Adelaide Figueiredo

Two-way analysis of variance for data from a concentrated bipolar Watson distribution

Abstract The bipolar Watson distribution is frequently used for modeling axial data. We extend the one-way analysis of variance based on this distribution to a two-way layout. We illustrate the method with directional data in three dimensions

Power of Tests of Uniformity Defined on the Hypersphere

Abstract In multivariate data with n individuals described by p variables it is usual to assume that the variables are fixed and the individuals are randomly selected; in the present study we assume that the individuals are fixed and the previously normalised variables are randomly selected from a population of variables. When we want to associate to the sample of variables a distribution, like the Bingham distribution, it is important to use beforehand a test of uniformity. In this article, we compare the power of Bingham and Giné tests of uniformity for some dimensions of the sphere, vs. a Bingham population or a mixture of Bingham populations.

Discriminant Analysis for the von Mises-Fisher Distribution

The von Mises-Fisher distribution is widely used for modeling directional data. In this article, we derive the discriminant rules based on this distribution to assign objects into pre-existing classes. We determine a distance between two von Mises-Fisher populations and we calculate estimates of the misclassification probabilities. We also analyze the behavior of the distance between two von Mises-Fisher populations and of the estimates of the misclassification probabilities when we modify the parameters of the populations or the samples size or the dimension of the sphere. Finally, we present an example with real spherical data available in the literature.

Discriminant analysis based on the Watson distribution defined on the hypersphere

The Watson distribution defined on the hypersphere is one of the most used distributions for modelling axial data. In this paper, we consider the discriminant analysis for axial data assumed to come from a mixture of Watson distributions defined on the hypersphere. We develop an optimal classification rule, which enables us to assign a new observation into one of several Watson subpopulations defined on the hypersphere. As the probabilities of misclassification cannot be calculated in closed form, we report on a simulation study to estimate, in some cases, the probabilities of misclassification and a distance between the two Watson subpopulations defined on the hypersphere. An illustration of our approach is provided using data defined on the sphere given in the literature.

Comparison of sampling plans by variables using the bootstrap and Monte Carlo simulations

We consider two sampling plans by variables to inspect batches of products from an industrial process under a context of unknown distribution underlying the measurements of the quality characteristic under study. Through the use of the bootstrap methodology and Monte Carlo simulations we evaluate and compare the performance of those sampling plans in terms of probability of acceptance of lots and average outgoing quality level.

Multi-Sample Likelihood Ratio Tests Based on Bipolar Watson Distributions Defined on the Hypersphere

We derive likelihood ratio tests for the equality of the directional parameters of k bipolar Watson distributions defined on the hypersphere with common concentration parameter. We analyze the power of these tests in the case of two distributions supposing in the alternative hypothesis two directional parameters forming an angle, which varies from 18° to 90°. We also compare the likelihood ratio tests with a high-concentration F-test.

Discordancy Tests Based on the Likelihood Ratio for the Bipolar Watson Distribution on the Hypersphere

As the Watson distribution is frequently used for modeling axial data, it is important to investigate the existence of possible outliers in samples from this distribution. Then, we develop for the bipolar Watson distribution defined on the hypersphere, some tests of discordancy of an outlier or several outliers en bloc based on the likelihood ratio, supposing an alternative model of contamination of slippage type. We evaluate the performance of these tests of discordancy of an outlier and we also compare some tests of discordancy of an outlier available for this distribution.

Discordancy Test for the Bipolar Watson Distribution Defined on the Hypersphere

ABSTRACT As the Watson distribution is much used for modeling axial data, it is important to investigate the existence of a possible outlier in samples from this distribution. So, we generalize to the hypersphere a test of discordancy of an outlier or several outliers en bloc available for the Watson distribution defined on the sphere. We also evaluate the performance of the test for some cases with the usual types of alternatives.

Acceptance-Sampling Plans for Reducing the Risk Associated with Chemical Compounds

In various manufacturing industries it is important to investigate the presence of some chemical or harmful substances in lots of raw material or final products, in order to evaluate if they are in conformity to requirements. In this work we highlight the adequacy of the inflated Pareto distribution to model measurements obtained by chromatography, and we define and evaluate acceptance-sampling plans under this distributional setup for lots of large dimension. Some technical results associated with the construction and evaluation of such sampling plans are provided as well as an algorithm for an easy implementation of the sampling plan that exhibits the best performance.

Clustering Directions Based on the Estimation of a Mixture of Von Mises-Fisher Distributions

Background : In the statistical analysis of directional data, the von Mises-Fisher distribution plays an important role to model unit vectors. The estimation of the parameters of a mixture of von Mises-Fisher distributions can be done through the Estimation-Maximization algorithm. Objective : In this paper we propose a dynamic clusters type algorithm based on the estimation of the parameters of a mixture of von Mises-Fisher distributions for clustering directions, and we compare this algorithm with the Estimation-Maximization algorithm. We also define the between-groups and within-groups variability measures to compare the solutions obtained with the algorithms through these measures. Results : The comparison of the clusters obtained with both algorithms is provided for a simulation study based on samples generated from a mixture of two Fisher distributions and for an illustrative example with spherical data.