Higher Order Cumulants Research Articles

An High Efficiency Angle and Range Estimation Method for Three-Dimensional Near-Field Sources Based on Higher-Order Cumulants

An High Efficiency Angle and Range Estimation Method for Three-Dimensional Near-Field Sources Based on Higher-Order Cumulants

A Cumulant-Based Method for Acquiring GNSS Signals.

Global Navigation Satellite Systems (GNSS) provide positioning, velocity, and time services for civilian applications. A critical step in the positioning process is the acquisition of visible satellites in the sky. Modern GNSS systems, such as Galileo-developed and maintained by the European Union-utilize a new modulation technique known as Binary Offset Carrier (BOC). However, BOC signals introduce multiple side-peaks in their autocorrelation function, which can lead to significant errors during the acquisition process. In this paper, we propose a novel acquisition method based on higher-order cumulants that effectively eliminates these side-peaks. This method is capable of simultaneously acquiring both conventional ranging signals, such as GPS C/A code, and BOC-modulated signals. The effectiveness of the proposed method is demonstrated through the acquisition of simulated signals, with a comparison to traditional methods. Additionally, we apply the proposed method to real satellite signals to further validate its performance. Our results show that the proposed method successfully suppresses side-peaks, improves acquisition accuracy in weak signal environments, and demonstrates potential for indoor GNSS applications. The study concludes that while the method may increase computational load, its performance in challenging conditions makes it a promising approach for future GNSS receiver designs.

Measurements of higher-order cumulants of multiplicity and net-electric charge distributions in inelastic proton–proton interactions by NA61/SHINE

This paper presents the energy dependence of multiplicity and net-electric charge fluctuations in p+p interactions at beam momenta 20, 31, 40, 80, and 158 GeV/c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{ Ge }\\hspace{-1.00006pt}\ ext{ V }\\!/\\!c$$\\end{document}. Results are corrected for the experimental biases and quantified with the use of cumulants and factorial cumulants. Cumulant ratios are an essential tool in the search for the critical point of strongly interacting matter in heavy ion collisions. Measurements performed in p+p interactions provide a vital baseline estimation in these studies. The measured signals are compared with the string hadronic models Epos1.99 and FTFP-BERT.

Novel filtering and regeneration technique with statistical feature extraction and machine learning for automatic modulation classification

Automatic modulation classification (AMC) plays a crucial role in the stages of processing signals from unknown sources and monitoring the airwaves. This paper presents an AMC method based on machine learning (ML) using constellation diagrams, distribution test function and high-order cumulants (HOCs) and novel filtering and regeneration technique. A data clustering approach based on the signal phase characteristics is utilized to perform filtering, regeneration, and testing distribution functions. NI PXIe-1065 was used as a source of modulated signals, which is a multifunctional experimental complex for generating signals with various types of modulation. In this work, 5 types of modulation schemes were employed, specifically BPSK, QPSK, PSK-8, PSK-16 and PSK-32. The proposed classification method consists of three main stages. At the first stage, the constellation diagram is partitioned into clusters. At the second stage, the phase distribution function of signals within each cluster is analyzed, HOCs were calculated and novel filtering and regeneration techniques were employed. The main concept of the proposed approach is compressing angular ranges depicted on the constellation diagram. This involves filtering points and subsequently regenerating, which means relocating points that fall within adjacent phase ranges. Also, a statistical analysis of the phase transition distribution function in clusters is conducted at this step. At the last stage the obtained data set was trained using XGBOOST ML method. The proposed approach shows excellent results and can be a strong contender to existing NN based AMC methods, achieving 95 % accuracy at an SNR value of 9 dB.

Measurements of charged-particle multiplicity dependence of higher-order net-proton cumulants in p + p collisions at [formula omitted] GeV from STAR at RHIC

We report on the charged-particle multiplicity dependence of net-proton cumulant ratios up to sixth order from s=200 GeV p+p collisions at the Relativistic Heavy Ion Collider (RHIC). The measured ratios C4/C2, C5/C1, and C6/C2 decrease with increased charged-particle multiplicity and rapidity acceptance. Neither the Skellam baselines nor PYTHIA8 calculations account for the observed multiplicity dependence. In addition, the ratios C5/C1 and C6/C2 approach negative values in the highest-multiplicity events, which implies that thermalized QCD matter may be formed in p+p collisions.

Statistical blade tip timing measurement, Part II: High-order cumulant architecture

Blade tip timing (BTT) is a potential non-contact vibration measurement technique for rotating blades owing to its high efficiency and long service life. However, probe layout design and vibration parameter identification are critical challenges in BTT due to its inherent undersampling. To address this issue, we have proposed the concept of statistical BTT measurement which promotes efficient layouts and parameter identification methods by shifting the recovery target from the signal to itself to its statistics. In the first paper of series papers, we have developed covariance architecture for BTT measurement and signal post-processing. In this second part of series papers, we set out to develop BTT measurement from a higher-order statistic perspective. Specifically, we find that the 2qth-order cumulant retains the vibration information (frequency and amplitude) of the signal itself and it can be recovered at lower sampling rates than what is prescribed by the covariance architecture. On this basis, we propose a higher-order cumulant architecture for BTT measurement, which contains two aspects: higher-order difference layout (HODL) and higher-order cumulant-based parameter identifications. HODL is a special family of layouts that physically guarantee the availability of consecutive higher-order cumulant estimations and thus leads to a satisfactory parameter identification. To facilitate application, a series of concrete HODLs are provided for users, including a four-level nested layout and its enhanced version, and optimal fourth-order difference layout. The quantitative comparison in the simulations and experiments shows the effectiveness of the proposed higher-order cumulant architecture. Compared with the architectures developed from covariance and signal itself perspective, the proposed higher-order cumulant architecture is not only more robust to noise but also further reduces the number of required probes.

Probability density function of roll amplitude for parametric rolling using moment equation

For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases.

Open-loop quantum control of small-size networks for high-order cumulants and cross-correlations sensing

Quantum control techniques are one of the most efficient tools for attaining high-fidelity quantum operations and a convenient approach for quantum sensing and quantum noise spectroscopy. In this work, we investigate dynamical decoupling while processing an entangling two-qubit gate based on an Ising-xx interaction, each qubit affected by pure dephasing classical correlated 1/f-noises. To evaluate the gate error, we used the Magnus expansion introducing generalized filter functions that describe decoupling while processing and allow us to derive an approximate analytic expression as a hierarchy of nested integrals of noise cumulants. The error is separated in contributions of Gaussian and non-Gaussian noise, with the corresponding generalized filter functions calculated up to the fourth order. By exploiting the properties of selected pulse sequences, we show that it is possible to extract the second-order statistics (spectrum and cross-spectrum) and to highlight non-Gaussian features contained in the fourth-order cumulant. We discuss the applicability of these results to state-of-the-art small networks based on solid-state platforms.

Higher-Order Spectral Analysis Combined with a Convolution Neural Network for Atrial Fibrillation Detection-Preliminary Study.

The global burden of atrial fibrillation (AFIB) is constantly increasing, and its early detection is still a challenge for public health and motivates researchers to improve methods for automatic AFIB prediction and management. This work proposes higher-order spectra analysis, especially the bispectrum of electrocardiogram (ECG) signals combined with the convolution neural network (CNN) for AFIB detection. Like other biomedical signals, ECG is non-stationary, non-linear, and non-Gaussian in nature, so the spectra of higher-order cumulants, in this case, bispectra, preserve valuable features. The two-dimensional (2D) bispectrum images were applied as input for the two CNN architectures with the output AFIB vs. no-AFIB: the pre-trained modified GoogLeNet and the proposed CNN called AFIB-NET. The MIT-BIH Atrial Fibrillation Database (AFDB) was used to evaluate the performance of the proposed methodology. AFIB-NET detected atrial fibrillation with a sensitivity of 95.3%, a specificity of 93.7%, and an area under the receiver operating characteristic (ROC) of 98.3%, while for GoogLeNet results for sensitivity and specificity were equal to 96.7%, 82%, respectively, and the area under ROC was equal to 96.7%. According to preliminary studies, bispectrum images as input to 2D CNN can be successfully used for AFIB rhythm detection.

Irreversibility of stochastic state transitions in Langevin systems with odd elasticity.

Active microscopic objects, such as an enzyme molecule, are modeled by the Langevin system with the odd elasticity, in which energy injection from the substrate to the enzyme is described by the antisymmetric part of the elastic matrix. By applying the Onsager-Machlup integral and large deviation theory to the Langevin system with odd elasticity, we can calculate the cumulant generating function of the irreversibility of the state transition. For an N-component system, we obtain a formal expression of the cumulant generating function and demonstrate that the oddness λ, which quantifies the antisymmetric part of the elastic matrix, leads to higher-order cumulants that do not appear in a passive elastic system. To demonstrate the effect of the oddness under the concrete parameter, we analyze the simplest two-component system and obtain the optimal transition path and cumulant generating function. The cumulants obtained from expansion of the cumulant generating function increase monotonically with the oddness. This implies that the oddness causes the uncertainty of stochastic state transitions.

What Insights Do Short-Maturity (7DTE) Return Predictive Regressions Offer about Risk Preferences in the Oil Market?

In this study, we investigate the ability of three higher-order risk-neutral return cumulants to predict short maturity (weekly) returns of oil futures. Our data includes weekly West Texas Crude Oil futures options that expire in 7 days (7DTE). Using a model-free approach, we estimate these risk-neutral return cumulants at the beginning of each options expiration cycle. Our results suggest that the third risk-neutral return cumulant consistently predicts the returns of various oil futures (including WTI, Brent, Dubai, Heating Oil, and RBOB Gasoline). We compare our findings with 14 other predictors and offer a theoretical explanation for the negative coefficient observed for the 7DTE third risk-neutral return cumulant. Our theory connects higher-order risk-neutral return cumulants with the risk premiums of oil futures. Furthermore, our quantitative investment strategy favors the predictability of oil futures returns.

Heavy tails and probability density functions to any nonlinear order for the surface elevation in irregular seas

The probability density function (PDF) for the free surface elevation in an irregular sea has an integral formulation when based on the cumulant generating function. To leading order, the result is Gaussian, whereas nonlinear extensions have long been limited to Gram–Charlier series approximations. As shown recently by Fuhrman et al. (J. Fluid Mech., vol. 970, 2023, A38), however, the second-order integral can be represented exactly in closed form. The present work extends this further, enabling determination of this PDF to even higher orders. Towards this end, a new ordinary differential equation (ODE) governing the PDF is first derived. Asymptotic solutions in the limit of large surface elevation are then found, utilizing the method of dominant balance. These provide new analytical forms for the positive tail of the PDF beyond second order. These likewise clarify how high-order cumulants (involving statistical moments such as the kurtosis) govern the tail, which is shown to get heavier with each successive order. The asymptotic solutions are finally utilized to generate boundary conditions, such that the governing ODE may be solved numerically, enabling novel determination of the PDF at third and higher order. Successful comparisons with challenging data sets confirm accuracy. The methodology thus enables the PDF of the surface elevation to be determined numerically, and the asymptotic tail analytically, to any desired order. Results are worked out explicitly up to fifth order. The theoretical probability of extreme surface elevations (typical of rogue waves) may thus be assessed quantitatively for highly nonlinear irregular seas, requiring only relevant statistical quantities as input.

Communication signal detection based on high-order cumulants time-frequency analysis: on the application of deep learning YOLOV5 network

The detection of communication signals in heterogeneous electromagnetic environments currently relies primarily on a one-dimensional statistical feature threshold method. However, this approach is highly sensitive to dynamic changes in the environment, fluctuations in signal-to-noise ratios, and complex noise. To address these limitations, this paper proposes a novel time-frequency diagram based on high-order accumulation for signal detection. Traditional time-frequency diagrams suffer from poor noise suppression ability and unclear features. However, higher-order cumulants can effectively overcome these shortcomings. Currently, methods based on higher-order cumulants are typically limited to one-dimensional signals. Yet, two-dimensional time-frequency signal diagrams can represent a broader array of features. This paper employs higher-order accumulation to extract time-frequency features from the received signal, thereby transforming the conventional radio detection problem into an image recognition challenge. By merging the advantages of higher-order accumulations and time-frequency diagrams, we propose the use of higher-order accumulation time-frequency diagrams for signal detection. Extensive experimental simulations demonstrate that the proposed time-frequency diagram exhibits strong anti-noise performance and effectively suppresses frequency bias from multiple perspectives. The performance of the Higher-Order Cumulant-Time Frequency (HOC-TF) indicated lower Root Mean Square Error (RMSE) compared with the Short-Time Fourier Transform-Time Frequency (STFT-TF) and Wavelet Transform-Time Frequency (WT-TF). Additionally, compared to the STFT-TF and WT-TF methodologies, the novel time-frequency diagram introduced demonstrates superior stability using the Singular Value Decomposition (SVD) method. Moreover, by combining the new time-frequency diagram with the deep learning YOLOV5 network, signal detection and modulation identification of communication signals can be achieved.

Multiparticle integral and differential correlation functions

This paper formalizes the use of integral and differential cumulants for measurements of multiparticle event-by-event transverse momentum fluctuations, rapidity fluctuations, as well as net-charge fluctuations. This enables the introduction of multiparticle balance functions, defined based on differential correlation functions (factorial cumulants), that suppress two- and three-prong resonance decays effects and enable measurements of underlying long-range correlations obeying quantum number conservation constraints. These multiparticle balance functions satisfy simple sum rules determined by quantum number conservation. It is additionally shown that these multiparticle balance functions arise as an intrinsic component of high-order net-charge cumulants. This implies that the magnitude of these cumulants, measured in a specific experimental acceptance, is strictly constrained by charge conservation and primarily determined by the rapidity and momentum width of these balance functions. The paper also presents techniques to reduce the computation time of differential correlation functions up to order n=10 based on the methods of moments. Published by the American Physical Society 2024

Identification of Causal Structure with Latent Variables Based on Higher Order Cumulants

Causal discovery with latent variables is a crucial but challenging task. Despite the emergence of numerous methods aimed at addressing this challenge, they are not fully identified to the structure that two observed variables are influenced by one latent variable and there might be a directed edge in between. Interestingly, we notice that this structure can be identified through the utilization of higher-order cumulants. By leveraging the higher-order cumulants of non-Gaussian data, we provide an analytical solution for estimating the causal coefficients or their ratios. With the estimated (ratios of) causal coefficients, we propose a novel approach to identify the existence of a causal edge between two observed variables subject to latent variable influence. In case when such a causal edge exits, we introduce an asymmetry criterion to determine the causal direction. The experimental results demonstrate the effectiveness of our proposed method.

Causal Discovery from Poisson Branching Structural Causal Model Using High-Order Cumulant with Path Analysis

Count data naturally arise in many fields, such as finance, neuroscience, and epidemiology, and discovering causal structure among count data is a crucial task in various scientific and industrial scenarios. One of the most common characteristics of count data is the inherent branching structure described by a binomial thinning operator and an independent Poisson distribution that captures both branching and noise. For instance, in a population count scenario, mortality and immigration contribute to the count, where survival follows a Bernoulli distribution, and immigration follows a Poisson distribution. However, causal discovery from such data is challenging due to the non-identifiability issue: a single causal pair is Markov equivalent, i.e., X->Y and Y->X are distributed equivalent. Fortunately, in this work, we found that the causal order from X to its child Y is identifiable if X is a root vertex and has at least two directed paths to Y, or the ancestor of X with the most directed path to X has a directed path to Y without passing X. Specifically, we propose a Poisson Branching Structure Causal Model (PB-SCM) and perform a path analysis on PB-SCM using high-order cumulants. Theoretical results establish the connection between the path and cumulant and demonstrate that the path information can be obtained from the cumulant. With the path information, causal order is identifiable under some graphical conditions. A practical algorithm for learning causal structure under PB-SCM is proposed and the experiments demonstrate and verify the effectiveness of the proposed method.

Higher-order spectral representation method: New algorithmic framework for simulating multi-dimensional non-Gaussian random physical fields

This study derives a novel higher-order spectral representation method (HOSRM) to represent and simulate multi-dimensional fourth-order non-Gaussian random physical fields. The method primarily extends the traditional second-order spectral representation method (SRM) for simulating non-Gaussian random physical fields by introducing higher-order cumulant function tensors and trispectrum tensors, thereby accomplishing the modeling of fourth-order non-Gaussian random physical fields (symmetric nonlinear physical fields) from a frequency domain perspective. In an endeavor to enhance the simulation efficiency of this theoretical framework, the Fast Fourier Transform (FFT) algorithm is astutely amalgamated into the simulation. This integration contributes significantly to the augmentation of computational efficiency. Furthermore, exhaustive derivations and proofs are presented for the first-order, second-order, and fourth-order ensemble properties of simulated fourth-order non-Gaussian random physical fields. Subsequently, the reliability and accuracy of the proposed algorithm framework are validated through numerical simulations of two two-dimensional and two three-dimensional non-Gaussian random physical fields. The findings demonstrate that the simulated sample function effectively captures the probability characteristics of the random field, including mean, variance, and kurtosis. Finally, an in-depth analysis of numerical simulation results highlights the difference between the proposed method and the traditional second-order spectral representation method, which further underscores the distinctive attributes and potential superiority of the proposed method.

Experimental and Theoretical Analysis of Bispectrum Characteristics of Phase Coupled Signals

Fourier transform points out that a certain function that meets certain conditions can be decomposed into a linear combination of trigonometric functions (sine or cosine functions). The functions of high-order cumulant include suppressing the Gaussian noise, eliminating independent signal components, and identifying the phase coupling phenomenon of the signal. To prove this hypothesis, this study constructs a cosine signal with the phase coupling phenomenon based on Fourier transform theory which substitutes it into the third-order cumulant expression and performs detailed reasoning. The constructed signal is extended to the complex signal domain and the same conclusion is obtained. The number of coupled signals is expanded from three to a higher value. The results of the study give definite mathematical and physical meaning to the bispectral peaks. The collected mechanical vibration signals are given to demonstrate this conclusion. The demonstrated characteristics of high-order cumulants have made them widely used in many fields.

Interferometric near-infrared spectroscopy (iNIRS) reveals that blood flow index depends on wavelength.

Blood flow index (BFI) is an optically accessible parameter, with unit distance-squared-over-time, that is widely used as a proxy for tissue perfusion. BFI is defined as the dynamic scattering probability (i.e. the ratio of dynamic to overall reduced scattering coefficients) times an effective Brownian diffusion coefficient that describes red blood cell (RBC) motion. Here, using a wavelength division multiplexed, time-of-flight- (TOF) - resolved iNIRS system, we obtain TOF-resolved field autocorrelations at 773 nm and 855 nm via the same source and collector. We measure the human forearm, comprising biological tissues with mixed static and dynamic scattering, as well as a purely dynamic scattering phantom. Our primary finding is that forearm BFI increases from 773 nm to 855 nm, though the magnitude of this increase varies across subjects (23% ± 19% for N = 3). However, BFI is wavelength-independent in the purely dynamic scattering phantom. From these data, we infer that the wavelength-dependence of BFI arises from the wavelength-dependence of the dynamic scattering probability. This inference is further supported by RBC scattering literature. Our secondary finding is that the higher-order cumulant terms of the mean squared displacement (MSD) of RBCs are significant, but decrease with wavelength. Thus, laser speckle and related modalities should exercise caution when interpreting field autocorrelations.

Non‐Invasive Super‐Resolution Imaging Through Scattering Media Using Object Fluctuation

AbstractIntroducing super‐resolution techniques to imaging through scattering media potentially revolutionizes the technical analysis for many exotic applications, such as cell structures behind biological tissues. The main challenge is scattering media's inhomogeneous structures, which scramble the light path and create noise‐like speckle patterns, hindering object's visualization even at a low‐resolution level. Here, a computational method is proposed relying on the object's spatial and temporal fluctuation to visualize nanoscale objects through scattering media non‐invasively. Taking advantage of the optical memory effect and multiple frames, the point spreading function (PSF) of scattering media is estimated. Multiple images of fluctuating objects are obtained by deconvolution; then, the super‐resolution image is achieved by computing the higher‐order cumulants. Non‐linearity of high order cumulant significantly suppresses artifacts in the resulting images and enhances resolution by a factor of , where N is the cumulant order. The proof‐of‐concept demonstrates a resolution of 266 nm at the 6th‐order cumulant with numerical aperture (NA) of 0.42, breaking the diffraction limit by a factor of 2.45. An adaptive approach is also demonstrated for imaging through dynamic scattering media. The non‐invasive super‐resolution speckle fluctuation imaging (NISFFI) presents a nanoscopy technique with straightforward imaging hardware configuration to visualize samples behind scattering media.