Ashis SenGupta

Topics in circular statistics

Circular Probability Distributions Some Sampling Distributions Estimation of Parameters Tests for Mean Direction and Concentration Tests for Uniformity Nonparametric Testing Procedures Circular Correlation and Regression Predictive Inference for Directional Data Outliers and Related Problems Change-Point Problems Miscellaneous Topics Some Facts on Bessel Functions How to Use the CircStats Package.

Asymmetric circular-linear multivariate regression models with applications to environmental data

We propose asymmetric angular-linear multivariate regression models, which were motivated by the need to predict some environmental characteristics based on some circular and linear predictors. A measure of fit is provided through the residual analysis. Some applications using data from solar energy radiation experiment and wind energy are given.

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.

Recent advances in the analyses of directional data in ecological and environmental sciences

Directional data (DD) generally refer to data on angular propagations or displacements, orientations, directional movements, etc. Periodic data, such as those recorded on hour of the day, day of the week, etc. can be viewed as and are also cast in the arena of DD through transformations to representative angles. The DD are encountered in all areas of applied sciences, presumably more prominently in ecological and environmental sciences (EES). Research on paleoenvironment often necessitates the hind-casting of directions of river flows obtained from the azimuthal direction of the current that formed the crossbeds. Significance of multimodality of cross-bedding orientations suggests multiple paleoperiods corresponding to varying transport conditions. Orientations of poles to beddings constitute three-dimensional directional data. Directions of remnant magnetism on rock cores are important data in paleontology and earth sciences, e.g. as in the study of reversal of polarity of earth. Studies on the directional movements of ice-floes and ice-bergs are indispensable for planning of transport and of routing of ocean-liners on the confused seas and oceans. Variations in the flight directions of migratory birds form the basis of the evolution of new migratory routes. Displacements in the wintering trajectory of migratory species in search of new breeding areas for colonisation or in their subsequent homing directions are important predictors of environmental and ecological changes. Such a change is often attributed to microevolutionary processes and also to possibly a selection from the earlier used list of genetically based migratory directions. These also lead to changes in the size of the wintering population as well as suggest the mode of the way of inheritance. Corresponding directional measurements may be obtained from ring recoveries, cage experiments or satellite-based radio telemetry. Peak directions of dissolved oxygen, pH value, algae concentration,

A review of optimality of multivariate tests

We consider unrestricted (unordered) parametric hypotheses for multivariate or multiparameter distributions and review some optimality aspects, both exact and asymptotic, for testing of hypotheses possibly in the presence of nuisance parameters. The aim is not to provide an exhaustive review but to represent the widely used classical approaches, expose some promising recent ones and present some interesting practical problems requiring the development of new methods.

A Bayesian analysis of the change-point problem for directional data

Abstract In this paper, we discuss a simple fully Bayesian analysis of the change-point problem for the directional data in the parametric framework with von Mises or circular normal distribution as the underlying distribution. We first discuss the problem of detecting change in the mean direction of the circular normal distribution using a latent variable approach when the concentration parameter is unknown. Then, a simpler approach, beginning with proper priors for all the unknown parameters – the sampling importance resampling technique – is used to obtain the posterior marginal distribution of the change-point. The method is illustrated using the wind data [E.P. Weijers, A. Van Delden, H.F. Vugts and A.G.C.A. Meesters, The composite horizontal wind field within convective structures of the atmospheric surface layer, J. Atmos. Sci. 52 (1995. 3866–3878]. The method can be adapted for a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts.In this paper we discuss a simple fully Bayesian analysis of the change point problem for the directional data in the parametric framework with circular normal distribution as the underlying distribution. We discuss the problem of detecting change in the mean direction of the circular normal distribution when the concentration parameter is unknown. Beginning with proper priors for all the unknown parameters, the sampling-importance-resampling (SIR) technique is used to obtain the posterior marginal distribution of the change point. The method is illustrated using the wind data (Weijer‘s et. al.(1995)). The method can be adapted to a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts. Note: For softcopy of this paper, please contact the authors - email: arnab@iima.ac.in

Probability distributions and statistical inference for axial data

Observations on axes which lack information on the direction of propagation are referred to as axial data. Such data are often encountered in enviromental sciences, e.g. observations on propagations of cracks or on faults in mining walls. Even though such observations are recorded as angles, circular probability models are inappropriate for such data since the constraint that observations lie only in [0, π) needs to be enforced. Probability models for such axial data are argued here to have a general structure stemming from that of wrapping a circular distribution on a semi-circle. In particular, we consider the most popular circular model, the von Mises or circular normal distribution, and derive the corresponding axial normal distribution. Certain properties of this distribution are established. Maximum likelihood estimation of its parameters are shown to be surprisingly, in contrast to trigonometric moment estimation, numerically quite appealing. Finally we illustrate our results by several real life axial data sets.

Modelling multi-stage processes through multivariate distributions

Abstract A new model combining parametric and semi-parametric approaches and following the lines of a semi-Markov model is developed for multi-stage processes. A Bivariate sojourn time distribution derived from the bivariate exponential distribution of Marshall & Olkin (1967) is adopted. The results compare favourably with the usual semi-parametric approaches that have been in use. Our approach also has several advantages over the models in use including its amenability to statistical inference. For example, the tests for symmetry and also for independence of the marginals of the sojourn time distributions, which were not available earlier, can now be conveniently derived and are enhanced in elegant forms. A unified Goodness-of-Fit test procedure for our proposed model is also presented. An application to the human resource planning involving real-life data from University of Nigeria is given.

Locally optimal tests for no contamination in standard symmetric multivariate normal mixtures

Mixture models are becoming increasingly popular in reliability studies and in survival analysis. Consider a mixture of two standard symmetric multivariate normal distributions with one of them having intraclass correlation coefficient, ϱ as 0. We are interested in testing for no contamination. To avoid the problem of identifiability, we assume that either ϱ≠0 or that the mixing proportion p<1 and then test for the other parameter. From practical and theoretical considerations based on statistical curvature, we advocate the test for H0: ϱ=0 in preference to that for H0: p=1. Serious complications still exist, so we concentrate on the problem when p is known. Assuming p known, the locally most powerful test is shown to be extremely simple, and the exact cut-off point is easily computed. The test is admissible, unbiased and possesses a globally monotone power function. Computation of the exact power values exhibit encouraging results. The test statistic is asymptotically normally distributed under both the null and alternative hypotheses. Finally, the test is shown to be also consistent.

Inverse circular–circular regression

The problem of determining the values of the independent variable given a value of the dependent variable is commonly referred to as the inverse regression problem. This problem is also encountered in real life with circular data and we refer to it in that context as the inverse circular regression problem. For such a problem, we develop distance-based methods, and parametric methods, where we use the von Mises (vM) error distribution and the asymmetric generalized von Mises (AGvM) error distribution. We then present a goodness of fit comparison among distance-based and parametric methods, utilizing a new criterion called the relative circular prediction bias (RCPB) criterion, with real and simulated examples. Real data applications are given from the biological and environmental sciences.