Correlated Noise Processes Research Articles

Characterizing Solar Center-to-limb Radial-velocity Variability with SDO

Stellar photospheric inhomogeneities are a significant source of noise, which currently precludes the discovery of Earth-mass planets orbiting Sun-like stars with the radial-velocity (RV) method. To complement several previous studies that have used ground- and spaced-based facilities to characterize the RV of the Sun, here we characterize the center-to-limb variability (CLV) of solar RVs arising from various solar-surface inhomogeneities observed by the Solar Dynamics Observatory (SDO)/Helioseismic Magnetic Imager and SDO/Atmospheric Imaging Assembly. By using various SDO observables to classify pixels and calculate line-of-sight velocities as a function of pixel classification and limb angle, we show that each identified feature type, including the umbrae and penumbrae of sunspots, quiet-Sun magnetoconvective cells, magnetic network, and plage, exhibit distinct and complex CLV signatures, including a notable limb-angle dependence in the observed suppression of convective blueshift for magnetically active regions. We discuss the observed distributions of velocities by identified region type and limb angle, offer interpretations of the physical phenomena that shape these distributions, and emphasize the need to understand the RV signatures of these regions as astrophysical signals, rather than simple (un)correlated noise processes.

PINT: Maximum-likelihood Estimation of Pulsar Timing Noise Parameters

PINT is a pure-Python framework for high-precision pulsar timing developed on top of widely used and well-tested Python libraries, supporting both interactive and programmatic data analysis workflows. We present a new frequentist framework within PINT to characterize the single-pulsar noise processes present in pulsar timing data sets. This framework enables parameter estimation for both uncorrelated and correlated noise processes, as well as model comparison between different timing and noise models in a computationally inexpensive way. We demonstrate the efficacy of the new framework by applying it to simulated data sets as well as a real data set of PSR B1855+09. We also describe the new features implemented in PINT since it was first described in the literature.

Signatures of non-Markovianity of a superconducting qubit

We describe temporally correlated noise processes that influence the idle evolution of a superconducting transmon qubit. To model the composite qubit-environment system we use quantum circuit theory, and we show how a circuit Hamiltonian can be derived for transverse noise affecting the qubit. Based on the time-convolutionless projection operator method, we construct a time-local master equation which, when transformed to its canonical Lindblad form, exhbitis a decay rate that is negative at all times, corresponding to eternally non-Markovian dynamics. By expressing the solution of the master equation in the Kraus representation, we identify two crucial non-Markovian phenomena: periodic revivals of coherence, and the appearance of additional frequencies far from the qubit frequency in the precession of the qubit state. When a single qubit gate acts on the qubit state, these extra frequency terms rotate undesirably and they effectively act as the memory of the state prior to the rotation around the Bloch sphere.

A two-stage Kalman filtering approach using GNSS and smartphone sensors for seismogeodetic applications

Recently, an enormous number of concepts have been introduced as new insights into seismogeodetic tools and algorithms. Integration of Global Navigation Satellite Systems (GNSS) with accelerometers is a well-known and widely used approach for estimating co-seismic displacements. In this study, we present a two-stage Kalman Filtering (TSKF) approach for seismogeodetic applications with low-cost accelerometers embedded in smartphones. The TSKF is initialized after a cost-effective and magnetometer-aided attitude estimation of accelerations in the sensor frame with respect to the local geodetic (topocentric) frame using the Singular Value Decomposition (SVD) method. In the first stage of TSKF, correlated noise processes in GNSS positions identified by the Allan Variance (AV) analysis are estimated with a linear Kalman Filter (LKF) and displacements are reflected in the innovation sequence of the LKF. In the second stage, the innovation sequence of the first stage LKF is integrated with the smartphone accelerations by a multi-rate Kalman Filter. The performance of TSKF was evaluated with harmonic motion and earthquake experiments using a single-axis shake table in a high multipath environment. Results of harmonic motion experiments showed that TSKF can produce mm-level amplitude deviations however the effect of phase shift decreases the performance and may cause mm to cm-level root mean square (RMS). On the other hand, the performance of the TSKF in earthquake experiments resulted in mm-level RMS values. In addition, comparisons of TSKF with an existing method i.e. the Multi-Rate Kalman Filter (MKF) show that TSKF may significantly increase the performance of displacement estimations.

Graphical modeling of stochastic processes driven by correlated noise

We study a class of graphs that represent local independence structures in stochastic processes allowing for correlated noise processes. Several graphs may encode the same local independencies and we characterize such equivalence classes of graphs. In the worst case, the number of conditions in our characterizations grows superpolynomially as a function of the size of the node set in the graph. We show that deciding Markov equivalence of graphs from this class is coNP-complete which suggests that our characterizations cannot be improved upon substantially. We prove a global Markov property in the case of a multivariate Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.

Experimental quantum verification in the presence of temporally correlated noise

Growth in the capabilities of quantum information hardware mandates access to techniques for performance verification that function under realistic laboratory conditions. Here we experimentally characterise the impact of common temporally correlated noise processes on both randomised benchmarking (RB) and gate-set tomography (GST). Our analysis highlights the role of sequence structure in enhancing or suppressing the sensitivity of quantum verification protocols to either slowly or rapidly varying noise, which we treat in the limiting cases of quasi-DC miscalibration and white noise power spectra. We perform experiments with a single trapped 171Yb+ ion-qubit and inject engineered noise left( { propto hat sigma _z} right) to probe protocol performance. Experiments on RB validate predictions that measured fidelities over sequences are described by a gamma distribution varying between approximately Gaussian, and a broad, highly skewed distribution for rapidly and slowly varying noise, respectively. Similarly we find a strong gate set dependence of default experimental GST procedures in the presence of correlated errors, leading to significant deviations between estimated and calculated diamond distances in the presence of correlated hat sigma _z errors. Numerical simulations demonstrate that expansion of the gate set to include negative rotations can suppress these discrepancies and increase reported diamond distances by orders of magnitude for the same error processes. Similar effects do not occur for correlated hat sigma _x or hat sigma _y errors or depolarising noise processes, highlighting the impact of the critical interplay of selected gate set and the gauge optimisation process on the meaning of the reported diamond norm in correlated noise environments.

Kramers problem for a dimer: Effect of noise correlations.

The Kramers problem for a dimer in a bistable piecewise linear potential is studied in the presence of correlated noise processes. The effect of such a correlation is to redistribute the thermal power between the dynamical degrees of freedom, and this leads to significant deviations in the dynamics of the system from the case of independent noise processes. The distribution of first passage times from one minima to the basin of attraction of the other minima is found to have exponentially decaying tails with the parameter dependent on the amount of correlation and the coupling between the particles. The strong coupling limit of the problem is analyzed using adiabatic elimination, where it is found that the initial probability density relaxes towards a stationary value on the same time scale as the mean escape time when the noise intensity of the system is low. For higher noise fluctuations, the relaxation towards the stationary state is slower in comparison to escape times. In the extreme limit of perfect anticorrelation, the random dynamical system behaves as a deterministic system in a steady state in which the center of mass starting from the unstable maxima moves down the hill and gets trapped in the potential minima. The implications for polymer dynamics in a potential are discussed.

High-Resolution Flicker-Noise-Free Frequency Measurements of Weak Microwave Signals

Amplification is usually necessary when measuring the frequency instability of microwave signals. In this work, we develop a flicker noise free frequency measurement system based on a common or shared amplifier. First, we show that correlated flicker phase noise can be cancelled in such a system. Then we compare the new system with the conventional by simultaneously measuring the beat frequency from two cryogenic sapphire oscillators with parts in 10^15 fractional frequency instability. We determine for low power, below -80 dBm, the measurements were not limited by correlated noise processes but by thermal noise of the readout amplifier. In this regime, we show that the new readout system performs as expected and at the same level as the standard system but with only half the number of amplifiers. We also show that, using a standard readout system, the next generation of cryogenic sapphire oscillators could be flicker phase noise limited when instability reaches parts in 10^16 or better

Spurious Mean-Reversion of Stock Prices in the State-Space Model

In order to explain the U-shaped pattern of autocorrelations of stock returns i.e., autocorrelations starting around 0 for short-term horizons and becoming negative and then moving toward 0 for long-term horizons, researchers suggested the use of a state-space model consisting of an I(1) permanent component and an AR(1) stationary component, where the two components are assumed to be independent. They concluded that auto-regression coefficients derived from the state-space model follow a U-shape pattern and thus there is mean-reversion in stock prices. In this paper, we show that only negative autocorrelations are feasible under the assumption that the permanent component and the stationary component are independent in the state-space model. When the two components are allowed to be correlated in the state-space model, we show that the sign of the auto-regression coefficients is not restricted as negative. Monthly return data for all NYSE stocks for the period from 1926 to 2007 support the state-space model with correlated noise processes. However, the auto-regression coefficients of the ARIMA process, equivalent to the state-space model with correlated noise processes, do not follow a U-shaped pattern, but are always positive.

Generalization of the escape rate from a metastable state driven by external cross-correlated noise processes

We propose generalization of the escape rate from a metastable state for externally driven correlated noise processes in one dimension. In addition to the internal non-Markovian thermal fluctuations, the external correlated noise processes we consider are Gaussian, stationary in nature and are of Ornstein-Uhlenbeck type. Based on a Fokker-Planck description of the effective noise processes with finite memory we derive the generalized escape rate from a metastable state in the moderate-to-large damping limit and investigate the effect of degree of correlation on the resulting rate. Comparison of the theoretical expression with numerical simulation gives a satisfactory agreement and shows that by increasing the degree of external noise correlation one can enhance the escape rate through the dressed effective noise strength.

A framework of incorporating variance estimates into wavelet analysis in PET metabolism and receptor studies

Recently, methods have been developed to estimate the variance of parameter estimates derived using wavelet methods 1. Previous methods of analysis in the wavelet domain 2, 3, 4 had only yielded estimates of the parameters after transformation back into the image domain. The development of methods which return the additional variance information present the opportunity for further analyses which incorporate the attractive properties of wavelet based methods. The underlying principle of the method of variance estimation is that whilst the variance parameters from wavelet space cannot be easily transformed back into image space, as they are a matrix (2nd order) as opposed to the intrinsic vector (first order) of the parameter estimates, there do exist first order estimates associated with the variance which can easily be transformed, namely the residuals. Thus by transforming back the residuals subject to the equivalent shrinkage of that of the parameter estimates, it is possible to determine the variance of the parameter estimates in the image domain. These methods have been derived and validated on linear models, and linearly transformed data. The figure 1 shows one such PET FDG study and its error variance after thresholding in the wavelet domain. The ability to use these methods with common non-linear PET compartmental models will be discussed further in the abstract by Su, HR et al. In neuropsychology, simple association has been considered the lowest level of evidence for functional segregation, the higher levels being respectively dissociation and double dissociation. The determination of dissociation requires the within-brain comparison of task related effects between two or more locations. Jernigan et al 5 suggested the use of t-statistics for point-wise comparisons between brain locations to assess dissociation. However, statistical maps are characterized by high noise levels and require regularization to allow a detailed assessment of regional variability. Wavelet methods are natural candidates for the regularization task. They provide a parsimonious multiresolution characterization of the signal and efficiently whiten the correlated noise processes of functional images. With the introduction of the error estimates for wavelet methods, dissociation investigation now becomes possible. The error variances allow surrogates for t-value maps to be derived. However, not only is it of importance to know the error variances but it is also of interest to characterise the underlying distribution with a view to integration of the methodology into the hypothesis testing framework. The error estimates are the first step in a characterisation of these distributions which can in turn be used to perform hypothesis testing.

Postfiltering versus prefiltering for signal recovery from noisy samples

We consider the extension of the Whittaker-Shannon (WS) reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited. Observing that in this situation the classical sampling expansion yields inconsistent reconstruction, we introduce a class of signal recovery methods with a smooth correction of the interpolation series. Two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing. We assess the accuracy of the methods by the global L/sub 2/ error. Both band-limited and non-band-limited signals are considered. A general class of correlated noise processes is taken into account. The weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed. The influence of noise memory and its moment structure on the accuracy is thoroughly examined.

Improved false nearest neighbor method to detect determinism in time series data.

The false nearest neighbor method introduced by Kennel et al. [Phys. Rev. A 45, 3403 (1992)] is revisited and modified in order to ensure a correct distinction between low-dimensional chaotic data and noise. Still, correlated noise processes can yield vanishing percentages of false nearest neighbors for rather low embedding dimensions and can be mistaken for deterministic signals. Therefore, the false nearest neighbors method has always to be combined with a surrogate data test.

Asymptotically optimal rate of convergence of smoothed stochastic recursive algorithms

This work is concerned with smoothed stochastic approximation/optimization algorithms. The main emphasis is placed on the asymptotic optimality issues. An algorithm with averaging in both state variables and observations is studied. Under correlated noise processes, it is shown that a scaled sequence of the iterates converges weakly to a Browman motion. As a result the algorithm is asymptotically optimal Numerical experiments are carried out. Comparisons are made among several algorithms for both linear and nonlinear functions. The numerical results yield good agreement with our analytical findings

A stopping rule for the Robbins-Monro method

A stopping rule for the multidimensional Robbins-Monro stochastic approximation method is developed in this paper. Both moving average and stationary φ-mixing type of correlated noise processes are treated. Sequentially determined confidence ellipsoids are constructed to fulfill the goal for the determination of the stopping rule. The limit behavior of the algorithm is investigated. It is shown that the stopped Robbins-Monro process is asymptotically normal. Such asymptotic normality is established by means of weak convergence methods.

Exact and approximate state estimation for nonlinear dynamic systems

This paper presents a modified approach to solve state estimation problems of nonlinear dynamic systems involving noise free, uncorrelated and correlated state and measurement noise processes. The basic approach makes use of the matrix minimum principle together with the Kolmogorov and Kushner's equations to minimize the error-variance, taken to be the estimation criterion. The filtering equations obtained for nonlinear systems with white noise process are exact, but for non-white noise processes the results obtained are approximate. For systems with polynomial or product types non-linearities, the proposed algorithms can be evaluated without the need of approximation under the assumption that the estimator errors are Gaussian. Such an assumption is significantly different from the most commonly used assumption that the state is Gaussian. Simulation results obtained from the proposed filtering algorithms are compared to various other approximate nonlinear filters. The results indicate the superiority of the proposed filter over those of other filters investigated.