Complex-valued Independent Component Analysis Research Articles

A complex-valued slow independent component analysis based incipient fault detection and diagnosis method with applications to wastewater treatment processes

Multivariate statistical process monitoring are the essential approaches to achieve better prognostics and health management (PHM) of process industries. However, incipient faults and complex behaviors (such as nonlinearity and dynamics) always render the traditional multivariate statistical process monitoring approaches inadequate. Thus, a complex-valued slow independent component analysis (CSICA) is proposed, which is able to extract optimized features from a complex-valued matrix containing both of raw data and their changing rates by resorting to a complex-valued independent component analysis operation and a batch of phase shifts. These features, named slow independent components (SICs), not only guarantee the statistical independence but also capture slowly-changing patterns, thus refining both dynamic and non-Gaussian information mostly related with incipient faults. The proposed algorithm together with novel statistics, Is2, If2 and SPE, as well as their control limits can sequentially detect incipient faults effectively. Then, together with the novel differential mapping reconstructed contribution plot (DM-RCP) and Granger causality analysis, the proposed method can accurately locate rooting causes of incipient faults. Finally, the proposed framework of process monitoring is validated through two data sets from a simulation platform and an oxidation-ditch-based wastewater treatment plant, respectively. The results demonstrate that the proposed method can achieve more accurate and efficient performances than conventional methods.

Convolutive Complex-Valued Independent Component Analysis for Nonlinear Radar Signal Processing and Maritime Weak Target Detection

This paper first establishes a new complex independent component analysis (cICA) algorithm based on the spatiotemporal extension of complex-valued entropy bound minimization (CEBM) to separate received complex-valued radar signals. Next, we propose a new cICA-based detector with an open structure to find Swerling model targets, lognormal targets, and sea-surface small floating targets in coherent high-resolution maritime surveillance radars. The detector encountered three major problems when adopting cICA for detection and solved them using three effective suggestions. After performing cICA on the time series received by the radar, we obtained two different sources. Using the first and second theoretical and empirical moment estimates of the K-distribution, the target was selected between these two output source signals. Detector performance was verified quantitatively and qualitatively using the real-life IPIX radar database. Comprehensive experiments on this database with synthetic injected targets showed promising results. The computational time and sample size dependency of the proposed cICA algorithm were also discussed. Finally, a comparison of the detector with several novel detectors for detecting sea-surface floating small targets of the IPIX radar database demonstrated the proposed detector’s superiority.

Performance analysis for complex-valued FastICA and its improvement based on the Tukey M-estimator

Independent component analysis (ICA) is increasingly utilized to modern digital signal processing. Complex-valued FastICA, a fast fixed-point algorithm for ICA, is one of the most non-trivial algorithms for solving the ICA problems in the complex domain. Hitherto, there have been several attempts to give performance analysis for complex-valued FastICA. Rigorous theoretical analysis, however, still has room for improvement further. Consequently, the purposes of this paper are threefold: Firstly, the uniformity of the complex-valued FastICA estimator is constructed for the first time. Secondly, the stability of the complex-valued ICA algorithm is rigorously deduced based on the augmented generating matrix. Meanwhile, the local convergence of complex-valued FastICA algorithm is derived based on circular source signals. Finally, for improving the performance of separation, we select a novel alternative for nonlinearity based on the Tukey M-estimator in the complex-valued FastICA algorithm. Further, we prove the existence of local optimal solution and stability of the complex ICA problem based on the Tukey M-estimator. Simulations are presented to demonstrate the accuracy of our analysis. Additionally, the experimental results with synthetic data and complex-valued wind signal show the superiorities of the improved method.

Polarimetric detection of maritime floating small target based on the Complex-valued Entropy Rate Bound Minimization

Detection of sea-surface small floating targets in maritime high-resolution surveillance radar has been an active area of research in recent years. In this paper, we propose a new detector based on a complex-valued independent component analysis (cICA) algorithm proposed by Geng-Shen Fu et al. called complex entropy rate bound minimization (CERBM), to look for targets in polarimetric radars. It uses received time series at cell under test (CUT) with different polarizations as distinct mixtures. The proposed detector can exploit all information of polarimetric radar for an accurate detection. It does separation on the mixtures using CERBM which results in two output sources, i.e. clutter and target. Finally, the target is detected using estimation of the parameters of K-distribution for outputted sources. Our experiments on the recognized IPIX radar database show that the proposed detector obtains better detection performance in comparison to the newly proposed detectors. The robustness of the detector is also investigated by experiment in either low and high sea state which shows its appropriate results.

Comparison of Complex-Valued Independent Component Analysis Algorithms for EEG Data

Independent Component Analysis (ICA) has been successfully applied to a variety of problems, from speaker identification and image processing to functional magnetic resonance imaging (fMRI) of the brain. In particular, it has been applied to analyze EEG data in order to estimate the sources form the measurements. However, it soon became clear that for EEG signals the solutions found by ICA often depends on the particular ICA algorithm, and that the solutions may not always have a physiologically plausible interpretation. Therefore, nowadays many researchers are using ICA largely for artifact detection and removal from EEG, but not for the actual analysis of signals from cortical sources. However, a recent modification of an ICA algorithm has been applied successfully to EEG signals from the resting state. The key idea was to perform a particular preprocessing and then apply a complexvalued ICA algorithm. In this paper, we consider multiple complex-valued ICA algorithms and compare their performance on real-world resting state EEG data. Such a comparison is problematic because the way of mixing the original sources (the “ground truth”) is not known. We address this by developing proper measures to compare the results from multiple algorithms. The comparisons consider the ability of an algorithm to find interesting independent sources, i.e. those related to brain activity and not to artifact activity. The performance of locating a dipole for each separated independent component is considered in the comparison as well. Our results suggest that when using complex-valued ICA algorithms on preprocessed signals the resting state EEG activity can be analyzed in terms of physiological properties. This reestablishes the suitability of ICA for EEG analysis beyond the detection and removal of artifacts with real-valued ICA applied to the signals in the time-domain.

Adaptive Complex-Valued Independent Component Analysis Based on Second-Order Statistics

This paper proposes a two-stage fast convergence adaptive complex-valued independent component analysis based on second-order statistics of complex-valued source signals. The first stage constructs a cost function by extending the real-valued whiten cost function to a complex-valued domain and optimizes the cost function using a complex-valued gradient. The second stage uses the restriction that the pseudocovariance matrix of the separated signal is a diagonal matrix to construct the cost function and the geodesic method is used to optimize the cost function. Compared with other adaptive complex-valued independent component analysis, the proposed method shows a faster convergence rate and smaller error. Computer simulations were performed on synthesized signals and communications signals. The simulation results demonstrate the validity of the proposed algorithm.

Complex Independent Component Analysis Using Three Types of Diversity: Non-Gaussianity, Nonwhiteness, and Noncircularity

By assuming latent sources are statistically independent, independent component analysis (ICA) separates underlying sources from a given linear mixture. Since in many applications, latent sources are non-Gaussian, noncircular, and have sample dependence, it is desirable to exploit all these properties jointly. Mutual information rate, which leads to the minimization of entropy rate, provides a natural cost for the task. In this paper, we establish the theory for complex-valued ICA giving Cramer-Rao lower bound and identification conditions, and present a new algorithm that takes all these properties into account. We propose an effective estimator of entropy rate and a complex-valued entropy rate bound minimization algorithm based on it. We show that the new method exploits all these properties effectively by comparing the estimation performance with the Cramer-Rao lower bound and by a number of examples.

Independent vector analysis using subband and subspace nonlinearity

Independent vector analysis (IVA) is a recently proposed technique, an application of which is to solve the frequency domain blind source separation problem. Compared with the traditional complex-valued independent component analysis plus permutation correction approach, the largest advantage of IVA is that the permutation problem is directly addressed by IVA rather than resorting to the use of an ad hoc permutation resolving algorithm after a separation of the sources in multiple frequency bands. In this article, two updates for IVA are presented. First, a novel subband construction method is introduced, IVA will be conducted in subbands from high frequency to low frequency rather than in the full frequency band, the fact that the inter-frequency dependencies in subbands are stronger allows a more efficient approach to the permutation problem. Second, to improve robustness and against noise, the IVA nonlinearity is calculated only in the signal subspace, which is defined by the eigenvector associated with the largest eigenvalue of the signal correlation matrix. Different experiments were carried out on a software suite developed by us, and dramatic performance improvements were observed using the proposed methods. Lastly, as an example of real-world application, IVA with the proposed updates was used to separate vibration components from high-speed train noise data.

De-noising, phase ambiguity correction and visualization techniques for complex-valued ICA of group fMRI data

Analysis of functional magnetic resonance imaging (fMRI) data in its native, complex form has been shown to increase the sensitivity both for data-driven techniques, such as independent component analysis (ICA), and for model-driven techniques. The promise of an increase in sensitivity and specificity in clinical studies, provides a powerful motivation for utilizing both the phase and magnitude data; however, the unknown and noisy nature of the phase poses a challenge. In addition, many complex-valued analysis algorithms, such as ICA, suffer from an inherent phase ambiguity, which introduces additional difficulty for group analysis. We present solutions for these issues, which have been among the main reasons phase information has been traditionally discarded, and show their effectiveness when used as part of a complex-valued group ICA algorithm application. The methods we present thus allow the development of new fully complex data-driven and semi-blind methods to process, analyze, and visualize fMRI data.We first introduce a phase ambiguity correction scheme that can be either applied subsequent to ICA of fMRI data or can be incorporated into the ICA algorithm in the form of prior information to eliminate the need for further processing for phase correction. We also present a Mahalanobis distance-based thresholding method, which incorporates both magnitude and phase information into a single threshold, that can be used to increase the sensitivity in the identification of voxels of interest. This method shows particular promise for identifying voxels with significant susceptibility changes but that are located in low magnitude (i.e. activation) areas. We demonstrate the performance gain of the introduced methods on actual fMRI data.

The Improved Method for Solving Permutation Problem in Frequency Domain Blind Source Separation of Speech Signals

The signals of convolutive mixture in time-domain can be transformed to instantaneous mixtures in frequency-domain and complex-valued independent component analysis (CICA) can separate efficiently the signals of instantaneous mixture in each frequency bin. However, since CICA is calculated in each frequency bin independently, the permutation ambiguity becomes a serious problem. The permutation ambiguity of CICA in each frequency bin should be aligned so that a separated signal in the time-domain contains frequency components of the same source signal. The paper presents a novel and efficient approach for solving the permutation problem in frequency domain blind source separation of speech signals. The new algorithm models the frequency-domain separated signals by means of Teager energy correlation between neighboring bins for the detection of correct permutations. Experimental results show that the proposed algorithm can efficiently solve the permutation ambiguity problem in each frequency bin.

Quality Map Thresholding for De-noising of Complex-Valued fMRI Data and Its Application to ICA of fMRI

Although functional magnetic resonance imaging (fMRI) data are acquired as complex-valued images, traditionally most fMRI studies only use the magnitude of the data. FMRI analysis in the complex domain promises to provide more statistically significant information; however, the noisy nature of the phase poses a challenge for successful study of fMRI by complex-valued signal processing algorithms. In this paper, we introduce a physiologically motivated de-noising method that uses phase quality maps to successfully identify and eliminate noisy areas in the fMRI data so they can be used in individual and group studies. Additionally, we show how the developed de-noising method improves the results of complex-valued independent component analysis of fMRI data, a very successful tool for blind source separation of biomedical data.

Complex-Valued Independent Component Analysis for Online Blind Speech Extraction

This paper presents a theoretical analysis of a certain criterion for complex-valued independent component analysis (ICA) with a focus on blind speech extraction (BSE) of a spatio-temporally nonstationary speech source. In the paper, the proposed criteria denoted KSICA is related to the well-known FastICA method with the Kurtosis contrast function. The proposed method is shown to share the important fixed-point feature with the FastICA method, although an improvement with the proposed method is that it does not exhibit the divergent behavior for a of Gaussian-only sources that the FastICA method tends to do, and it shows better performance in online implementations. Compared to the FastICA, the KSICA method provides a 10 dB higher source extraction performance and a 10 dB lower standard deviation in a data batch approach when the data batch size is less than 100 samples. For larger batch sizes, the KSICA metod performs equally well. In an online application with spatially stationary sources the KSICA method provides around 10 dB higher interference suppression, and 1 MOS-unit lower speech distortion compared to the FastICA for 0.15 s time constant in the algorithm update parameter. Thus, the FastICA performance matches the KSICA performance for a time constant above 1 s. Finally, in an online application with a moving speech source, the KSICA method provides 10 dB higher interference suppression, compared to the FastICA for the same algorithm settings. All in all, the proposed KSICA method is shown to be a viable alternative for online BSE of complex-valued signal mixtures.

Complex ICA using generalized uncorrelating transform

An extension of the whitening transformation for complex random vectors, called the generalized uncorrelating transformation (GUT), is introduced. GUT is a generalization of the strong-uncorrelating transform [J. Eriksson, V. Koivunen, Complex-valued ICA using 2nd-order statistics, in: Proceedings of the IEEE Workshop on Machine Learning for Signal Processing (MLSP’04), Sao Luis, Brazil, 2004] based upon generalized estimators of the covariance and pseudo-covariance matrix, called the scatter matrix and spatial pseudo-scatter matrix, respectively. Depending on the selected scatter and spatial pseudo-scatter matrix, GUT estimators can have largely different statistical properties. Special emphasis is put on robust GUT estimators. We show that GUT is a separating matrix estimator for complex-valued independent component analysis (ICA) when at most one source random variable possess circularly symmetric distribution and sources do not have identical distribution. In the context of ICA, our approach is computationally attractive as it is based on straightforward matrix computations. Simulations and examples are used to confirm reliable performance of our method.

LEARNING BY CRITERION OPTIMIZATION ON A UNITARY UNIMODULAR MATRIX GROUP

The present manuscript aims at illustrating fundamental challenges and solutions arising in the design of learning theories by optimization on manifolds in the context of complex-valued neural systems. The special case of a unitary unimodular group of matrices is dealt with. The unitary unimodular group under analysis is a low dimensional and easy-to-handle matrix group. Notwithstanding, it exhibits a rich geometrical structure and gives rise to interesting speculations about methods to solve optimization problems on manifolds. Also, its low dimension allows us to treat most of the quantities involved in computation in closed form as well as to render them in graphical format. Some numerical experiments are presented and discussed within the paper, which deal with complex-valued independent component analysis.

Complex-valued ICA based on a pair of generalized covariance matrices

It is shown that any pair of scatter and spatial scatter matrices yields an estimator of the separating matrix for complex-valued independent component analysis (ICA). Scatter (resp. spatial scatter) matrix is a generalized covariance matrix in the sense that it is a positive definite hermitian matrix functional that satisfies the same affine (resp. unitary) equivariance property as does the covariance matrix and possesses an additional IC-property, namely, it reduces to a diagonal matrix at distributions with independent marginals. Scatter matrix is used to decorrelate the data and the eigenvalue decomposition of the spatial scatter matrix is used to find the unitary mixing matrix of the uncorrelated data. The method is a generalization of the FOBI algorithm, where a conventional covariance matrix and a certain fourth-order moment matrix take the place of the scatter and spatial scatter matrices, respectively. Emphasis is put on estimators employing robust scatter and spatial scatter matrices. The proposed approach is one among the computationally most attractive ones, and a new efficient algorithm that avoids decorrelation of the data is also proposed. Moreover, the method does not rely upon the commonly made assumption of complex circularity of the sources. Simulations and examples are used to confirm the reliable performance of our method.

Complex-valued ICA utilizing signal-subspace demixing for robust DOA estimation and blind signal separation

This paper deals with direction of arrival (DOA) estimation and blind signal separation (BSS) based on independent component analysis (ICA) with robust capabilities. An efficient demixing procedure of complex-valued ICA is presented here, which combines the signal-subspace demixing procedure exploiting individual signal-subspace projection and Newton's iteration algorithm based on maximization of the approximate negentropy of non-Gaussian signal for array signal processing. It resolves the problems of order ambiguity and identifiability of traditional ICA for time-domain BSS. The proposed method could be directly applied to radar, sonar, radio surveillance, and communications systems for separating signals and estimating relative DOAs of signals. Several computer simulation examples for perturbations to the array manifold, unknown noise environments, and Rayleigh fading channel are provided to illustrate the effectiveness of the proposed method.

A General Approach for Image and Co-Channel Interference Suppression in Diversity Wireless Receivers Employing ICA

In this contribution, a general approach to suppress image signal and co-channel interference in a wireless receiver is presented. The technique employs diversity reception, and the algorithm is based on complex-valued independent component analysis (ICA). The proposed ICA method increases the channel capacity through greater frequency reuse, and it simplifies the receiver’s front end by eliminating the need for analog image filtering. Also, the new technique is computationally efficient and has attractive implementation features in terms of the requirement for A/D converters’ speed and bandpass filters’ selectivity. Simulation results show that the performance is robust over a wide range of input signal-to-interference ratios, and the algorithm has fast convergence.

Independent component analysis of fMRI data in the complex domain

In BOLD fMRI a series of MR images is acquired and examined for task-related amplitude changes. These functional changes are small, so it is important to maximize detection efficiency. Virtually all fMRI processing strategies utilize magnitude information and ignore the phase, resulting in an unnecessary loss of efficiency. As the optimum way to model the phase information is not clear, a flexible modeling technique is useful. To analyze complex data sets, independent component analysis (ICA), a data-driven approach, is proposed. In ICA, the data are modeled as spatially independent components multiplied by their respective time-courses. There are thus three possible approaches: 1) the time-courses can be complex-valued, 2) the images can be complex-valued, or 3) both the time-courses and the images can be complex-valued. These analytic approaches are applied to data from a visual stimulation paradigm, and results from three complex analysis models are presented and compared with magnitude-only results. Using the criterion of the number of contiguous activated voxels at a given threshold, an average of 12-23% more voxels are detected by complex-valued ICA estimation at a threshold of /Z/ > 2.5. Additionally, preliminary results from the complex models reveal a phase modulation similar to the magnitude time-course in some voxels, and oppositely modulated in other voxels.