Ana R. Teixeira

KPCA denoising and the pre-image problem revisited

Kernel principal component analysis (KPCA) is widely used in classification, feature extraction and denoising applications. In the latter it is unavoidable to deal with the pre-image problem which constitutes the most complex step in the whole processing chain. One of the methods to tackle this problem is an iterative solution based on a fixed-point algorithm. An alternative strategy considers an algebraic approach that relies on the solution of an under-determined system of equations. In this work we present a method that uses this algebraic approach to estimate a good starting point to the fixed-point iteration. We will demonstrate that this hybrid solution for the pre-image shows better performance than the other two methods. Further we extend the applicability of KPCA to one-dimensional signals which occur in many signal processing applications. We show that artefact removal from such data can be treated on the same footing as denoising. We finally apply the algorithm to denoise the famous USPS data set and to extract EOG interferences from single channel EEG recordings.

How to Apply Nonlinear Subspace Techniques to Univariate Biomedical Time Series

In this paper, we propose an embedding technique for univariate single-channel biomedical signals to apply projective subspace techniques. Biomedical signals are often recorded as 1-D time series; hence, they need to be transformed to multidimensional signal vectors for subspace techniques to be applicable. The transformation can be achieved by embedding an observed signal in its delayed coordinates. We propose the application of two nonlinear subspace techniques to embedded multidimensional signals and discuss their relation. The techniques consist of modified versions of singular-spectrum analysis (SSA) and kernel principal component analysis (KPCA). For illustrative purposes, both nonlinear subspace projection techniques are applied to an electroencephalogram (EEG) signal recorded in the frontal channel to extract its dominant electrooculogram (EOG) interference. Furthermore, to evaluate the performance of the algorithms, an experimental study with artificially mixed signals is presented and discussed.

dAMUSE---A new tool for denoising and blind source separation

In this work a generalized version of AMUSE, called dAMUSE is proposed. The main modification consists in embedding the observed mixed signals in a high-dimensional feature space of delayed coordinates. With the embedded signals a matrix pencil is formed and its generalized eigendecomposition is computed similar to the algorithm AMUSE. We show that in this case the uncorrelated output signals are filtered versions of the unknown source signals. Further, denoising the data can be achieved conveniently in parallel with the signal separation. Numerical simulations using artificially mixed signals are presented to show the performance of the method. Further results of a heart rate variability (HRV) study are discussed showing that the output signals are related with LF (low frequency) and HF (high frequency) fluctuations. Finally, an application to separate artifacts from 2D NOESY NMR spectra and to denoise the reconstructed artefact-free spectra is presented also.

Denoising Using Local ICA and a Generalized Eigendecomposition with Time-Delayed Signals

We present denoising algorithms based on either local independent component analysis (ICA) and a minimum description length (MDL) estimator or a generalized eigenvalue decomposition (GEVD) using a matrix pencil of time-delayed signals. Both methods are applied to signals embedded in delayed coordinates in a high-dim feature space Ω and denoising is achieved by projecting onto a lower dimensional signal subspace. We discuss the algorithms and provide applications to the analysis of 2D NOESY protein NMR spectra.

Unsupervised feature extraction via kernel subspace techniques

This paper provides a new insight into unsupervised feature extraction techniques based on kernel subspace models. The data projected onto kernel subspace models are new data representations which might be better suited for classification. The kernel subspace models are always described exploiting the dual form for the basis vectors which requires that the training data must be available even during the test phase. By exploiting an incomplete Cholesky decomposition of the kernel matrix, a computationally less demanding implementation is proposed. Online benchmark data sets allow the evaluation of these feature extraction methods comparing the performance of two classifiers which both have as input either the raw data or the new representations.

Delayed AMUSE - A tool for blind source separation and denoising

In this work we propose a generalized eigendecomposition (GEVD) of a matrix pencil computed after embedding the data into a high-dim feature space of delayed coordinates. The matrix pencil is computed like in AMUSE but in the feature space of delayed coordinates. Its GEVD yields filtered versions of the source signals as output signals. The algorithm is implemented in two EVD steps. Numerical simulations study the influence of the number of delays and the noise level on the performance.

Single-channel electroencephalogram analysis using non-linear subspace techniques

In this work, we propose the correction of univariate, single channel EEGs using projective subspace techniques. The biomedical signals which often represent one dimensional time series, need to be transformed to multi-dimensional signal vectors for the latter techniques to be applicable. The transformation can be achieved by embedding an observed signal in its delayed coordinates. We propose the application of two non-linear subspace techniques to the obtained multidimensional signal. One of the techniques consists in a modified version of Singular Spectrum Analysis (SSA) and the other is kernel Principal Component Analysis (KPCA) implemented using a reduced rank approximation of the kernel matrix. Both nonlinear subspace projection techniques are applied to an electroencephalogram (EEG) signal recorded in the frontal channel to extract its prominent electrooculogram (EOG) interference.

On the use of simulated annealing to automatically assign decorrelated components in second-order blind source separation

In this paper, an automatic assignment tool, called BSS-AutoAssign,for artifact-related decorrelated components within a second-order blind source separation (BSS) is presented. The latter is based on the recently proposed algorithm dAMUSE, which provides an elegant solution to both the BSS and the denoising problem simultaneously. BSS-AutoAssign uses a local principal component analysis (PCA)to approximate the artifact signal and defines a suitable cost function which is optimized using simulated annealing. The algorithms dAMUSE plus BSS-AutoAssign are illustrated by applying them to the separation of water artifacts from two-dimensional nuclear overhauser enhancement (2-D NOESY)spectroscopy signals of proteins dissolved in water

Feature Extraction Using Low-Rank Approximations of the Kernel Matrix

In this work we use kernel subspace techniques to perform feature extraction. The projections of the data onto the coordinates of the high-dimensional space created by the kernel function are called features. The basis vectors to project the data depend on the eigendecomposition of the kernel matrix which might become very high-dimensional in case of a large training set. Nevertheless only the largest eigenvalues and corresponding eigenvectors are used to extract relevant features. In this work, we present low-rank approximations to the kernel matrix based on the Nystrom method. Numerical simulations will then be used to demonstrate the Nystrom extension method applied to feature extraction and classification. The performance of the presented methods is demonstrated using the USPS data set.

Greedy KPCA in biomedical signal processing

Biomedical signals are generally contaminated with artifacts and noise. In case artifacts dominate, the useful signal can easily be extracted with projective subspace techniques. Then, biomedical signals which often represent one dimensional time series, need to be transformed to multi-dimensional signal vectors for the latter techniques to be applicable. In this work we propose the application of a greedy kernel Principal Component Analysis(KPCA) which allows to decompose the multidimensional vectors into components, and we will show that the one related with the largest eigenvalues correspond to an high-amplitude artifact that can be subtracted from the original.