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

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

Digital Single-Cell Analysis of Plant Organ Development Using 3DCellAtlas

Diverse molecular networks underlying plant growth and development are rapidly being uncovered. Integrating these data into the spatial and temporal context of dynamic organ growth remains a technical challenge. We developed 3DCellAtlas, an integrative computational pipeline that semiautomatically identifies cell types and quantifies both 3D cellular anisotropy and reporter abundance at single-cell resolution across whole plant organs. Cell identification is no less than 97.8% accurate and does not require transgenic lineage markers or reference atlases. Cell positions within organs are defined using an internal indexing system generating cellular level organ atlases where data from multiple samples can be integrated. Using this approach, we quantified the organ-wide cell-type-specific 3D cellular anisotropy driving Arabidopsis thaliana hypocotyl elongation. The impact ethylene has on hypocotyl 3D cell anisotropy identified the preferential growth of endodermis in response to this hormone. The spatiotemporal dynamics of the endogenous DELLA protein RGA, expansin gene EXPA3, and cell expansion was quantified within distinct cell types of Arabidopsis roots. A significant regulatory relationship between RGA, EXPA3, and growth was present in the epidermis and endodermis. The use of single-cell analyses of plant development enables the dynamics of diverse regulatory networks to be integrated with 3D organ growth.</p

Are treatment effect assumptions in orthodontic studies overoptimistic?

BACKGROUND At the clinical trial design stage, assumptions regarding the treatment effects to be detected should be appropriate so that the required sample size can be calculated. There is evidence in the medical literature that sample size assumption can be overoptimistic. The aim of this study was to compare the distribution of the assumed effects versus that of the observed effects as a proxy for overoptimistic treatment effect assumptions at the study design stage. MATERIALS AND METHOD Systematic reviews (SRs) published between 1 January 2010 and 31 December 2019 containing at least one meta-analysis on continuous outcomes were identified electronically. SR and primary study level characteristics were extracted from the SRs and the individual trials. Details on the sample size calculation process and assumptions and the observed treatment effects were extracted. RESULTS Eighty-five SRs with meta-analysis containing 347 primary trials were included. The median number of SR authors was 5 (interquartile range: 4-7). At the primary study level, the majority were single centre (78.1%), utilized a parallel design (52%), and rated as an unclear/moderate level of risk of bias (34.3%). A sample size was described in only 31.7% (110/347) of studies. From this cohort of 110 studies, in only 37 studies was the assumed clinical difference that the study was designed to detect reported (37/110). The assumed treatment effect was recalculated for the remaining 73 studies (73/110). The one-sided exact signed rank test showed a significant difference between the assumed and observed treatment effects (P < 0.001) suggesting greater values for the assumed effect sizes. CONCLUSIONS Careful consideration of the assumptions at the design stage of orthodontic studies are necessary in order to reduce the unreliability of clinical study results and research waste

Robust principal component analysis for power transformed compositional data

Geochemical surveys collect sediment or rock samples, measure the concentration of chemical elements, and report these typically either in weight percent or in parts per million (ppm). There are usually a large number of elements measured and the distributions are often skewed, containing many potential outliers. We present a new robust principal component analysis (PCA) method for geochemical survey data, that involves first transforming the compositional data onto a manifold using a relative power transformation. A flexible set of moment assumptions are made which take the special geometry of the manifold into account. The Kent distribution moment structure arises as a special case when the chosen manifold is the hypersphere. We derive simple moment and robust estimators (RO) of the parameters which are also applicable in high-dimensional settings. The resulting PCA based on these estimators is done in the tangent space and is related to the power transformation method used in correspondence analysis. To illustrate, we analyze major oxide data from the National Geochemical Survey of Australia. When compared with the traditional approach in the literature based on the centered log-ratio transformation, the new PCA method is shown to be more successful at dimension reduction and gives interpretable results