Probability Distribution Of Noise Research Articles

Improved Student’s t-Based Unscented Filter and its Application to Trajectory Estimation for Maneuvering Target

To estimate the systems with one-step randomly delayed measurement and time-correlated heavy-tailed measurement noises, on the basis of robust Student’s t based unscented filter (RSTUF), an improved Student’s t based unscented filter (ISTUF) is proposed. Referring to the measurement differencing method, a reformed measurement model was built. Then, by augmenting the system state vector, the conditional probability distribution of the measurement noise with respect to delayed measurement was taken into consideration. Based on the reformed measurement model and the augment state vector, a novel estimator was designed to solve the one-step randomly delayed problem. Maneuvering target tracking systems were used for simulation. Compared with unscented Kalman filter (UKF) or RSTUF, ISTUF had higher accuracy.

Finite time stable attitude estimation of rigid bodies with unknown dynamics

AbstractAn almost global finite time stable attitude estimation scheme for a rigid body that does not require the knowledge of the dynamics or the probability distribution of the sensor noise is presented. The attitude of the rigid body is estimated from measurements of at least two linearly independent known vectors and the angular velocity in the body‐fixed frame. The estimation scheme is shown to be almost globally finite time stable using a Lyapunov analysis in the absence of measurement errors. A generalized Wahba's cost function, designed to be a Morse function on SO(3), is used to derive the nonlinear estimation scheme and show its stability. The proposed scheme is discretized as a geometric variational integrator for digital implementation. The stability and convergence of the estimation scheme are shown analytically, and validated through numerical simulations.

Cauchy Matrix Factorization for Tag-Based Social Image Retrieval

User-provided tags associated with social images are essential information for social image retrieval. Unfortunately, these tags are often imperfect to describe the visual contents, which severely degrades the performance of image retrieval. Tag relevance learning models are proposed to improve the descriptive powers of tags mostly based on the Gaussian noise assumption. However, the intrinsic probability distribution of the noise is unknown and other probability distributions may be much better. Towards this end, this paper investigates the applicable probability distributions of tag noise and proposes a novel Cauchy Matrix Factorization (CMF) method for tag-based image retrieval. The Cauchy probability distribution is robust to all kinds of noise and more suitable to model the tagging noise of social images. Therefore, we utilize Cauchy distribution to model noise under the matrix factorization framework. Besides, other five probability density functions, i.e., Gaussian, Laplacian, Poisson, Student-t and Logistic, are investigated to model noise of social tags. To evaluate the performance of different probability distributions, extensive experiments on two widely-used datasets are conducted and results show the robustness of CMF to noisy tags of social images.

Estimation of Noise Using Non-local Regularization Frameworks for Image Denoising and Analysis

In this paper, we propose a novel model which adaptively estimates the noise probability distribution and noise parameters from the input image and restores the data accordingly choosing appropriate regularization model designed for it. In most imaging applications the noise characteristics are assumed prior to the restoration process. This assumption is generally based on the previous experimental study of the images from a specific modality. The adaptive detection of the noise distribution from the data makes it robust and highly suitable for automated signal and image restoration systems. The non-local framework implemented using fast numerical solvers catalyzes the convergence rate of the model. Here we analyze three different noise distributions such as Gamma, Poisson, and Gaussian. Among this Gaussian is additive and source independent, Gamma is multiplicative and source dependent, and finally Poisson is data dependent (neither multiplicative nor additive). The model can be extended to the other source-dependent distributions such as Rayleigh and Rician by appropriately tuning it. The experimental results conform to the assumption regarding the noise distribution and noise parameters estimation capability of the model.

A Bayesian measure of association that utilizes the underlying distributions of noise and information.

We propose a new approach, Bayesian Probability of Association (BPA) which takes into account the probability distributions of information and noise in the variables and uses Bayesian statistics to predict associations better than existing approaches. Our approach overcomes the limitations of linearity of the relationship and normality of the data, assumed by the Pearson correlation coefficient. It is different from the current measures of association because considering information separately from noise helps identify the association in information more accurately, makes the approach less sensitive to noise and also helps identify causal directions. We tested the approach on 15 datasets with no underlying association and on 75 datasets with known causal relationships and compared the results with other measures of association. No false associations were detected and true associations were predicted in more than 90% cases whereas the Pearson correlation coefficient and mutual information content predicted associations for less than half of the datasets.

Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise.

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.

An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers.

Tracking error estimation is of great importance in global navigation satellite system (GNSS) receivers. Any inaccurate estimation for tracking error will decrease the signal tracking ability of signal tracking loops and the accuracies of position fixing, velocity determination, and timing. Tracking error estimation can be done by traditional discriminator, or Kalman filter-based pre-filter. The pre-filter can be divided into two categories: coherent and non-coherent. This paper focuses on the performance improvements of non-coherent pre-filter. Firstly, the signal characteristics of coherent and non-coherent integration—which are the basis of tracking error estimation—are analyzed in detail. After that, the probability distribution of estimation noise of four-quadrant arctangent (ATAN2) discriminator is derived according to the mathematical model of coherent integration. Secondly, the statistical property of observation noise of non-coherent pre-filter is studied through Monte Carlo simulation to set the observation noise variance matrix correctly. Thirdly, a simple fault detection and exclusion (FDE) structure is introduced to the non-coherent pre-filter design, and thus its effective working range for carrier phase error estimation extends from (−0.25 cycle, 0.25 cycle) to (−0.5 cycle, 0.5 cycle). Finally, the estimation accuracies of discriminator, coherent pre-filter, and the enhanced non-coherent pre-filter are evaluated comprehensively through the carefully designed experiment scenario. The pre-filter outperforms traditional discriminator in estimation accuracy. In a highly dynamic scenario, the enhanced non-coherent pre-filter provides accuracy improvements of 41.6%, 46.4%, and 50.36% for carrier phase error, carrier frequency error, and code phase error estimation, respectively, when compared with coherent pre-filter. The enhanced non-coherent pre-filter outperforms the coherent pre-filter in code phase error estimation when carrier-to-noise density ratio is less than 28.8 dB-Hz, in carrier frequency error estimation when carrier-to-noise density ratio is less than 20 dB-Hz, and in carrier phase error estimation when carrier-to-noise density belongs to (15, 23) dB-Hz (26, 50) dB-Hz.

A state estimation approach based on stochastic expansions

This paper presents a new approach for state estimation problems. It is based on the representation of random variables using stochastic functions. Its main idea is to expand the state variables in terms of the noise variables of the system, and then estimate the unnoisy value of the state variables by taking the mean value of the stochastic expansion. Moreover, it was shown that in some situations, the proposed approach may be adapted to the determination of the probability distribution of the state noise. For the determination of the coefficients of the expansions, we present three approaches: moment matching (MM), collocation (COL) and variational (VAR). In the numerical analysis section, three examples are analyzed including a discrete linear system, the Influenza in a boarding school and the state estimation problem in the Hodgkin–Huxley’s model. In all these examples, the proposed approach was able to estimate the values of the state variables with precision, i.e., with very low RMS values.

Experimental and analytic comparison of the accuracy of different estimates of parameters in a linear regression model

We consider LS-, LAD-, R-, M-, S-, LMS-, LTS-, MM-, and HBR-estimates for the parameters of a linear regression model with unknown noise distribution. With computer modeling for medium sized samples, we compare the accuracy of the considered estimates for the most popular probability distributions of noise in a regression model. For different noise distributions, we analytically compute asymptotic efficiencies of LS-, LAD-, R-, M-, S-, and LTS- estimates. We give recommendations for practical applications of these methods for different noise distributions in the model. We show examples on real datasets that support the advantages of robust estimates.

A comprehensive statistical investigation of schlieren image velocimetry (SIV) using high-velocity helium jet

A detailed statistical assessment of seedless velocity measurement using Schlieren Image Velocimetry (SIV) was explored using open source Robust Phase Correlation (RPC) algorithm. A well-known flow field, an axisymmetric turbulent helium jet, was analyzed near and intermediate region $$(0\le x/d\le 20)$$ for two different Reynolds numbers, Re d = 11,000 and Re d = 22,000 using schlieren with horizontal knife-edge, schlieren with vertical knife-edge and shadowgraph technique, and the resulted velocity fields from SIV techniques were compared to traditional Particle Image Velocimetry (PIV) measurements. A novel, inexpensive, easy to setup two-camera SIV technique had been demonstrated to measure high-velocity turbulent jet, with jet exit velocities 304 m/s (Mach = 0.3) and 611 m/s (Mach = 0.6), respectively. Several image restoration and enhancement techniques were tested to improve signal to noise ratio (SNR) in schlieren and shadowgraph images. Processing and post-processing parameters for SIV techniques were examined in detail. A quantitative comparison between self-seeded SIV techniques and traditional PIV had been made using correlation statistics. While the resulted flow field from schlieren with horizontal knife-edge and shadowgraph showed excellent agreement with PIV measurements, schlieren with vertical knife-edge performed poorly. The performance of spatial cross-correlations at different jet locations using SIV techniques and PIV was evaluated. Turbulence quantities like turbulence intensity, mean velocity fields, Reynolds shear stress influenced spatial correlations and correlation plane SNR heavily. Several performance metrics such as primary peak ratio (PPR), peak to correlation energy (PCE), the probability distribution of signal and noise were used to compare capability and potential of different SIV techniques.

The Optimal Noise-Adding Mechanism in Differential Privacy

Differential privacy is a framework to quantify to what extent individual privacy in a statistical database is preserved while releasing useful aggregate information about the database. In this paper, within the classes of mechanisms oblivious of the database and the queries beyond the global sensitivity, we characterize the fundamental tradeoff between privacy and utility in differential privacy, and derive the optimal $\epsilon $ -differentially private mechanism for a single real-valued query function under a very general utility-maximization (or cost-minimization) framework. The class of noise probability distributions in the optimal mechanism has staircase-shaped probability density functions which are symmetric (around the origin), monotonically decreasing and geometrically decaying. The staircase mechanism can be viewed as a geometric mixture of uniform probability distributions, providing a simple algorithmic description for the mechanism. Furthermore, the staircase mechanism naturally generalizes to discrete query output settings as well as more abstract settings. We explicitly derive the parameter of the optimal staircase mechanism for $\ell _{1}$ and $\ell _{2}$ cost functions. Comparing the optimal performances with those of the usual Laplacian mechanism, we show that in the high privacy regime ( $\epsilon $ is small), the Laplacian mechanism is asymptotically optimal as $\epsilon \to 0$ ; in the low privacy regime ( $\epsilon $ is large), the minimum magnitude and second moment of noise are $\Theta (\Delta e^{(-{\epsilon }/{2})})$ and $\Theta (\Delta ^{2} e^{(-{2\epsilon }/{3})})$ as $\epsilon \to +\infty $ , respectively, while the corresponding figures when using the Laplacian mechanism are ${\Delta }/{\epsilon }$ and ${2\Delta ^{2}}/{\epsilon ^{2}}$ , where $\Delta $ is the sensitivity of the query function. We conclude that the gains of the staircase mechanism are more pronounced in the moderate-low privacy regime.

Modelling Dependence between Traffic Noise and Traffic Flow through An Entropy-Copula Method

In this study, an entropy-copula method is proposed for modelling dependence between traffic volume and traffic noise on the Trans-Canada Highway (#1 highway of Canada) in the City of Regina based on a series of field experiment measurements. The proposed entropy-copula method combines the maximum entropy and copula methods into a general framework. The marginal distributions of traffic volume and traffic noise are estimated through the principle of maximum entropy (POME) theory, and the joint probabilities are derived through the Gaussian and Student t copulas. The underlying assumptions of the coupled entropy-copula method are that: i) the entropy variables are mutually independent from each other, and ii) the marginal distributions of traffic flow and traffic noise are continuous. The proposed method is applied to two field experiment sites on the Trans-Canada Highway. Based on the K-S and A-D tests and RMSE value, the entropy method shows well performance in quantifying the probability distributions of traffic volume and traffic noise. Meanwhile, both the Gaussian and Student t copulas can well model the joint probability distributions of the traffic volume and traffic noise at the both experiment sites, which is demonstrated by the CramA©r von Mises statistics and the RMSE value. Furthermore, the conditional CDFs of the traffic noise at the two experiment sites are derived based on the established copulas with respect to different traffic volume scenarios. These conditional CDFs indicate positive structures between traffic volume and traffic noise at the both experiment sites. The conditional PDFs of the traffic noise under different traffic flow scenarios are also generated, indicating the potential reduction effect of traffic noise due to the decrease of the traffic volume. This proposed approach can quantify the dependence between traffic flow and traffic noise, and reveal the inherent uncertain relationship between these two variables. Moreover, the obtained results can provide useful information for traffic noise reduction through traffic flow management.

Analysis of electron multiplying charge coupled device and scientific CMOS readout noise models for Shack–Hartmann wavefront sensor accuracy

In recent years, detectors with subelectron readout noise have been used very effectively in astronomical adaptive optics systems. Here, we compare readout noise models for the two key faint flux level detector technologies that are commonly used: electron multiplying charge coupled device (EMCCD) and scientific CMOS (sCMOS) detectors. We find that in almost all situations, EMCCD technology is advantageous, and that the commonly used simplified model for EMCCD readout is appropriate. We also find that the commonly used simple models for sCMOS readout noise are optimistic, and we recommend that a proper treatment of the sCMOS root mean square readout noise probability distribution should be considered during instrument performance modeling and development.

Effects of stochastic resonance for linear–quadratic detector

In this paper, the stochastic resonance (SR) effects of the fixed linear–quadratic (L–Q) detector in binary hypothesis testing problems is investigated. The effects include the improvement of the overall performance of the detector wherein the output signal-to-noise ratio (SNR) of receiver and the detection probability (PD) are increased and the false-alarm probability (PFA) is reduced at the same time. The SNR of a noise modified detector is derived. Since the improvement of the SNR does not mean the improvement of detection performance, the noise enhanced detection performance in terms of PFA and PD are discussed graphically and elaborated through an example of Gaussian background noise. Furthermore, the conditions are deduced to determine whether the overall performance of the detector can be improved or not. In addition, the form of the suitable noise probability distribution function (PDF) is determined and the PDF of noise that satisfies the conditions to improve the overall performance of detector are derived. Finally, an illustrative example is presented to verify the theoretical results.

Closed-Form Relations for Resonance Detection Error Using Statistical Analysis of Amplitude Noise

The optimization of resonance-tracking sensors relies critically on proper estimation of resonance detection error. We study this error in two common resonance detection algorithms: the absolute minimum method and the linear regression method. Closed-form relations for the error originating from additive noise are presented. The formulation accommodates the majority of lineshapes of practical interest and a wide variety of noise statistics. Lorentzian and Fano line shapes are studied here with further detail for their practical importance. It is discussed that while the performance of the absolute minimum method depends on the tail of the noise probability distribution function, for the linear regression method the total noise power is the determining characteristic of the noise. This fact is explained for the specific case of quantization noise. The results of this study enable a quantitative comparison of the performance of resonance-based sensors, which is a center piece in the optimization of their limit of detection.

Target tracking in a collaborative sensor network

In a collaborative sensor network (CSN), the conventional target tracking algorithms employed are Kalman filtering (KF) or extended Kalman filtering (EKF). However, these techniques have a presumed probability distribution of the system noise and prediction noise. They also need some a priori information that may be unavailable in some circumstances. Therefore, the system is not flexible for a complicated scenario. With the help of a machine learning technique called expert prediction (EP), a novel target tracking approach for CSNs is developed. This scheme makes use of the aforementioned EP in parameter estimation course for the target of interest, instead of exploiting the filtering method as typically found in available literature. This idea is further unfolded with comparisons regarding the CSN using Kalman filters, extended Kalman filters, and decentralized sigma-point information filters (DSPIFs). The new tracking algorithm is investigated with both linear and nonlinear prediction methods. Simulation results demonstrate that this proposed measure will deliver forecasting output with more precision because of the built-in multimodel mode among different experts, the learning ability, and the self-perfection characteristic. Not only does this performance occur in a more robust way than those of the existing approaches - particularly in the presence of heavy clutter, highly maneuvering targets, and/or multiple targets - but it simultaneously requires the least a priori information and imposes the least limitation on the observation model.

Set-membership identification and fault detection using a Bayesian framework

This paper deals with the problem of set-membership identification and fault detection using a Bayesian framework. The paper presents how the set-membership model estimation problem can be reformulated from the Bayesian viewpoint in order to, first, determine the feasible parameter set in the identification stage and, second, check the consistency between the measurement data and the model in the fault-detection stage. The paper shows that, assuming uniform distributed measurement noise and uniform model prior probability distributions, the Bayesian approach leads to the same feasible parameter set than the well-known set-membership technique based on approximating the feasible parameter set using sets. Additionally, it can deal with models that are nonlinear in the parameters. The single-output and multiple-output cases are addressed as well. The procedure and results are illustrated by means of the application to a quadruple-tank process.

Statistical Characters Analysis of LF Atmospheric Noise

In low frequency (LF) communication, the main factor of affecting property of LF communication system is the atmospheric noise caused by lightning phenomenon. Amplitude probability distribution of this atmospheric noise is with non-Gaussian characters seriously. Due to influence of various factors, such as occurrence time, physical location and seasons, making analysis of LF atmospheric noise on the receiver communication performance becomes more complex. Therefore, the study of characteristics of atmospheric noise amplitude probability distribution is necessary. In this paper, Firstly, we are on the assumption that the amplitude probability distribution characteristics of atmospheric noise obey Alpha stable distribution; then introduce the Alpha stable distribution model; Finally, we use probability distribution function (pdf) and Quantile-Quantile (Q-Q) plot fitting for Alpha stable distribution according to real LF channel noise data acquired by a high sensitive superconducting quantum interference device (SQUID). Simulation results show the assumption is correct.

Local trajectory parameters estimation and detection of moving targets in rayleigh noise

The problem of detection of moving targets and estimation of local trajectory parameters based on the analysis of sensor data in the form of two-dimensional image is considered. In accordance with the target and sensor models, probability distribution of noise at the output of the detector is Rayleigh distribution, while probability distribution of signal is Rice distribution. Two trajectory parameters estimation techniques are considered: ordinary least squares and Hough transform. A detection stage based on the integration of an input signal along estimated trajectory is proposed. Statistical modeling was performed and detection characteristics were obtained.

Signal Detection in Generalized Gaussian Noise by Nonlinear Wavelet Denoising

In this paper, a nonlinear suboptimal detector whose performance in heavy-tailed noise is significantly better than that of the matched filter is proposed. The detector consists of a nonlinear wavelet denoising filter to enhance the signal-to-noise ratio, followed by a replica correlator. Performance of the detector is investigated through an asymptotic theoretical analysis as well as Monte Carlo simulations. The proposed detector offers the following advantages over the optimal (in the Neyman-Pearson sense) detector: it is easier to implement, and it is more robust with respect to error in modeling the probability distribution of noise.