Showing papers on "White noise published in 2012"

Paracontrolled distributions and singular PDEs

Abstract: We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential.

Enhancing Source Camera Identification Performance With a Camera Reference Phase Sensor Pattern Noise

Abstract: Sensor pattern noise (SPN) extracted from digital images has been proved to be a unique fingerprint of digital cameras. However, SPN can be contaminated largely in the frequency domain by image content and nonunique artefacts of JPEG compression, on-sensor signal transfer, sensor design, color interpolation. The source camera identification (CI) performance based on SPN needs to be improved for small sizes of images and especially in resisting JPEG compression. Because the SPN is modelled as an additive white Gaussian noise (AWGN) in its extraction process from an image, it is reasonable to assume the camera reference SPN to be a white noise signal in order to remove the interference mentioned above. The noise residues (SPN) extracted from the original images are whitened first, then they are averaged to generate the camera reference SPN. Motivated by Goljan 's test statistic peak to correlation energy (PCE), we propose to use correlation to circular correlation norm (CCN) as the test statistic, which can lower the false positive rate to be a half of that with PCE. Theoretical analysis shows that the proposed CI method can remove the interference and raise the CCN value of a positive sample and thus achieve greater CI performance, CCN values of the negative sample class with the proposed method follow the normal distribution N (0,1) and the false positive rate can be calculated. Compared with the existing state of the art on seven cameras, 1400 photos totally (200 for each camera), the experimental results show that the proposed CI method achieves the best receiver operating characteristic (ROC) performance among all CI methods in all cases and especially achieves much better resistance to JPEG compression than all of the existing state-of-the-art CI methods.

Explicit solutions of some linear-quadratic mean field games

Abstract: We consider $N$-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number $N$ of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.

Stability of Stochastic Differential Equations

Abstract: In Chap. 1 we studied problems of stability under random perturbations of the parameters. We noted there that no significant results can be expected unless the random perturbations possess sufficiently favorable mixing properties. Fortunately, in practical applications one may often assume that the “noise” has a “short memory interval.” The natural limiting case of such noise is of course white noise. Thus it is very important to study the stability of solutions of Ito equations since this is equivalent to the study of stability of systems perturbed by white noise. Generalization of well known results on stability and instability for the deterministic ODE in terms of the Lyapunov functions are proven for SDE. Conditions for stability and instability of moments are also proven.

Mixed-state evolution in the presence of gain and loss.

Abstract: A model is proposed that describes the evolution of a mixed state of a quantum system for which gain and loss of energy or amplitude are present. Properties of the model are worked out in detail. In particular, invariant subspaces of the space of density matrices corresponding to the fixed points of the dynamics are identified, and the existence of a transition between the phase in which gain and loss are balanced and the phase in which this balance is lost is illustrated in terms of the time average of observables. The model is extended to include a noise term that results from a uniform random perturbation generated by white noise. Numerical studies of example systems show the emergence of equilibrium states that suppress the phase transition.

Variational Algorithms to Remove Stationary Noise: Applications to Microscopy Imaging

Abstract: A framework and an algorithm are presented in order to remove stationary noise from images. This algorithm is called variational stationary noise remover. It can be interpreted both as a restoration method in a Bayesian framework and as a cartoon+texture decomposition method. In numerous denoising applications, the white noise assumption fails. For example, structured patterns such as stripes appear in the images. The model described here addresses these cases. Applications are presented with images acquired using different modalities: scanning electron microscope, FIB-nanotomography, and an emerging fluorescence microscopy technique called selective plane illumination microscopy.

Time-Varying Combinations of Predictive Densities Using Nonlinear Filtering

Abstract: We propose a Bayesian combination approach for multivariate predictive densities which relies upon a distributional state space representation of the combination weights. Several specifications of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models can be individually misspecified. A Sequential Monte Carlo method is proposed to approximate the filtering and predictive densities. The combination approach is assessed using statistical and utility-based performance measures for evaluating density forecasts. Simulation results indicate that, for a set of linear autoregressive models, the combination strategy is successful in selecting, with probability close to one, the true model when the model set is complete and it is able to detect parameter i nstability when the model set includes the true model that has generated subsamples of data. For the macro series we find that incompleteness of the models is relatively large in the 70's, the beginning of the 80's and during the recent financial crisis, and lower during the Great Moderation. With respect to returns of the S&P 500 series, we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 90's and switches to giving more weight to the professional forecasts over time.

Nonparametric Bernstein-von Mises theorems in Gaussian white noise

Abstract: Bernstein-von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference procedures in a variety of concrete nonparametric problems. Particularly Bayesian credible sets are constructed that have asymptotically exact $1-\alpha$ frequentist coverage level and whose $L^2$-diameter shrinks at the minimax rate of convergence (within logarithmic factors) over H\"{o}lder balls. Other applications include general classes of linear and nonlinear functionals and credible bands for auto-convolutions. The assumptions cover nonconjugate product priors defined on general orthonormal bases of $L^2$ satisfying weak conditions.

A New Adaptive Line Enhancer Based on Singular Spectrum Analysis

Abstract: Original adaptive line enhancer (ALE) is used for denoising periodic signals from white noise. ALE, however, relies mainly on second order similarity between the signal and its delayed version and is more effective when the signal is narrowband. A new ALE based on singular spectrum analysis (SSA) is proposed here. In this approach in the reconstruction stage of SSA, the eigentriples are adaptively selected (filtered) using the delayed version of the data. Unlike the conventional ALE where (second) order statistics are taken into account, here the full eigen-spectrum of the embedding matrix is exploited. Consequently, the system works for non-Gaussian noise and wideband periodic signals. By performing some experiments on synthetic signals it is demonstrated that the proposed system is very effective for separation of biomedical data, which often have some periodic or quasi-periodic components, such as EMG affected by ECG artefacts. This data are examined here.

The Continuum Directed Random Polymer

Abstract: Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Holder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).

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

Abstract: 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.

Localization of distinct reflections in rooms using spherical microphone array eigenbeam processing.

Abstract: This paper presents an experimental and comparative study of several spherical microphone array eigenbeam (EB) processing techniques for localization of early reflections in room acoustic environments, which is a relevant research topic in both audio signal processing and room acoustics. This paper focuses on steered beamformer-based and subspace-based localization techniques implemented in the spherical EB domain, including the plane-wave decomposition, eigenbeam delay and sum, eigenbeam minimum variance distortionless response, eigenbeam multiple signal classification (EB-MUSIC), and eigenbeam estimation of signal parameters via rotational invariance techniques (EB-ESPRIT) methods. The directions of arrival of the original sound source and the associated reflection signals in acoustic environments are estimated from acoustic maps of the rooms, which are obtained using a spherical microphone array. The EB-domain-based frequency smoothing and white noise gain control techniques are derived and employed to improve the performance and robustness of reflection localization. The applicability of the presented methods in practice is confirmed by experiments carried out in real rooms.

Triviality of the 2D stochastic Allen-Cahn equation

Abstract: We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term.

A New Robust Filtering Strategy for Linear Systems

Abstract: For linear and well-defined estimation problems with Gaussian white noise, the Kalman filter (KF) yields the best result in terms of estimation accuracy. However, the KF performance degrades and can fail in cases involving large uncertainties such as modeling errors in the estimation process. The smooth variable structure filter (SVSF) is a relatively new estimation strategy based on sliding mode theory and has been shown to be robust to modeling uncertainties. The SVSF makes use of an existence subspace and of a smoothing boundary layer to keep the estimates bounded within a region of the true state trajectory. Currently, the width of the smoothing boundary layer is chosen based on designer knowledge of the upper bound of modeling uncertainties, such as maximum noise levels and parametric errors. This is a conservative choice, as a more well-defined smoothing boundary layer will yield more accurate results. In this paper, the state error covariance matrix of the SVSF is used for the derivation of an optimal time-varying smoothing boundary layer. The robustness and accuracy of the new form of the SVSF was validated and compared with the KF and the standard SVSF by testing it on a linear electrohydrostatic actuator (EHA).

Stochastic resonance phenomenon in an underdamped bistable system driven by weak asymmetric dichotomous noise

Abstract: Stochastic resonance in an underdamped bistable system subjected to a weak asymmetric dichotomous noise is investigated numerically. Dichotomous noise is a non-Gaussian color noise and more complex than Gaussian white noise, whose waiting time complies with the exponential distribution. Utilizing an efficiently numerical algorithm, we acquire the asymmetric dichotomous noise accurately. Then the system responses and the averaged power spectrum as the signatures of the stochastic resonance are calculated by the fourth-order Runge–Kutta algorithm. The effects of the noise strength, the forcing frequency, and the asymmetry of dichotomous noise on the system responses and the effects of the forcing frequency on the averaged power spectrum are discussed, respectively. It is found that the increasing of the noise strength or the forcing frequency could strengthen the passage between the stable points of the system, and the system responses also display the asymmetry for the asymmetric dichotomous noise, which has not been discovered in other investigated results. Additionally, the averaged power spectrum exhibits the sharp peaks, which indicates the occurrence of stochastic resonance, and we also discover two critical forcing frequencies: one denoting the transformation of the peaks and another for the optimum on stochastic resonance.

Channel Noise from Both Slow Adaptation Currents and Fast Currents Is Required to Explain Spike-Response Variability in a Sensory Neuron

Abstract: Spike-timing variability has a large effect on neural information processing. However, for many systems little is known about the noise sources causing the spike-response variability. Here we investigate potential sources of spike-response variability in auditory receptor neurons of locusts, a classic insect model system. At low-spike frequencies, our data show negative interspike-interval (ISI) correlations and ISI distributions that match the inverse Gaussian distribution. These findings can be explained by a white-noise source that interacts with an adaptation current. At higher spike frequencies, more strongly peaked distributions and positive ISI correlations appear, as expected from a canonical model of suprathreshold firing driven by temporally correlated (i.e., colored) noise. Simulations of a minimal conductance-based model of the auditory receptor neuron with stochastic ion channels exclude the delayed rectifier as a possible noise source. Our analysis suggests channel noise from an adaptation current and the receptor or sodium current as main sources for the colored and white noise, respectively. By comparing the ISI statistics with generic models, we find strong evidence for two distinct noise sources. Our approach does not involve any dendritic or somatic recordings that may harm the delicate workings of many sensory systems. It could be applied to various other types of neurons, in which channel noise dominates the fluctuations that shape the neuron's spike statistics.

On Multiple Antenna Spectrum Sensing Under Noise Variance Uncertainty and Flat Fading

Abstract: We investigate a spectrum sensing method based on asymptotic analysis of the discrete Fourier transform of the received multiantenna signal, possibly non-Gaussian, for flat-fading primary user signals in white noise under noise variance uncertainty. The proposed approach is based on the generalized likelihood ratio test (GLRT) paradigm for a restricted version of the problem obtained by ignoring the spatial structure of the primary users' received signals, and it permits the noise variances to be different at different antennas without requiring knowledge of their values. Simulation examples show the efficacy of the proposed approach compared with the energy detector and some existing time-domain GLRT approaches. A performance analysis of the proposed detector is carried out and verified via simulations. It is also shown that the proposed test statistic is equivalent to an existing time-domain GLRT statistic except that the latter has been derived under the assumption that received signal is Gaussian whereas we make no such assumption.

Robust Finite-Horizon Kalman Filtering for Uncertain Discrete-Time Systems

Abstract: In this note, we propose a design for a robust finite-horizon Kalman filtering for discrete-time systems suffering from uncertainties in the modeling parameters and uncertainties in the observations process (missing measurements). The system parameter uncertainties are expected in the state, output and white noise covariance matrices. We find the upper-bound on the estimation error covariance and we minimize the proposed upper-bound.

DWT and LPC based feature extraction methods for isolated word recognition

Abstract: In this article, new feature extraction methods, which utilize wavelet decomposition and reduced order linear predictive coding (LPC) coefficients, have been proposed for speech recognition. The coefficients have been derived from the speech frames decomposed using discrete wavelet transform. LPC coefficients derived from subband decomposition (abbreviated as WLPC) of speech frame provide better representation than modeling the frame directly. The WLPC coefficients have been further normalized in cepstrum domain to get new set of features denoted as wavelet subband cepstral mean normalized features. The proposed approaches provide effective (better recognition rate), efficient (reduced feature vector dimension), and noise robust features. The performance of these techniques have been evaluated on the TI-46 isolated word database and own created Marathi digits database in a white noise environment using the continuous density hidden Markov model. The experimental results also show the superiority of the proposed techniques over the conventional methods like linear predictive cepstral coefficients, Mel-frequency cepstral coefficients, spectral subtraction, and cepstral mean normalization in presence of additive white Gaussian noise.

Nonwhite Noise Reduction in Hyperspectral Images

Abstract: Noise reduction is an important preprocessing step to analyze the information in the hyperspectral image (HSI). Because the common filtering methods for HSIs are based on the data vectorization or matricization while ignoring the related information between image planes, there are new approaches considering multidimensional data as whole entities, for example, multidimensional Wiener filtering (MWF) based on Tucker3 tensor decomposition. However, if HSIs are not disturbed by white noise, MWF cannot effectively remove the nonwhite noise and obtain the expected signal. To reduce nonwhite noise from HSIs, a new method is proposed in this letter. The first step of this method is to whiten the noise in HSIs through a prewhitening procedure. Then, MWF can help to denoise the prewhitened data. At last, an inverse prewhitening process can rebuild the estimated signal. Comparative studies with existing denoising methods show that the proposed approach has promising prospects in this field.

Stochastic Perturbations to Dynamical Systems: A Response Theory Approach

Abstract: Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.