Trigonometric Moments Research Articles

Wrapped Lindley distribution

ABSTRACTIn this article, we introduce a new circular distribution to be called as wrapped Lindley distribution and derive expressions for characteristic function, trigonometric moments, coefficients of skewness, and kurtosis. Method of maximum likelihood estimation is used for the estimation of parameters. We carry out a simulation study to show that the obtained maximum likelihood estimator is consistent. The proposed model is also applied to a real-life dataset, and its performance is compared with that of wrapped exponential distribution.

Transformation of circular random variables based on circular distribution functions

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

Prony’s Method Under an Almost Sharp Multivariate Ingham Inequality

The parameter reconstruction problem in a sum of Dirac measures from its low frequency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham inequality which improves the previously best known dimension-dependent constant from square-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham inequality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function.

Deterministic Sampling on the Torus for Bivariate Circular Estimation

Many modern approaches to nonlinear filtering employ sample-based density approximations. These approximations are generated via random (Monte Carlo methods) or deterministic sampling [say, the unscented Kalman filter (UKF)]. The advantages of deterministic techniques are their reproducibility and that they require fewer samples. While the UKF is designed for real vector spaces, we present an approach for deterministic sampling applicable to two-dimensional periodic manifolds. This approach employs five weighted samples and matches trigonometric moments and a circular-circular correlation coefficient.

Nonparametric estimating equations for circular probability density functions and their derivatives

We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.

Estimation for Wrapped Zero Inflated Poisson and Wrapped Poisson Distributions

There has been a growing interest in discrete circular models such as wrapped zero inflated Poisson and wrapped Poisson distributions and the trigonometric moments (see Brobbey et al., 2016 and Girija et al., 2014). Also, characteristic functions of stable processes have been used to study the estimation of the model parameters using estimating function approach (see Thavaneswaran et al., 2013). One difficulty in estimating the circular mean and the resultant mean length parameter of wrapped Poisson (WP) or wrapped zero inflated Poisson (WZIP) is that neither the likelihood of WP/WZIP random variable nor the score function is available in closed form, which leads one to use either trigonometric method of moment estimation (TMME) or an estimating function approach. In this paper, we study the estimation of WZIP distribution and WP distribution using estimating functions and obtain the closed form expression of the information matrix. We also derive the asymptotic distribution of the tangent of the mean direction for both the WZIP and WP distributions.

Wrapped Zero-inflated Poisson Distribution and Its Properties

Recently, there has been a growing interest in discrete valued wrapped distributions and the trigonometric moments.<br />Characteristic functions of stable processes have been used to study the estimation of the model parameters using<br />estimating function approach (Thavaneswaran et al., 2013). In this paper, we introduce a new discrete circular distribution,<br />the wrapped zero-inflated Poisson distribution and derive its population characteristics.<br /><br />

Symmetric Circular Model Induced by Inverse Stereographic Projection On Double Weibull Distribution with Application

In practical life usually we come across angular or periodic data which needs distributions that are invariant of zero direction and sense of rotation for analysis. Inverse stereographic projection/ Bilinear transformation defined by a one to one mapping generate a class of probability distributions on unit circle that are flexible to analyse circular data. The probability density f unction of many life testing models ranges from 0 to ∞ . Aiming to construct new circular model by applying Inverse Stereographic Projection, we consider Double Weibull distribution and derive density, distribution and characteristic functions of new circular model coined by us Stereographic Double Weibull distribution. Also an attempt is made to study the mathematical aspects in evaluating trigonometric moments of new circular model and present the population characteristics. Goodness of fit to a live data is verified.

A tractable and interpretable four-parameter family of unimodal distributions on the circle

This article presents a class of four-parameter distributions for circular data that are unimodal, possess simple characteristic and density functions and a tractable distribution function, can be interpretably parameterized directly in terms of their trigonometric moments, afford a very wide range of skewness and kurtosis, envelop numerous interesting submodels including the wrapped Cauchy and cardioid distributions, allow straightforward parameter estimation by both method of moments and maximum likelihood, and are closed under convolution. This class of distributions exhibits the widest range of attractive properties yet available while retaining unimodality.

New Circular model induced by Inverse Stereographic projection on Double Exponential Model - Application to Birds Migration Data

Abstract This paper introduces Stereographic Double Exponential Model based on inverse Stereographic Projection or Bilinear (Mobius) Transformation, [Minh and Farnum (2003)]. onsidering the data set of 13 homing pigeons were released singly in the Toggenburg Valley in Switzerland under sub Alpine conditions (data quoted in Batschelet (1981)), it is shown that the said model is a good fit by most of the tests at various level of significance. The derivation of the characteristic function for Stereographic Double Exponential Model and its trigonometric moments are presented. Relative performance of Stereographic Logistic [Phani (2013], Wrapped Logistic [Dattatreya Rao et al (2007)] and Stereographic Double Exponential models for the live data of 13 birds is studied. Also graphs of pdf of this new Model for various combinations of the parameters are drawn.

Using the model of a mixture of a uniform distribution and a von Mises distribution for segmentation of anisotropic images

An algorithm is proposed for segmenting anisotropic image based on estimates of the local orientation of textures obtained using a gradient structural tensor. A mixture of a uniform distribution and a von Mises distribution is used as the estimate model. An estimation algorithm for the unknown parameters of the mixture based on trigonometric moments is obtained. Examples are given demonstrating the effectiveness of the segmentation algorithm in identifying textures with one dominant direction.

Distributions for spherical data based on nonnegative trigonometric sums

A family of distributions for a random pair of angles that determine a point on the surface of a three-dimensional unit sphere (three-dimensional directions) is proposed. It is based on the use of nonnegative double trigonometric (Fourier) sums (series). Using this family of distributions, data that possess rotational symmetry, asymmetry or one or more modes can be modeled. In addition, the joint trigonometric moments are expressed in terms of the model parameters. An efficient Newton-like optimization algorithm on manifolds is developed to obtain the maximum likelihood estimates of the parameters. The proposed family is applied to two real data sets studied previously in the literature. The first data set is related to the measurements of magnetic remanence in samples of Precambrian volcanics in Australia and the second to the arrival directions of low mu showers of cosmic rays.

An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion

We introduce a four-parameter extended family of distributions related to the wrapped Cauchy distribution on the circle. The proposed family can be derived by altering the settings of a problem in Brownian motion which generates the wrapped Cauchy. The densities of this family have a closed form and can be symmetric or asymmetric depending on the choice of the parameters. Trigonometric moments are available, and they are shown to have a simple form. Further tractable properties of the model are obtained, many by utilizing the trigonometric moments. Other topics related to the model, including alternative derivations and M\"{o}bius transformation, are considered. Discussion of the symmetric submodels is given. Finally, generalization to a family of distributions on the sphere is briefly made.

Wrapped hypo-exponential distribution

If a country has a satellite, the trafficking of the uses of that satellite by other countries could be modeled as a hypo-exponential distribution for axial data, as such a new circular distribution named wrapped hypo-exponential distribution along with its probability density function has been derived. Some of the various shaped wrapped distributions have also been unfolded through proper specification of the concern parameters. Explicit forms of trigonometric moments, related parameters and some other properties of the same distribution are also obtained.

On the trigonometric moment problem in two dimensions

A finite sequence c0,…,ck of complex numbers is the set of trigonometric moments of a measure dμ=∣Pk(eiθ)∣−2dθ with Pk(z) a polynomial of degree k, zero-free in |z|≤1, providing only that the Hermitian Toeplitz matrix having c0,…,ck as its top row is positive-definite. The analogous representation in two dimensions for a rectangular array of numbers requires additional conditions, discovered by J.S. Geronimo and H.J. Woerdeman. We use orthogonal decomposition to give an alternate proof of their important result.

Sine-skewed axial distributions with an application for fallen tree data

An axis is an undirected line where there is no reason to distinguish one end of the line from the other. Phenomena in nature that can be described as axial data are numerous. In this paper a method of trigonometric moments for the axial normal or axial von Mises distribution as an alternative to the method by Arnold and SenGupta (Environ Ecol Stat 13:271–285, 2006) is discussed. Sine-skewed axial Jones–Pewsey, von Mises and wrapped Cauchy distributions are introduced as special cases of a more general construction of skew axial distributions. As an example the methods are applied to a data set which consists of the orientations of logs on the floor of a primeval spruce forest.

On characteristic functions of the wrapped lognormal and the wrapped Weibull distributions

The characteristic function plays a prominent role in determining the pdf of a circular model using the trigonometric moments. The characteristic functions of the wrapped lognormal and the wrapped Weibull distributions cannot be expressed in a closed form. Hence, numerical evaluation of the same is presented along with graphs. Also, certain population characteristics of the wrapped lognormal and the wrapped Weibull distributions are presented.

A Projected Exponential Family for Modeling Semicircular Data

For modeling(skewed) semicircular data, we derive a new exponential family of distributions. We extend it to the l-axial exponential family of distributions by a projection for modeling any arc of arbitrary length. It is straightforward to generate samples from the l-axial exponential family of distributions. Asymptotic result reveals that the linear exponential family of distributions can be used to approximate the l-axial exponential family of distributions. Some trigonometric moments are also derived in closed forms. The maximum likelihood estimation is adopted to estimate model parameters. Some hypotheses tests and confidence intervals are also developed. The Kolmogorov-Smirnov test is adopted for a goodness of t test of the l-axial exponential family of distributions. Samples of orientations are used to demonstrate the proposed model.

Symmetric Unimodal Models for Directional Data Motivated by Inverse Stereographic Projection

In this paper, a modified inverse stereographic projection, from the real line to the circle, is used as the motivation for a means of resolving a discontinuity in the Minh–Farnum family of circular distributions. A four-parameter family of symmetric unimodal distributions which extends both the Minh–Farnum and Jones–Pewsey families is proposed. The normalizing constant of the density can be expressed in terms of Appell’s function or, equivalently, the Gauss hypergeometric function. Important special cases of the family are identified, expressions for its trigonometric moments are obtained, and methods for simulating random variates from it are described. Parameter estimation based on method of moments and maximum likelihood techniques is discussed, and the latter approach is used to fit the family of distributions to an illustrative data set. A further extension to a family of rotationally symmetric distributions on the sphere is briefly made.

Information theoretic results for circular distributions

The broad class of generalized von Mises (GvM) circular distributions has certain optimal properties with respect to information theoretic quantities. It is shown that, under constraints on the trigonometric moments, and using the Kullback–Leibler information as the measure, the closest circular distribution to any other is of the GvM form. The lower bounds for the Kullback–Leibler information in this situation are also provided. The same problem is also considered using a modified version of the Kullback–Leibler information. Finally, series expansions are given for the entropy and the normalizing constants of the GvM distribution.