0000000000314205
AUTHOR
Jari Miettinen
A more efficient second order blind identification method for separation of uncorrelated stationary time series
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. peerReviewed
Fast equivariant JADE
Independent component analysis (ICA) is a widely used signal processing tool having applications in various fields of science. In this paper we focus on affine equivariant ICA methods. Two such well-established estimation methods, FOBI and JADE, diagonalize certain fourth order cumulant matrices to extract the independent components. FOBI uses one cumulant matrix only, and is therefore computationally very fast. However, it is not able to separate identically distributed components which is a major drawback. JADE overcomes this restriction. Unfortunately, JADE uses a huge number of cumulant matrices and is computationally very heavy in high-dimensional cases. In this paper, we hybridize the…
Deflation-Based FastICA With Adaptive Choices of Nonlinearities
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 consist…
Dimension reduction for time series in a blind source separation context using r
Funding Information: The work of KN was supported by the CRoNoS COST Action IC1408 and the Austrian Science Fund P31881-N32. The work of ST was supported by the CRoNoS COST Action IC1408. The work of JV was supported by Academy of Finland (grant 321883). We would like to thank the anonymous reviewers for their comments which improved the paper and package considerably. Publisher Copyright: © 2021, American Statistical Association. All rights reserved. Multivariate time series observations are increasingly common in multiple fields of science but the complex dependencies of such data often translate into intractable models with large number of parameters. An alternative is given by first red…
Blind Source Separation Based on Joint Diagonalization in R: The Packages JADE and BSSasymp
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 es…
On Independent Component Analysis with Stochastic Volatility Models
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…
fICA : FastICA Algorithms and Their Improved Variants
Abstract In independent component analysis (ICA) one searches for mutually independent non gaussian latent variables when the components of the multivariate data are assumed to be linear combinations of them. Arguably, the most popular method to perform ICA is FastICA. There are two classical versions, the deflation-based FastICA where the components are found one by one, and the symmetric FastICA where the components are found simultaneously. These methods have been implemented previously in two R packages, fastICA and ica. We present the R package fICA and compare it to the other packages. Additional features in fICA include optimization of the extraction order in the deflation-based vers…
Deflation-based separation of uncorrelated stationary time series
In this paper we assume that the observed pp time series are linear combinations of pp latent uncorrelated weakly stationary time series. The problem is then to find an estimate for an unmixing matrix that transforms the observed time series back to uncorrelated time series. The so called SOBI (Second Order Blind Identification) estimate aims at a joint diagonalization of the covariance matrix and several autocovariance matrices with varying lags. In this paper, we propose a novel procedure that extracts the latent time series one by one. The limiting distribution of this deflation-based SOBI is found under general conditions, and we show how the results can be used for the comparison of es…
Alternative Diagonality Criteria for SOBI
Blind source separation (BSS) is a multivariate data analysis method, whose roots are in the signal processing community. BSS is applied in diverse fields, including, for example, brain imaging and economic time series analysis. In the BSS model there are interesting latent uncorrelated variables, and the aim is to estimate the latent variables from multiple linear combinations of them. In this article we assume that these variables are weakly stationary time series, and we consider estimation methods which are based on approximate joint diagonalization of autocovariance matrices. In the popular SOBI estimator, a set of matrices is most diagonal when the sum of squares of their diagonal ele…
Separation of Uncorrelated Stationary time series using Autocovariance Matrices
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 deriv…
Statistical properties of a blind source separation estimator for stationary time series
Abstract In this paper, we assume that the observed p time series are linear combinations of p latent uncorrelated weakly stationary time series. The problem is then, using the observed p -variate time series, to find an estimate for a mixing or unmixing matrix for the combinations. The estimated uncorrelated time series may then have nice interpretations and can be used in a further analysis. The popular AMUSE algorithm finds an estimate of an unmixing matrix using covariances and autocovariances of the observed time series. In this paper, we derive the limiting distribution of the AMUSE estimator under general conditions, and show how the results can be used for the comparison of estimate…
On the Computation of Symmetrized M-Estimators of Scatter
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…
A more efficient second order blind identification method for separation of uncorrelated stationary time series
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.
Model selection using limiting distributions of second-order blind source separation algorithms
Signals, recorded over time, are often observed as mixtures of multiple source signals. To extract relevant information from such measurements one needs to determine the mixing coefficients. In case of weakly stationary time series with uncorrelated source signals, this separation can be achieved by jointly diagonalizing sample autocovariances at different lags, and several algorithms address this task. Often the mixing estimates contain close-to-zero entries and one wants to decide whether the corresponding source signals have a relevant impact on the observations or not. To address this question of model selection we consider the recently published second-order blind identification proced…
Fourth Moments and Independent Component Analysis
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 wi…
The squared symmetric FastICA estimator
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…
ICA and stochastic volatility models
We consider multivariate time series where each component series is an unknown linear combination of latent mutually independent stationary time series. Multivariate financial time series have often periods of low volatility followed by periods of high volatility. This kind of time series have typically non-Gaussian stationary distributions, and therefore standard independent component analysis (ICA) tools such as fastICA can be used to extract independent component series even though they do not utilize any information on temporal dependence. In this paper we review some ICA methods used in the context of stochastic volatility models. We also suggest their modifications which use nonlinear…
Extracting Conditionally Heteroskedastic Components using Independent Component Analysis
In the independent component model, the multivariate data are assumed to be a mixture of mutually independent latent components. The independent component analysis (ICA) then aims at estimating these latent components. In this article, we study an ICA method which combines the use of linear and quadratic autocorrelations to enable efficient estimation of various kinds of stationary time series. Statistical properties of the estimator are studied by finding its limiting distribution under general conditions, and the asymptotic variances are derived in the case of ARMA-GARCH model. We use the asymptotic results and a finite sample simulation study to compare different choices of a weight coef…