Nonnegative Trigonometric Sums Research Articles

Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions

The sum of independent circular uniformly distributed random variables is also circular uniformly distributed. In this study, it is shown that a family of circular distributions based on nonnegative trigonometric sums (NNTS) is also closed under summation. Given the flexibility of NNTS circular distributions to model multimodality and skewness, these are good candidates for use as alternative models to test for circular uniformity to detect different deviations from the null hypothesis of circular uniformity. The circular uniform distribution is a member of the NNTS family, but in the NNTS parameter space, it corresponds to a point on the boundary of the parameter space, implying that the regularity conditions are not satisfied when the parameters are estimated by using the maximum likelihood method. Two NNTS tests for circular uniformity were developed by considering the standardised maximum likelihood estimator and the generalised likelihood ratio. Given the nonregularity condition, the critical values of the proposed NNTS circular uniformity tests were obtained via simulation and interpolated for any sample size by the fitting of regression models. The validity of the proposed NNTS circular uniformity tests was evaluated by generating NNTS models close to the circular uniformity null hypothesis.

Test of bivariate independence based on angular probability integral transform with emphasis on circular-circular and circular-linear data

Abstract The probability integral transform of a continuous random variable X X with distribution function F X {F}_{X} is a uniformly distributed random variable U = F X ( X ) U={F}_{X}\left(X) . We define the angular probability integral transform (APIT) as θ U = 2 π U = 2 π F X ( X ) {\theta }_{U}=2\pi U=2\pi {F}_{X}\left(X) , which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum modulus 2 π \pi of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation modulus 2 π \pi , and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the APITs of two random variables, X 1 {X}_{1} and X 2 {X}_{2} , and test for the circular uniformity of their sum (difference) modulus 2 π \pi , this is equivalent to test of independence of the original variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the family, to evaluate the power of the proposed independence test by generating samples from NNTS alternative distributions that could be at a closer proximity with respect to the circular uniform null distribution.

Regression models for circular data based on nonnegative trigonometric sums

The parameter space of nonnegative trigonometric sums (NNTS) models for circular data is the surface of a hypersphere; thus, constructing regression models for a circular-dependent variable using NNTS models can comprise fitting great (small) circles on the parameter hypersphere that can identify different regions (rotations) along the great (small) circle. We propose regression models for circular- (angular-) dependent random variables in which the original circular random variable, which is assumed to be distributed (marginally) as an NNTS model, is transformed into a linear random variable such that common methods for linear regression can be applied. The usefulness of NNTS models with skewness and multimodality is shown in examples with simulated and real data.

On the Study of Trigonometric Polynomials Using Strum Sequence

This article constructs trigonometric polynomials of the sine and cosine whose sums are nonnegative. As an application, those nonnegative trigonometric sums are used to study the geometric properties of complex polynomials in the unit disk. The Strum sequences are used to prove the main outcome.

Consumer Segmentation Based on Use Patterns

Recent technological advances have enabled the easy collection of consumer behavior data in real time. Typically, these data contain the time at which a consumer engages in a particular activity such as entering a store, buying a product, or making a call. The occurrence time of certain events must be analyzed as circular random variables, with 24:00 corresponding to 0:00. To effectively implement a marketing strategy (pricing, promotion, or product design), consumers should be segmented into homogeneous groups. This paper proposes a methodology based on circular statistical models from which we construct a clustering algorithm based on the use patterns of consumers. In particular, we model temporal patterns as circular distributions based on nonnegative trigonometric sums (NNTSs). Consumers are clustered into homogeneous groups based on their vectors of parameter estimates by using a spherical k-means clustering algorithm. For this purpose, we define the parameter space of NNTS models as a hypersphere. The methodology is applied to three real datasets comprising the times at which individuals send short-service messages and start voice calls and the check-in times of the users of a mobile application Foursquare.

Discrete Circular Distributions with Applications to Shared Orthologs of Paired Circular Genomes

For structural comparisons of paired prokaryotic genomes, an important topic in synthetic and evolutionary biology, the locations of shared orthologous genes (henceforth orthologs) are observed as binned data. This and other data, e.g., wind directions recorded at monitoring sites and intensive care unit arrival times on the 24-hour clock, are counted in binned circular arcs, thus modeling them by discrete circular distributions (DCDs) is required. We propose a novel method to construct a DCD from a base continuous circular distribution (CCD). The probability mass function is defined to take the normalized values of the probability density function at some pre-fixed equidistant points on the circle. Five families of constructed DCDs which have normalizing constants in closed form are presented. Simulation studies show that DCDs outperform the corresponding CCDs in modeling grouped (discrete) circular data, and minimum chi-square estimation outperforms maximum likelihood estimation for parameters. We apply the constructed DCDs, invariant wrapped Poisson and wrapped discrete skew Laplace to compare the structures of paired bacterial genomes. Specifically, discrete four-parameter wrapped Cauchy (nonnegative trigonometric sums) distribution models multi-modal shared orthologs in <i>Clostridium</i> (<i>Sulfolobus</i>) better than the others considered, in terms of AIC and Freedman’s goodness-of-fit test. The result that different DCDs fit the shared orthologs is consistent with the fact they belong to two kingdoms. Nevertheless, these prokaryotes have a common favored site around 70° on the unit circle; this finding is important for building synthetic prokaryotic genomes in synthetic biology. These DCDs can also be applied to other binned circular data.

Bayesian analysis of circular distributions based on non-negative trigonometric sums

ABSTRACTFernández-Durán [Circular distributions based on nonnegative trigonometric sums. Biometrics. 2004;60:499–503] developed a new family of circular distributions based on non-negative trigonometric sums that is suitable for modelling data sets that present skewness and/or multimodality. In this paper, a Bayesian approach to deriving estimates of the unknown parameters of this family of distributions is presented. Because the parameter space is the surface of a hypersphere and the dimension of the hypersphere is an unknown parameter of the distribution, the Bayesian inference must be based on transdimensional Markov Chain Monte Carlo (MCMC) algorithms to obtain samples from the high-dimensional posterior distribution. The MCMC algorithm explores the parameter space by moving along great circles on the surface of the hypersphere. The methodology is illustrated with real and simulated data sets.

CircNNTSR: AnRPackage for the Statistical Analysis of Circular, Multivariate Circular, and Spherical Data Using Nonnegative Trigonometric Sums

The statistical analysis of circular, multivariate circular, and spherical data is very important in different areas, such as paleomagnetism, astronomy and biology. The use of nonnegative trigonometric sums allows for the construction of flexible probability models for these types of data to model datasets with skewness and multiple modes. The R package CircNNTSR includes functions to plot, fit by maximum likelihood, and simulate models based on nonnegative trigonometric sums for circular, multivariate circular, and spherical data. For maximum likelihood estimation of the models for the three different types of data an efficient Newton-like algorithm on a hypersphere is used. Examples of applications of the functions provided in the CircNNTSR package to actual and simulated datasets are presented and it is shown how the package can be used to test for uniformity, homogeneity, and independence using likelihood ratio tests.

Modeling a Multi-modal Distribution of Wind Direction Data in Kudat, Malaysia

Wind direction is the direction from which the wind is blowing. It is expressed in terms of degrees measured clockwise from geographical direction. The knowledge of the wind direction can be used to obtain information about the wind energy potential, dispersion of particulate matter in the air, the effects of engineering structures on the building, maritime study, and etc. This study provides a suitable model for the wind direction that indicates multi-modal distributional properties. A case study involves with a data from Kudat, Malaysia has been analysed. The statistical models known as a Finite Mixture of von Mises Fisher (mvMF) and Circular Distribution based on Nonnegative Trigonometric Sums (NNTS) has been fitted to the data. Then, the suitability of mvMF and NNTS models were judged based on a graphical representation and goodness-of-fit statistics. The results found that the mvMF model with components is sufficient to provide a best model.

Modeling seasonal effects in the Bass Forecasting Diffusion Model

The Bass Forecasting Diffusion Model is one of the most used models to forecast the sales of a new product. It is based on the idea that the probability of an initial sale is a function of the number of previous buyers. Almost all products exhibit seasonality in their sales patterns and these seasonal effects can be influential in forecasting the weekly/monthly/quarterly sales of a new product, which can also be relevant to making different decisions concerning production and advertising. The objective of this paper is to estimate these seasonal effects using a new family of distributions for circular random variables based on nonnegative trigonometric sums and to use this family of circular distributions to define a seasonal Bass model. Additionally, comparisons in terms of one-step-ahead forecasts between the Bass model and the proposed seasonal Bass model for products such as iPods, DVD players, and Wii Play video game are included.

Modeling angles in proteins and circular genomes using multivariate angular distributions based on multiple nonnegative trigonometric sums

Fernández-Durán, J. J. (2004): "Circular distributions based on nonnegative trigonometric sums," Biometrics, 60, 499-503, developed a family of univariate circular distributions based on nonnegative trigonometric sums. In this work, we extend this family of distributions to the multivariate case by using multiple nonnegative trigonometric sums to model the joint distribution of a vector of angular random variables. Practical examples of vectors of angular random variables include the wind direction at different monitoring stations, the directions taken by an animal on different occasions, the times at which a person performs different daily activities, and the dihedral angles of a protein molecule. We apply the proposed new family of multivariate distributions to three real data-sets: two for the study of protein structure and one for genomics. The first is related to the study of a bivariate vector of dihedral angles in proteins. In the second real data-set, we compare the fit of the proposed multivariate model with the bivariate generalized von Mises model of [Shieh, G. S., S. Zheng, R. A. Johnson, Y.-F. Chang, K. Shimizu, C.-C. Wang, and S.-L. Tang (2011): "Modeling and comparing the organization of circular genomes," Bioinformatics, 27(7), 912-918.] in a problem related to orthologous genes in pairs of circular genomes. The third real data-set consists of observed values of three dihedral angles in γ-turns in a protein and serves as an example of trivariate angular data. In addition, a simulation algorithm is presented to generate realizations from the proposed multivariate angular distribution.

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.

Testing for seasonality using circular distributions based on non-negative trigonometric sums as alternative hypotheses

In medical and epidemiological studies, the importance of detecting seasonal patterns in the occurrence of diseases makes testing for seasonality highly relevant. There are different parametric and non-parametric tests for seasonality. One of the most widely used parametric tests in the medical literature is the Edwards test. The Edwards test considers a parametric alternative that is a sinusoidal curve with one peak and one trough. The Cave and Freedman test is an extension of the Edwards test that is also frequently applied and considers a sinusoidal curve with two peaks and two troughs as the alternative hypothesis. The Kuiper, Hewitt and David and Newell are common non-parametric tests. Fernández-Durán (2004) developed a family of univariate circular distributions based on non-negative trigonometric (Fourier) sums (series) (NNTS) that can account for an arbitrary number of peaks and troughs. In this article, this family of distributions is used to construct a likelihood ratio test for seasonality considering parametric alternative hypotheses that are NNTS distributions.

Maximum likelihood estimation of nonnegative trigonometric sum models using a Newton-like algorithm on manifolds

Department of Actuarial ScienceInstituto Tecnolo´gico Auto´nomo de M´exicoR´io Hondo No. 1, Col. Progreso Tizap´an, C.P. 01080, M´exico D.F., M´exicoe-mail: mercedes@itam.mxAbstract: In Fern´andez-Duran [4], a new family of circular distributionsbased on nonnegative trigonometric sums (NNTS models) is developed.Because the parameter space of this family is the surface of the hypersphere,an efficient Newton-like algorithm on manifolds is generated in order toobtain the maximum likelihood estimates of the parameters.AMS 2000 subject classifications: Primary 49M15, 62G07; secondary49Q99.Keywords and phrases: Differential geometry, maximum likelihood es-timation, Newton algorithm, nonnegative Fourier series, smooth Riemannmanifold.Received November 2010.

A Likelihood Ratio Test for Homogeneity in Circular Data

Testing for the homogeneity of density functions of circular random variables is useful in many settings including the study of wind patterns, paleocurrents trends, the seasonality in human-related events such as homicides and suicides and the seasonality in the appearance of diseases. In this paper, we considered that the density functions are members of the flexible family of circular distributions based on nonnegative trigonometric (Fourier) sums (series) developed by Fernandez-Duran (1). We constructed a test based on the likelihood ratio and we applied the proposed test to simulated and real datasets.

Models for Circular–Linear and Circular–Circular Data Constructed from Circular Distributions Based on Nonnegative Trigonometric Sums

Johnson and Wehrly (1978, Journal of the American Statistical Association 73, 602-606) and Wehrly and Johnson (1980, Biometrika 67, 255-256) show one way to construct the joint distribution of a circular and a linear random variable, or the joint distribution of a pair of circular random variables from their marginal distributions and the density of a circular random variable, which in this article is referred to as joining circular density. To construct flexible models, it is necessary that the joining circular density be able to present multimodality and/or skewness in order to model different dependence patterns. Fernández-Durán (2004, Biometrics 60, 499-503) constructed circular distributions based on nonnegative trigonometric sums that can present multimodality and/or skewness. Furthermore, they can be conveniently used as a model for circular-linear or circular-circular joint distributions. In the current work, joint distributions for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums are presented and applied to two data sets, one for circular-linear data related to the air pollution patterns in Mexico City and the other for circular-circular data related to the pair of dihedral angles between consecutive amino acids in a protein.

Circular distributions based on nonnegative trigonometric sums.

A new family of distributions for circular random variables is proposed. It is based on nonnegative trigonometric sums and can be used to model data sets which present skewness and/or multimodality. In this family of distributions, the trigonometric moments are easily expressed in terms of the parameters of the distribution. The proposed family is applied to two data sets, one related with the directions taken by ants and the other with the directions taken by turtles, to compare their goodness of fit versus common distributions used in the literature.

Stable functions and Vietoris' theorem

An analytic function f( z) in the unit disc D is called stable if s n ( f,·)/ f≺1/ f holds for all for n∈ N 0 . Here s n stands for the nth partial sum of the Taylor expansion about the origin of f, and ≺ denotes the subordination of analytic functions in D . We prove that (1− z) λ , λ∈[−1,1], are stable. The stability of (1+z)/(1−z) turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials.

Univalent Polynomials and Non-Negative Trigonometric Sums

(1998). Univalent Polynomials and Non-Negative Trigonometric Sums. The American Mathematical Monthly: Vol. 105, No. 6, pp. 508-522.

Univalent Polynomials and Non-Negative Trigonometric Sums

Univalent Polynomials and Non-Negative Trigonometric Sums