Cauchy Distribution Research Articles

Enhancing Sine–Cosine mutation strategy with Lorentz distribution for solving engineering design problems

Enhancing Sine–Cosine mutation strategy with Lorentz distribution for solving engineering design problems

Daughter Coloured Noises: The Legacy of Their Mother White Noises Drawn from Different Probability Distributions

White noise is fundamentally linked to many processes; it has a flat power spectral density and a delta-correlated autocorrelation. Operators acting on white noise can result in coloured noise, whether they operate in the time domain, like fractional calculus, or in the frequency domain, like spectral processing. We investigate whether any of the white noise properties remain in the coloured noises produced by the action of an operator. For a coloured noise, which drives a physical system, we provide evidence to pinpoint the mother process from which it came. We demonstrate the existence of two indices, that is, kurtosis and codifference, whose values can categorise coloured noises according to their mother process. Four different mother processes are used in this study: Gaussian, Laplace, Cauchy, and Uniform white noise distributions. The mother process determines the kurtosis value of the coloured noises that are produced. It maintains its value for Gaussian, never converges for Cauchy, and takes values for Laplace and Uniform that are within a range of its white noise value. In addition, the codifference function maintains its value for zero lag-time essentially constant around the value of the corresponding white noise.

A New Dagum-Cauchy{Exponential} Distribution

This study proposed a four-parameter continuous distribution, called the Dagum-Cauchy{Exponential} Distribution (DCED) for modelling financial time series returns using the generalized family of Cauchy distribution by Alzaatreh et al. [1]. Some structural properties of this new distribution such as quantile function, reliability measures and hazard function, and order statistics are obtained. The method of maximum likelihood estimation was proposed in estimating its parameters. Finally, the distribution was used to model some financial datasets adequately.

Synchronization transitions and sensitivity to asymmetry in the bimodal Kuramoto systems with Cauchy noise

We analyze the synchronization dynamics of the thermodynamically large systems of globally coupled phase oscillators under Cauchy noise forcings with a bimodal distribution of frequencies and asymmetry between two distribution components. The systems with the Cauchy noise admit the application of the Ott–Antonsen ansatz, which has allowed us to study analytically synchronization transitions both in the symmetric and asymmetric cases. The dynamics and the transitions between various synchronous and asynchronous regimes are shown to be very sensitive to the asymmetry degree, whereas the scenario of the symmetry breaking is universal and does not depend on the particular way to introduce asymmetry, be it the unequal populations of modes in a bimodal distribution, the phase delay of the Kuramoto–Sakaguchi model, the different values of the coupling constants, or the unequal noise levels in two modes. In particular, we found that even small asymmetry may stabilize the stationary partially synchronized state, and this may happen even for an arbitrarily large frequency difference between two distribution modes (oscillator subgroups). This effect also results in the new type of bistability between two stationary partially synchronized states: one with a large level of global synchronization and synchronization parity between two subgroups and another with lower synchronization where the one subgroup is dominant, having a higher internal (subgroup) synchronization level and enforcing its oscillation frequency on the second subgroup. For the four asymmetry types, the critical values of asymmetry parameters were found analytically above which the bistability between incoherent and partially synchronized states is no longer possible.

Confidence Disc and Square for the Cauchy Distributions

Confidence Disc and Square for the Cauchy Distributions

Characterization-based approach for construction of goodness-of-fit test for Lévy distribution

The Lévy distribution, alongside the Normal and Cauchy distributions, is one of the only three stable distributions whose density can be obtained in a closed form. However, there are only a few specific goodness-of-fit tests for the Lévy distribution. In this paper, two novel classes of goodness-of-fit tests for the Lévy distribution are proposed. Both tests are based on V-empirical Laplace transforms. New tests are scale free under the null hypothesis, which makes them suitable for testing the composite hypothesis. The finite sample and limiting properties of test statistics are obtained. In addition, a generalization of the recent Bhati–Kattumannil goodness-of-fit test to the Lévy distribution is considered. For assessing the quality of novel and competitor tests, the local Bahadur efficiencies are computed, and a wide power study is conducted. Both criteria clearly demonstrate the quality of the new tests. The applicability of the novel tests is demonstrated with two real-data examples.

A New Prediction Model of Surface Subsidence with Cauchy Distribution in the Coal Mine of Thick Topsoil Condition

A New Prediction Model of Surface Subsidence with Cauchy Distribution in the Coal Mine of Thick Topsoil Condition

Non-commutative wormhole in non-minimal curvature–matter coupling of f(R) gravity with Gaussian and Lorentzian distributions

In this work, we construct two new wormhole solutions in the theory dealing with non-minimal coupling between curvature and matter. We take into account an explicitly non-minimal coupling between an arbitrary function of scalar curvature [Formula: see text] and the Lagrangian density of matter. For this purpose, we discuss the Wormhole geometries inspired by non-minimal curvature coupling in [Formula: see text] gravity for linear model in [Formula: see text] as well as nonlinear model in [Formula: see text]. To derive these solutions, we choose the Gaussian and Lorentzian density distributions. To check the viability of these solutions, we plot the graphs for energy conditions and wormhole parameters. It is found that obtained wormhole solutions in both distributions satisfy the energy condition. The resulting wormhole solutions for both non-commutative distributions are determined to be physically stable when we evaluate the stability of these wormhole solutions graphically. It is concluded that wormhole solutions exist with viable physical properties in the non-minimal curvature–matter coupling of [Formula: see text] gravity with Gaussian and Lorentzian distributions.

Existence of wormhole solutions in [formula omitted] gravity under non-commutative geometries

In this paper, we have systematically discussed the existence of the spherically symmetric wormhole solutions in the framework of f(Q,T) gravity under two interesting non-commutative geometries such as Gaussian and Lorentzian distributions of the string theory. Also, to find the solutions, we consider two f(Q,T) models such as linear f(Q,T)=αQ+βT and non-linear f(Q,T)=Q+λQ2+ηT models in our study. We obtained analytic and numerical solutions for the above models in the presence of both non-commutative distributions. We discussed wormhole solutions analytically for the first model and numerically for the second model and graphically showed their behaviors with the appropriate choice of free parameters. We noticed that the obtained shape function is compatible with the flare-out conditions under asymptotic background. Further, we checked energy conditions at the wormhole throat with throat radius r0 and found that NEC is violated for both models under non-commutative background. At last, we examine the gravitational lensing phenomenon for the precise wormhole model and determine that the deflection angle diverges at the wormhole throat.

Etalon-Assisted Study of the Strong CO Ligand Vibrations of the fac-[Re(CO)3(bpy)(CH3CN)]+ Octahedral Complex.

The strong CO ligand vibrations of an octahedral complex, fac-[Re (CO)3(bpy)(CH3CN)]+, in acetonitrile are observed at 2040 and 1932 cm-1. Facial rhenium tricarbonyl systems offer very strong and isolated CO vibrations with the potential for interactions between these vibrations. This work first identifies the dominant ion-pair species using attenuated total reflection infrared (ATR-IR) absorption spectra on a dilution series and then determines the strength of these CO ligand vibrations (as isolated vibrations) with a combination of ATR-IR and etalon-based measurements that determine the absolute complex index of refraction of the solution. Finally, the etalon experiments are modeled to study the interaction between vibrations, which is a property not embedded in the solution's complex index of refraction. The ATR-IR spectra are accomplished on a dilution series as well as a larger set of spectra as these solutions evaporated. The A'(1) CO ligand band at 2040 cm-1 is fit with a sum of three Lorentzian functions characterizing the distribution of free, solvent-separated, and contact ion pairs of this octahedral complex vs concentration. The other CO ligand band at 1932 cm-1 is broader and complicated by the dynamics of vibrational interactions, the unresolved splitting of the A'(2) and A″ CO vibrations, and ion-pair speciation. The etalon transmission measurements vs angle were on a 0.029 M solution, and Rabi splittings of 19 and 38 cm-1 were observed for the A'(1) CO vibration and the unresolved A'(2) + A″ CO vibrations, respectively. The great strength of the CO ligand vibrations is evident despite the use of a dilute solution. Integrated band intensities are reported in comparison to hybrid density functional calculations for isolated vibrations. Then, the observed Rabi splittings are modeled to obtain the coupling strength of the CO ligand vibration with etalon cavity modes and with each other. In summary, this work develops a method to determine the concentration of these solutions from the ATR-IR spectrum, characterizes the ion-pairing, shows that the index of refraction is not constant in the IR spectral region of interest, and develops an interaction Hamiltonian that characterizes cavity-vibration and vibration-vibration coupling.

A Modified BRDF Model Based on Cauchy-Lorentz Distribution Theory for Metal and Coating Materials

This paper presents a modified Bidirectional Reflectance Distribution Function (BRDF) model based on the Cauchy–Lorentz distribution that accurately characterizes the reflected energy distribution of typical materials, such as metals and coatings in hemispherical space. The proposed model overcomes the problem of large errors in classical models when detecting angles far away from the specular reflection angle by dividing the reflected light into specular reflection, directional diffuse reflection, and ideal diffuse reflection components. The newly added directional diffuse reflection component is represented by the Cauchy–Lorentz distribution, and its parameters are directly obtained from experimental measurement curves without distribution fitting. Surface morphology and model parameters are determined through measurements, and the comparison between simulation and actual measurement results shows that the modified BRDF model is in excellent agreement with the measured data. The proposed model not only achieves higher accuracy and universality, but it also represents a significant advancement in the field of BRDF modeling research. Its contributions have profound implications for advancing the state of the art in BRDF modeling, as well as having a broader impact on computer graphics and computer vision domains.

Local-and-Nonlocal Spectral Prior Regularized Tensor Recovery for Cauchy Noise Removal

Cauchy noise removal in multidimension visual data is a fundamental problem in image processing and data science communities. Recently, this problem can be effectively solved by resorting to an appropriate tensor recovery approach and has attracted increasing attention. However, it is not extensively reported whether tensor recovery approaches with multiple priors can achieve even better denosing performance. To address this issue, we propose a novel tensor recovery approach in this paper based on the maximum a posteriori estimation (MAP). Our approach is a nonconvex optimization model that includes a composite regularization term and a data fidelity term. The composite regularization term incorporates nonlocal self-similarity in the spatial and gradient domains of multidimensional visual data, preserving both local image details and non-local structures. The data fidelity term is obtained by computing the negative log-likelihood of Cauchy noise distribution. To solve this proposed tensor recovery model, we develop an effective denoising algorithm with a global convergence guarantee via the well-known nonconvex alternating direction method of multipliers (ADMM). Finally, we conduct extensive numerical experiments to demonstrate that our proposed recovery approach can achieve superior denoising performance, both quantitatively and qualitatively, compared to existing common methods.

How can machine learning be used for accurate representations and predictions of fracture nucleation in zirconium alloys with hydride populations?

Zirconium alloys are critical material components of systems subjected to harsh environments such as high temperatures, irradiation, and corrosion. When exposed to water in high temperature environments, these alloys can thermo-mechanically degrade by forming hydrides that have a crystalline structure that is different from that of zirconium. Cracks can nucleate near these hydrides; hence, these hydrides are a direct link to fracture failure and overall large inelastic strain deformation modes. To fundamentally understand and predict these microstructural failure modes, we interrogated a finite-element database that was deterministically tailored and generated for large strain-dislocation-density crystalline plasticity and fracture modes. A database of 210 simulations was created to randomly sample from a group of microstructural fingerprints that encompass hydride volume fraction, hydride orientation, grain orientation, hydride length, and hydride spacing for a hydride that is physically representative of an aggregate of a hydride population. Machine learning approaches were then used to understand, identify, and characterize the dominant microstructural mechanisms and characteristics. We first used fat-tailed Cauchy distributions to determine the extreme events. A multilayer perceptron was used to learn the mechanistic characteristics of the material response to predefined strain levels and accurately determine the critical fracture stress response and the accumulated shear slips in critical regions. The predictions indicate that hydride volume fraction, a population-level parameter, had a significant effect on localized parameters, such as fracture stress distribution regions, and on the accumulated immobile dislocation densities both within the face centered cubic hydrides and the hexagonal cubic packed h.c.p. matrix.

Differential evolution for population diversity mechanism based on covariance matrix

Differential evolution (DE) is a heuristic global search algorithm based on population. It has exhibited great adaptability in solving continuous-domain problems, but sometimes suffered from insufficient local search ability and being trapped in local optimum when dealing with complicated optimization problems. To solve these problems, an improved differential evolution algorithm with population diversity mechanism based on covariance matrix (CM-DE) is proposed. First, a new parameter adaptation strategy is used to adapt the control parameters, in which the scale factor F is updated according to the improved wavelet basis function in the early stage and Cauchy distribution in the later stage and the crossover rate CR is generated according to normal distribution. The diversity of population and convergence speed are improved by employing the method above. Second, the perturbation strategy is incorporated into crossover operator to enhance the search ability of DE. Finally, the covariance matrix of the population is constructed, where the variance in the covariance matrix is used as indicator to measure the similarity between individuals in the population in order to prevent the algorithm from falling into local optimum resulted by low population diversity. The CM-DE is compared with the state-of-art DE variants including LSHADE (Tanabe and Fukunaga, 2014), jSO [1], LPalmDE [2], PaDE [3] and LSHADE-cnEpSin [4] under 88 test functions from CEC2013 [5], CEC2014 [6] and CEC2017 (Wu et al., 2017) test suites. From the experiment results, it is obvious that among 30 benchmark functions from CEC2017 on 50D optimization, the CM-DE algorithm has 22, 20, 24, 23, 28 better performances comparing with LSHADE, jSO, LPalmDE, PaDE, and LSHADE-cnEpsin. For CEC2017 on 30D optimization, the proposed algorithm secures better performance on 19 out of 30 benchmark functions in terms of convergence speed. In addition, a real-world application is also used to verify the feasibility of the proposed algorithm. The experiment results validate the highly competitive performance in terms of solution accuracy and convergence speed.

Image Restoration under Cauchy Noise: A Group Sparse Representation and Multidirectional Total Generalized Variation Approach

Image Restoration under Cauchy Noise: A Group Sparse Representation and Multidirectional Total Generalized Variation Approach

Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend: comment.

To estimate the intercore crosstalk in uncoupled multicore fibers with cores' propagation constants perturbed by bending, twist, and structure fluctuations, in the paper [Opt. Express21, 5401 (2013)10.1364/OE.21.005401] entitled "Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend," Hayashi et al. derived an intuitively interpretable expression of the average power-coupling coefficient as the convolution of the arcsine and Lorentzian distributions, and calculated the derived expression numerically by performing the fast Fourier transform. In the present paper, we derive a closed-form solution of the convolution of the arcsine and Lorentzian distributions, and point out that the closed-form analytical expression derived here is exactly equivalent to the previously reported closed-form expression.

Modeling of Stress Distribution and Fracture in ABS, PLA, and Alumina-Filled PLA Filaments and FDM-Printed Specimens

In the presented work, a probabilistic approach to fracture analysis of commercially available polyacrylonitrile butadiene styrene (ABS) and polylactide (PLA) filaments and fused deposition modeling (FDM)-printed tensile samples produced from them, as well as PLA filled with alumina (Al2O3) specimens, is discussed. The results of the analysis reveal the presence of symmetry between the stress distribution in the plastic–elastic and fracture stress regions. The value of the transition point from elastoplastic stress distribution to fracture distribution was established. It was shown that the probability density of the destruction of samples in a bound state obeys the Cauchy distribution. The resulting model well describes the stress distributions of tensile specimens, with a minimum deviation of 0.021% and a maximum deviation of 0.048% for the ceramic–polylactide (60Al2O3/40PLA) filament.

MCE-Net: polyp segmentation with multiple branch series-parallel attention and channel interaction via edge distribution guidance

Objective. Accurate polyp segmentation is vital for diagnosing colorectal cancer. However, it is still challenging for accurate polyp segmentation and several bottlenecks exist, such as incomplete boundary, localization bias and lack of micro blocks along with large fragmented boundaries in uncertain regions. Approach. To address the above issues, a novel polyp segmentation network with multiple branch series-parallel attention (MBSA) and channel interaction via edge distribution guidance is proposed. Initially, the edge distribution guidance strategy is proposed to generate the edge distribution following Cauchy distribution to capture complementary edges with sufficient details. Subsequently, a MBSA module is put forward to extract features from various receptive fields to pinpoint tiny polyps by a multiple kernel dilated convolution block, while combining semantics of different dimensions to filter out noise and refining the details of micro target. Ultimately, the channel interaction model is proposed to improve the segmentation accuracy of the polyps in uncertain area by splitting channels into groups and conducts group-wise interaction to excavate subtle clues contained in different channels. Main results. Extensive experimental results demonstrate that the proposed method is superior over the state-of-the-art methods with the mean dice of 0.8972, 0.9420, 0.8312, 0.8064 and 0.9214 on five public polyp datasets. Significance. The proposed method improves the integrity of the margins and internal details for polyp segmentation, which will provide a powerful aid for doctors to achieve accurate judgments, reducing the likelihood of colorectal cancer and improving the survival chances of patients.

Maximum likelihood estimation for [formula omitted]-stable double autoregressive models

The paper investigates the maximum likelihood estimation (MLE) for a first-order double autoregressive model with standardized non-Gaussian symmetric α-stable innovation (sDAR) within a unified framework of stationary and explosive cases. It is shown that the MLE of all parameters, including the stable exponent in the innovation, are strongly consistent and asymptotically normal (with the exception of the intercept for the explosive case). Particularly, the MLE of the parameter in the conditional location is always asymptotically normal, regardless of the stationary or explosive case. This point totally differs from that for linear α-stable AR models in Andrews et al. (2009). Furthermore, it is the first time to provide exact values of the quantities related to the innovation in the asymptotic covariance matrices when the true innovation is the standard Cauchy distribution. Additionally, a modified Kolmogorov-type test statistic is proposed for model diagnostic checking in the stationary case. Monte Carlo simulation studies are conducted to confirm our theoretical findings and assess the finite-sample performance of the MLE and the modified Kolmogorov-type test. An empirical example is analyzed to illustrate the usefulness of sDAR models.

Effect of Cauchy noise on a network of quadratic integrate-and-fire neurons with non-Cauchy heterogeneities

We analyze the dynamics of large networks of pulse-coupled quadratic integrate-and-fire neurons driven by Cauchy noise and non-Cauchy heterogeneous inputs. Two types of heterogeneities defined by families of q-Gaussian and flat distributions are considered. Both families are parametrized by an integer n, so that as n increases, the first family tends to a normal distribution, and the second tends to a uniform distribution. For both families, exact systems of mean-field equations are derived and their bifurcation analysis is carried out. We show that noise and heterogeneity can have qualitatively different effects on the collective dynamics of neurons.