Bivariate Cauchy Distribution Research Articles

Probabilistic analysis of the instantaneous center of rotation of a floating structure under regular and irregular beam waves

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 (y1, y2) 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 y1 and y2 follow the Cauchy distribution, for which the mean and variance are undefined. For irregular waves, it is shown that the joint probability density of y1 and y2 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.

Image Denoising Based on Bivariate Distribution

The literature has shown that the performance of the de-noising algorithm was greatly influenced by the dependencies between wavelet coefficients. In this paper, the bivariate probability density function (PDF) was proposed which was symmetric, and the dependencies between the coefficients were considered. The bivariate Cauchy distribution and the bivariate Student’s distribution are special cases of the proposed bivariate PDF. One of the parameters in the probability density function gave the estimation method, and the other parameter can take any real number greater than 2. The algorithm adopted a maximum a posteriori estimator employing the dual-tree complex wavelet transform (DTCWT). Compared with the existing best results, the method is faster and more efficient than the previous numerical integration techniques. The bivariate shrinkage function of the proposed algorithm can be expressed explicitly. The proposed method is simple to implement.

Asymptotic behaviour of non-isotropic random walks with heavy tails

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.

Automatic contrast enhancement of low-light images based on local statistics of wavelet coefficients

This paper describes a new method for contrast enhancement in images and image sequences of low-light or unevenly illuminated scenes based on statistical modelling of wavelet coefficients of the image. A non-linear enhancement function has been designed based on the local dispersion of the wavelet coefficients modelled as a bivariate Cauchy distribution. Within the same statistical framework, a simultaneous noise reduction in the image is performed by means of a shrinkage function, thus preventing noise amplification. The proposed enhancement method has been shown to perform very well with insufficiently illuminated and noisy imagery, outperforming other conventional methods, in terms of contrast enhancement and noise reduction in the output data.

On Bimodal Offset Cauchy Distribution

Abstract The bivariate Cauchy distribution has received applications in many areas, including biological analyses, clinical trials, stochastic modeling of decreasing failure rate life components, study of labour turnover, queuing theory and reliability (Nayak (1987) and Lee and Gross (1991)). In the study of biological analyses, clinical trials and reliability circular distributions will yield suitable results. Circular data arises in a number of different areas such as geological, meteorological, biological and industrial sciences. It is not suggestive to use standard statistical techniques to model circular data, due to the circular geometry of the sample space (p.2 Jammalamadaka and Sen Gupta (2001). It is possible to construct a circular model by transforming a bivariate linear random variate to just its directional component and the resultant model is called ‘offset distribution’. In the literature most of the available circular models were constructed by wrapping a linear model. In recent years some wrapped models were constructed by Dattatreya Rao et al (2007). Here an attempt is made to exploit method of offsetting on Bivariate Cauchy distribution to construct a circular model named by us “OFFSET CAUCHY DISTRIBUTION (OC)”. The characteristic function of the Offset Cauchy model is derived and its characteristics are discussed.

On multivariate truncated generalized Cauchy distribution

In this paper, a multivariate form of truncated generalized Cauchy distribution (TGCD), which is denoted by (MVTGCD), is introduced. The joint density function, conditional density function, moment generating function and mixed moments of order $${b=\sum_{i=1}^{k}b_{i}}$$ are obtained. Making use of the mixed moments formula, skewness and kurtosis in case of the bivariate case are obtained. Also, all parameters of the distribution are estimated using the maximum likelihood and Bayes methods. A real data set is introduced and analyzed using three models. The first model is the bivariate Cauchy distribution, the second is the truncated bivariate Cauchy distribution and the third is the bivariate truncated generalized Cauchy distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the bivariate truncated generalized Cauchy model fits the data better than the other models.

A holographic method to measure the source size broadening in STEM

Source size broadening is an important resolution limiting effect in modern STEM experiments. Here, we propose an alternative method to measure the source size broadening making use of a holographic biprism to create interference patterns in an 'empty' Ronchigram. This allows us to measure the exact shape of the source size broadening with a much better sampling than previously possible. We find that the shape of the demagnified source deviates considerably from a Gaussian profile that is often assumed. We fit the profile with a linear combination of a Gaussian and a bivariate Cauchy distribution showing that even though the full width at half maximum is similar to previously reported measurements, the tails of the profile are considerable wider. This is of fundamental importance for quantitative comparison of STEM simulations with experiments as these tails make the image contrast dependent on the interatomic distance, an effect that cannot be reproduced by a single Gaussian profile of fixed width alone.

Segmentation of noisy colour images using Cauchy distribution in the complex wavelet domain

This study proposes a novel image segmentation technique for noisy colour images, in which the heavy-tailed characteristics of the image are modelled by Cauchy distributions. First, the RGB colour bands of the noisy image are decomposed into multiresolution representations using the dual-tree complex wavelet transform. For each wavelet subband, a model is built assuming that the input coefficients are contaminated with signal-independent additive white Gaussian noise. Hence, the authors derive an estimation rule in the wavelet domain to obtain the noise-free coefficients based on the bivariate Cauchy distribution. The bivariate model makes it possible to exploit the inter-scale dependencies of wavelet coefficients. Subsequently, the image is roughly segmented into textured and non-textured regions using the bivariate model parameters corresponding to the denoised coefficients. A multiscale segmentation is then applied to the resulting regions. Finally, a novel statistical region merging algorithm is introduced by measuring the Kullback-Leibler distance between the estimated Cauchy models for the neighbouring segments. The experiments demonstrate that the authors algorithm yields robust segmentation results for noisy images containing artificial or natural noise.

On a Generalization of Bivariate Cauchy Distribution

This paper addresses a generalization of the bivariate Cauchy distribution discussed by Fang et al. (1990), derived from a trivariate normal distribution with a general correlation matrix. We obtain explicit expressions for the joint distribution function and joint density function, and show that they reduce in a special case to the corresponding expressions of Fang et al. (1990). Finally, we show that this generalized distribution is useful in determining the orthant probability of a bivariate skew-normal distribution of Azzalini and Dalla Valle (1996).

A new class of skew-Cauchy distributions

We discuss here a new class of skew-Cauchy distributions, which is related to Azzalini's [1985. A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178] skew-normal distribution denoted by Z λ ∼ SN ( λ ) . A random variable W λ is said to have a skew-Cauchy distribution (denoted by SC ( λ ) ) with parameter λ ∈ R if W λ = d Z λ / | X | , where Z λ ∼ SN ( λ ) and X ∼ N ( 0 , 1 ) are independent. In this paper, we discuss some simple properties of W λ , such as its density, distribution function, quantiles and a measure of skewness. Next, a bivariate Cauchy distribution is introduced using which some representations and important characteristics of W λ are presented.

Identification of probability measures via distribution of quotients

Straightforward generalizations of the classical Kotlarski characterization of normality using bivariate Cauchy distribution of quotients of independent r.v.'s are given. The symmetry assumption in Kotlarski's result is omitted. Two larger families of bivariate distributions are considered: symmetric second kind beta and elliptically contoured measures.

A Bivariate $t$ Distribution

In this note we consider the joint distribution of Student variates $(t_1, t_2)$, where $t_1$ corresponds to the $x$-observations and $t_2$ to $y$-observations from a bivariate normal distribution. No applications are suggested as Hotelling's $T^2$ is more appropriate whenever estimation of covariance matrix is necessary. Possibly on occasions, when the correlation coefficient, $\rho$, between $x$ and $y$ may be assumed known, for example from past records, the bivariate $(t_1, t_2)$ may be useful. The main interest in this distribution is theoretical. First, because this type of bivariate $(t_1, t_2)$ has never been worked out before while the joint distribution of $(\bar{x}, \bar{y}, s_1, s_2, r)$ is commonly known. Second, for degrees of freedom $n = 1$ (sample size $N = 2$) the bivariate $t$ distribution is an example of a bivariate Cauchy distribution. Lastly, the asymptotic approximation obtained in Section 3 is an application of the method of steepest descent, which has some methodological interest and can be used in other situations. There is no loss of generality, as far as the distribution of $(t_1, t_2, r)$ is concerned, in assuming the means of $x$ and $y$ to be zero and variances to be unity. The only parameter which enters into the joint distribution of $(t_1, t_2, r)$ or into that of $(t_1, t_2)$ is $\rho$. Because of the simplicity of the limiting distribution and the asymptotic approximation we will present them first, while the exact distribution are evaluated only for $n = 1$, and 3 $(N = 2$ and 4). The exact distribution for arbitrary $n$ can be worked out, in double or triple sums, following the method given for $n = 3$

A Representation of the Symmetric Bivariate Cauchy Distribution

A Representation of the Symmetric Bivariate Cauchy Distribution