structures and binarized power-law noise added to a limited area of the phantom qualitatively improves its realism.

Estimating rate uncertainty with maximum likelihood: differences between power-law and flicker–random-walk models

Recent studies have documented that global positioning system (GPS) time series of position estimates have temporal correlations which have been modeled as a combination of power-law and white noise processes. When estimating quantities such as a constant rate from GPS time series data, the estimated uncertainties on these quantities are more realistic when using a noise model that includes temporal correlations than simply assuming temporally uncorrelated noise. However, the choice of the specific representation of correlated noise can affect the estimate of uncertainty. For many GPS time series, the background noise can be represented by either: (1) a sum of flicker and random-walk noise or, (2) as a power-law noise model that represents an average of the flicker and random-walk noise. For instance, if the underlying noise model is a combination of flicker and random-walk noise, then incorrectly choosing the power-law model could underestimate the rate uncertainty by a factor of two. Distinguishing between the two alternate noise models is difficult since the flicker component can dominate the assessment of the noise properties because it is spread over a significant portion of the measurable frequency band. But, although not necessarily detectable, the random-walk component can be a major constituent of the estimated rate uncertainty. None the less, it is possible to determine the upper bound on the random-walk noise.

Noise analysis of continuous GPS coordinate time series for CMONOC

The Crustal Movement Observation Network of China (CMONOC) is one of the major scientific infrastructures, mainly using Global Positioning System (GPS) measurements, to monitor crustal deformation in the Chinese mainland. In this paper, decade-long coordinate time series of 26 continuous GPS sites of CMONOC are analyzed for their noise content using maximum likelihood estimation (MLE). We study the noise properties of continuous GPS time series of CMONOC for the unfiltered, filtered solutions and also the common mode signals in terms of power law plus white noise model. In the spatial filtering, we remove for every time series a common mode error that was estimated from a modified stacking of position residuals from other sites within ∼1000km of the selected site. We find that the common mode signal in our network has a combination of spatially correlated flicker noise and a common white noise with large spatial extent. We demonstrate that for the unfiltered solutions of CMONOC continuous GPS sites the main colored noise is a flicker process, with a mean spectral index of ∼1. For the filtered solutions, the main colored noise is a general power law process, indicating that a major number of the filtered regional solutions have a combination of noise sources or local effects. The velocity uncertainties from CMONOC continuous GPS coordinate time series may be underestimated by factors of 8–16 if a pure white noise model is assumed. In addition, using a white plus flicker noise model, the median values of velocity errors for the unfiltered solutions are 0.16 (north), 0.17 (east) and 0.58 (vertical)mm/yr, and the median values for the filtered solutions are 0.09 (north), 0.10 (east) and 0.40 (vertical)mm/yr.

Estimation of velocity uncertainties from GPS time series: Examples from the analysis of the South African TrigNet network

[1] We present a method to derive velocity uncertainties from GPS position time series that are affected by time-correlated noise. This method is based on the Allan variance, which is widely used in the estimation of oscillator stability and requires neither spectral analysis nor maximum likelihood estimation (MLE). The Allan variance of the rate (AVR) is calculated in the time domain and hence is not too sensitive to gaps in the time series. We derived analytical expressions of the AVR for different kinds of noises like power law noise, white noise, flicker noise, and random walk and found an expression for the variance produced by an annual signal. These functional relations form the basis of error models that have to be fitted to the AVR in order to estimate the velocity uncertainty. Finally, we applied the method to the South Africa GPS network TrigNet. Most time series show noise characteristics that can be modeled by a power law noise plus an annual signal. The method is computationally very cheap, and the results are in good agreement with the ones obtained by methods based on MLE.

Spectro-timing analysis of Cygnus X-1 during a fast state transition

We present the analysis of two long, quasi-uninterrupted RXTE observations of Cygnus X-1 that span several days within a 10 d interval. The spectral characteristics during this observation cover the region where previous observations have shown the source to be most dynamic. Despite that the source behavior on time scales of hours and days is remarkably similar to that on year time scales. This includes a variety of spectral/temporal correlations that previously had only been observed over Cyg X-1's long-term evolution. Furthermore, we observe a full transition from a hard to a soft spectral state that occurs within less than 2.5 hours - shorter than previously reported for any other similar Cyg X-1 transition. We describe the spectra with a phenomenological model dominated by a broken power law, and we fit the X-ray variability power spectra with a combination of a cutoff power law and Lorentzian components. The spectral and timing properties are correlated: the power spectrum Lorentzian components have an energy-dependent amplitude, and their peak frequencies increase with photon spectral index. Averaged over 3.2-10 Hz, the time lag between the variability in the 4.5-5.7 keV and 9.5-15 keV bands increases with decreasing hardness when the variability is dominated by the Lorentzian components during the hard state. The lag is small when there is a large power law noise contribution, shortly after the transition to the soft state. Interestingly, the soft state not only shows the shortest lags, but also the longest lags when the spectrum is at its softest and faintest. We discuss our results in terms of emission models for black hole binaries.

Correlated errors in GPS position time series: Implications for velocity estimates

[1] This study focuses on the effects of time correlation in weekly GPS position time series on velocity estimates. Time series 2.5 to 13 years long from a homogeneously reprocessed solution of 275 globally distributed stations are analyzed in terms of noise content and velocity uncertainty assessment. Several noise models were tested, including power law and Gauss-Markov processes. The best noise model describing our global data set was a combination of variable white noise and power law noise models with mean amplitudes of ∼2 mm and ∼6 mm, respectively, for the sites considered. This noise model provided a mean vertical velocity uncertainty of ∼0.3 mm/yr, 4–5 times larger than the uncorrelated data assumption. We demonstrated that correlated noise content with homogeneously reprocessed data is dependent on time series length and, especially, on data time period. Time series of 2–3 years of the oldest data contain noise amplitude similar to that found for time series of 12 years. The data time period should be taken into account when estimating correlated noise content, when comparing different noise estimations, or when applying an external noise estimation to assess velocity uncertainty. We showed that the data period dependency cannot be explained by the increasing tracking network or the ambiguity fixation rate but is probably related to the amount and quality of recorded data.

Predictions of Taylor's power law, density dependence and pink noise from a neutrally modeled time series

There has recently been increasing interest in neutral models of biodiversity and their ability to reproduce the patterns observed in nature, such as species abundance distributions. Here we investigate the ability of a neutral model to predict phenomena observed in single-population time series, a study complementary to most existing work that concentrates on snapshots in time of the whole community. We consider tests for density dependence, the dominant frequencies of population fluctuation (spectral density) and a relationship between the mean and variance of a fluctuating population (Taylor's power law). We simulated an archipelago model of a set of interconnected local communities with variable mortality rate, migration rate, speciation rate, size of local community and number of local communities. Our spectral analysis showed ‘pink noise’: a departure from a standard random walk dynamics in favor of the higher frequency fluctuations which is partly consistent with empirical data. We detected density dependence in local community time series but not in metacommunity time series. The slope of the Taylor's power law in the model was similar to the slopes observed in natural populations, but the fit to the power law was worse. Our observations of pink noise and density dependence can be attributed to the presence of an upper limit to community sizes and to the effect of migration which distorts temporal autocorrelation in local time series. We conclude that some of the phenomena observed in natural time series can emerge from neutral processes, as a result of random zero-sum birth, death and migration. This suggests the neutral model would be a parsimonious null model for future studies of time series data.

The profound impact of negative power law noise on statistical estimation

This paper investigates the profound impact of negative power law (neg-p) noise - that is, noise with a power spectral density L(p)(f) proportional variant | f |(p) for p < 0 - on the ability of practical implementations of statistical estimation or fitting techniques, such as a least squares fit (LSQF) or a Kalman filter, to generate valid results. It demonstrates that such negp noise behaves more like systematic error than conventional noise, because neg-p noise is highly correlated, non-stationary, non-mean ergodic, and has an infinite correlation time tau(c). It is further demonstrated that stationary but correlated noise will also cause invalid estimation behavior when the condition T >> tau(c) is not met, where T is the data collection interval for estimation. Thus, it is shown that neg-p noise, with its infinite Tau(c), can generate anomalous estimation results for all values of T, except in certain circumstances. A covariant theory is developed explaining much of this anomalous estimation behavior. However, simulations of the estimation behavior of neg-p noise demonstrate that the subject cannot be fully understood in terms of covariant theory or mean ergodicity. It is finally conjectured that one must investigate the variance ergodicity properties of neg-p noise through the use of 4th order correlation theory to fully explain such simulated behavior.

The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series

Abstract Most GPS time-series exhibit a seasonal signal that can have an amplitude of a few millimetres. This seasonal signal can be removed by fitting an extra sinusoidal signal with a period of one year to the GPS data during the estimation of the linear trend. However, Blewitt and Lavallee (2002) showed that including an annual signal in the estimation process still can give a larger linear trend error than the trend error estimated from data from which the annual signal has been removed by other means. They assumed that the GPS data only contained white noise and we extend their result to the case of power-law plus white noise which is known to exist in most GPS observations. For the GPS stations CASC, LAGO, PDEL and TETN the difference in trend error between having or not having an annual signal in the data is around ten times larger when a power-law plus white noise model is used instead of a pure white noise model. Next, our methodology can be used to estimate for any station how much the accuracy of the linear trend will improve when one tries to subtract the annual signal from the GPS time-series by using a physical model. Finally, we demonstrate that for short time-series the trend error is more influenced by the fact that the noise properties also need to be estimated from the data. This causes on average an underestimation of the trend error.

An Empirical Study on Time-Correlation of GSM Telephone Traffic

In this paper, we investigate possible time correlation of answered call arrivals in sets of real GSM telephone traffic data. Instead of attempting to model the empirical distribution of the call interarrival time, as done in several previous works in literature, we emphasize results obtained by the modified Allan variance (MAVAR), a widely used time-domain quantity with excellent capability of discriminating power-law noise types. The call arrival rate exhibits a diurnal trend, with peak hours in the morning and late afternoon. Besides this diurnal rate change, the number of call arrivals in a second is found perfectly uncorrelated to the number of arrivals in other seconds and Poisson distributed, with good consistency by X/sup 2/- test evaluation. Uniform and accurate whiteness of call arrivals per second is verified in all hours, regardless the time of the day. In all series analyzed, the empirical statistics of both originated and terminated call arrivals proves excellent consistency with the ideal Poisson model with diurnal variable rate lambda(t). This study may be valuable to researchers concerned about realistic traffic modelling in planning and performance evaluation of cellular networks.

Instantaneous deformation from continuous GPS: contributions from quasi-periodic loads

SUMMARY Continuous GPS (CGPS) coordinate time-series are known to experience repeating deformation signals with seasonal and other periods. It is unlikely that these signals represent perfect sinusoids with temporally constant amplitude. We develop an analysis method that accommodates temporal variations in the amplitudes of sinusoidal signals. We apply the method to simulated coordinate time-series to numerically explore the potential consequences of neglecting decadal variation in amplitude of annual motions on the residual-error spectra of CGPS measurements, as well as potential bias in estimates for secular site velocity. We find that secular velocity bias can be appreciable for shorter time-series, and that residual-error time-series of longer duration may contain significant power in a broad band centred on semiannual frequency if temporal variation in the amplitude of annual motions is not accounted for in the model used to reduce the observations to residuals. It may be difficult to differentiate the bandpass filtered signature of mismodelled loading signals from power-law noise, using residual-error spectra for shorter time-series. We provide an example application to a ∼9-yr coordinate time-series for a CGPS station located in southern California at Carbon Creek Control Structure (CCCS), which is known to experience large amplitude seasonal motions associated with the Santa Ana aquifer system.

Noise in GPS displacement measurements from Southern California and Southern Nevada

Time series of position changes estimated from data from 236 continuously recording GPS receivers operating in Southern California and Southern Nevada are evaluated for noise models that characterize their temporal correlations. The lengths of the time series range between 3.5 and 10 years. After adjusting these data for postseismic deformation, offsets, and annual periodicities, I find that about one‐half of the time series have temporal correlations that are categorized as either flicker or random‐walk noise. The remaining time series can be best categorized as either a combination of flicker and random‐walk; power law noise; first‐order Gauss‐Markov plus random‐walk noise; or power law plus broadband, seasonal noise. A variety of geodetic monuments are used in Southern California and Nevada, including deeply braced designs, cement piers, pins drilled in outcrop, and buildings. When I evaluate the noise for each time series in terms of an estimate of the standard error in velocity, I find that the sites with the smallest errors are those located in Nevada using deeply braced monuments. Sites that are installed within regions of active pumping, both for groundwater and oil, had the largest standard errors in velocity. Comparison of monument stability, as measured by standard error in rate, with average, annual rainfall nearby indicates a marginally significant correlation. In addition, even though regional filtering removed much of the common‐mode signals in these time series, there still remains a common‐mode seasonal signal which can and should be removed.

Simulation of majority vote disturbed by power-law noise

Simulations are reported on the Ising two-dimensional ferromagnetin the presence of a special kind of noise. The noise spectrumP(n) follows a powerlaw, where P(n) is theprobability of flipping n randomly selected spins at each timestep. This is introduced to mimic the self-organizedcriticality as a model influence of a complex environment. We reproduced the phasetransition in a way that is similar to the 1992 model of de Oliveira. Above some value ofthe noise amplitude the magnetization tends to zero; otherwise it remains constant aftersome relaxation. Information of the initial spin orientation remains preserved to someextent by short-range spin–spin correlations. The distribution of the times between flips isexponential. The results are discussed as a step towards modelling of social systems.

NONLINEAR TIME SERIES PREDICTION BASED ON A POWER-LAW NOISE MODEL

In this paper we investigate the influence of a power-law noise model, also called Pareto noise, on the performance of a feed-forward neural network used to predict nonlinear time series. We introduce an optimization procedure that optimizes the parameters of the neural networks by maximizing the likelihood function based on the power-law noise model. We show that our optimization procedure minimizes the mean squared error leading to an optimal prediction. Further, we present numerical results applying our method to time series from the logistic map and the annual number of sunspots and demonstrate that a power-law noise model gives better results than a Gaussian noise model.