Demixing Matrix Research Articles

Two-step wavelet-based estimation for Gaussian mixed fractional processes

A Gaussian mixed fractional process $$\{Y(t)\}_{t \in {\mathbb {R}}} = \{PX(t)\}_{t \in {\mathbb {R}}}$$ is a multivariate stochastic process obtained by pre-multiplying a vector of independent, Gaussian fractional process entries X by a nonsingular matrix P. It is interpreted that Y is observable, while X is a hidden process occurring in an (unknown) system of coordinates P. Mixed processes naturally arise as approximations to solutions of physically relevant classes of multivariate fractional stochastic differential equations under aggregation. We propose a semiparametric two-step wavelet-based method for estimating both the demixing matrix $$P^{-1}$$ and the memory parameters of X. The asymptotic normality of the estimator is established both in continuous and discrete time. Monte Carlo experiments show that the estimator is accurate over finite samples, while being very computationally efficient. As an application, we model a bivariate time series of annual tree ring width measurements.

Blind Source Separation for Complex-Valued Signals Using Generalized Autocorrelation

We introduce a new complex-valued blind source separation approach, based on generalized autocorrelations of sources, to improve the spectrum efficiency for the next-generation wireless communications system. The proposed algorithm considers the temporal structures of communication signals and the natural gradient-based method is used to optimize the demixing matrix. In addition, the local stability condition is proved. Simulation results are presented showing the superior performance of the proposed algorithm in the intersymbol interference of the estimated signals.

Gradient Algorithms for Complex Non-Gaussian Independent Component/Vector Extraction, Question of Convergence

We revise the problem of extracting one independent component from an instantaneous linear mixture of signals. The mixing matrix is parameterized by two vectors, one column of the mixing matrix and one row of the de-mixing matrix. The separation is based on the non-Gaussianity of the source of interest, while the other background signals are assumed to be Gaussian. Three gradient-based estimation algorithms are derived using the maximum likelihood principle and are compared with the Natural Gradient algorithm for Independent Component Analysis and with One-unit FastICA based on negentropy maximization. The ideas and algorithms are also generalized for the extraction of a vector component when the extraction proceeds jointly from a set of instantaneous mixtures. Throughout the paper, we address the problem of the size of the region of convergence for which the algorithms guarantee the extraction of the desired source. We show how that size is influenced by the ratio of powers of the sources within the mixture. Simulations confirm this observation where several algorithms are compared. They show various convergence behavior in a scenario where the source of interest is dominant or weak. Here, our proposed modifications of the gradient methods taking into account the dominance/weakness of the source show improved global convergence property.

Common Interferences Removal from Dense Multichannel EEG Using Independent Component Decomposition.

To improve the spatial resolution, dense multichannel electroencephalogram with more than 32 leads has gained more and more applications. However, strong common interference will not only conceal the weak components generated from the specific isolated neural source, but also lead to severe spurious correlation between different brain regions, which results in great distortion on brain connectivity or brain network analysis. Starting from the fast independent component analysis algorithm, we first derive the mixing matrix of independent source components based on the baseline signals prior to tasks. Then, we identify the common interferences as those components whose mixing vectors span the minimum angles with respect to the unitary vector. By assuming that both the common interferences and their corresponding mixing vectors stay consistent during the entire experiment, we apply the demixing and mixing matrix to the task signals and remove the inferred common interferences. Subsequently, we validate the method using simulation. Finally, the index of global coherence is calculated for validation. It turns out that the proposed method can successfully remove the common interferences so that the prominent coherence of mu rhythms in motor imagery tasks is unmasked. The proposed method can gain wide applications because it reveals the true correlation between the local sources in spite of the low signal-to-noise ratio.

Super-Gaussian BSS Using Fast-ICA with Chebyshev–Pade Approximant

Independent component analysis (ICA) is one of the most powerful techniques to solve the problem of blind source separation (BSS), and the well-known Fast-ICA is excellent for large-scale BSS. Fast-ICA tries to find the demixing matrix by optimizing the nonlinear objective (cost or contrast) functions. For Fast-ICA, there are three built-in contrast functions to separate non-Gaussian sources, and their derivatives are similar to nonlinearities used in neural networks. However, the contrast functions and their nonlinearities for separating super-Gaussian sources are not optimal owing to their high computational cost, which greatly affect the convergence rate of Fast-ICA. To address this issue, this paper proposes two rational nonlinearities to replace the built-in (original) nonlinearities. The rational nonlinearities are derived by the Chebyshev–Pade approximant from Chebyshev polynomials (series) of the original nonlinearities. To speed up the convergence of Fast-ICA, the degrees in both numerator and denominator of rational functions are designed to be relatively low, and thus the rational polynomials can be quickly evaluated. Simulation results of audio BSS show that the proposed rational nonlinearities can not only improve the convergence rate of Fast-ICA, but also improve the separation performance index; Experiment results of real electrocardiogram data also verify the validity of the proposed approach.

A Nonorthogonal Joint Block Diagonalization Algorithm Avoiding Degenerate Solutions

The joint block diagonalization algorithm is instrumental to the convolution blind signal separation. However, the nonorthogonal joint block diagonalization algorithm based on a Weighted least-squares (WLS) criterion sometimes converges to the degenerate or trivial solution. It is simple to avoid the trivial solution by adding some constraint on the de-mixing matrix. But the degenerate solution is hard to be excluded. This paper probes the phenomenon of the degenerate solution. By adding a new penalty function to the WLS criterion, an efficient algorithm which can avoid the degenerate solution is developed. Through the theoretical derivation and numerical simulation, we can make the conclusion that the proposed algorithm is more robust and not sensitive to the square or non-square mixing matrix.

Large‐scale super‐Gaussian sources separation using Fast‐ICA with rational nonlinearities

SummaryIndependent component analysis (ICA) is one of the most powerful methods for solving blind source separation problem. In various ICA methods, the Fast‐ICA is an excellent algorithm, and it finds the demixing matrix that optimizes the nonlinear contrast function. There are three original contrast functions in the Fast‐ICA to separate super‐Gaussian and sub‐Gaussian sources, and their respective derivatives are similar to nonlinearities used in neural networks. For the separation of large‐scale super‐Gaussian sources, however, the contrast functions and the nonlinearities are not optimal owing to high computational cost. To solve this potential problem, this paper proposes four rational polynomial functions to replace the original nonlinearities. Because the rational polynomials can be quickly evaluated, when they are used in the Fast‐ICA, the computational load of the algorithms can be effectively reduced. The proposed rational functions are derived by the Pade approximant from Taylor series expansion of the original nonlinearities. To reduce the error of approximation, we make the behaviors of rational functions approach that of the original ones within an effective range as well as possible. The simulation results show that the Fast‐ICA algorithms with rational nonlinearities not only can speed up the convergence but also improve the separation performance of super‐Gaussian blind source separation. Copyright © 2016 John Wiley & Sons, Ltd.

Independent component analysis model utilizing de-mixing information for improved non-Gaussian process monitoring

The de-mixing matrix generated from independent component analysis (ICA) can reveal information about the relations between variables and independent components, but the traditional ICA model does not preserve the whole de-mixing information for the purpose of feature extraction and dimensionality reduction, so that some important information may be abandoned. Multi-block strategy has been improved to be an efficient method to deal with numerous data. However, the manner of dividing original data is still subject for discussion and the priori knowledge is necessary for process division. This paper proposes a totally data-driven ICA model that divides de-mixing matrix based on the Generalized Dice’s coefficient and combines the results from sub-blocks using Bayesian inference. All information in de-mixing matrix is fully utilized and the ability of monitoring non-Gaussian process is improved. Meanwhile, a corresponding contribution plot is developed for fault diagnosis to find the root causes. The performance of the proposed method is illustrated through a numerical example and the Tennessee Eastman benchmark case study.

A blind source separation method for chaotic signals based on artificial bee colony algorithm

The inherent features, such as non-periodic, wide band spectrum, and extreflely sensitive to initial values etc. make it quite a challenge to blindly separate the mixed chaotic signals. A new blind source separation method based on the artificial bee colony algorithm is proposed in this paper. This method can recover chaotic sources from noisy observations on their linear mixtures without any prior information about the source equations. The proposed method is structured in the phase space of the demixed signals, which is reconstructed from the observations by using delay-embedding method. An objective function in the reconstructed phase space is designed so that the blind source separation problem is transformed into an optimization problem. The optimal demixing matrix is obtained by maximize the objective function with an artificial bee colony optimizer and the chaotic sources are then recovered by multiplying the observed mixtures and the optimal demixing matrix. Before the optimization procedure is made, a pre-whitening should be employed. Additionally, the parameterized repreflentation of orthogonal matrices through principal rotation is adopted to reduce the dimension of the optimization procedure so that the proposed blind source separation algorithm can converge quickly. Different from the traditional independent component analysis approaches which concern mainly the statistical features, the proposed blind source separation method utilizes the dynamics in the observed mixtures by means of phase space reconstruction. Therefore, better performance can be achieved when it is used to deal with chaotic signals. In computer simulation, two cases are taken into consideration: namely, the mixture is noiseless or not contaminated by noise. The correlation coefficient criterion and the performance index criterion are adopted to evaluate the separation performances. Simulation result shows that in most cases the proposed approach converges within a few tens of iterations and the chaotic sources can be accurately recovered. The impact of noise level and signal length on the separation performance is investigated in detail. The overall performance of the proposed approach is much better than the traditional independent component analysis approaches. Moreover, the capability of separating the mixed chaotic and Gaussian signals reflealed in the simulation indicates that the proposed approach has the potential to be applied in a wider range of applications.

General nonunitary constrained ICA and its application to complex-valued fMRI data.

Constrained independent component analysis (C-ICA) algorithms provide an effective way to introduce prior information into the complex- and real-valued ICA framework. The work in this area has focus on adding constraints to the objective function of algorithms that assume a unitary demixing matrix. The unitary condition is required in order to decouple-isolate-the constraints applied for each individual source. This assumption limits the optimization space and, therefore, the separation performance of C-ICA algorithms. We generalize the existing C-ICA framework by using a novel decoupling method that preserves the larger optimization space for the demixing matrix. This framework allows for the constraining of either the sources or the mixing coefficients. A constrained version of the nonunitary entropy bound minimization algorithm is introduced and applied to actual complex-valued fMRI data. We show that constraining the mixing parameters using a temporal constraint improves the estimation of the spatial map and timecourses of task-related components.

General Non-Orthogonal Constrained ICA

Constrained independent component analysis (C-ICA) algorithms have been an effective way to introduce prior information into the ICA framework. The work in this area has focus on adding constraints to the objective function of algorithms that assume an orthogonal demixing matrix. Orthogonality is required in order to decouple—isolate—the constraints applied for each individual source. This assumption limits the optimization space and therefore the separation performance of C-ICA algorithms. We generalize the existing C-ICA framework by using a novel decoupling method that preserves the larger optimization space for the demixing matrix. In addition, this framework allows for the constraining of either the sources or the mixing coefficients. A constrained version of the extended Infomax algorithm is used as an example to show the benefits obtained from the non-orthogonal constrained framework we introduce.

A Bayesian inverse solution using independent component analysis

We present new results about the simultaneous linear inverse problems using independent component analysis (ICA), which can be used to separate the data into statistically independent components. The idea of using ICA in solving such inverse problems, especially in EEG/MEG context, has been a known topic for at least more than a decade, but the known results have been justified heuristically, and their relationships are not understood properly. Here we show how to obtain a Bayesian posterior for a spatial source distribution, by using an ICA demixing matrix as an input. The posterior enables us to rederive and reinterpret the previously known methods, and also provides completely new methods.

Uniqueness Analysis of Non-Unitary Matrix Joint Diagonalization

Matrix Joint Diagonalization (MJD) is a powerful approach for solving the Blind Source Separation (BSS) problem. It relies on the construction of matrices which are diagonalized by the unknown demixing matrix. Their joint diagonalizer serves as a correct estimate of this demixing matrix only if it is uniquely determined. Thus, a critical question is under what conditions is a joint diagonalizer unique. In the present work we fully answer this question about the identifiability of MJD based BSS approaches and provide a general result on uniqueness conditions of matrix joint diagonalization. It unifies all existing results which exploit the concepts of non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. As a corollary, we propose a solution for complex BSS, which can be formulated in closed form in terms of an eigen and a singular value decomposition of two matrices.

Cramér-Rao Bound for Circular and Noncircular Complex Independent Component Analysis

Despite an increased interest in complex independent component analysis (ICA) during the last two decades, a closed form expression for the Cramér-Rao bound (CRB) for the demixing matrix is not known yet. In this paper, we fill this gap by deriving a closed-form expression for the CRB of the demixing matrix for instantaneous noncircular complex ICA. It contains the CRB for circular complex ICA and noncircular complex Gaussian ICA as two special cases. We also study the CRB numerically for the family of noncircular complex generalized Gaussian distributions and compare it to simulation results of two ICA estimators. Furthermore, we show how to extend the CRB to the case where the source signals are not temporally independent and identically distributed.

A Novel Method for Surface Defect Detection of Photovoltaic Module Based on Independent Component Analysis

This paper proposed a new method for surface defect detection of photovoltaic module based on independent component analysis (ICA) reconstruction algorithm. Firstly, a faultless image is used as the training image. The demixing matrix and corresponding ICs are obtained by applying the ICA in the training image. Then we reorder the ICs according to the range values and reform the de-mixing matrix. Then the reformed de-mixing matrix is used to reconstruct the defect image. The resulting image can remove the background structures and enhance the local anomalies. Experimental results have shown that the proposed method can effectively detect the presence of defects in periodically patterned surfaces.

ECG-based gating in ultra high field cardiovascular magnetic resonance using an independent component analysis approach.

BackgroundIn Cardiovascular Magnetic Resonance (CMR), the synchronization of image acquisition with heart motion is performed in clinical practice by processing the electrocardiogram (ECG). The ECG-based synchronization is well established for MR scanners with magnetic fields up to 3 T. However, this technique is prone to errors in ultra high field environments, e.g. in 7 T MR scanners as used in research applications. The high magnetic fields cause severe magnetohydrodynamic (MHD) effects which disturb the ECG signal. Image synchronization is thus less reliable and yields artefacts in CMR images.MethodsA strategy based on Independent Component Analysis (ICA) was pursued in this work to enhance the ECG contribution and attenuate the MHD effect. ICA was applied to 12-lead ECG signals recorded inside a 7 T MR scanner. An automatic source identification procedure was proposed to identify an independent component (IC) dominated by the ECG signal. The identified IC was then used for detecting the R-peaks. The presented ICA-based method was compared to other R-peak detection methods using 1) the raw ECG signal, 2) the raw vectorcardiogram (VCG), 3) the state-of-the-art gating technique based on the VCG, 4) an updated version of the VCG-based approach and 5) the ICA of the VCG.ResultsECG signals from eight volunteers were recorded inside the MR scanner. Recordings with an overall length of 87 min accounting for 5457 QRS complexes were available for the analysis. The records were divided into a training and a test dataset. In terms of R-peak detection within the test dataset, the proposed ICA-based algorithm achieved a detection performance with an average sensitivity (Se) of 99.2%, a positive predictive value (+P) of 99.1%, with an average trigger delay and jitter of 5.8 ms and 5.0 ms, respectively. Long term stability of the demixing matrix was shown based on two measurements of the same subject, each being separated by one year, whereas an averaged detection performance of Se = 99.4% and +P = 99.7% was achieved.Compared to the state-of-the-art VCG-based gating technique at 7 T, the proposed method increased the sensitivity and positive predictive value within the test dataset by 27.1% and 42.7%, respectively.ConclusionsThe presented ICA-based method allows the estimation and identification of an IC dominated by the ECG signal. R-peak detection based on this IC outperforms the state-of-the-art VCG-based technique in a 7 T MR scanner environment.

An Independent Component Analysis Based Defect Detection for the OLED Display

Organic Light Emitting Displays (OLED) is a new type of display device which has become increasingly attractive and popular. Due to the complex manufacturing process, various defects may exist on the OLED panel. These defects have the characteristics of fuzzy boundaries, irregular in shape, low contrast with background and they are mixed with the texture background increasing the difficulty of a rapid identification. In this paper, we proposed an approach to detect these defects based on the model of independent component analysis (ICA). The ICA model is applied to a perfect OLED image to determine the de-mixing matrix and its corresponding independent components (ICs). Through the choice of a proper ICi row vector, the new de-mixing matrix is generated which contains only uniform information and is used to reconstruct the OLED background image. The defect result can be obtained by the subtracting operation between the reconstructed background and the source images. The detection system is implemented in the Labview and the testing results show that the ICA based OLED defect detection method is feasible and effective.

OLED Defect Inspection System Development through Independent Component Analysis

Organic Light Emitting Displays (OLED) is a new type of display device which has become increasingly attractive and popular. Due to the complex manufacturing process, various defects may exist on the OLED panel affecting the quality and life of the display panels. These defects have the characteristics of fuzzy boundaries, irregular in shape, low contrast with background and especially, they are mixed with the pixel texture background increasing the difficulty of a rapid recognition. In this paper, we proposed an approach to detect the defects based on the model of independent component analysis (ICA). The ICA model is applied to a perfect OLED image to estimate its corresponding independent components (ICs) and create the de-mixing matrix. Through estimation and determination of a proper ICi row vector of the faultless image, a new de-mixing matrix can be generated which constitutes only uniform information and is then applied to reconstruct the texture background of source OLED images. Through the operation of subtraction of the reconstructed background from the source images and the binary segmentation, the defects can be detected. Based on the algorithms, a defect detection system for the OLED panels was implemented and the testing work was performed in this study. The testing results show that the proposed method is feasible and effective. DOI: http://dx.doi.org/10.11591/telkomnika.v10i8.1968

Convex Divergence ICA for Blind Source Separation

Independent component analysis (ICA) is vital for unsupervised learning and blind source separation (BSS). The ICA unsupervised learning procedure attempts to demix the observation vectors and identify the salient features or mixture sources. This work presents a novel contrast function for evaluating the dependence among sources. A convex divergence measure is developed by applying the convex functions to the Jensen's inequality. Adjustable with a convexity parameter, this inequality-based divergence measure has a wide range of the steepest descents to reach its minimum value. A convex divergence ICA (C-ICA) is constructed and a nonparametric C-ICA algorithm is derived with different convexity parameters where the non-Gaussianity of source signals is characterized by the Parzen window-based distribution. Experimental results indicate that the specialized C-ICA significantly reduces the number of learning epochs during estimation of the demixing matrix. The convergence speed is improved by using the scaled natural gradient algorithm. Experiments on the BSS of instantaneous, noisy and convolutive mixtures of speech and music signals further demonstrate the superiority of the proposed C-ICA to JADE, Fast-ICA, and the nonparametric ICA based on mutual information.

A New Approximate Joint Diagonalization Algorithm Based on the Fused Complex-Valued Cost Function

A new algorithm to solve the fused complex-valued cost function for approximate joint diagonalization (AJD), named CVAJD (Complex-Valued Approximate Joint Diagonalization), is presented. The CVAJD algorithm adopts an iterative scheme to update the demixing matrix through the strictly diagonally-dominant residual mixing matrix obtained in each of iterations. Due to the relaxation of several constraints on the target matrices, it has more general utilizations. Besides, it is also easy to implement. A numerical simulation illustrates fast convergence and good performance of the CVAJD.