Lower Order Statistics Research Articles

Statistical evaluation of road traffic noise in complex sound environment by using fuzzy inference

In order to evaluate road traffic noise in a sound environment, it is necessary to predict the sound levels at evaluation points based on the observations at a reference point. In actual sound environment, road traffic noise shows various types of probability distribution of non-Gaussian distribution and nonstationary property. Furthermore, there potentially exist various nonlinear correlations in addition to the linear correlation between the sound levels at evaluation and reference points. Consequently, the relationship between both points cannot be represented by a simple linear model based on only the linear correlation and lower order statistics. In this study, a complex sound environment difficult to analyze by using usual structural method is considered. By introducing a nonlinear model based on conditional probability distribution with various correlations between the sound levels, and applying fuzzy inference, an evaluation method for road traffic noise is theoretically proposed in a suitable form for the complex sound environment with nonlinear, non-Gaussian and nonstationary properties. The effectiveness of the proposed method is experimentally confirmed by applying it to the observed road traffic noise data in a complex sound environment.

Analyzing symmetric distributions by utilizing extropy measures based on order statistics

Quantification of the uncertainty of distribution functions, by using entropy and extropy, is important in many statistical analyses. Inspired by this, our study uses extropy and several related measures (including the cumulative residual extropy, cumulative past extropy, and extropy-inaccuracy measure) of order statistics (OSs) to offer multiple characterizations of symmetric continuous distributions. We demonstrate that a defining feature of symmetric distributions is the equality of these measures of upper and lower OSs. Using concomitants of OSs based on the bivariate distributions belonging to the Farlie-Gumbel-Morgenstern (FGM) family, the same characteristic is demonstrated for these measures. Finally, a real data set is used to illustrate the applicability of the suggested test.

Cumulative entropies and sums of moments of order statistics

Generalized weighted cumulative residual entropies and generalized weighted cumulative entropies are represented by means of weighted sums of moments of upper and lower order statistics, respectively. A variety of examples is shown by specifying the weight function.

UFNet: Joint U-Net and Fully Connected Neural Network to Bias Correct Precipitation Predictions from Climate Models

Abstract Precipitation values produced by climate models are biased due to the parameterization of physical processes and limited spatial resolution. Current bias-correction approaches usually focus on correcting lower-order statistics (mean and standard deviation), which make it difficult to capture precipitation extremes. However, accurate modeling of extremes is critical for policymaking to mitigate and adapt to the effects of climate change. We develop a deep learning framework, leveraging information from key dynamical variables impacting precipitation to also match higher-order statistics (skewness and kurtosis) for the entire precipitation distribution, including extremes. The deep learning framework consists of a two-part architecture: a U-Net convolutional network to capture the spatiotemporal distribution of precipitation and a fully connected network to capture the distribution of higher-order statistics. The joint network, termed UFNet, can simultaneously improve the spatial structure of the modeled precipitation and capture the distribution of extreme precipitation values. Using climate model simulation data and observations that are climatologically similar but not strictly paired, the UFNet identifies and corrects the climate model biases, significantly improving the estimation of daily precipitation as measured by a broad range of spatiotemporal statistics. In particular, UFNet significantly improves the underestimation of extreme precipitation values seen with current bias-correction methods. Our approach constitutes a generalized framework for correcting other climate model variables which improves the accuracy of the climate model predictions, while utilizing a simpler and more stable training process.

Characterizations of continuous log-symmetric distributions based on properties of order statistics

The class of log-symmetric distributions is a generalization of log-normal distribution and includes some well-known distributions such as log-normal, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-student-t, log-slash, and Birnbaum-Saunders distributions. In this paper, several characterization results are obtained for log-symmetric distributions based on moments of some functions of the parent distribution and also on the basis of some properties of order statistics. Specifically, when X is identical in distribution with a decreasing continuous function h ( X ) , then a relationship is established between upper and lower order statistics which is then utilized to construct characterization results for log-symmetric distributions in terms of functions of order statistics. The established results can be used for constructing a goodness-of-fit test for log-symmetric distributions.

Robust multimodulus blind equalization algorithm for multilevel QAM signals in impulsive noise

As a blind equalization algorithm, the fractional lower-order statistics based multimodulus algorithm (FLOS-MMA) is suitable for equalizing multilevel quadrature amplitude modulation (QAM) signals in impulsive noise. However, the equalization performance of FLOS-MMA is severely degraded in strong impulsive noise because the first-order gradient of the cost function constrains the ability of FLOS to suppress large outliers in the equalizer output signal and is sensitive to large outliers in the equalizer input signal. To enhance the robustness of multimodulus blind equalization algorithm against impulsive noise, a FLOS based momentum fractional-order multimodulus algorithm (FLOS-MFOMMA) is proposed in this paper. The proposed algorithm mitigates the adverse effects of impulsive noise on the updating of equalizer coefficients by utilizing the fractional-order gradient and accelerates the convergence speed by incorporating a momentum term. Then, the steady-state excess mean square error and the computational complexity of FLOS-MFOMMA are analyzed theoretically. Finally, numerical simulations, for equalizing 16-QAM signals under impulsive noise environments, show that the proposed algorithm is superior to FLOS-MMA in terms of robustness and convergence speed.

Modeling Lower-Order Statistics to Enable Decoy-Free FDR Estimation in Proteomics.

One of the chief objectives in mass spectrometry-based peptide identification in proteomics is the statistical validation of top-scoring peptide-spectrum matches (PSMs) in the form of false discovery rate (FDR) estimation. Existing methods construct a null model that captures the characteristics of incorrect target PSMs to estimate the FDR, most often with the help of decoys. Decoy-based methods, however, increase the computational cost and rely on the difficult-to-verify assumption that decoy PSMs constitute a sufficient and representative sample of the population of possible incorrect target PSMs. On the other hand, the possibility of FDR estimation assisted by the plentiful non-top-scoring PSMs, which are almost always incorrect, has been scarcely explored. In this work, we propose a novel decoy-free procedure for developing null models for top-scoring PSMs using the transformed e-value (TEV) score and the distributions of non-top-scoring target PSMs. The method relies on a theoretically derivable relationship between the parameters of the distributions of lower-order statistics of the TEV score and a necessary empirical optimization to fit a single parameter to actual data. The framework was tested on multiple different data sets and two search engines. We present evidence that our method is comparable to and occasionally outperforms popular decoy-free and decoy-based methods in FDR estimation.

Leakage Detection and Localization of Water Pipeline Using Multi-features and Adaptive Time Delay Estimation

The leakage of water in pipelines severely affects the environment and economy. However, there are limitations in the effectiveness of existing leak detection and localization techniques and methodologies. In this paper, we propose a novel leakage detection and localization method based on the multiple time-frequency features, a neural network, and an adaptive time delay estimation algorithm. First, we use spectral subtraction and wavelet denoising to reduce the effects of noise. In addition, to ensure and improve the accuracy of leakage detection in complex realistic environments, we propose the use of multi time-frequency features that can comprehensively represent the leak signal and make the neural network more robust to train a radial basis function (RBF)neural network to detect the leak signal. Further, we extract multiple features of the leakage signal and input into the RBF neural network to train. Moreover, to prevent the impulsive components of environmental noise and improve localization accuracy, we further propose the use of a fractional lower-order statistics (FLOS) based adaptive time delay estimation algorithm to estimate the time delay and locate the leakage. The simulation results show that the detection and localization performance of the proposed method is superior to those of existing schemes.

Studying Interstellar Turbulence Driving Scales Using the Bispectrum

We demonstrate the utility of the bispectrum, the Fourier three-point correlation function, for studying driving scales of magnetohydrodynamic (MHD) turbulence in the interstellar medium. We calculate the bispectrum by implementing a parallelized Monte Carlo direct measurement method, which we have made publicly available. In previous works, the bispectrum has been used to identify nonlinear scaling correlations and break degeneracies in lower-order statistics like the power spectrum. We find that the bicoherence, a related statistic which measures phase coupling of Fourier modes, identifies turbulence-driving scales using density and column density fields. In particular, it shows that the driving scale is phase-coupled to scales present in the turbulent cascade. We also find that the presence of an ordered magnetic field at large scales enhances phase coupling as compared to a pure hydrodynamic case. We therefore suggest the bispectrum and bicoherence as tools for searching for non-locality for wave interactions in MHD turbulence.

A higher-order moment matching based fine-grained adversarial domain adaptation method for intelligent bearing fault diagnosis

Due to the discrepancy in data distribution caused by the time-varying working conditions, intelligent diagnostic methods fail to achieve accurate fault classification in engineering scenarios. This paper presents a novel higher-order moment matching-based adversarial domain adaptation method (HMMADA) for intelligent bearing fault diagnosis. First, a deep one-dimensional convolution neural network is constructed as the feature extractor to learn the discriminative features of each category through different domains. Then, the distribution discrepancy across domains is significantly reduced by using joint higher-order moment statistics (HMS) and adversarial learning. In particular, HMS integrates the first-order and second-order statistics into a unified framework and achieves a fine-grained distribution adaptation between different domains. Finally, the feasibility and effectiveness of HMMADA are validated by several transfer experiments constructed on two different bearing datasets. The results demonstrate that HMS is more effective than lower-order statistics.

The roles of lower- and higher-order surface statistics in tactile texture perception.

Humans can haptically discriminate surface textures when there is a significant difference in the statistics of the surface profile. Previous studies on tactile texture discrimination have emphasized the perceptual effects of lower-order statistical features such as carving depth, inter-ridge distance, and anisotropy, which can be characterized by local amplitude spectra or spatial-frequency/orientation subband histograms. However, the real-world surfaces we encounter in everyday life also differ in the higher-order statistics, such as statistics about correlations of nearby spatial-frequencies/orientations. For another modality, vision, the human brain has the ability to use the textural differences in both higher- and lower-order image statistics. In this work, we examined whether the haptic texture perception can use higher-order surface statistics as visual texture perception does, by three-dimensional (3-D)-printing textured surfaces transcribed from different "photos" of natural scenes such as stones and leaves. Even though the maximum carving depth was well above the haptic detection threshold, some texture pairs were hard to discriminate. Specifically, those texture pairs with similar amplitude spectra were difficult to discriminate, which suggests that the lower-order statistics have the dominant effect on tactile texture discrimination. To directly test the poor sensitivity of the tactile texture perception to higher-order surface statistics, we matched the lower-order statistics across different textures using a texture synthesis algorithm and found that haptic discrimination of the matched textures was nearly impossible unless the stimuli contained salient local features. We found no evidence for the ability of the human tactile system to use higher-order surface statistics for texture discrimination.NEW & NOTEWORTHY Humans can discriminate subtle spatial patterns differences in the surrounding world through their hands, but the underlying computation remains poorly understood. Here, we 3-D-printed textured surfaces and analyzed the tactile discrimination performance regarding the sensitivity to surface statistics. The results suggest that observers have sensitivity to lower-order statistics whereas not to higher-order statistics. That is, touch differs from vision not only in spatiotemporal resolution but also in (in)sensitivity to high-level surface statistics.

Recursive dynamic functional connectivity reveals a characteristic correlation structure in human scalp EEG

Time-varying neurophysiological activity has been classically explored using correlation based sliding window analysis. However, this method employs only lower order statistics to track dynamic functional connectivity of the brain. We introduce recursive dynamic functional connectivity (rdFC) that incorporates higher order statistics to generate a multi-order connectivity pattern by analyzing neurophysiological data at multiple time scales. The technique builds a hierarchical graph between various temporal scales as opposed to traditional approaches that analyze each scale independently. We examined more than a million rdFC patterns obtained from morphologically diverse EEGs of 2378 subjects of varied age and neurological health. Spatiotemporal evaluation of these patterns revealed three dominant connectivity patterns that represent a universal underlying correlation structure seen across subjects and scalp locations. The three patterns are both mathematically equivalent and observed with equal prevalence in the data. The patterns were observed across a range of distances on the scalp indicating that they represent a spatially scale-invariant correlation structure. Moreover, the number of patterns representing the correlation structure has been shown to be linked with the number of nodes used to generate them. We also show evidence that temporal changes in the rdFC patterns are linked with seizure dynamics.

Robust Signals Detection Algorithm Based on Cyclostationarity in Impulsive Noise

A robust method for detecting the communication signals impinging on an antenna with interference and non-Gaussian impulsive noise is introduced in this paper. Degradation of the conventional cyclic detector which based on max-output-SNR criterion in impulsive noise is shown both theoretically and experimentally. By fusing second-order cyclostationarity and fractional lower-order statistics, a type of cyclic fractional lower-order statistics is developed which is defined for exploiting cyclostationarity property. Then, a new robust type of detection algorithm is developed using the theory of optimal filtering based on max-output-SNR criterion and alpha-stable distribution, including the fractional lower-order cyclic matched filter, which is formulated for detecting the communication signals in the presence of interference and non-Gaussian alpha-stable distribution impulsive noise. It is shown that the new method is robust to Gaussian and non-Gaussian impulsive noises, and is immune to the interfering signals which occupy the same spectral band as that of the received signal. Simulation results show the robustness and effectiveness of the proposed algorithm.

Threshold selection and trimming in extremes

We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. Trajectories of these trimmed statistics as a function of the extent of trimming turn out to be quite flat near the optimal threshold value. For the regularly varying case, the classical threshold selection problem in tail estimation is then revisited, both visually via trimmed Hill plots and, for the Hall class, also mathematically via minimizing the expected empirical variance. This leads to a simple threshold selection procedure for the classical Hill estimator which circumvents the estimation of some of the tail characteristics, a problem which is usually the bottleneck in threshold selection. As a by-product, we derive an alternative estimator of the tail index, which assigns more weight to large observations, and works particularly well for relatively lighter tails. A simple ratio statistic routine is suggested to evaluate the goodness of the implied selection of the threshold. We illustrate the favourable performance and the potential of the proposed method with simulation studies and real insurance data.

Dimensional reduction of direct statistical simulation

Direct statistical simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by direct numerical simulation (DNS). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low-order statistics typically have higher dimension than the underlying fields. Here we show that it is in some cases possible to go much further by using a form of unsupervised learning, proper orthogonal decomposition, to address the ‘curse of dimensionality’. We apply proper orthogonal decomposition directly to DSS in the form of expansions in equal-time cumulants to second order. We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models of fluid on a rotating sphere (a jet that relaxes deterministically towards an unstable profile and a stochastically driven flow that spontaneously organizes into jets). We show that the method offers the possibility of parameter continuation, in the reduced basis, for the lower-order statistics of the flow. Order-of-magnitude savings in computational cost are sometimes obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS.

Characterizations of symmetric distributions using equi-distributions and moment properties of functions of order statistics

Several characterizations of symmetry of a probability distribution are provided. These include the equality in distributions of symmetrically chosen upper and lower order statistics, symmetric spacings of order statistics, and quasi-midranges. Characterizations of symmetry based on the moment properties of order statistics, quasi-midranges and spacings are established. Some characterizations in terms of moments of the underlying distribution are also given.

$${\mathcal {L}}_p$$-norm of the LMS algorithm for bilinear forms with $$\alpha $$-stable processes

In this paper, a novel bilinear algorithm is proposed for multiple-input–single-output (MISO) system identification, which is based on a modified $${\mathcal {L}}_p$$-norm cost function with fractional lower order statistics. To model the MISO system, we employ the bilinear form (BF) which is defined with respect to the impulse responses of a spatiotemporal model. Considering the non-Gaussian behavior in practical MISO system with BF, $$\alpha $$-stable distribution, whose density function decays in the tails less rapidly than the Gaussian density function, is used to model the interference noise. As an added contribution, we extend the least mean pth power (LMP) and normalized LMP (NLMP) algorithms to BF, resulting in the LMP-BF and NLMP-BF algorithms. Simulation results demonstrate the effectiveness of the proposed algorithm.

Cyclostationarity-based DOA estimation algorithms for coherent signals in impulsive noise environments

Estimating direction of arrival (DOA) is important in a variety of practical applications. Conventional cyclostationarity-based coherent DOA estimation algorithms are not robust to non-Gaussian α-stable impulsive noise. Additionally, fractional lower-order statistics (FLOS)-based algorithms are tolerant to impulsive noise; however, they experience performance degradation for coherent signals and interference. To overcome these drawbacks, two types of fractional lower-order cyclostationarity-based subspace DOA estimation methods are proposed for coherent signals in the presence of interference and α-stable impulsive noise. The new proposed algorithms exploit the fractional lower-order cyclostationarity properties of the signals and are immune to the impulsive noise and interference. Moreover, they can provide more accurate DOA estimates of coherent signals than conventional cyclostationarity-based and FLOS-based methods. The simulation results illustrate the robustness and effectiveness of the proposed methods for coherent signals based on a comparison with traditional methods. The new algorithms can be used in the presence of a wide range of interference, Gaussian noise, and α-stable distribution impulsive noise environments.

Joint Estimation of Doppler Stretch and Time Delay of Wideband Echoes for LFM Pulse Radar Based on Sigmoid-FRFT Transform under the Impulsive Noise Environment

To overcome the limitation of performance degradation of existing methods based on fractional Fourier transform in impulsive noise, and fractional lower-order statistics based method dependence on a priori knowledge of the noise, a novel Sigmoid fractional Fourier transform (Sigmoid-FRFT) is presented in this paper. This novel approach is then used to estimate the Doppler stretch and time delay. Furthermore, the properties of the Sigmoid transform, robustness and boundedness of the Sigmoid-FRFT to the S α S noise, and the computation complexity of the Sigmoid-FRFT method are presented to evaluate the performance of the proposed method. Simulation results and theoretical analysis are presented to demonstrate the applicability of the forgoing method. It is shown that the proposed method not only can effectively suppress impulsive noise interference but also does not need a priori knowledge of the noise, with higher estimation accuracy and lower computational complexity in impulsive noise environments.

Characterization of symmetric distributions based on concomitants of ordered variables from FGMs family of bivariate distributions

Several characterization results of a symmetric distribution based on concomitants of order statistics as well as k-records from Farlie-Gumbel-Morgenstern (FGM) family of bivariate distributions are established. These include characterizations of a symmetric distribution on the basis of equality in distribution, moments, R?nyi and Tsallis entropies of concomitants of upper and lower order statistics, also in terms of the same properties of concomitants of upper and lower k-records.