Spherical Normal Distribution Research Articles

Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere

In directional statistics, the von Mises-Fisher (vMF) distribution is one of the most basic and popular probability distributions for data on the unit hypersphere. Recently, the spherical normal (SN) distribution was proposed as an intrinsic counterpart to the vMF distribution by replacing the standard Euclidean norm with the great-circle distance, which is length of the shortest path joining two points on the unit sphere. Focusing on an isotropic version of SN distribution, it is shown that maximum likelihood estimators uniquely exist under mild support conditions. Since no analytic formula are available for the estimation, efficient numerical routines are proposed for parameter estimation. The estimation is considered in a general setting where non-negative weights are assigned to observations. This leads to a more interesting contribution for model-based clustering on the unit hypersphere by finite mixture model with SN distributions. Efficiency of optimization-based estimation procedures and effectiveness of SN mixture model are validated using simulated and real data examples.

Robust formulations for clustering-based large-scale classification

Chebyshev-inequality-based convex relaxations of Chance-Constrained Programs (CCPs) are shown to be useful for learning classifiers on massive datasets. In particular, an algorithm that integrates efficient clustering procedures and CCP approaches for computing classifiers on large datasets is proposed. The key idea is to identify high density regions or clusters from individual class conditional densities and then use a CCP formulation to learn a classifier on the clusters. The CCP formulation ensures that most of the data points in a cluster are correctly classified by employing a Chebyshev-inequality-based convex relaxation. This relaxation is heavily dependent on the second-order statistics. However, this formulation and in general such relaxations that depend on the second-order moments are susceptible to moment estimation errors. One of the contributions of the paper is to propose several formulations that are robust to such errors. In particular a generic way of making such formulations robust to moment estimation errors is illustrated using two novel confidence sets. An important contribution is to show that when either of the confidence sets is employed, for the special case of a spherical normal distribution of clusters, the robust variant of the formulation can be posed as a second-order cone program. Empirical results show that the robust formulations achieve accuracies comparable to that with true moments, even when moment estimates are erroneous. Results also illustrate the benefits of employing the proposed methodology for robust classification of large-scale datasets.

Generalised Model for Aircraft Vulnerability by Different Weapon Systems

The authors earlier model for the vulnerability of aircraft where aircraft was considered as a combination of cylinder, cones and wedges has been extended to the case when structural data of aircraft as well as its vital parts are given in the form of three-dimensional curvilinear triangles. In the case of VT -fused ammunition, spherical normal distribution has been used to estimate the landing probability of the shell in a cylindrical vicinity region around the aircraft. Kill criteria of vital parts have been redefined.

Projection indices and multimodality for mixtures of normal distributions

Abstract Projection pursuit indices are intended to give ‘interesting’ one- or two-dimensional projections of data points in higher-dimensional spaces. This paper compares the performance of seven such indices. They are all minimum for a normal distribution, and are standardized to be zero for that case. They are compared by considering projections of a mixture of four trivariate spherical normal distributions with means at the vertices of a regular tetrahedron.

Sensitive and Sturdy $p$-Values

We introduce new criteria for evaluating test statistics based on the $p$-values of the statistics. Given a set of test statistics, a good statistic is one which is robust in being reasonably sensitive to all departures from the null implied by that set. We present a constructive approach to finding the optimal statistic. We apply the criteria to two-sided problems; combining independent tests; testing that the mean of a spherical normal distribution is 0, and extensions to other spherically symmetric and exponential distributions; Bartlett's problem of testing the equality of several normal variances; and testing for one outlier in a normal linear model. For the most part, the optimal statistic is quite easy to use. Often, but not always, it is the likelihood ratio statistic.

The Chebyshev center: A multidimensional estimate of location

In this paper the center of the smallest k-dimensional sphere enclosing a data set, hereinafter called the Chebyshev center, is introduced as a multidimensional measure of location. The distribution of this estimator, which is a multidimensional generalization of the univariate midrange, is derived in the general case and its properties investigated for a host of distributions. The Chebyshev center is shown to be a maximum likelihood estimator for the center of a uniform distribution over a k-sphere and both unbiased and consistent for the multivariate spherical normal distribution and any spherical finite range distribution.

On the similarity of the entropy power inequality and the Brunn- Minkowski inequality (Corresp.)

The entropy power inequality states that the effective variance (entropy power) of the sum of two independent random variables is greater than the sum of their effective variances. The Brunn-Minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii. Both these inequalities are recast in a form that enhances their similarity. In spite of this similarity, there is as yet no common proof of the inequalities. Nevertheless, their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found. Two applications of this reasoning are presented. First, an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the Fisher information matrix given an entropy constraint--just as a sphere minimizes the surface area given a volume constraint. Second, a theorem involving the effective radii of growing convex sets is proved.

On Ali's Characterization of the Spherical Normal Distribution

Summary Recently Ali (1980) showed that the independence of the sample mean and the sample variance characterizes the spherical normal distribution in the class of absolutely continuous spherically symmetric distributions. Here, using invariance considerations, we give a proof of this characterization without the absolute continuity assumption.

Limiting behaviour of the von Mises-Fisher distribution

AbstractThe von Mises-Fisher density of parameter K is shown to be very similar to the spherical normal distribution of parameter υ = 1/K as k → ∞. In fact the two densities are closer by factor of K than one might expect.

Discriminant Analysis of Directional Data

Expressions are obtained for discriminant statistics when sampling from circular and spherical normal distributions. Some moments and sampling distributions are obtained for these statistics. Also some expressions for the theoretical error probabilities are evaluated.

Probability Content of Regions Under Spherical Normal Distributions, IV: The Distribution of Homogeneous and Non-Homogeneous Quadratic Functions of Normal Variables

Probability Content of Regions Under Spherical Normal Distributions, IV: The Distribution of Homogeneous and Non-Homogeneous Quadratic Functions of Normal Variables

On the Probability of Capturing a Randomly Selected Point in Three Dimensions

Previous article Next article On the Probability of Capturing a Randomly Selected Point in Three DimensionsWilliam C. GuentherWilliam C. Guentherhttps://doi.org/10.1137/1003045PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. H. Germond, The Circular Coverage Function, Report, RM-330, Rand, Santa Monica, Calif., 1950 Google Scholar[2] Andre G. Laurent, Bombing problems—a statistical approach, Operations Res., 5 (1957), 75–89 MR0087293 0173.48003 CrossrefISIGoogle Scholar[3] Harold Ruben, Probability content of regions under spherical normal distributions. I, Ann. Math. Statist., 31 (1960), 598–618 MR0117828 0108.32501 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails The application of the Hankel transformation to spherically symmetric coverage problemsJournal of Applied Probability, Vol. 17, No. 04 | 14 July 2016 Cross Ref The application of the Hankel transformation to spherically symmetric coverage problemsJournal of Applied Probability, Vol. 17, No. 4 | 14 July 2016 Cross Ref A Generalization of the Integral of the Circular Coverage FunctionThe American Mathematical Monthly, Vol. 71, No. 3 | 11 April 2018 Cross Ref Volume 3, Issue 3| 1961SIAM Review211-292 History Submitted:28 October 1960Published online:18 July 2006 InformationCopyright © 1961 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1003045Article page range:pp. 247-251ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Correction Notes: Corrections to "Probability Content of Regions Under Spherical Normal Distributions, II: The Distribution of the Range in Normal Samples"

Correction Notes: Corrections to "Probability Content of Regions Under Spherical Normal Distributions, II: The Distribution of the Range in Normal Samples"

Probability Content of Regions Under Spherical Normal Distributions, II: The Distribution of the Range in Normal Samples

Numerous investigations, both theoretical and numerical, have been made of the distribution of the range in normal samples. One of the first investigators was Student [1] who examined the distribution on an empirical basis. Somewhat earlier, Tippett [2] presented tables and charts for the mean, standard deviation, and of the measures of skewness and kurtosis $(\beta_1 \text{and} \beta_2)$ for the range; further studies of the moments were made by E. S. Pearson [3], Hartley and Pearson [6] and by Ruben [7]. Tables of the moment constants are available in [6] and in Pearson and Hartley's book of statistical tables [10] (Tables 20 and 27), while, more recently, tables of the moment constants were provided by Harter and Clemm [11]. Approximations to the probability integral and percentage points of the distribution were suggested by E. S. Pearson [4], Cox [12], Patnaik [14] and Tukey [15]. Pearson's approximations were based on Pearsonian distributions of Type I and VI, while Cox used the Gamma Function (i.e., a multiple of $\chi^2$ with fractional degrees of freedom) and Patnaik a multiple of $\chi$ with fractional degrees of freedom as the basis for their approximations. These last two approximations have been compared by Pearson [5]. An approximation of a different type was derived by Johnson [16] who gave a series expansion for the probability integral suitable for low sample sizes and low values of the range. The behavior of the distribution for large sample size has been studied by Gumbel [17], [18], [19], Elfving [20], Cox [13] and Harley and Pearson [21]. For theoretical studies of the exact distribution reference is made to McKay and Pearson [22], Pillai [23], [24] and Cadwell [25] (see also Hartley [26]). Finally, tables of the probability integral and percentage points of the distribution have been provided by Hartley and Pearson [27], [10] (Tables 23 and 22), as well as by Harter and Clemm [11]. In the present paper, the following new results relating to the range distribution will be obtained: (i) The latter function may be expressed as the product of the sample size and the probability content of a certain parallelotope relative to a hyperspherical normal distribution, and (ii) the function can be evaluated as an infinite series involving the even moments of a sum of independent truncated normal variables. The distribution function may also be directly related to the moment generating function of the square of the sum of these truncated variables.

Probability Content of Regions Under Spherical Normal Distributions, I

Probability Content of Regions Under Spherical Normal Distributions, I