arXiv subjects

Andrew T. A. Wood

Publications and source records attributed to Andrew T. A. Wood.

26 records · Page 2Linked to original sources

arXiv2020

Score matching for compositional distributions

Compositional data and multivariate count data with known totals are challenging to analyse due to the non-negativity and sum-to-one constraints on the sample space. It is often the case that many of the compositional components are highly right-skewed, with large numbers of zeros. A major limitation of currently available estimators for compositional models is that they either cannot handle many zeros in the data or are not computationally feasible in moderate to high dimensions. We derive a new set of novel score matching estimators applicable to distributions on a Riemannian manifold with boundary, of which the standard simplex is a special case. The score matching method is applied to estimate the parameters in a new flexible truncation model for compositional data and we show that the estimators are scalable and available in closed form. Through extensive simulation studies, the scoring methodology is demonstrated to work well for estimating the parameters in the new truncation model and also for the Dirichlet distribution. We apply the new model and estimators to real microbiome compositional data and show that the model provides a good fit to the data.

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arXiv2019

Gaussian asymptotic limits for the $α$-transformation in the analysis of compositional data

Compositional data consists of vectors of proportions whose components sum to 1. Such vectors lie in the standard simplex, which is a manifold with boundary. One issue that has been rather controversial within the field of compositional data analysis is the choice of metric on the simplex. One popular possibility has been to use the metric implied by logtransforming the data, as proposed by Aitchison [1, 2]; and another popular approach has been to use the standard Euclidean metric inherited from the ambient space. Tsagris et al. [21] proposed a one-parameter family of power transformations, the $α$-transformations, which include both the metric implied by Aitchison's transformation and the Euclidean metric as particular cases. Our underlying philosophy is that, with many datasets, it may make sense to use the data to help us determine a suitable metric. A related possibility is to apply the $α$-transformations to a parametric family of distributions, and then estimate a along with the other parameters. However, as we shall see, when one follows this last approach with the Dirichlet family, some care is needed in a certain limiting case which arises $(α\neq 0)$, as we found out when fitting this model to real and simulated data. Specifically, when the maximum likelihood estimator of a is close to 0, the other parameters tend to be large. The main purpose of the paper is to study this limiting case both theoretically and numerically and to provide insight into these numerical findings.

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arXiv2017

The extended power distribution: A new distribution on $(0, 1)$

We propose a two-parameter bounded probability distribution called the extended power distribution. This distribution on $(0, 1)$ is similar to the beta distribution, however there are some advantages which we explore. We define the moments and quantiles of this distribution and show that it is possible to give an $r$-parameter extension of this distribution ($r>2$). We also consider its complementary distribution and show that it has some flexibility advantages over the Kumaraswamy and beta distributions. This distribution can be used as an alternative to the Kumaraswamy distribution since it has a closed form for its cumulative function. However, it can be fitted to data where there are some samples that are exactly equal to 1, unlike the Kumaraswamy and beta distributions which cannot be fitted to such data or may require some censoring. Applications considered show the extended power distribution performs favourably against the Kumaraswamy distribution in most cases.

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arXiv2016

Nonparametric hypothesis testing for equality of means on the simplex

In the context of data that lie on the simplex, we investigate use of empirical and exponential empirical likelihood, and Hotelling and James statistics, to test the null hypothesis of equal population means based on two independent samples. We perform an extensive numerical study using data simulated from various distributions on the simplex. The results, taken together with practical considerations regarding implementation, support the use of bootstrap-calibrated James statistic.

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arXiv2015

Improved classification for compositional data using the $α$-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 $α$-transformation to transform the data and then to classify the transformed data via regularised discriminant analysis and the k-nearest neighbours algorithm. Using the $α$-transformation generalises two rival approaches in compositional data analysis, one (when $α=1$) that treats the data as though they were Euclidean, ignoring the compositional constraint, and another (when $α=0$) that employs Aitchison's centred log-ratio transformation. A numerical study with several real datasets shows that whether using $α=1$ or $α=0$ gives better classification performance depends on the dataset, and moreover that using an intermediate value of $α$ can sometimes give better performance than using either 1 or 0.

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arXiv2011

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, α. 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 α, 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 α-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 α-transformation and consider the corresponding family of Frechet means.

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arXiv2005

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.

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arXiv2004

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 function

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