Search

Many geological phenomena are regularly measured over time to follow developments and changes. For many of these phenomena, the absolute values are not of interest, but rather the relative information, which means that the data are compositional time series. Thus, the serial nature and the compositional geometry should be considered when analyzing the data. Multivariate time series are already challenging, especially if they are higher dimensional, and latent variable models are a popular way to deal with this kind of data. Blind source separation techniques are well-established latent factor models for time series, with many variants covering quite different time series models. Here, several such methods and their assumptions are reviewed, and it is shown how they can be applied to high-dimensional compositional time series. Also, a novel blind source separation method is suggested which is quite flexible regarding the assumptions of the latent time series. The methodology is illustrated using simulations and in an application to light absorbance data from water samples taken from a small stream in Lower Austria.

Scatter matrices generalize the covariance matrix and are useful in many multivariate data analysis methods, including well-known principal component analysis (PCA), which is based on the diagonalization of the covariance matrix. The simultaneous diagonalization of two or more scatter matrices goes beyond PCA and is used more and more often. In this paper, we offer an overview of many methods that are based on a joint diagonalization. These methods range from the unsupervised context with invariant coordinate selection and blind source separation, which includes independent component analysis, to the supervised context with discriminant analysis and sliced inverse regression. They also encompass methods that handle dependent data such as time series or spatial data. peerReviewed

Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li (2016). The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are compared in simulations and illustrated in real data examples.

Sliced inverse regression is one of the most popular sufficient dimension reduction methods. Originally, it was designed for independent and identically distributed data and recently extend to the case of serially and spatially dependent data. In this work we extend it to the case of spatially dependent data where the response might depend also on neighbouring covariates when the observations are taken on a grid-like structure as it is often the case in econometric spatial regression applications. We suggest guidelines on how to decide upon the dimension of the subspace of interest and also which spatial lag might be of interest when modeling the response. These guidelines are supported by a conducted simulation study.

Regional data analysis is concerned with the analysis and modeling of measurements that are spatially separated by specifically accounting for typical features of such data. Namely, measurements in close proximity tend to be more similar than the ones further separated. This might hold also true for cross-dependencies when multivariate spatial data is considered. Often, scientists are interested in linear transformations of such data which are easy to interpret and might be used as dimension reduction. Recently, for that purpose spatial blind source separation (SBSS) was introduced which assumes that the observed data are formed by a linear mixture of uncorrelated, weakly stationary random fields. However, in practical applications, it is well-known that when the spatial domain increases in size the weak stationarity assumptions can be violated in the sense that the second order dependency is varying over the domain which leads to non-stationary analysis. In our work we extend the SBSS model to adjust for these stationarity violations, present three novel estimators and establish the identifiability and affine equivariance property of the unmixing matrix functionals defining these estimators. In an extensive simulation study, we investigate the performance of our estimators and also show their use in the analysis of a geochemical dataset which is derived from the GEMAS geochemical mapping project. peerReviewed

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 to 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 latter and demonstrate the usefulness of this procedure on a metabolomics dataset.

Multivariate measurements taken at irregularly sampled locations are a common form of data, for example in geochemical analysis of soil. In practical considerations predictions of these measurements at unobserved locations are of great interest. For standard multivariate spatial prediction methods it is mandatory to not only model spatial dependencies but also cross-dependencies which makes it a demanding task. Recently, a blind source separation approach for spatial data was suggested. When using this spatial blind source separation method prior the actual spatial prediction, modelling of spatial cross-dependencies is avoided, which in turn simplifies the spatial prediction task significantly. In this paper we investigate the use of spatial blind source separation as a pre-processing tool for spatial prediction and compare it with predictions from Cokriging and neural networks in an extensive simulation study as well as a geochemical dataset.

Partial orderings and measures of information for continuous univariate random variables with special roles of Gaussian and uniform distributions are discussed. The information measures and measures of non-Gaussianity including the third and fourth cumulants are generally used as projection indices in the projection pursuit approach for the independent component analysis. The connections between information, non-Gaussianity and statistical independence in the context of independent component analysis is discussed in detail.

The Oja median is one of several extensions of the univariate median to the multivariate case. It has many desirable properties, but is computationally demanding. In this paper, we first review the properties of the Oja median and compare it to other multivariate medians. Then, we discuss four algorithms to compute the Oja median, which are implemented in our R package OjaNP. Besides these algorithms, the package contains also functions to compute Oja signs, Oja signed ranks, Oja ranks, and the related scatter concepts. To illustrate their use, the corresponding multivariate one- and C-sample location tests are implemented.

Dimension reduction is a common strategy in multivariate data analysis which seeks a subspace which contains all interesting features needed for the subsequent analysis. Non-Gaussian component analysis attempts for this purpose to divide the data into a non-Gaussian part, the signal, and a Gaussian part, the noise. We will show that the simultaneous use of two scatter functionals can be used for this purpose and suggest a bootstrap test to test the dimension of the non-Gaussian subspace. Sequential application of the test can then for example be used to estimate the signal dimension.

In this article, we provide a personal review of the literature on nonparametric and robust tools in the standard univariate and multivariate location and scatter, as well as linear regression problems, with a special focus on sign and rank methods, their equivariance and invariance properties, and their robustness and efficiency. Beyond parametric models, the population quantities of interest are often formulated as location, scatter, skewness, kurtosis and other functionals. Some old and recent tools for model checking, dimension reduction, and subspace estimation in wide semiparametric models are discussed. We also discuss recent extensions of procedures in certain nonstandard semiparametric cases including clustered and matrix-valued data. Our personal list of important unsolved and future issues is provided.

Independent component analysis is a standard tool in modern data analysis and numerous different techniques for applying it exist. The standard methods however quickly lose their effectiveness when the data are made up of structures of higher order than vectors, namely matrices or tensors (for example, images or videos), being unable to handle the high amounts of noise. Recently, an extension of the classic fourth order blind identification (FOBI) specifically suited for tensor-valued observations was proposed and showed to outperform its vector version for tensor data. In this paper we extend another popular independent component analysis method, the joint approximate diagonalization of eigen-matrices (JADE), for tensor observations. In addition to the theoretical background we also provide the asymptotic properties of the proposed estimator and use both simulations and real data to show its usefulness and superiority over its competitors., 10 pages, 3 figures

Blind source separation (BSS) is a well-known signal processing tool which is used to solve practical data analysis problems in various fields of science. In BSS, we assume that the observed data consists of linear mixtures of latent variables. The mixing system and the distributions of the latent variables are unknown. The aim is to find an estimate of an unmixing matrix which then transforms the observed data back to latent sources. In this paper we present the R packages JADE and BSSasymp. The package JADE offers several BSS methods which are based on joint diagonalization. Package BSSasymp contains functions for computing the asymptotic covariance matrices as well as their data-based estimates for most of the BSS estimators included in package JADE. Several simulated and real datasets are used to illustrate the functions in these two packages. peerReviewed

The blind source separation model for multivariate time series generally assumes that the observed series is a linear transformation of an unobserved series with temporally uncorrelated or independent components. Given the observations, the objective is to find a linear transformation that recovers the latent series. Several methods for accomplishing this exist and three particular ones are the classic SOBI and the recently proposed generalized FOBI (gFOBI) and generalized JADE (gJADE), each based on the use of joint lagged moments. In this paper we generalize the methodologies behind these algorithms for tensor-valued time series. We assume that our data consists of a tensor observed at each time point and that the observations are linear transformations of latent tensors we wish to estimate. The tensorial generalizations are shown to have particularly elegant forms and we show that each of them is Fisher consistent and orthogonal equivariant. Comparing the new methods with the original ones in various settings shows that the tensorial extensions are superior to both their vector-valued counterparts and to two existing tensorial dimension reduction methods for i.i.d. data. Finally, applications to fMRI-data and video processing show that the methods are capable of extracting relevant information from noisy high-dimensional data., 26 pages, 6 figures

Summary Recently a blind source separation model was suggested for spatial data, along with an estimator based on the simultaneous diagonalization of two scatter matrices. The asymptotic properties of this estimator are derived here, and a new estimator based on the joint diagonalization of more than two scatter matrices is proposed. The asymptotic properties and merits of the novel estimator are verified in simulation studies. A real-data example illustrates application of the method.

The classical second order source separation methods use approximate joint diagonalization of autocovariance matrices with several lags to estimate the unmixing matrix. Based on recent asymptotic results, we propose a novel unmixing matrix estimator which selects the best lag set from a finite set of candidate sets specified by the user. The theory is illustrated by a simulation study.

We generalize the Minimum Distance (MD) index to be applicable in complex valued ICA. To illustrate the use of the MD index, we present a complex version of AMUSE and compare it to complex FOBI in a simulation study.

We extend the theory of M-estimation to incomplete and dependent multivariate data. ML-estimation can still be considered a special case of M-estimation in this context. We notice that the unobserved data must be missing completely at random but not only missing at random, which is a typical assumption of ML-estimation, to guarantee the consistency of an M-estimator. Further, we show that the weight functions for scatter must satisfy a critical scaling condition, which is implicitly fulfilled both by the Gaussian and by Tyler’s weight function. We generalize this principal result by introducing the class of power weight functions, which contains the two aforementioned weight functions as limiting cases. A simulation study confirms our theoretical findings. If the data are heavy tailed or contaminated, the M-estimators turn out to be favorable compared to the ML-estimators that are based on the normal-distribution assumption.

We extend two methods of independent component analysis, fourth order blind identification and joint approximate diagonalization of eigen-matrices, to vector-valued functional data. Multivariate functional data occur naturally and frequently in modern applications, and extending independent component analysis to this setting allows us to distill important information from this type of data, going a step further than the functional principal component analysis. To allow the inversion of the covariance operator we make the assumption that the dependency between the component functions lies in a finite-dimensional subspace. In this subspace we define fourth cross-cumulant operators and use them to construct the two novel, Fisher consistent methods for solving the independent component problem for vector-valued functions. Both simulations and an application on a hand gesture data set show the usefulness and advantages of the proposed methods over functional principal component analysis., Comment: 39 pages, 3 figures

In this work, 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., 44 pages, 11 figures. Note: the title of the manuscript was earlier "Asymptotically and computationally efficient tensorial JADE"

Schematic representation of various location and scatter models. Independent component (IC) model and elliptical model are diverse semiparametric extensions of Gaussian model. (Credits to Elina Var...

There is an increased need for integrative analyses of multi-omic data. Although several algorithms for analysing multi-omic data exist, no study has yet performed a detailed comparison of these methods in biologically relevant contexts. Here we benchmark a novel tensorial independent component analysis (tICA) algorithm against current state-of-the-art methods. Using simulated and real multi-omic data, we find that tICA outperforms established methods in identifying biological sources of data variation at a significantly reduced computational cost. Using two independent multi cell-type EWAS, we further demonstrate how tICA can identify, in the absence of genotype information, mQTLs at a higher sensitivity than competing multi-way algorithms. We validate mQTLs found with tICA in an independent set, and demonstrate that approximately 75% of mQTLs are independent of blood cell subtype. In an application to multi-omic cancer data, tICA identifies many gene modules whose expression variation across tumors is driven by copy number or DNA methylation changes, but whose deregulation relative to the normal state is independent such alterations, an important finding that we confirm by direct analysis of individual data types. In summary, tICA is a powerful novel algorithm for decomposing multi-omic data, which will be of great value to the research community.

We consider a second order source separation model where a set of latent signals is internally mixed with several channels of noise and the goal is to estimate the number of signals. For the purpose we extend the ladle estimator which has been so far considered only for iid methods such as PCA, CCA or FOBI. Using time series bootstrapping methods ladle estima-tors based on AMUSE and SOBI are presented and a simulation study demonstrates that especially SOBI works well if the time series are sufficiently long.

While an important topic in practice, the estimation of the number of non-noise components in blind source separation has received little attention in the literature. Recently, two bootstrap-based techniques for estimating the dimension were proposed, and although very efficient, they suffer from the long computation times caused by the resampling. We approach the problem from a large sample viewpoint and develop an asymptotic test for the true dimension. Our test statistic based on second-order temporal information has a very simple limiting distribution under the null hypothesis and requires no parameters to estimate. Comparisons to the resampling-based estimates show that the asymptotic test provides comparable error rates with significantly faster computation time. An application to sound recording data is used to illustrate the method in practice., Comment: 60 pages, 5 figures

We assume a second-order source separation model where the observed multivariate time series is a linear mixture of latent, temporally uncorrelated time series with some components pure white noise. To avoid the modelling of noise, we extract the non-noise latent components using some standard method, allowing the modelling of the extracted univariate time series individually. An important question is the determination of which of the latent components are of interest in modelling and which can be considered as noise. Bootstrap-based methods have recently been used in determining the latent dimension in various methods of unsupervised and supervised dimension reduction and we propose a set of similar estimation strategies for second-order stationary time series. Simulation studies and a sound wave example are used to show the method's effectiveness.

Blind source separation (BSS) is a signal processing tool, which is widely used in various fields. Examples include biomedical signal separation, brain imaging and economic time series applications. In BSS, one assumes that the observed $p$ time series are linear combinations of $p$ latent uncorrelated weakly stationary time series. The aim is then to find an estimate for an unmixing matrix, which transforms the observed time series back to uncorrelated latent time series. In SOBI (Second Order Blind Identification) joint diagonalization of the covariance matrix and autocovariance matrices with several lags is used to estimate the unmixing matrix. The rows of an unmixing matrix can be derived either one by one (deflation-based approach) or simultaneously (symmetric approach). The latter of these approaches is well-known especially in signal processing literature, however, the rigorous analysis of its statistical properties has been missing so far. In this paper, we fill this gap and investigate the statistical properties of the symmetric SOBI estimate in detail and find its limiting distribution under general conditions. The asymptotical efficiencies of symmetric SOBI estimate are compared to those of recently introduced deflation-based SOBI estimate under general multivariate MA$(\infty)$ processes. The theory is illustrated by some finite-sample simulation studies as well as a real EEG data example.

In independent component analysis we assume that the observed vector is a linear transformation of a latent vector of independent components, our objective being the estimation of the latter. Deflation-based FastICA estimates the components one-by-one by repeatedly maximizing the expected value of some function measuring non-Gaussianity, the derivative of which is called the non-linearity. Under some weak assumptions, the asymptotically optimal non-linearity for extracting sources with a specific density is given by the location score function of the density. In this paper we look into the consequences of this result from the viewpoint of estimating Gaussian location and scale mixtures. As one of our results we justify the common use of hyperbolic tangent, tanh, as a non-linearity in blind clustering by showing that it is optimal for estimating certain Gaussian mixtures. Finally, simulations are used to show that the asymptotic optimality results hold in various settings also for finite samples.

Complex valued random variables and time series are common in various applications, for example in wireless communications, radar applications and magnetic resonance imaging. These applications often involve the famous blind source separation problem. However, the observations rarely fully follow specific models and robust methods that allow deviations from the model assumptions and endure outliers are required. We propose a new algorithm, robust multidimensional eSAM-SOBI, for complex valued blind source separation. The algorithm takes into account possible multidimensional spatial or temporal dependencies, whereas traditional SOBI-like procedures only consider dependencies in a single direction. In applications like functional magnetic resonance imaging, the dependencies are indeed not only one-dimensional. We provide a simulation study with complex valued data to illustrate the better performance of the methods that utilize multidimensional autocovariance in the presence of two-dimensional dependency. Moreover, we also examine the performance of the multidimensional eSAM-SOBI in the presence of outliers.

Two standard assumptions of the classical blind source separation (BSS) theory are frequently violated by modern data sets. First, the majority of the existing methodology assumes vector-valued signals while data exhibiting a natural tensor structure is frequently observed. Second, many typical BSS applications exhibit serial dependence which is usually modeled using second order stationarity assumptions, which is however often quite unrealistic. To address these two issues we extend three existing methods of nonstationary blind source separation to tensor-valued time series. The resulting methods naturally factor in the tensor form of the observations without resorting to vectorization of the signals. Additionally, the methods allow for two types of nonstationarity, either the source series are blockwise second order weak stationary or their variances change smoothly in time. A simulation study and an application to video data show that the proposed extensions outperform their vectorial counterparts and successfully identify source series of interest.

Consider a multivariate time series where each component series is assumed to be a linear mixture of latent mutually independent stationary time series. Classical independent component analysis (ICA) tools, such as fastICA, are often used to extract latent series, but they don't utilize any information on temporal dependence. Also financial time series often have periods of low and high volatility. In such settings second order source separation methods, such as SOBI, fail. We review here some classical methods used for time series with stochastic volatility, and suggest modifications of them by proposing a family of vSOBI estimators. These estimators use different nonlinearity functions to capture nonlinear autocorrelation of the time series and extract the independent components. Simulation study shows that the proposed method outperforms the existing methods when latent components follow GARCH and SV models. This paper is an invited extended version of the paper presented at the CDAM 2016 conference.

A regression model where the response as well as the explaining variables are time series is considered. A general model which allows supervised dimension reduction in this context is suggested without considering the form of dependence. The method for this purpose combines ideas from sliced inverse regression (SIR) and blind source separation methods to obtain linear combinations of the explaining time series which are ordered according to their relevance with respect to the response. The method gives also an indication of which lags of the linear combinations are of importance. The method is demonstrated using simulations and a real data example. ispartof: Econometrics and Statistics vol:4 pages:57-69 status: published

Dimension reduction is often a preliminary step in the analysis of large data sets. The so-called non-Gaussian component analysis searches for a projection onto the non-Gaussian part of the data, and it is then important to know the correct dimension of the non-Gaussian signal subspace. In this paper we develop asymptotic as well as bootstrap tests for the dimension based on the popular fourth order blind identification (FOBI) method.

Deflation-based FastICA is a popular method for independent component analysis. In the standard deflation-base d approach the row vectors of the unmixing matrix are extracted one after another always using the same nonlinearities. In prac- tice the user has to choose the nonlinearities and the efficiency and robustness of the estimation procedure then strongly depends on this choice as well as on the order in which the components are extracted. In this paper we propose a novel adaptive two- stage deflation-based FastICA algorithm that (i) allows one to use different nonlinearities for different components and (ii) optimizes the order in which the components are extracted. Based on a consistent preliminary unmixing matrix estimate and our theoretical results, the algorithm selects in an optimal way the order and the nonlinearities for each component from a finite set of candidates specified by the user. It is also shown that, for each component, the best possible nonlinearity is obtained by using the log-density function. The resulting ICA estimate is affine equivariant with a known asymptotic distribution. The excellent performance of the new procedure is shown with asymptotic efficiency and finite-sample simulation studies.

The R-package REPPlab is designed to explore multivariate data sets using one-dimensional unsupervised projection pursuit. It is useful in practice as a preprocessing step to find clusters or as an outlier detection tool for multivariate numerical data. Except from the package tourr that implements smooth sequences of projection matrices and rggobi that provides an interface to a dynamic graphics package called GGobi, there is no implementation of exploratory projection pursuit tools available in R especially in the context of outlier detection. REPPlab is an R interface for the Java program EPPlab that implements four projection indices and three biologically inspired optimization algorithms. The implemented indices are either adapted to cluster or to outlier detection and the optimization algorithms have at most one parameter to tune. Following the original software EPPlab, the exploration strategy in REPPlab is divided into two steps. Many potentially interesting projections are calculated at the first step and examined at the second step. For this second step, different tools for plotting and combining the results are proposed with specific tools for outlier detection. Compared to EPPlab, some of these tools are new and their performance is illustrated through some simulations and using some real data sets in a clustering context. The functionalities of the package are also illustrated for outlier detection on a new data set that is provided with the package.

There are two aspects in functional magnetic resonance imaging (fMRI) data that make them awkward to analyse with traditional multivariate methods - high order and high dimension. The first of these refers to the tensorial nature of observations as array-valued elements instead of vectors. Although this can be circumvented by vectorizing the array, doing so simultaneously loses all the structural information in the original observations. The second aspect refers to the high dimensionality along each dimension making the concept of dimension reduction a valuable tool in the processing of fMRI data. Different methods of tensor dimension reduction are currently gaining popUlarity in literature, and in this paper we apply two recently proposed methods of tensorial independent component analysis to simulated task-based fMRI data. Additionally, as a preprocessing step we introduce a novel extension of PCA for tensors. The simulations show that when extracting a sufficiently large number of principal components, the tensor methods find the task signals very reliably, something the standard temporal independent component analysis (tICA) fails in. peerReviewed

This paper focuses on the computational aspects of symmetrized Mestimators of scatter, i.e. the multivariate M-estimators of scatter computed on the pairwise differences of the data. Such estimators do not require a location estimate, and more importantly, they possess the important block and joint independence properties. These properties are needed, for example, when solving the independent component analysis problem. Classical and recently developed algorithms for computing the M-estimators and the symmetrized M-estimators are discussed. The effect of parallelization is considered as well as new computational approach based on using only a subset of pairwise differences. Efficiencies and computation time comparisons are made using simulation studies under multivariate elliptically symmetric models and under independent component models. peerReviewed

In preprocessing tensor-valued data, e.g. images and videos, a common procedure is to vectorize the observations and subject the resulting vectors to one of the many methods used for independent component analysis (ICA). However, the tensor structure of the original data is lost in the vectorization and, as a more suitable alternative, we propose the matrix- and tensor fourth order blind identification (MFOBI and TFOBI). In these tensorial extensions of the classic fourth order blind identification (FOBI) we assume a Kronecker structure for the mixing and perform FOBI simultaneously on each direction of the observed tensors. We discuss the theory and assumptions behind MFOBI and TFOBI and provide two different algorithms and related estimates of the unmixing matrices along with their asymptotic properties. Finally, simulations are used to compare the method's performance with that of classical FOBI for vectorized data and we end with a real data clustering example., Comment: 26 pages, 4 figures

In high reliability standards fields such as automotive, avionics or aerospace, the detection of anomalies is crucial. An efficient methodology for automatically detecting multivariate outliers is introduced. It takes advantage of the remarkable properties of the Invariant Coordinate Selection (ICS) method. Based on the simultaneous spectral decomposition of two scatter matrices, ICS leads to an affine invariant coordinate system in which the Euclidian distance corresponds to a Mahalanobis Distance (MD) in the original coordinates. The limitations of MD are highlighted using theoretical arguments in a context where the dimension of the data is large. Unlike MD, ICS makes it possible to select relevant components which removes the limitations. Owing to the resulting dimension reduction, the method is expected to improve the power of outlier detection rules such as MD-based criteria. It also greatly simplifies outliers interpretation. The paper includes practical guidelines for using ICS in the context of a small proportion of outliers which is relevant in high reliability standards fields. The choice of scatter matrices together with the selection of relevant invariant components through parallel analysis and normality tests are addressed. The use of the regular covariance matrix and the so called matrix of fourth moments as the scatter pair is recommended. This choice combines the simplicity of implementation together with the possibility to derive theoretical results. A simulation study confirms the good properties of the proposal and compares it with other scatter pairs. This study also provides a comparison with Principal Component Analysis and MD. The performance of our proposal is also evaluated on several real data sets using a user-friendly R package accompanying the paper.

In this paper we study the theoretical properties of the deflation-based FastICA method, the original symmetric FastICA method, and a modified symmetric FastICA method, here called the squared symmetric FastICA. This modification is obtained by replacing the absolute values in the FastICA objective function by their squares. In the deflation-based case this replacement has no effect on the estimate since the maximization problem stays the same. However, in the symmetric case we obtain a different estimate which has been mentioned in the literature, but its theoretical properties have not been studied at all. In the paper we review the classic deflation-based and symmetric FastICA approaches and contrast these with the squared symmetric version of FastICA in a unified way. We find the estimating equations and derive the asymptotical properties of the squared symmetric FastICA estimator with an arbitrary choice of nonlinearity. This allows the main contribution of the paper, i.e., efficiency comparison of the estimates in a wide variety of situations using asymptotic variances of the unmixing matrix estimates. HighlightsThe formal definition of the squared symmetric FastICA estimator is given.The asymptotical properties of the squared symmetric FastICA are derived.Asymptotic variances of different FastICA estimators are compared.

In independent component analysis it is assumed that the components of the observed random vector are linear combinations of latent independent random variables, and the aim is then to find an estimate for a transformation matrix back to these independent components. In the engineering literature, there are several traditional estimation procedures based on the use of fourth moments, such as FOBI (fourth order blind identification), JADE (joint approximate diagonalization of eigenmatrices), and FastICA, but the statistical properties of these estimates are not well known. In this paper various independent component functionals based on the fourth moments are discussed in detail, starting with the corresponding optimization problems, deriving the estimating equations and estimation algorithms, and finding asymptotic statistical properties of the estimates. Comparisons of the asymptotic variances of the estimates in wide independent component models show that in most cases JADE and the symmetric version of FastICA perform better than their competitors., Published at http://dx.doi.org/10.1214/15-STS520 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Independent component analysis is a popular approach in search of latent variables and structures in high-dimensional data. We propose extensions of classical FOBI and JADE estimates for multivariate time series, with a special focus on time series with stochastic volatility.

We present new algorithms for $M$-estimators of multivariate scatter and location and for symmetrized $M$-estimators of multivariate scatter. The new algorithms are considerably faster than currently used fixed-point and related algorithms. The main idea is to utilize a second order Taylor expansion of the target functional and to devise a partial Newton-Raphson procedure. In connection with symmetrized $M$-estimators we work with incomplete $U$-statistics to accelerate our procedures initially.

Independent component models have gained increasing interest in various fields of applications in recent years. The basic independent component model is a semiparametric model assuming that a p-variate observed random vector is a linear transformation of an unobserved vector of p independent latent variables. This linear transformation is given by an unknown mixing matrix, and one of the main objectives of independent component analysis (ICA) is to estimate an unmixing matrix by means of which the latent variables can be recovered. In this article, we discuss the basic independent component model in detail, define the concepts and analysis tools carefully, and consider two families of ICA estimates. The statistical properties (consistency, asymptotic normality, efficiency, robustness) of the estimates can be analyzed and compared via the so called gain matrices. Some extensions of the basic independent component model, such as models with additive noise or models with dependent observations, are briefly discussed. The article ends with a short example. Keywords: blind source separation; fastICA; independent component model; independent subspace analysis; mixing matrix; overcomplete ICA; undercomplete ICA; unmixing matrix

Invariant coordinate selection (ICS) has recently been introduced as a method for exploring multivariate data. It includes as a special case a method for recovering the unmixing matrix in independent components analysis (ICA). It also serves as a basis for classes of multivariate nonparametric tests, and as a tool in cluster analysis or blind discrimination. The aim of this paper is to briefly explain the (ICS) method and to illustrate how various applications can be implemented using the R package ICS. Several examples are used to show how the ICS method and ICS package can be used in analyzing a multivariate data set.

Background The current gold standard in dimension reduction methods for high-throughput genotype data is the Principle Component Analysis (PCA). The presence of PCA is so dominant, that other methods usually cannot be found in the analyst’s toolbox and hence are only rarely applied. Results We present a modern dimension reduction method called ’Invariant Coordinate Selection’ (ICS) and its application to high-throughput genotype data. The more commonly known Independent Component Analysis (ICA) is in this framework just a special case of ICS. We use ICS on both, a simulated and a real dataset to demonstrate first some deficiencies of PCA and how ICS is capable to recover the correct subgroups within the simulated data. Second, we apply the ICS method on a chicken dataset and also detect there two subgroups. These subgroups are then further investigated with respect to their genotype to provide further evidence of the biological relevance of the detected subgroup division. Further, we compare the performance of ICS also to five other popular dimension reduction methods. Conclusion The ICS method was able to detect subgroups in data where the PCA fails to detect anything. Hence, we promote the application of ICS to high-throughput genotype data in addition to the established PCA. Especially in statistical programming environments like e.g. R, its application does not add any computational burden to the analysis pipeline. Electronic supplementary material The online version of this article (doi:10.1186/s12859-017-1589-9) contains supplementary material, which is available to authorized users.

Oja, Sirkiä, and Eriksson (2006) and Ollila, Oja, and Koivunen (2007) showed that, under general assumptions, any two scatter matrices with the so called independent components property can be used to estimate the unmixing matrix for the independent component analysis (ICA). The method is a generalization of Cardoso’s (Cardoso, 1989) FOBI estimate which uses the regular covariance matrix and a scatter matrix based on fourth moments. Different choices of the two scatter matrices are compared in a simulation study. Based on the study, we recommend always the use of two robust scatter matrices. For possible asymmetric independent components, symmetrized versions of the scatter matrix estimates should be used.