Statistical inference for homologous gene pairs between two circular genomes: a new circular–circular regression model

In this paper, we investigate the problem of determining the relationship, represented by similarity of the homologous gene configuration, between paired circular genomes using a regression analysis. We propose a new regression model for studying two circular genomes, where the Möbius transformation naturally arises and is taken as the link function, and propose the least circular distance estimation method, as an appropriate method for analyzing circular variables. The main utility of the new regression model is in identification of a new angular location of one of a homologous gene pair between two circular genomes, for various types of possible gene mutations, given that of the other gene. Furthermore, we demonstrate the utility of our new regression model for grouping of various genomes based on closeness of their relationship. Using angular locations of homologous genes from the five pairs of circular genomes (Horimoto et al. in Bioinformatics 14:789–802, 1998), the new model is compared with the existing models.     

Bayesian inference for circular distributions with unknown normalising constants

Very often, the likelihoods for circular data sets are of quite complicated forms, and the functional forms of the normalising constants, which depend upon the unknown parameters, are unknown. This latter problem generally precludes rigorous, exact inference (both classical and Bayesian) for circular data.Noting the paucity of literature on Bayesian circular data analysis, and also because realistic data analysis is naturally permitted by the Bayesian paradigm, we address the above problem taking a Bayesian perspective. In particular, we propose a methodology that combines importance sampling and Markov chain Monte Carlo (MCMC) in a very effective manner to sample from the posterior distribution of the parameters, given the circular data. With simulation study and real data analysis, we demonstrate the considerable reliability and flexibility of our proposed methodology in analysing circular data.     

On optimal tests for isotropy against the symmetric wrapped stable-circular uniform mixture family

The family of Symmetric Wrapped Stable (SWS) distributions can be widely used for modelling circular data. Mixtures of Circular Uniform (CU) with the former also have applications as a larger family of circular distributions to incorporate possible outliers. Restricting ourselves to such a mixture, we derive the locally most powerful invariant (LMPI) test for the hypothesis of isotropy or randomness of directions-expressed in terms of the null value of the mixing proportion, p, in the model. Global monotonicity of the power function of the test is established. The test is also consistent. Power values of the test for some selected parameter combinations, obtained through simulation reveal quite encouraging performances even for moderate sample sizes. The P 3 approach (SenGupta, 1991; Pal & SenGupta, 2000) for unknown p and rho and the non-regular case of unknown a, the index parameter, are also discussed. A real-life example is presented to illustrate the inadequacy of the circular normal distribution as a circular model. This example is also used to demonstrate the applications of the LMPI test, optimal P 3 test and a Daviesmotivated test (Davies, 1977, 1987). Finally, a goodness-of-fit test performed on the data establishes the plausibility of the above SWS-CU mixture model for real-life problems encountered in practical situations.     

Multivariate-multiple circular regression

We introduce a fully model-based approach of studying functional relationships between a multivariate circular-dependent variable and several circular covariates, enabling inference regarding all model parameters and related prediction. Two multiple circular regression models are presented for this approach. First, for an univariate circular-dependent variable, we propose the least circular mean-square error (LCMSE) estimation method, and asymptotic properties of the LCMSE estimators and inferential methods are developed and illustrated. Second, using a simulation study, we provide some practical suggestions for model selection between the two models. An illustrative example is given using a real data set from protein structure prediction problem. Finally, a straightforward extension to the case with a multivariate-dependent circular variable is provided.     

A three-parameter generalized von Mises distribution

In this paper, we propose a three-parameter generalized von Mises distribution, called the asymmetric generalized von Mises (AGvM) distribution, which is an extension of the von Mises (vM) distribution, and a subclass of the generalized von Mises (GvM) distribution introduced by Gatto and Jammalamadaka (Stat Methodol 4:341–353, 2007). The three parameter model belongs to an exponential family of distributions and can be used to model both asymmetric and bimodal data. Some properties are studied and interpretation of the parameters is discussed in detail. It is shown that the parameters of the AGvM distribution are particularly easy to interpret and contain a skewness measure as one of its three parameters. A real environmental data set example is provided to illustrate the goodness of fit for AGvM distribution.     

A Likelihood Integrated Method for Exploratory Graphical Analysis of Change Point Problem with Directional Data

In this article we introduce a new likelihood based method, called the likelihood integrated method, which is distinct from the well-known integrated likelihood method. We use the likelihood integrated to propose a simple exploratory graphical analysis for the change point problem in the context of directional data. The method is applied to analysis of two real life data sets. The results obtained by application of this simple method are seen to be quite similar to those obtained earlier by different formal methods in most cases. A small simulation study is conducted to assess the effectiveness of this procedure in indicating presence of change point.     

A Unified Approach for Construction of Probability Models for Bivariate Linear and Directional Data

This article considers a unified approach based on the mixture method to construct linear bivariate models and those on the cylinder and torus involving the exponential and cardioid distributions with the truncated exponential distribution as the mixing distribution. Parameter estimation of the bivariate model on the torus is considered for the data set of phase angles of circadian-related genes in heart and liver tissues.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.