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We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution. [ABSTRACT FROM AUTHOR]

The notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coincides with the spectrum of the integral operator on with kernel . An example in which there is no deterministic spectral measure is constructed. [ABSTRACT FROM AUTHOR]

We show an elementary way to calculate the exact Lyapunov exponent of affine Boole transformations using the interchangeability theorem for differentiation and integration due to Leibniz. [ABSTRACT FROM AUTHOR]

Combining individual p-values to perform an overall test is often encountered in statistical applications. The Cauchy combination test (CCT) (Journal of the American Statistical Association, 2020, 115, 393–402) is a powerful and computationally efficient approach to integrate individual p-values under arbitrary dependence structures for sparse signals. We revisit this test to additionally show that (i) the tail probability of the CCT can be approximated just as well when more relaxed assumptions are imposed on individual p-values compared to those of the original test statistics; (ii) such assumptions are satisfied by six popular copula distributions; and (iii) the power of the CCT is no less than that of the minimum p-value test when the number of p-values goes to infinity under some regularity conditions. These findings are confirmed by both simulations and applications in two real datasets, thus, further broadening the theory and applications of the CCT. [ABSTRACT FROM AUTHOR]

The Gaussian law reigns supreme in the information theory of analog random variables. This paper showcases a number of information theoretic results which find elegant counterparts for Cauchy distributions. New concepts such as that of equivalent pairs of probability measures and the strength of real-valued random variables are introduced here and shown to be of particular relevance to Cauchy distributions. [ABSTRACT FROM AUTHOR]

Randomized dimensionality reduction has been recognized as one of the cornerstones in handling high-dimensional data, originating in various foundational works such as the celebrated Johnson-Lindenstrauss Lemma. More specifically, nearest neighbor-preserving embeddings exist for ℓ 2 (Euclidean) and ℓ 1 (Manhattan) metrics, as well as doubling subsets of ℓ 2 , where doubling dimension is today the most effective way of capturing intrinsic dimensionality, as well as input structure in various applications. These randomized embeddings bound the distortion only for distances between the query point and a point set. Motivated by the foundational character of fast Approximate Nearest Neighbor search in ℓ 1 , this paper settles an important missing case, namely that of doubling subsets of ℓ 1. In particular, we introduce a randomized dimensionality reduction by means of a near neighbor-preserving embedding, which is related to the decision-with-witness problem. The input set gets represented with a carefully chosen covering point set; in a second step, the algorithm randomly projects the latter. In order to obtain the covering point sets, we leverage either approximate r -nets or randomly shifted grids, with different tradeoffs between preprocessing time and target dimension. We exploit Cauchy random variables, and derive a concentration bound of independent interest. Our algorithms are rather simple and should therefore be useful in practice. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mildly explosive autoregression y t = ρ n y t − 1 + u t , 1 ≤ t ≤ n , where ρ n = 1 + c / n ν , c > 0 , ν ∈ (0 , 1) , and u 1 , ... , u n are arithmetically α -mixing errors. Under some weak conditions, such as E u 1 = 0 , E | u 1 | 4 + δ < ∞ for some δ > 0 and mixing coefficients α (n) = O (n − (2 + 8 / δ)) , the Cauchy limiting distribution is established for the least squares (LS) estimator ρ ^ n of ρ n , which extends the cases of independent errors and geometrically α -mixing errors. Some simulations for ρ n , such as the empirical probability of the confidence interval and the empirical density, are presented to illustrate the Cauchy limiting distribution, which have good finite sample performances. In addition, we use the Cauchy limiting distribution of the LS estimator ρ ^ n to illustrate real data from the NASDAQ composite index from April 2011 to April 2021. [ABSTRACT FROM AUTHOR]

Some quasi-arithmetic means of random variables easily give unbiased strongly consistent closed-form estimators of the joint of the location and scale parameters of the Cauchy distribution. The one-step estimators of those quasi-arithmetic means of the Cauchy distribution are considered. We establish the Bahadur efficiency of the maximum likelihood estimator and the one-step estimators. We also show that the rate of the convergence of the mean-squared errors achieves the Cramér–Rao bound. Our results are also applicable to the circular Cauchy distribution. [ABSTRACT FROM AUTHOR]

Bridge failures due to scour occur more frequently than expected, although bridge design considers conservative scenarios, evidencing a stochastic scour behaviour and the need for improvements in current design procedures. In this paper, a new method for determination of design local scour depth at bridge piers is proposed. The results show that peaks-over-threshold series are better suited than annual-maximum series of maximum scour depths for frequency analysis. Extreme scour depths follow the heavy-tailed Cauchy distribution. Scour depths with a 100-year return period resulted in 22 to 202% of the equilibrium scour depth currently used in design, evidencing the need for a different criterion to select the design scour depth. The proposed method computes the design local scour depth based on its probability of occurrence, introducing a novel probabilistic approach in bridge design for enhanced infrastructure risk management compatible with multi-hazard analysis. [ABSTRACT FROM AUTHOR]

In this paper, the concept of a neutrosophic stable random variable is introduced. Two definitions of a neutrosophic random variable are presented. We introduced both the neutrosophic probability distribution function and the neutrosophic probability density function, and the convolution with the neutrosophic concept. In addition, we proved some properties of a neutrosophic stable random variable, and three examples are discussed. [ABSTRACT FROM AUTHOR]

In the class of discontinuous unbounded initial functions with an integrable singularity, we consider the Cauchy problem for the pseudodifferential equation of local action of moving objects in the corresponding Riesz gravitational field. The fundamental solution to this problem is the Cauchy probability distribution of the force of local interaction between these objects. An explicit form of this solution is obtained, and the correct solvability of this problem is established. In this case, the form of the classical solution of the Cauchy problem is found, and the properties of its smoothness and behavior at infinity are studied. [ABSTRACT FROM AUTHOR]

By analyzing the performance of biogeography-based optimization (BBO), this paper proposed a dual biogeographybased optimization based on hybrid convex migration and optimal Cauchy mutation (DuBBO). The algorithm used the dynamic mixed convex migration operator made it converge to the optimal solution quickly. In the mutation mechanism, the algorithm used optimal mutation and Cauchy distribution random number to help the algorithm jump out of the local optimal solution. Finally, the algorithm added the dual learning strategy to speed up the convergence speed and improve ability. The experimental results on 23 benchmark functions show the effectiveness and necessity of the three strategies. Compared DuBBO with BBO and other six excellent improved algorithms. The experimental results show that DuBBO has the best overall performance, faster convergence speed and higher convergence accuracy. [ABSTRACT FROM AUTHOR]

This paper deals with functional equations in the form of f (x) + g (y) = h (x , y) where h is given and f and g are unknown. We will show that if h is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation. [ABSTRACT FROM AUTHOR]

Outliers have been of constant concern for statistician. These outliers must be handled carefully while analyzing data, particularly, when the data points are measurements based on experiments conducted in scientific labs. Various studies have been conducted on the comparison of outliers detection tests when the data points were assumed to follow normal distribution or skewed exponential or gamma distribution. However, the data may follow some heavy tailed symmetric distribution. One important heavy tailed distribution is Cauchy distribution which has applications in different fields of science including Physics, engineering etc. In the present study, we introduce two tests, in univariate case, for detecting one or more outliers while sampling is done from Cauchy distribution. The proposed tests make use of the robust statistics, namely, median, quartile deviation and the variance based on "sample free of suspected outlier(s)". We simulate critical values and powers of the tests to compare them with the tests available in the literature. We compare their efficiency using power criteria. [ABSTRACT FROM AUTHOR]

Analysis of single-cell RNA sequencing (scRNA-seq) data is often complicate due to the sparsity and high data dimensionality. In this work, we proposed Fuzzy C-means based linear stable-exponential distribution (LSED) model for analyzing scRNA-seq count data of chronic myeloid leukemia (CML) patients. We propose pipelines stages for analysis in which noisy and inconsistent data form sequencing is removed during data preprocessing, this process data then form the cluster of gene feature using fuzzy c-means (FCM) clustering, relevant features are extracted during feature extraction approach. These extracted features are then fed into LSED model in order to difference feature data of gene expression. Finally we evaluate the performance for proposed analysis model based on parameter estimation, distribution comparison and parameter analysis. From the result analysis it was observed that proposed analysis model parameter reflect change in condition of patient more effectively as well as this model fits difference data of gene expression in more better way in comparison to Cauchy and stable distribution. Additional, the results of Gene-set enrichment analysis specify the affinity of proposed model can replicate the distinct enhancement of BCR-ABL+ stem cell as well as BCR-ABL- stem cells. Significantly, proposed FCM based LSED analysis model studies CML from the perspective of statistical models, which present a new sight for CML scientific research. [ABSTRACT FROM AUTHOR]

The optimal parameter estimation problem is considered. The optimization problem is solved in the general problem statement. A model-free approach is applied and supposes no knowledge of the model that the parameter to be estimated belongs to. Optimality of the considered estimators in the sense of a special type risk function is established. The considered risk function makes it possible to optimize the asymptotic variances of the estimators and is used for sample size estimation. Applications for optimization of the truncated parameter estimators of heavy-tailed indexes of distributions, such as Pareto type, Cauchy, and log-gamma, are presented. A class of these estimators is introduced having guaranteed accuracy based on a sample of fixed size. Simulation results confirm theoretical results. [ABSTRACT FROM AUTHOR]

Accurate prediction of roll motion is crucial to the safety of ships and floating structures. The "roll center" mentioned in the literature is ambiguous and various definitions exist. Hence, the instantaneous center of rotation (ICR) is preferred, being an established kinematic concept. Understanding the ICR is important on account of its significant influence on roll damping. A previous study postulated that the ICR moves in a straight line for regular beam waves, based on experimental observations. In this paper, the probability distribution of the ICR coordinates (y 1 , y 2) is derived analytically for regular and irregular beam waves. For regular waves, it is proven a fortiori that the ICR lies on a straight line, and that y 1 and y 2 follow the Cauchy distribution, for which the mean and variance are undefined. For irregular waves, it is shown that the joint probability density of y 1 and y 2 is a bivariate Cauchy distribution, thus the iso-density contour lines are concentric ellipses. Regular and irregular wave experiments are performed to investigate realistic conditions. The analytical solutions are found to be quite accurate, notwithstanding some discrepancies arising from nonlinearities and experimental imperfections. The theoretical results presented herein have potential practical applications, for example since the ICR governs damping, the ICR distribution allows damping to be modeled as a stochastic process. • ICR is the point of pure rotation; important for understanding roll motion. • For regular wave, ICR proven to move in a straight line under linear assumption. • Density of ICR coordinates derived and shown to follow Cauchy distribution. • For irregular wave, ICR coordinates shown to follow bivariate Cauchy distribution. • Cauchy has no mean or variance, characterized by mode and interquartile range. [ABSTRACT FROM AUTHOR]

Motivated by the particle swarm optimization (PSO) and quantum computing theory, we have presented a quantum variant of PSO (QPSO) mutated with Cauchy operator and natural selection mechanism (QPSO-CD) from evolutionary computations. The performance of proposed hybrid quantum-behaved particle swarm optimization with Cauchy distribution (QPSO-CD) is investigated and compared with its counterparts based on a set of benchmark problems. Moreover, QPSO-CD is employed in well-studied constrained engineering problems to investigate its applicability. Further, the correctness and time complexity of QPSO-CD are analyzed and compared with the classical PSO. It has been proved that QPSO-CD handles such real-life problems efficiently and can attain superior solutions in most of the problems. The experimental results shown that QPSO associated with Cauchy distribution and natural selection strategy outperforms other variants in context of stability and convergence. [ABSTRACT FROM AUTHOR]

Abstract. Some estimates of the parameters of a nonlinear regression between the variables of X and Y are constructed for the arctangent as a regression function. The obtained estimates are used to evaluate the unknown parameters of the Cauchy distribution. Computer simulations are performed, and the estimates are compared with another estimates such as the quantile ones, maximum liklyhood estimates, and some others. The confidence intervals for parameters of the Cauchy distribution are obtained. [ABSTRACT FROM AUTHOR]

Speckle suppression and elimination are very important to improve the visual quality of ultrasound image and the diagnostic ability of the diseases. An effective technique of image denoising based on discrete wavelet transform is to employ a Bayesian maximum a posteriori (MAP) estimator. To suppress and remove the speckle noise using MAP estimator effectively, it must assign correctly the shrinkage function based on appropriate probability density functions (PDFs) for the wavelet coefficients of logarithmically transformed noise-free ultrasound image and speckle noise. In this paper, we introduce a new closed-form shrinkage function that is an analytical solution of a Bayesian MAP estimator for despeckling of the ultrasound images effectively in wavelet domain. We employ a Cauchy prior and Gaussian PDF to model the wavelet coefficients of logarithmically transformed noise-free ultrasound image and speckle noise, respectively. Firstly, we derive the CauchyShrinkGMAP that is a closed-form shrinkage function. In addition, we estimate the noise variance and parameter of MAP estimator. Next, we evaluate the despeckling performance of wavelet image denoising method using the CauchyShrinkGMAP compared to various despeckling method using median and Wiener filters, hard and soft thresholding and GaussShrinkGMAP and MCMAP3N shrinkage function. The experiment results show that PSNR of new closed-form shrinkage function is highest, MSE is smallest, and the correlation coefficient (ρ) and SSIM are closer to one than the other existing image denoising methods for noisy synthetic ultrasound images at different speckle noise levels. Also, experiment results show that ENL of new closed-form shrinkage function is highest and that of EN and SD is smallest than the other existing image denoising methods for noisy real ultrasound image. [ABSTRACT FROM AUTHOR]

This paper proposes an improved version of antlion optimizer (ALO) to solve data clustering problem. In this work, Cauchy distribution-based random walk is employed in place of uniform distribution to jump out of local optima as a first strategy. Then opposition-based learning model is utilized in conjunction with acceleration coefficient to overcome the slow convergence of classical ALO as second strategy to propose opposition-based ALO using Cauchy distribution (OB-C-ALO). The performance of the proposed OB-C-ALO is evaluated over a set of benchmark problems of different varieties of characteristics and analysed statistically by performing Wilcoxon rank-sum test. The proposed version then utilizes K-means clustering by refining the clusters formed using K-means as objective function. The algorithm is evaluated on six data sets of UCI machine learning repository and compared with classical ALO and recently developed version of ALO, namely OB-L-ALO, over benchmark test problems as well as data clustering problem and proved to be better in terms of performance achieved. [ABSTRACT FROM AUTHOR]

We demonstrate that, in a regression setting with a Hilbertian predictor, a response variable is more likely to be more highly correlated with the leading principal components of the predictor than with trailing ones. This is despite the extraction procedure being unsupervised. Our results are established under the conditional independence model, which includes linear regression and single-index models as special cases, with some assumptions on the regression vector. These results are a generalisation of earlier work which showed that this phenomenon holds for predictors which are real random vectors. A simulation study is used to quantify the phenomenon. [ABSTRACT FROM AUTHOR]

Abstract–Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a nonasymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests. for this article are available online. [ABSTRACT FROM AUTHOR]

This paper studies maximum likelihood estimation of autoregressive models of order 1 with a near unit root and Cauchy errors. Autoregressive models with an intercept and with an intercept and a linear time trend are also considered. The maximum likelihood estimator (MLE) for the autoregressive coefficient is n 3 / 2 -consistent with n denoting the sample size and has a mixture-normal distribution in the limit. The MLE for the scale parameter of Cauchy distribution is n 1 / 2 -consistent, and its limiting distribution is normal. The MLEs of the intercept and the linear time trend are n 1 / 2 - and n 3 / 2 -consistent, respectively. It is also shown that the t statistic for the null hypothesis of a unit root based on the MLE has a standard normal distribution in the limit. In addition, finite-sample properties of the MLE are compared with those of the least square estimator (LSE). It is found that the MLE is more efficient than the LSE when the errors have a Cauchy distribution or a distribution which is a mixture of Cauchy and normal distributions. It is also shown that empirical power of the MLE-based t test for a unit root is much higher than that of the Dickey–Fuller t test. [ABSTRACT FROM AUTHOR]

We consider inverse problems in which the unknown target includes sharp edges, for example interfaces between different materials. Such problems are typical in image reconstruction, tomography, and other inverse problems algorithms. A common solution for edge-preserving inversion is to use total variation (TV) priors. However, as shown by Lassas and Siltanen 2004, TV-prior is not discretization-invariant: the edge-preserving property is lost when the computational mesh is made denser and denser. In this paper we propose another class of priors for edge-preserving Bayesian inversion, the Cauchy difference priors. We construct Cauchy priors starting from continuous one-dimensional Cauchy motion, and show that its discretized version, Cauchy random walk, can be used as a non-Gaussian prior for edge-preserving Bayesian inversion. We generalize the methodology to two-dimensional Cauchy fields, and briefly consider a generalization of the Cauchy priors to Lévy α-stable random field priors. We develop a suitable posterior distribution sampling algorithm for conditional mean estimates with single-component Metropolis–Hastings. We apply the methodology to one-dimensional deconvolution and two-dimensional X-ray tomography problems. [ABSTRACT FROM AUTHOR]

Grey Wolf Optimiser (GWO) is a recently developed optimisation approach to solve complex non-linear optimisation problems. It is relatively simple and leadership-hierarchy based approach in the class of SwarmIntelligence based algorithms. For solving complex real-world non-linear optimisation problems, the search equation provided in GWO is not of sufficient explorative behaviour. Therefore, in the present paper, an attempt has been made to increase the exploration capability along with the exploitation of a search space by proposing an improved version of classical GWO. The proposed algorithm is named as Cauchy-GWO. In Cauchy-GWO Cauchy operator has been integrated in which first two new wolves are generated with the help of Cauchy distributed random numbers and then another new wolf is generated by taking the convex combination of these new wolves. The performance of Cauchy-GWO is exhibited on standard IEEE CEC 2014 benchmark problem set. Statistical analysis of the results on CEC 2014 benchmark set and popular evaluation criteria, Performance Index (PI) proves that Cauchy-GWO outperforms GWO in terms of error values defined in IEEE CEC 2014 benchmarks collection. Later on in the paper, GWO and Cauchy-GWO algorithms have been used to solve three well-known engineering application problems and two problems of reliability. From the analysis conducted in the present paper, it can be concluded that the proposed algorithm, Cauchy-GWO is reliable and efficient algorithm to solve continuous benchmark test problems, as well as real-life applications problems. [ABSTRACT FROM AUTHOR]

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game. [ABSTRACT FROM AUTHOR]

This paper investigates stochastic market clearing solutions in peer-to-peer (P2P) energy markets as two parameters of prosumers' cost functions are flexibly and randomly chosen in certain intervals to compensate uncertainties and guarantee their energy preferences. For tractability, the scenarios in which one parameter is varied while the other is fixed are considered. It is then shown that a few distributions, namely normal, Cauchy, and gamma distributions, are invariant for P2P energy markets, i.e., if prosumer cost function parameters follow one of those distributions, then market clearing solutions will follow the same type of distribution. Explicit and analytical formulas for probability density functions of market clearing price and trading powers in P2P energy markets are derived for each type of the above-mentioned distributions. As such, variations on P2P energy market clearing solutions can be assessed and predicted when parameters in the stochastic distribution of prosumer cost function parameters are changed. Simulations are carried out for a modified IEEE European Low Voltage Test Feeder system whose results validate the obtained theoretical characterizations. • Varied prosumer cost function parameters to reflex uncertainties and energy preferences. • Tractable cases include normal, Cauchy and gamma distributions of prosumer parameters. • Analytical formulas for the p.d.f of market-clearing price and trading powers. • Assessment and prediction of trading price and trading power p.d.f variations. [ABSTRACT FROM AUTHOR]

A novel framework for compressive sensing (CS) data acquisition and reconstruction in quantitative acoustic microscopy (QAM) is presented. Three different CS patterns, adapted to the specifics of QAM systems, were investigated as an alternative to the current raster-scanning approach. They consist of diagonal sampling, a row random, and a spiral scanning pattern and can all significantly reduce both the acquisition time and the amount of sampled data. For subsequent image reconstruction, we design and implement an innovative technique, whereby a recently proposed approximate message passing method is adapted to account for the underlying data statistics. A Cauchy maximum a posteriori image denoising algorithm is thus employed to account for the non-Gaussianity of QAM wavelet coefficients. The proposed methods were tested retrospectively on experimental data acquired with a 250- or 500-MHz QAM system. The experimental data were obtained from a human lymph node sample (250 MHz) and human cornea (500 MHz). Reconstruction results showed that the best performance is obtained using a spiral sensing pattern combined with the Cauchy denoiser in the wavelet domain. The spiral sensing matrix reduced the number of spatial samples by a factor of 2 and led to an excellent peak signal-to-noise ratio of 43.21 dB when reconstructing QAM speed-of-sound images of a human lymph node. These results demonstrate that the CS approach could significantly improve scanning time, while reducing costs of future QAM systems. [ABSTRACT FROM AUTHOR]

Optimal operation of a practical blast furnace (BF) iron-making process depends largely on a good measurement of molten iron quality (MIQ) indices. However, measuring the MIQ online is not feasible using the available techniques. In this paper, a novel data-driven robust modeling is proposed for an online estimation of MIQ using improved random vector functional-link networks (RVFLNs). Since the output weights of traditional RVFLNs are obtained by the least squares approach, a robustness problem may occur when the training dataset is contaminated with outliers. This affects the modeling accuracy of RVFLNs. To solve this problem, a Cauchy distribution weighted M-estimation-based robust RFVLNs are proposed. Since the weights of different outlier data are properly determined by the Cauchy distribution, their corresponding contribution on modeling can be properly distinguished. Thus, robust and better modeling results can be achieved. Moreover, given that the BF is a complex nonlinear system with numerous coupling variables, the data-driven canonical correlation analysis is employed to identify the most influential components from multitudinous factors that affect the MIQ indices to reduce the model dimension. Finally, experiments using industrial data and comparative studies have demonstrated that the obtained model produces a better modeling and estimating accuracy and stronger robustness than other modeling methods. [ABSTRACT FROM PUBLISHER]

Based on the study of alpha-stable distribution and its score function, by combining M-estimation theory and maximum likelihood estimation theory, this paper proposes a novel Cauchy score function based cost function of the residual fitting error matrix. And then, this cost function is employed as a substitute for the ℓ p -norm based cost function of the residual fitting error matrix which is utilized in the ℓ p -MUSIC algorithm. To solve the cost function, alternating convex optimization is applied. And the complex-valued Newton's method with optimal step size is developed to solve the resulting convex problem. With the obtained signal subspace, the direction of arrival (DOA) estimates are retrieved by the MUSIC technique. Comprehensive simulation results demonstrate that, the proposed algorithm can achieve more robust performance than its counterpart both in terms of resolution probability and root mean square error (RMSE), especially when the generalized signal to noise ratio (GSNR) is fairly low, or the underlying noise is extremely impulsive. [ABSTRACT FROM AUTHOR]

An alternative derivation of the yield curve based on entropy or the loss of information as it is communicated through time is introduced. Given this focus on entropy growth in communication the Shannon entropy will be utilized. Additionally, Shannon entropy’s close relationship to the Kullback–Leibler divergence is used to provide a more precise understanding of this new yield curve. The derivation of the entropic yield curve is completed with the use of the Burnashev reliability function which serves as a weighting between the true and error distributions. The deep connections between the entropic yield curve and the popular Nelson–Siegel specification are also examined. Finally, this entropically derived yield curve is used to provide an estimate of the economy’s implied information processing ratio. This information theoretic ratio offers a new causal link between bond and equity markets, and is a valuable new tool for the modeling and prediction of stock market behavior. [ABSTRACT FROM AUTHOR]

Tensor analysis has reached a celebrity status in the areas of machine learning, computer vision, and artificial intelligence. Completing and recovering tensor is an important problem for tensor analysis. It involves recovering a tensor from either a subset of its entries or the whole entries contaminated by noise. Classical tensor completion/recovery methods are based mainly on l 2 - norm (Gaussian distribution noise) models, and are sensitive to noise of large magnitude. While l 1 - norm (Laplacian distribution noise) based methods are robust to noise of large magnitude, they do not deal with dense noise effectively. In this paper, we present a novel Cauchy tensor decomposition method for simultaneously recovering and completing a low rank tensor with both missing data and complex noise. We utilize the Cauchy distribution to model noise and derive the objective function of Cauchy tensor decomposition under the maximum likelihood estimation (MLE) framework. Then we developed a robust tensor decomposition framework that used first-order optimization approaches to optimize the objective function. Extensive numerical experiments are conducted and show that our method is able to successfully recover and complete tensors with large/dense noise and missing data. We further demonstrate the usefulness of Cauchy tensor decomposition on three real-world applications, image inpainting, traffic data process and foreground/background separation. The experimental results show that the method is applicable to a wide range of problems. [ABSTRACT FROM AUTHOR]

Shirvani and Soltani suggested a method for characterizing and simulating one-sided truncated Cauchy random variables. For simulation, that involves solving a nonlinear transformation in each iteration. Here, we propose an alternative method to generate Cauchy random variables. Our method is more general (e.g., incorporates one-sided and two-sided truncations) and does not involve solving any equation. It can be implemented even using a hand calculator. [ABSTRACT FROM AUTHOR]

Recently, Ristić and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets. [ABSTRACT FROM AUTHOR]

Epochs are instants of significant excitation of vocal-tract system in speech production process. In this paper, we attempt to extract information about epochs from phase spectra of speech signals. The phase spectrum of speech is modelled as the response of an allpass (AP) filter, and the resulting error signal is used for epoch extraction. The parameters of AP model are estimated by imposing sparsity constraints on the error signal. The error signal, thus obtained, exhibits prominent peaks at epoch locations. The epochal candidates obtained from the error signal are refined using a dynamic programming algorithm. The performance of the proposed method is consistent across genders and is comparable with the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Scene depth estimation from a single image is a challenging problem in the field of computer vision and computer graphics. We present an approach to estimate the amount of defocus blur at edges based on the Cauchy distribution model for the DSF (points spread function). The input image was re-blurred twice respectively using two Cauchy distribution kernels, and the amount of defocus blur at the edges was obtained by the two scale parameters and the ratio between the gradients of the two re-blurred images. By propagating the blur amount at edges to the entire image, a full depth map was obtained. Experimental results on several real images demonstrate the effectiveness of our method in providing reliable depth estimation. Our method is robust to image noise, inaccurate edge location, and interference from neighboring edges, and its performance is better than existing methods based on a Gaussian DSF model. The results verify that a non-Gaussian model for DSF is feasible. [ABSTRACT FROM AUTHOR]

The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of fitting a known distribution function to the marginal distribution of a stationary long memory moving average random field observed on increasing $$\nu $$ -dimensional 'cubic' domains when its mean $$\mu $$ and scale $$\sigma $$ are known or unknown. Using two suitable estimators of $$\mu $$ and a classical estimate of $$\sigma $$ , a modification of the Kolmogorov-Smirnov statistic is defined based on the residual empirical process and having a Cauchy-type limit distribution, independent of $$\mu ,\sigma $$ and the long memory parameter d. Based on this result, a simple goodness-of-fit test for the marginal distribution is constructed, which does not require the estimation of d or any other underlying nuisance parameters. The result is new even for the case of time series, i.e., when $$\nu =1$$ . Findings of a simulation study investigating the finite sample behavior of size and power of the proposed test is also included in this paper. [ABSTRACT FROM AUTHOR]

Attribute Reduction (AR) is an important preprocessing step for data mining. AR based on rough set is an efficient method. Its reduction performance has been verified to be better or comparable with other methods in large amount of works, but existing reduction algorithms have some problems such as slow convergent speed and probably converging to a local optimum. A novel attribute reduction algorithm based on Artificial Fish Swarm Algorithm (AFSA) and rough set is proposed. For AFSA has a slow convergence rate in the later phase of iterations, normal distribution function, Cauchy distribution function, multi-parent crossover operator, mutation operator and modified minimal generation gap model are adopted to improve AFSA. The attribute reduction algorithm based on improved AFSA and rough set takes full advantages of the improved AFSA and rough set,which are faster, more efficient, simpler, and easier to be implemented. Datasets in the UC Irvine (UCI) Machine Learning Repository are selected to verify the aforementioned new method. The results show that above algorithm can search the attribute reduction set effectively, and it has low time complexity and the excellent global search ability. [ABSTRACT FROM AUTHOR]

In the standard SVM algorithm still exists the problem of low accuracy when demand forecast, this article puts forward a kind of talent demand forecasting model based on simulated annealing algorithm to optimize the SVM algorithm. First of all, introduce the simulated annealing algorithm, use Cauchy distribution as the distance of disturbance to disturbance optimization it, then changing the annealing schedule, properly controlling the cooling process, combining with the method of success - failure and the method of variable metric to optimize convergence. At last, we introduced the improved simulated annealing algorithm into the parameter selection of support vector machines, improving accuracy which is supporting vector machine algorithm. According to the simulation results, the improved simulated annealing algorithm from this paper has better convergence.sa for accuracy, the talent demand forecasting model based on simulated annealing algorithm to optimize the SVM algorithm is higher than the original SVM algorithm. [ABSTRACT FROM AUTHOR]

Differential evolution algorithms are weak in local searching and easy to dropping into the local optimal solutions at the same time. The search performance of these algorithms is mainly based on the parameter setting of their crossover probability and mutation factors. To improve the above shortcomings of differential evolution algorithms, we propose an adaptive differential evolution algorithm called ZJADE on the basis of in-depth research and analysis of the adaptive differential evolution with optional external archive (JADE). Skew tent chaotic mapping is used to initialize the population in order to generate uniformly dispersed population. During each generation, the crossover probability of each individual is generated according to the normal distribution and the Cauchy distribution while the mutation factors are independently generated according to the normal distribution. The crossover probability and mutation factors of successful individuals are saved, and the statistical crossover probability is employed. The ZJADE algorithm is compared with multiple state-of-the-art adaptive differential evolution algorithms through thirteen classical test functions. The results show that the ZJADE obtains better solution accuracy and quicker convergence speed, thus having a better search performance. [ABSTRACT FROM AUTHOR]

• A hybrid dilated convolution spatial attention module, namely, HDCSA, is proposed. • HDCSA pays attention to the feature information of different distance scales. • A Cauchy rotation invariance loss function, namely, CRI, is proposed. • CRI constrains networks to learn rotations in skin lesion targets. [Display omitted] Dermoscopic image retrieval technology can provide dermatologists with valuable information such as similar confirmed skin disease cases and diagnosis reports to assist doctors in their diagnosis. In this study, we design a dermoscopic image retrieval algorithm using convolutional neural networks (CNNs) and hash coding. A hybrid dilated convolution spatial attention module is proposed, which can focus on important information and suppress irrelevant information based on the complex morphological characteristics of dermoscopic images. Furthermore, we also propose a Cauchy rotation invariance loss function in view of the skin lesion target without the main direction. This function constrains CNNs to learn output differences in samples from different angles and to make CNNs obtain a certain rotation invariance. Extensive experiments are conducted on dermoscopic image datasets to verify the effectiveness and versatility of the proposed module, algorithm, and loss function. Experiment results show that the rotation-invariance deep hashing network with the proposed spatial attention module obtains better performance on the task of dermoscopic image retrieval. [ABSTRACT FROM AUTHOR]

To prevent unauthorized access to a person's medical images, it is widely acknowledged that the security features of confidentiality, availability and integrity should be in place. This paper proposes the watermarking of ultrasonic echo images together with optimal watermark detection, in which pseudorandom noise is added to the images for integrity. The optimum watermark detection is the integration of the generalized Gaussian distribution ( ρ-GGD) and the Cauchy distribution. The results show that the proposed method gives good detection performance. The proposed method not merely achieves optimum detection using the Rao test but also leads to the highest detection probability with JPEG2000 compression. Compared with other detection methods, our proposed method exhibits better watermark detection performance even when the watermark-to-document ratio (WDR) is −50 dB. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

This paper is devoted to the diffusion limit of the Fokker-Planck equation of plasma physics, in which the equilibrium function decays towards zero at infinity like a negative power function. We prove that for an appropriate time scale, in a suitable weighted Sobolev space, the small mean free path limit gives rise to a diffusion equation. [ABSTRACT FROM AUTHOR]

We study the estimation problem of linear regression in the presence of a new impulsive noise model, which is a sum of Cauchy and Gaussian random variables in time domain. The probability density function (PDF) of this mixture noise, referred to as the Voigt profile, is derived from the convolution of the Cauchy and Gaussian PDFs. To determine the linear regression parameters, the maximum likelihood estimator (MLE) is developed first. Since the Voigt profile suffers from a complicated analytical form, an M-estimator with the pseudo-Voigt function is also derived. In our algorithm development, both scenarios of known and unknown density parameters are considered. For the latter case, we estimate the density parameters by utilizing the empirical characteristic function prior to applying the MLE. Simulation results show that the performance of both proposed methods can attain the Cramér–Rao lower bound. [ABSTRACT FROM AUTHOR]

The compressive sensing (CS) based 4-D synthetic aperture radar (SAR) imaging method performs well in the case of high signal-to-noise ratios (SNR). However, in the presence of strong noises, the performance of CS-based method degrades and the number of false targets increases rapidly. In this paper, a novel 4-D SAR imaging method is proposed based on Bayesian compressive sensing (BCS). Assume that the target scattering field follows the Cauchy distribution, the 4-D SAR imaging is transformed into signal reconstruction via maximum a posteriori estimation. In addition, Poisson disk sampling is utilized to generate the radar positions of 4-D SAR in the baseline-time plane. Experimental results show that the proposed method is capable of effective suppression of the noise by exploiting the sparseness prior distribution of the image scene, and a well-focused image could also be achieved even under the condition of low SNR. [ABSTRACT FROM AUTHOR]

A dual Cauchy rate-distortion model is proposed for video coding. In our approach, the coefficient distribution of the integer transform is first studied. Then, based on the observation that the rate-distortion model of the luminance and that of the chrominance can be well expressed by separate Cauchy functions, a dual Cauchy rate-distortion model is presented. Furthermore, the simplified rate-distortion formulas are deduced to reduce the computational complexity of the proposed model without losing the accuracy. Experimental results have shown that the proposed model is better able to approximate the actual rate-distortion curve for various sequences with different motion activities. [ABSTRACT FROM AUTHOR]