Estimation of fractal dimension for a class of Non-Gaussian stationary processes and fields

We present the asymptotic distribution theory for a class of increment-based estimators of the fractal dimension of a random field of the form g{X(t)}, where g:R\to R is an unknown smooth function and X(t) is a real-valued stationary Gaussian field on R^d, d=1 or 2, whose covariance function obeys a power law at the origin. The relevant theoretical framework here is ``fixed domain'' (or ``infill'') asymptotics. Surprisingly, the limit theory in this non-Gaussian case is somewhat richer than in the Gaussian case (the latter is recovered when g is affine), in part because estimators of the type considered may have an asymptotic variance which is random in the limit. Broadly, when g is smooth and nonaffine, three types of limit distributions can arise, types (i), (ii) and (iii), say. Each type can be represented as a random integral. More specifically, type (i) can be represented as the integral of a certain random function with respect to Lebesgue measure; type (ii) can be represented as the integral of a second random functio

Saddlepoint approximation for moment generating functions of truncated random variables

We consider the problem of approximating the moment generating function (MGF) of a truncated random variable in terms of the MGF of the underlying (i.e., untruncated) random variable. The purpose of approximating the MGF is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables. Two important statistical applications are the following: the approximation of certain multivariate cumulative distribution functions; and the approximation of passage time distributions in ion channel models which incorporate time interval omission. We derive two types of representation for the MGF of a truncated random variable. One of these representations is obtained by exponential tilting. The second type of representation, which has two versions, is referred to as an exponential convolution representation. Each representation motivates a different approximation. It turns out that each of the three approximations is extremely accurate in those cases ``to which it is suited.'' Moreover, there is a simple rule of thumb for deciding which approximation to use in a given case, and if this rule is followed, then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate.Comment: Published at http://dx.doi.org/10.1214/009053604000000689 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A data-based power transformation for compositional data

Compositional data analysis is carried out either by neglecting the compositional constraint and applying standard multivariate data analysis, or by transforming the data using the logs of the ratios of the components. In this work we examine a more general transformation which includes both approaches as special cases. It is a power transformation and involves a single parameter, {\alpha}. The transformation has two equivalent versions. The first is the stay-in-the-simplex version, which is the power transformation as defined by Aitchison in 1986. The second version, which is a linear transformation of the power transformation, is a Box-Cox type transformation. We discuss a parametric way of estimating the value of {\alpha}, which is maximization of its profile likelihood (assuming multivariate normality of the transformed data) and the equivalence between the two versions is exhibited. Other ways include maximization of the correct classification probability in discriminant analysis and maximization of the pseudo R-squared (as defined by Aitchison in 1986) in linear regression. We examine the relationship between the {\alpha}-transformation, the raw data approach and the isometric log-ratio transformation. Furthermore, we also define a suitable family of metrics corresponding to the family of {\alpha}-transformation and consider the corresponding family of Frechet means.Comment: Published in the proceddings of the 4th international workshop on Compositional Data Analysis. http://congress.cimne.com/codawork11/frontal/default.as

Improved classification for compositional data using the α\alpha-transformation

In compositional data analysis an observation is a vector containing non-negative values, only the relative sizes of which are considered to be of interest. Without loss of generality, a compositional vector can be taken to be a vector of proportions that sum to one. Data of this type arise in many areas including geology, archaeology, biology, economics and political science. In this paper we investigate methods for classification of compositional data. Our approach centres on the idea of using the $\alpha$-transformation to transform the data and then to classify the transformed data via regularised discriminant analysis and the k-nearest neighbours algorithm. Using the $\alpha$-transformation generalises two rival approaches in compositional data analysis, one (when $\alpha=1$) that treats the data as though they were Euclidean, ignoring the compositional constraint, and another (when $\alpha=0$) that employs Aitchison's centred log-ratio transformation. A numerical study with several real datasets shows that whether using $\alpha=1$ or $\alpha=0$ gives better classification performance depends on the dataset, and moreover that using an intermediate value of $\alpha$ can sometimes give better performance than using either 1 or 0.Comment: This is a 17-page preprint and has been accepted for publication at the Journal of Classificatio

Scaled von Mises-Fisher distributions and regression models for paleomagnetic directional data

We propose a new distribution for analysing paleomagnetic directional data that is a novel transformation of the von Mises-Fisher distribution. The new distribution has ellipse-like symmetry, as does the Kent distribution; however, unlike the Kent distribution the normalising constant in the new density is easy to compute and estimation of the shape parameters is straightforward. To accommodate outliers, the model also incorporates an additional shape parameter which controls the tail-weight of the distribution. We also develop a general regression model framework that allows both the mean direction and the shape parameters of the error distribution to depend on covariates. The proposed regression procedure is shown to be equivariant with respect to the choice of coordinate system for the directional response. To illustrate, we analyse paleomagnetic directional data from the GEOMAGIA50.v3 database (Brown et al. 2015). We predict the mean direction at various geological 1 time points and show that there is significant heteroscedasticity present. It is envisaged that the regression structures and error distribution proposed here will also prove useful when covariate information is available with (i) other types of directional response data; and (ii) square-root transformed compositional data of general dimension

Cauchy robust principal component analysis with applications to high-deimensional data sets

Principal component analysis (PCA) is a standard dimensionality reduction technique used in various research and applied fields. From an algorithmic point of view, classical PCA can be formulated in terms of operations on a multivariate Gaussian likelihood. As a consequence of the implied Gaussian formulation, the principal components are not robust to outliers. In this paper, we propose a modified formulation, based on the use of a multivariate Cauchy likelihood instead of the Gaussian likelihood, which has the effect of robustifying the principal components. We present an algorithm to compute these robustified principal components. We additionally derive the relevant influence function of the first component and examine its theoretical properties. Simulation experiments on high-dimensional datasets demonstrate that the estimated principal components based on the Cauchy likelihood outperform or are on par with existing robust PCA techniques

Network analysis of host-virus communities in bats and rodents reveals determinants of cross-species transmission.

Bats are natural reservoirs of several important emerging viruses. Cross-species transmission appears to be quite common among bats, which may contribute to their unique reservoir potential. Therefore, understanding the importance of bats as reservoirs requires examining them in a community context rather than concentrating on individual species. Here, we use a network approach to identify ecological and biological correlates of cross-species virus transmission in bats and rodents, another important host group. We show that given our current knowledge the bat viral sharing network is more connected than the rodent network, suggesting viruses may pass more easily between bat species. We identify host traits associated with important reservoir species: gregarious bats are more likely to share more viruses and bats which migrate regionally are important for spreading viruses through the network. We identify multiple communities of viral sharing within bats and rodents and highlight potential species traits that can help guide studies of novel pathogen emergence.This work was supported by the Research and Policy for Infectious Disease Dynamics (RAPIDD) program of the Science and Technology Directorate (US Department of Homeland Security) and the Fogarty International Center (National Institutes of Health). D.T.S.H. acknowledges funding from a David H. Smith post-doctoral fellowship. A.A.C. is partially funded by a Royal Society Wolfson Research Merit award, and J.L.N.W. is supported by the Alborada Trust. Thanks to Paul Cryan and Michael O'Donnell of the USGS Fort Collins Science Center for help with species distribution analyses.This is the final version of the article. It first appeared from Wiley via http://dx.doi.org/10.1111/ele.1249

Network analysis of host–virus communities in bats and rodents reveals determinants of cross-species transmission

Bats are natural reservoirs of several important emerging viruses. Cross-species transmission appears to be quite common among bats, which may contribute to their unique reservoir potential. Therefore, understanding the importance of bats as reservoirs requires examining them in a community context rather than concentrating on individual species. Here, we use a network approach to identify ecological and biological correlates of cross-species virus transmission in bats and rodents, another important host group. We show that given our current knowledge the bat viral sharing network is more connected than the rodent network, suggesting viruses may pass more easily between bat species. We identify host traits associated with important reservoir species: gregarious bats are more likely to share more viruses and bats which migrate regionally are important for spreading viruses through the network. We identify multiple communities of viral sharing within bats and rodents and highlight potential species traits that can help guide studies of novel pathogen emergence