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Register a new userIndependent Component Analysis for Compositional Data
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Added by
Christoph Muehlmann
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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Compositional data represent a specific family of multivariate data, where the information of interest is contained in the ratios between parts rather than in absolute values of single parts. The analysis of such specific data is challenging as the application of standard multivariate analysis tools on the raw observations can lead to spurious results. Hence, it is appropriate to apply certain transformations prior further analysis. One popular multivariate data analysis tool is independent component analysis. Independent component analysis aims to find statistically independent components in the data and as such might be seen as an extension to principal component analysis. In this paper we examine an approach of how to apply independent component analysis on compositional data by respecting the nature of the former and demonstrate the usefulness of this procedure on a metabolomic data set.
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Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately identifiable with the diverging dimensionality under some subspace sparsity assumption. The interesting blessing-of-dimensionality phenomenon enables us to propose the principal subspace estimation methods by using the sample centered log-ratio covariance. We also derive nonasymptotic error bounds for the subspace estimators, which exhibits a tradeoff between identification and estimation. Moreover, we develop efficient proximal alternating direction method of multipliers algorithms to solve the nonconvex and nonsmooth optimization problems. Simulation results demonstrate that the proposed methods perform as well as the oracle methods with known basis. Their usefulness is illustrated through an analysis of word usage pattern for statisticians.
Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on M-estimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but in-depth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a straightforward estimate based on the efficient score function by using B-spline approximations. The estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ICA methods in a variety of simulations.
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.
Independent component analysis (ICA) has been shown to be useful in many applications. However, most ICA methods are sensitive to data contamination and outliers. In this article we introduce a general minimum U-divergence framework for ICA, which covers some standard ICA methods as special cases. Within the U-family we further focus on the gamma-divergence due to its desirable property of super robustness, which gives the proposed method gamma-ICA. Statistical properties and technical conditions for the consistency of gamma-ICA are rigorously studied. In the limiting case, it leads to a necessary and sufficient condition for the consistency of MLE-ICA. This necessary and sufficient condition is weaker than the condition known in the literature. Since the parameter of interest in ICA is an orthogonal matrix, a geometrical algorithm based on gradient flows on special orthogonal group is introduced to implement gamma-ICA. Furthermore, a data-driven selection for the gamma value, which is critical to the achievement of gamma-ICA, is developed. The performance, especially the robustness, of gamma-ICA in comparison with standard ICA methods is demonstrated through experimental studies using simulated data and image data.
Functional principal component analysis (FPCA) could become invalid when data involve non-Gaussian features. Therefore, we aim to develop a general FPCA method to adapt to such non-Gaussian cases. A Kenalls $tau$ function, which possesses identical eigenfunctions as covariance function, is constructed. The particular formulation of Kendalls $tau$ function makes it less insensitive to data distribution. We further apply it to the estimation of FPCA and study the corresponding asymptotic consistency. Moreover, the effectiveness of the proposed method is demonstrated through a comprehensive simulation study and an application to the physical activity data collected by a wearable accelerometer monitor.
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