Trigonometric Moments Research Articles

New Circular Distribution with an Application to Biology

In this paper, we introduce a new circular distribution known as the wrapped new weighted exponential distribution (WNWE) is introduced. We derive an explicit expression for its probability density function and establish closed-form expressions for the distribution function, characteristic function, and trigonometric moments. Furthermore, we discuss the properties of the proposed model. We employ the method of maximum likelihood estimation to estimate the parameters. To demonstrate the applicability of the proposed distribution, we analyze a real dataset consisting of 50 starhead top minnows.

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

AbstractComplex valued measures of finite total variation are a powerful signal model in many applications. Restricting to the d-dimensional torus, finitely supported measures can be exactly recovered from their trigonometric moments up to some order if this order is large enough. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the 1-Wasserstein distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the indicator function on the support of the measure and to converge to zero outside.

A note on the Hendrickson-Lattman phase probability distribution and its equivalence to the generalized von Mises distribution.

Hendrickson & Lattman [Acta Cryst. (1970), B26, 136-143] introduced a method for representing crystallographic phase probabilities defined on the unit circle. Their approach could model the bimodal phase probability distributions that can result from experimental phase determination procedures. It also provided simple and highly effective means to combine independent sources of phase information. The present work discusses the equivalence of the Hendrickson-Lattman distribution and the generalized von Mises distribution of order two, which has been studied in the statistical literature. Recognizing this connection allows the Hendrickson-Lattman distribution to be expressed in an alternative form which is easier to interpret, as it involves the location and concentration parameters of the component von Mises distributions. It also allows clarification of the conditions for bimodality and access to a simplified analytical method for evaluating the trigonometric moments of the distribution, the first of which is required for computing the best Fourier synthesis in the presence of phase, but not amplitude, uncertainty.

Nonparametric Tests of Independence for Circular Data Based on Trigonometric Moments

We introduce nonparametric tests of independence for bivariate circular data based on trigonometric moments. Our contributions lie in (i) proposing nonparametric tests that are locally and asymptotically optimal against bivariate cosine von Mises alternatives and (ii) extending these tests, via the empirical characteristic function, to obtain consistent tests against broader sets of alternatives, eventually being omnibus. We thus provide a collection of trigonometric-based tests of varying generality and known optimalities. The large-sample behaviours of the tests under the null and alternative hypotheses are obtained, while simulations show that the new tests are competitive against previous proposals. Two data applications in astronomy and forest science illustrate the usage of the tests.

The wrapped XLindley distribution

In the study of many environmental phenomena, circular statistical analysis and applications to directional data are crucial. In this paper, we introduce the wrapped XLindley distribution, a novel circular distribution with a single parameter. We derive expressions for the distribution’s characteristic function, trigonometric moments, and other associated descriptive measures. The unknown distributional parameter is estimated using the maximum likelihood, least squares, and weighted least squares approach, and the accuracy of these estimates is tested using a simulation study. Finally, to clarify the suggested distribution’s modeling potential, we fit the proposed model to two circular real-world data sets and evaluate the goodness of fit of this distribution in comparison to the wrapped Lindley, wrapped modified Lindley, Von Mises, Jones–Pewsey, and Kato–Jones distributions.

Two Doubly Truncated Semicircular Distributions: Some Important Properties

In this paper, two doubly truncated semicircular distributions, doubly truncated Semicircular Exponentiated Weibull ( ) and doubly truncated Semicircular Generalized Gompertz are presented. The most important statistical properties, including moments, characteristic function, trigonometric moments, quantile function, simulated data, reliability stress strength model, Shannon entropy, and relative entropy, are obtained.

Wrapped Birnbaum-Saunders distribution: definition and estimation

In this work a new circular distribution called wrapped Birnbaum-Saunders was proposed. It was obtained by wrapping the classical Birnbaum Saunders distribution in a reparameterized form. For this distribution we have found expressions for theirprobability density function, distribution function and trigonometric moments. We show some properties of this new distribution and obtained the maximum likelihood estimators of its two parameters, in addition, we conducted a Monte Carlo simulation study to evaluate the performance of the maximum likelihood estimators of the parameters. We also made an application to a real dataset from the Rudolf Jander’s experiments concerning the direction chosen by ants in response to a stimulus and compare its estimates via Kuiper’s statistic with those obtained from the Von Mises and Asymmetric Von Mises models. This distribution is very promising as model for asymmetric directional data.

Identifiability of Asymmetric Circular and Cylindrical Distributions

Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sine-skewed circular distributions, including the sine-skewed von Mises and sine-skewed wrapped Cauchy distributions, and that of a M\"{o}bius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sine-skewed wrapped Cauchy distribution, and conditional distributions of a random variable on the non-negative real line given the circular random variable are a Weibull distribution and a generalized Pareto-type distribution, respectively.

Transformed Semicircular Exponentiated Weibull Distribution

In this paper, a new transformed Semicircular Exponentiated Weibull ( SEW ) distribution of three parameters is constructed by applying the transformation of the inverse stereographic projection that maps a real line to the unit circle. The cumulative distribution function, probability density function, expanded probability density function, and reliability measures are presented. Furthermore, the most important statistical aspects of this new distribution, such as moments, trigonometric moments, characteristic function, simulated data, quantile function, reliability stress strength model, Shannon entropy, and relative entropy are obtained. The maximum likelihood estimation method is implemented to estimate the unknown three parameters. A simulation study is conducted to detect empirically the behaviors of the maximum likelihood estimates of the SEW parameters for different default values of the parameters and different samples size.

The stereographic generalized inverse Weibull distribution

When the unit circle is taken as a sample space, the circular data manifest in different aspects of real life. Minh and Farnum (2003) used a stereographic projection to generate new probability distributions on real line [1]. Here, we present an asymmetric distribution called Stereographic generalized inverse Weibull distribution (SGIWD) to represent circular data with focus on the inverse Stereographic Projection. The graph of probability density function is done. We consider here some statistical properties of SGIW distribution, survival and reverse hazard functions, the rth raw moments function, trigonometric moments, stress-strength reliability model, Tsallis, Renyi Shannon entropies and the Kullback–Leibler divergence measure. A simulation study is conducted to evaluate the behavior of the maximum likelihood method to estimate parameters of SGIWD.

Image-based Visualization of Large Volumetric Data Using Moments.

We present a novel image-based representation to interactively visualize large and arbitrarily structured volumetric data. This image-based representation is created from a fixed view and models the scalar densities along each viewing ray. Then, any transfer function can be applied and changed interactively to visualize the data. In detail, we transform the density in each pixel to the Fourier basis and store Fourier coefficients of a bounded signal, i.e. bounded trigonometric moments. To keep this image-based representation compact, we adaptively determine the number of moments in each pixel and present a novel coding and quantization strategy. Additionally, we perform spatial and temporal interpolation of our image representation and discuss the visualization of introduced uncertainties. Moreover, we use our representation to add single scattering illumination. Lastly, we achieve accurate results even with changes in the view configuration. We evaluate our approach on two large volume datasets and a time-dependent SPH dataset.

Simple construction of a toroidal distribution from independent circular distributions

In this paper, we propose a new method for constructing a toroidal distribution from independent circular distributions by specifying marginal distributions. The constructed probability density function is expressed without additional normalizing constants, and its conditional distributions belong to the family of sine-skewed circular distributions. The simple expressions for trigonometric moments, especially for correlation measures, help characterize the proposed model. Maximum likelihood estimation is computationally tractable, and the correlation parameter is orthogonal to the location and concentration parameters when the marginal distributions are symmetric. Two illustrative applications are shown comparing genome structures between paired bacteria.

Exploration of some distributional properties, parameter estimates and applications of the wrapped quasi Lindley distribution

A circular distribution called Wrapped Quasi Lindley distribution with two parameters has been recently proposed, but apart from the expressions for pdf, cdf, their circular representations, characteristic function and the maximum likelihood equations for the proposed distribution, no other properties of the distribution as well as the characteristics of the parameter estimates were explored by the authors. A slight error has also been observed in the expression for pdf of the distribution. Also, the application of the distribution in modeling real life data was not exhibited. Further, the form of the characteristic function in the paper is not compact and there is no closed form of the expression of the trigonometric moments. This paper thus aims to rectify the expression for pdf and explore a few descriptive measures and distributional properties of the Wrapped Quasi Lindley distribution and derive closed form expressions for the characteristic function and hence the trigonometric moments using an identity. it is found that the operations of wrapping and convoluting linear distributions around unit circle are commutative. The maximum likelihood estimates of the parameters of the distribution are shown to be consistent through a simulation study. The utility of the Wrapped Quasi Lindley model to a real-life data set on orientations is shown and the goodness-of-fit of the distribution is assessed and compared to that of the Wrapped Exponential and Wrapped Lindley distribution with the help of the log-likelihood, AIC and BIC measures. Further, the probabilities of the orientations to lie in a certain interval are estimated on the basis of the fitted Wrapped Quasi Lindley distribution. The distribution is found to be more appropriate in modeling the situations where the directions having lower magnitude have higher likelihood of occurrence.

A class of goodness-of-fit tests for circular distributions based on trigonometric moments

We propose a class of goodness–of–fit test procedures for arbitrary parametric families of circular distributions with unknown parameters. The tests make use of the specific form of the characteristic function of the family being tested, and are shown to be consistent. We derive the asymptotic null distribution and suggest that the new method be implemented using a bootstrap resampling technique that approximates this distribution consistently. As an illustration, we then specialize this method to testing whether a given data set is from the von Mises distribution, a model that is commonly used and for which considerable theory has been developed. An extensive Monte Carlo study is carried out to compare the new tests with other existing omnibus tests for this model. An application involving five real data sets is provided in order to illustrate the new procedure.

Two-dimensional active motion.

The diffusion in two dimensions of noninteracting active particles that follow an arbitrary motility pattern is considered for analysis. A Fokker-Planck-like equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of active motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided, and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. The analysis is focused in the intermediate-time regime, where the effects of the specific pattern of active motion are conspicuous. For this, it is shown that only the expectation value of the first two harmonics of the scattering angle of the direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.

On Wrapping of Quasi Lindley Distribution

In this paper, as an extension of Wrapping Lindley Distribution (WLD), we suggest a new circular distribution called the Wrapping Quasi Lindley Distribution (WQLD). We obtain the probability density function and derive the formula of a cumulative distribution function, characteristic function, trigonometric moments, and some related parameters for this WQLD. The maximum likelihood estimation method is used for the estimation of parameters.

Prony’s method on the sphere

Eigenvalue analysis based methods are well suited for the reconstruction of finitely supported measures from their moments up to a certain degree. We give a precise description when Prony's method succeeds in terms of an interpolation condition. In particular, this allows for the unique reconstruction of a measure from its trigonometric moments whenever its support is separated and also for the reconstruction of a measure on the unit sphere from its moments with respect to spherical harmonics. Both results hold in arbitrary dimensions and also yield a certificate for popular semidefinite relaxations of these reconstruction problems.

Wrapped hb-skewed laplace distribution and its application in meteorology

In this paper,we introduce a new circular distribution called Wrapped Holla and Bhattacharya’s (HB) Skewed Laplace distribution.We obtain explicit form for the density and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for estimation of parameters. The methods are applied to a data set which consists of the wind directions of a Black Mountain ACT and compared with that of wrapped variance gamma distribution (WvG) and generalised von Mises (GvM) distribution.

Moment-Based Order-Independent Transparency

Compositing transparent surfaces rendered in an arbitrary order requires techniques for order-independent transparency. Each surface color needs to be multiplied by the appropriate transmittance to the eye to incorporate occlusion. Building upon moment shadow mapping, we present a moment-based method for compact storage and fast reconstruction of this depth-dependent function per pixel. We work with the logarithm of the transmittance such that the function may be accumulated additively rather than multiplicatively. Then an additive rendering pass for all transparent surfaces yields moments. Moment-based reconstruction algorithms provide approximations to the original function, which are used for compositing in a second additive pass. We utilize existing algorithms with four or six power moments and develop new algorithms using eight power moments or up to four trigonometric moments. The resulting techniques are completely order-independent, work well for participating media as well as transparent surfaces and come in many variants providing different tradeoffs. We also utilize the same approach for the closely related problem of computing shadows for transparent surfaces.

Erlang circular model motivated by inverse stereographic projection

The Erlang distribution is a special case of the Gamma distribution with the shape parameter is an integer. This paper proposed a new circular model used inverse stereographic projection. The inverse stereographic projection which is a mapping that projects a random variable from a real line onto a circle can be used in circular statistics to construct a distribution on the circle from real domain. From the circular model, then can be derived the characteristics of the Erlang circular model such as the mean resultant length, mean direction, circular variance and trigonometric moments of the distribution.