arXiv subjects

Ionas Erb

Publications and source records attributed to Ionas Erb.

6 recordsLinked to original sources

arXiv2023

Compositional Covariance Shrinkage and Regularised Partial Correlations

We propose an estimation procedure for covariation in wide compositional data sets. For compositions, widely-used logratio variables are interdependent due to a common reference. Logratio uncorrelated compositions are linearly independent before the unit-sum constraint is imposed. We show how they are used to construct bespoke shrinkage targets for logratio covariance matrices and test a simple procedure for partial correlation estimates on both a simulated and a single-cell gene expression data set. For the underlying counts, different zero imputations are evaluated. The partial correlation induced by the closure is derived analytically. Data and code are available from GitHub.

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arXiv2023

Power Transformations of Relative Count Data as a Shrinkage Problem

Here we show an application of our recently proposed information-geometric approach to compositional data analysis (CoDA). This application regards relative count data, which are, e.g., obtained from sequencing experiments. First we review in some detail a variety of necessary concepts ranging from basic count distributions and their information-geometric description over the link between Bayesian statistics and shrinkage to the use of power transformations in CoDA. We then show that powering, i.e., the equivalent to scalar multiplication on the simplex, can be understood as a shrinkage problem on the tangent space of the simplex. In information-geometric terms, traditional shrinkage corresponds to an optimization along a mixture (or m-) geodesic, while powering (or, as we call it, exponential shrinkage) can be optimized along an exponential (or e-) geodesic. While the m-geodesic corresponds to the posterior mean of the multinomial counts using a conjugate prior, the e-geodesic corresponds to an alternative parametrization of the posterior where prior and data contributions are weighted by geometric rather than arithmetic means. To optimize the exponential shrinkage parameter, we use mean-squared error as a cost function on the tangent space. This is just the expected squared Aitchison distance from the true parameter. We derive an analytic solution for its minimum based on the delta method and test it via simulations. We also discuss exponential shrinkage as an alternative to zero imputation for dimension reduction and data normalization.

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arXiv2023

Aitchison's Compositional Data Analysis 40 Years On: A Reappraisal

The development of John Aitchison's approach to compositional data analysis is followed since his paper read to the Royal Statistical Society in 1982. Aitchison's logratio approach, which was proposed to solve the problematic aspects of working with data with a fixed sum constraint, is summarized and reappraised. It is maintained that the properties on which this approach was originally built, the main one being subcompositional coherence, are not required to be satisfied exactly -- quasi-coherence is sufficient, that is near enough to being coherent for all practical purposes. This opens up the field to using simpler data transformations, such as power transformations, that permit zero values in the data. The additional property of exact isometry, which was subsequently introduced and not in Aitchison's original conception, imposed the use of isometric logratio transformations, but these are complicated and problematic to interpret, involving ratios of geometric means. If this property is regarded as important in certain analytical contexts, for example unsupervised learning, it can be relaxed by showing that regular pairwise logratios, as well as the alternative quasi-coherent transformations, can also be quasi-isometric, meaning they are close enough to exact isometry for all practical purposes. It is concluded that the isometric and related logratio transformations such as pivot logratios are not a prerequisite for good practice, although many authors insist on their obligatory use. This conclusion is fully supported here by case studies in geochemistry and in genomics, where the good performance is demonstrated of pairwise logratios, as originally proposed by Aitchison, or Box-Cox power transforms of the original compositions where no zero replacements are necessary.

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arXiv2021

A Critique of Differential Abundance Analysis, and Advocacy for an Alternative

It is largely taken for granted that differential abundance analysis is, by default, the best first step when analyzing genomic data. We argue that this is not necessarily the case. In this article, we identify key limitations that are intrinsic to differential abundance analysis: it is (a) dependent on unverifiable assumptions, (b) an unreliable construct, and (c) overly reductionist. We formulate an alternative framework called ratio-based biomarker analysis which does not suffer from the identified limitations. Moreover, ratio-based biomarkers are highly flexible. Beyond replacing DAA, they can also be used for many other bespoke analyses, including dimension reduction and multi-omics data integration.

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arXiv2021

The Information-Geometric Perspective of Compositional Data Analysis

Information geometry uses the formal tools of differential geometry to describe the space of probability distributions as a Riemannian manifold with an additional dual structure. The formal equivalence of compositional data with discrete probability distributions makes it possible to apply the same description to the sample space of Compositional Data Analysis (CoDA). The latter has been formally described as a Euclidean space with an orthonormal basis featuring components that are suitable combinations of the original parts. In contrast to the Euclidean metric, the information-geometric description singles out the Fisher information metric as the only one keeping the manifold's geometric structure invariant under equivalent representations of the underlying random variables. Well-known concepts that are valid in Euclidean coordinates, e.g., the Pythogorean theorem, are generalized by information geometry to corresponding notions that hold for more general coordinates. In briefly reviewing Euclidean CoDA and, in more detail, the information-geometric approach, we show how the latter justifies the use of distance measures and divergences that so far have received little attention in CoDA as they do not fit the Euclidean geometry favored by current thinking. We also show how entropy and relative entropy can describe amalgamations in a simple way, while Aitchison distance requires the use of geometric means to obtain more succinct relationships. We proceed to prove the information monotonicity property for Aitchison distance. We close with some thoughts about new directions in CoDA where the rich structure that is provided by information geometry could be exploited.

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arXiv2019

Partial Correlations in Compositional Data Analysis

Partial correlations quantify linear association between two variables adjusting for the influence of the remaining variables. They form the backbone for graphical models and are readily obtained from the inverse of the covariance matrix. For compositional data, the covariance structure is specified from log ratios of variables, so unless we try to "open" the data via a normalization, this implies changes in the definition and interpretation of partial correlations. In the present work, we elucidate how results derived by Aitchison (1986) lead to a natural definition of partial correlation that has a number of advantages over current measures of association. For this, we show that the residuals of log-ratios between a variable with a reference, when adjusting for all remaining variables including the reference, are reference-independent. Since the reference itself can be controlled for, correlations between residuals are defined for the variables directly without the necessity to recur to ratios except when specifying which variables are partialled out. Thus, perhaps surprisingly, partial correlations do not have the problems commonly found with measures of pairwise association on compositional data. They are well-defined between two variables, are properly scaled, and allow for negative association. By design, they are subcompositionally incoherent, but they share this property with conventional partial correlations (where results change when adjusting for the influence of fewer variables). We discuss the equivalence with normalization-based approaches whenever the normalizing variables are controlled for. We also discuss the partial variances and correlations we obtain from a previously studied data set of Roman glass cups.

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