JADE for Tensor-Valued Observations

A review of second-order blind identification methods

Second-order source separation (SOS) is a data analysis tool which can be used for revealing hidden structures in multivariate time series data or as a tool for dimension reduction. Such methods are nowadays increasingly important as more and more high-dimensional multivariate time series data are measured in numerous fields of applied science. Dimension reduction is crucial, as modeling such high-dimensional data with multivariate time series models is often impractical as the number of parameters describing dependencies between the component time series is usually too high. SOS methods have their roots in the signal processing literature, where they were first used to separate source signals from an observed signal mixture. The SOS model assumes that the observed time series (signals) is a linear mixture of latent time series (sources) with uncorrelated components. The methods make use of the second-order statistics-hence the name "second-order source separation." In this review, we discuss the classical SOS methods and their extensions to more complex settings. An example illustrates how SOS can be performed. This article is categorized under:Statistical Models > Time Series ModelsStatistical and Graphical Methods of Data Analysis > Dimension ReductionData: Types and Structure > Time Series, Stochastic Processes, and Functional Data.

Fast tensorial JADE

We propose a novel method for tensorial-independent component analysis. Our approach is based on TJADE and k-JADE, two recently proposed generalizations of the classical JADE algorithm. Our novel method achieves the consistency and the limiting distribution of TJADE under mild assumptions and at the same time offers notable improvement in computational speed. Detailed mathematical proofs of the statistical properties of our method are given and, as a special case, a conjecture on the properties of k-JADE is resolved. Simulations and timing comparisons demonstrate remarkable gain in speed. Moreover, the desired efficiency is obtained approximately for finite samples. The method is applied successfully to large-scale video data, for which neither TJADE nor k-JADE is feasible. Finally, an experimental procedure is proposed to select the values of a set of tuning parameters. Supplementary material including the R-code for running the examples and the proofs of the theoretical results is available online.

Sampling Properties of color Independent Component Analysis

Independent Component Analysis (ICA) offers an effective data-driven approach for blind source extraction encountered in many signal and image processing problems. Although many ICA methods have been developed, they have received relatively little attention in the statistics literature, especially in terms of rigorous theoretical investigation for statistical inference. The current paper aims at narrowing this gap and investigates the statistical sampling properties of the colorICA (cICA) method. The cICA incorporates the correlation structure within sources through parametric time series models in the frequency domain and outperforms several existing ICA alternatives numerically. We establish the consistency and asymptotic normality of the cICA estimates, which then enables statistical inference based on the estimates. These asymptotic properties are further validated using simulation studies.

Tensorial blind source separation for improved analysis of multi-omic data

There is an increased need for integrative analyses of multi-omic data. We present and benchmark a novel tensorial independent component analysis (tICA) algorithm against current state-of-the-art methods. We find that tICA outperforms competing methods in identifying biological sources of data variation at a reduced computational cost. On epigenetic data, tICA can identify methylation quantitative trait loci at high sensitivity. In the cancer context, tICA identifies gene modules whose expression variation across tumours is driven by copy-number or DNA methylation changes, but whose deregulation relative to normal tissue is independent of such alterations, a result we validate by direct analysis of individual data types.