Eigenvector Decomposition Research Articles

Fast Colour2Grey

A standard approach to generating a greyscale equivalent to an input colour image involves calculating the so-called structure tensor at each image pixel. Defining contrast as associated with the maximum-change direction of this matrix, the grey gradient is identified with the first eigenvector direction, with gradient strength given by the square root of its eigenvalue. However, aside from the inherent complexity of such an approach, each pixel's gradient still possesses a sign ambiguity, since an eigenvector is given only up to a sign. This is ostensibly resolved by looking at how one of the R,G,B colour channels behaves, or how the the luminance changes. Instead, we would like to circumvent the sign problem in the first place, and also avoid calculating the costly eigenvector decomposition. So here we suggest replacing the eigenvector approach by generating a greyscale gradient equal to the maximum gradient amongst the R,G,B gradients, in each of x, y. But in order not to neglect the tensor approach, we consider the relationship between the complex and the simple approaches. We also note that, at each pixel, we have both forward-facing and backward-facing derivatives, which are different, and we consider a tensor formed from both. Then, over a standard training set, we ask for an optimum set of weights for all the maximum gradients such that the simple maxima scheme generates a greyscale structure tensor to best match the original, colour, one. We find that a simple scheme that facilitates fast solutions is best. Greyscale results are shown to be excellent, and the algorithm is very fast.

Source localization and interference suppression using mode space estimation

Weak target detection and localization in the presence of loud surface ship noise is a critical problem for matched field processing in shallow water. For stationary sources, each signal component of a received signal can be separated, and interference can be suppressed using eigen-space analysis schemes. However, source motion, in realistic cases, causes spreading of signal energies in their subspace. In this case, eigenvalues of target and interference signal components are mixed and hard to be separated with usual phone-space eigenvector decomposition (EVD) approaches. Our technique is based on mode space and utilizes the difference in their physical characteristics of surface and submerged sources. Performing EVD for modal cross-spectral density matrix, interference components in the mode amplitude subspace can be classified and eliminated. This technique is demonstrated with data obtained from an L-shaped (vertical and horizontal) line array during the MAPLE IV experiment conducted in the East Sea, Korea, and results will be discussed.

Light scattering by an ensemble of interacting dipolar particles with both electric and magnetic polarizabilities

We have studied the problem of light scattering by an ensemble of dipoles with both electric and magnetic polarizabilities. Using the coupled electric and magnetic dipole method as the formal base, we have generalized the eigenvector decomposition of the local dipole moments previously derived for purely electric particles to the case of both electric and magnetic dipoles. We have analyzed the properties of eigenvalues and eigenvectors in the most elementary case of two particles. In the purely electric case, the eigenvalues correspond to the resonance modes of the system due to the electromagnetic coupling of its components. For a two-dipole system with both electric and magnetic responses, purely electric, purely magnetic, and mixed states can be distinguished. The resonance spectrum is analyzed as a function of the magnetic permeability, and it is shown that the latter can be fitted quite accurately by the eigenmode decomposition.

Synchronization Measurement of Multiple Neuronal Populations

The purpose of the present paper is to develop a method, based on equal-time correlation, correlation matrix analysis and surrogate resampling, that is able to quantify and describe properties of synchronization of population neuronal activity recorded simultaneously from multiple sites. Initially, Lorenz-type oscillators were used to model multiple time series with different patterns of synchronization. Eigenvalue and eigenvector decomposition was then applied to identify "clusters" of locally synchronized activity and to calculate a "global synchronization index." This method was then applied to multichannel data recorded from an in vitro model of epileptic seizures. The results demonstrate that this novel method can be successfully used to analyze synchronization between multiple neuronal population series.

Efficiently updating and tracking the dominant kernel principal components

The dominant set of eigenvectors of the symmetrical kernel Gram matrix is used in many important kernel methods (like e.g. kernel Principal Component Analysis, feature approximation, denoising, compression, prediction) in the machine learning area. Yet in the case of dynamic and/or large-scale data, the batch calculation nature and computational demands of the eigenvector decomposition limit these methods in numerous applications. In this paper we present an efficient incremental approach for fast calculation of the dominant kernel eigenbasis, which allows us to track the kernel eigenspace dynamically. Experiments show that our updating scheme delivers a numerically stable and accurate approximation for eigenvalues and eigenvectors at every iteration in comparison to the batch algorithm.

Covariance intersection-based sensor fusion for sounding rocket tracking and impact area prediction

The estimation of object motion from radar measurements is critical in applications such as air traffic control, airborne surveillance systems, launching of sounding rockets and orbital vehicles, and impact point prediction (IPP). This paper investigates the use of covariance intersection (CI) for the fusion of data from a pair of distinct radar sites at Alcântara Launch Center (ALC) to track a sounding rocket and predict the impact point and its uncertainty area on the ground in compliance with safety-of-flight issues. Debiased measurement transformation from spherical to cartesian coordinates, boost and free-fall models embedded in Kalman filters, and multiple hypothesis testing for multiple-model adaptive estimation are employed by the processing node at each radar site to locally estimate position, velocity, and acceleration. The local estimates and the corresponding computed covariance matrices from the radar sites are transformed to a common reference frame at the launch-pad and CI-fused. For the purpose of comparison, measurement fusion and track-to-track fusion are also evaluated. Prediction of the impact area with a given probability assumes free fall and considers the uncertainties inferred from the eigenvalue–eigenvector decomposition of the computed covariance matrix. All simulations presented herein make use of actual radar data of a Brazilian VS30 sounding rocket launched from ALC in February 2000. The automated fusion approach presents a much improved performance relative to the procedure then used at ALC for trajectory tracking and IPP, and qualitatively reproduces the expertise of the safety-of-flight officer.

Link function selection in stochastic multicriteria decision making models

A stochastic formulation of the Analytic Hierarchy Process (AHP) using an approach based on Bayesian categorical data models has been developed. However, in categorical data models it is known that the selection of the link function may have an impact on the model estimates. In particular, the selection of the probit link implies an assumption that model error terms are normally distributed and this normality assumption is regularly utilized in other related methods such as the multiplicative AHP. We examine model performance with respect to the choice of two model link functions. With regard to point estimates, it is found that the logit formulation is better able to replicate the estimates obtained by the eigenvector decomposition associated with the original formulation of the AHP. By contrast, the probit link produces priorities which are consistently more moderate than those of the AHP. The results suggest that the logit formulation will be preferred by decision makers who wish to replicate the AHP priorities as closely as possible. The results also suggest that the unexamined use of the normality assumption in other stochastic AHP methods may have an impact on priority estimates and thus is worthy of further attention.

Considering noise orbital deviations on the evaluation of power density spectrum of oscillators

This brief presents a new application of the theory of noise in free running oscillators based on the Floquet eigenvector decomposition. In oscillators, all orbital deviations contribute to the power density spectrum (PDS) as much as the "phase" term, usually considered. Each orbital deviation component shows a time evolution depending on the related Floquet eigenvalue, which thus characterizes statistical properties related to that component. Orbital deviations are partially correlated, due to their common origin from noise sources, thus also correlation terms are considered in the evaluation of the PDS. In this brief, we introduce a simplified method of calculation of PDS and apply it to an example of RLC negative resistance oscillator. Results show the relevance of orbital deviations in PDS in presence of stationary noise, these contributions becomes particularly relevant when noise is cyclostationary

Using Eigenvectors to Partition Circuits

Many fields, ranging from bioinformatics to databases to large-scale integrated circuits, deal with the interrelation between elementary objects. Objects are represented as vertices and the relationships between them are represented as nets (edges that connect two or more vertices) in the form of a hypergraph. We seek a placement of vertices that groups like objects and separates unlike objects. This involves separating related objects into a few, possibly disjoint, blocks. These hypergraph-partitioning problems are NP-hard so cannot be solved exactly, except for very small instances. We develop and use a numerical technique based on eigenvector decomposition of the connectivity matrix associated with the circuit netlist to partition hypergraphs emanating from circuit netlists. The eigenvector components of the circuit connectivity matrix are then used to determine vertex coordinates in one dimension that are then rounded in some fashion to determine block assignments. The inherent difficulty with eigenvector techniques is that the eigenvector components tend to cluster, making it difficult to determine correct block assignments. Our technique uses weights on nets, vertices, and fixed vertices to obtain a more “discrete” placement of vertices, making it easier to determine correct block assignments.

A functional-PCA approach for analyzing and reducing complex chemical mechanisms

In industrial reactive flow systems such as furnaces and gas turbines, there are considerable variations in the temperature and concentrations of species along different spatial directions. Functional principal component analysis (fPCA) can be used to study the temporal (or spatial) evolution of reactions in a reactive flow system, and to develop simplified kinetic models to describe this behaviour. A comprehensive kinetic mechanism for CO oxidation is used to demonstrate application of fPCA to identify important reactions as a function of time. In conventional PCA, the eigenvalue–eigenvector decomposition specifically transforms the variations associated with the time (or spatial directions) and species into loadings that represent only the reactions. However, fPCA produces functional loading vectors ξ → l ( t ) which are functions of time or distance, whose elements are referred to as functional loadings. The functional loading vectors are the eigenfunctions of the covariance matrix associated with the sensitivity trajectories. The functional loadings are used to identify reactions playing a significant role, possibly as a function of time, and are used to develop a reduced kinetic scheme from a detailed kinetic mechanism.

Strategies for reducing large fMRI data sets for independent component analysis

In independent component analysis (ICA), principal component analysis (PCA) is generally used to reduce the raw data to a few principal components (PCs) through eigenvector decomposition (EVD) on the data covariance matrix. Although this works for spatial ICA (sICA) on moderately sized fMRI data, it is intractable for temporal ICA (tICA), since typical fMRI data have a high spatial dimension, resulting in an unmanageable data covariance matrix. To solve this problem, two practical data reduction methods are presented in this paper. The first solution is to calculate the PCs of tICA from the PCs of sICA. This approach works well for moderately sized fMRI data; however, it is highly computationally intensive, even intractable, when the number of scans increases. The second solution proposed is to perform PCA decomposition via a cascade recursive least squared (CRLS) network, which provides a uniform data reduction solution for both sICA and tICA. Without the need to calculate the covariance matrix, CRLS extracts PCs directly from the raw data, and the PC extraction can be terminated after computing an arbitrary number of PCs without the need to estimate the whole set of PCs. Moreover, when the whole data set becomes too large to be loaded into the machine memory, CRLS-PCA can save data retrieval time by reading the data once, while the conventional PCA requires numerous data retrieval steps for both covariance matrix calculation and PC extractions. Real fMRI data were used to evaluate the PC extraction precision, computational expense, and memory usage of the presented methods.

The effect of principal component analysis on machine learning accuracy with high-dimensional spectral data

This paper presents the results of an investigation into the use of machine learning methods for the identification of narcotics from Raman spectra. The classification of spectral data and other high-dimensional data, such as images, gene-expression data and spectral data, poses an interesting challenge to machine learning, as the presence of high numbers of redundant or highly correlated attributes can seriously degrade classification accuracy. This paper investigates the use of principal component analysis (PCA) to reduce high-dimensional spectral data and to improve the predictive performance of some well-known machine learning methods. Experiments are carried out on a high-dimensional spectral dataset. These experiments employ the NIPALS (Non-Linear Iterative Partial Least Squares) PCA method, a method that has been used in the field of chemometrics for spectral classification, and is a more efficient alternative than the widely used eigenvector decomposition approach. The experiments show that the use of this PCA method can improve the performance of machine learning in the classification of high-dimensional data.

Kinetic analysis of NO-sensitized methane oxidation

The role of NO-sensitized oxidation during the product-gas entrainment of a low-NO x , multi-jet, natural gas burner is investigated. A detailed kinetic mechanism for the NO-sensitized oxidation of CH 4 , consisting of 483 reactions and 69 species, is used for the kinetic analysis. An eigenvalue–eigenvector decomposition is performed on normalized sensitivity coefficients to study the important reactions using principal component analysis (PCA), and the loadings corresponding to the largest eigenvalue are used to identify the reaction pathways of NO-sensitized oxidation. The main reaction pathway is most strongly affected by the temperature profile and equivalence ratio. Also, a reduced kinetic scheme of 110 reactions and 47 species is developed by eliminating reactions with small loadings. The temporal evolution of reactions is investigated using functional PCA, in which the functional loadings reflect the importance of reactions as a function of time. A discretization approach is used to perform the functional PCA.

Time-dependent frequency domain principal components analysis of multichannel non-stationary signals

High dimensional multi-channel signals often exhibit multi-collinearities. This suggests that such signals can be decomposed into uncorrelated principal components with possibly lower dimension than that of the original signal. A time-localized frequency domain principal components analysis method is proposed for signals that exhibit locally stationary behavior. The first step is to form a mean square consistent estimate of the time-varying spectrum matrix by smoothing the time-localized periodograms using a kernel defined on the frequency axis whose span is selected automatically using a generalized cross-validation procedure that is based on the asymptotic gamma distribution. The eigenvalues of the spectral density estimate are then computed which are the estimated spectra of the principal components. In addition, one may apply a formal statistical procedure for testing whether the weights (components of an eigenvector) at a particular channel change over time. The proposed method can be easily implemented because it only requires the fast Fourier transform (FFT) and eigenvalue–eigenvector decomposition routines. An illustration is presented using a multi-channel brain waves data set recorded during an epileptic seizure.

Theoretical evaluation of microwave vegetation scattering through an eigen-analysis of the coherency matrix

To ascertain the physical scattering properties of the polarimetric imaging radar, one has to investigate actual scattering processes by modeling the physical interaction mechanisms of microwaves in vegetated terrain. In this paper, a recent development in the polarimetric processing technique of imaging radar data, namely the target decomposition method, is investigated for the purpose of analyzing uncertain scattering information without having complex knowledge of the target environment in vegetated terrain. Although it is difficult to interpret the consistent physical scattering processes in each eigen-component of the coherency matrix, the theoretical relations between eigen-parameters and vegetation scattering mechanisms are acquired by using the nonsymmetric vegetation scattering model. The extended interpretation of the eigenvalue–eigenvector decomposition of the reflection symmetric coherency matrix to the nonsymmetric general scattering situation can be further used to retrieve more reliable ground or biomass information in vegetated terrain.

An efficient approach for computing non-Gaussian ARMA model coefficients using Pisarenko's method

This paper addresses the problem of estimating the coefficients of a general autoregressive moving average (ARMA) model from only third order cumulants (TOCs) of the noisy observations of the system output. The observed signal may be corrupted by additive coloured Gaussian noise. The system is driven by a zero-mean independent and identically distributed (i.i.d.) non-Gaussian sequence. The input is not observed. The unknown model coefficients are obtained using eigenvalue–eigenvector decomposition. The derivation of this procedure is an extension of Pisarenko harmonic autocorrelation-based (PHA) method to third order statistics. It will be shown that the desired ARMA coefficients vector corresponds to the eigenvector associated with the minimum eigenvalue of a data covariance matrix of TOCs. The proposed method is also compared with well-known algorithms as well as with the PHA method. Copyright © 2005 John Wiley & Sons, Ltd.

Blind multipath channel equalization based on eigenvector decomposition

A new direct blind equalization algorithm is presented in this paper. It relies on the auto‐correlation matrix of the channel, and adopts its eigenvector decomposition for designing an equalizer, which exploits the correlation characteristics of the channel. The proposed equalization is insensitive to the inaccuracy of the channel order estimation, and its data sequence length required for estimating the channel's correlation matrix is shorter than that of the other eigenvector decomposition methods that assumed that the channel order was known or had been estimated. Simulation results show that this algorithm is capable of acquiring a good performance under lower SNR in a communication system employing 16‐QAM modulation.

Granular avalanches across irregular three‐dimensional terrain: 1. Theory and computation

To establish a theoretical basis for predicting and interpreting the behavior of rapid mass movements on Earth's surface, we develop and test a new computational model for gravity‐driven motion of granular avalanches across irregular, three‐dimensional (3‐D) terrain. The principles embodied in the model are simple and few: continuum mass and momentum conservation and intergranular stress generation governed by Coulomb friction. However, significant challenges result from the necessity of satisfying these principles when deforming avalanches interact with steep and highly variable 3‐D terrain. We address these challenges in four ways. (1) We formulate depth‐averaged governing equations that are referenced to a rectangular Cartesian coordinate system (with z vertical) and that account explicitly for the effect of nonzero vertical accelerations on depth‐averaged mass and momentum fluxes and stress states. (2) We compute fluxes of mass and momentum across vertical cell boundaries using a high‐resolution finite volume method and Roe‐type Riemann solver. Our algorithm incorporates flux difference splitting, an entropy correction for the flux, and eigenvector decomposition to embed the effects of driving and resisting forces in Riemann solutions. (3) We use a finite element method and avalanche displacements predicted by Riemann solutions to compute Coulomb stresses conjugate to the displacements in 3‐D stress space. (4) We test the model output against analytical solutions, a sand cone conceptual experiment, and (in a companion paper) data from detailed laboratory experiments. Model results illustrate a complex interplay of basal traction and internal stress, and they successfully predict not only the gross behavior but also many details of avalanche motion from initiation to deposition.

Revisiting iterative time reversal processing: application to detection of multiple targets.

The iterative time reversal processing represents a high speed and easy way to self-focus on the strongest scatterer in a multitarget medium. However, finding weaker scatterers is a more difficult task that can be solved by computing the eigenvalue and eigenvector decomposition of the time reversal operator, the so-called DORT method. Nevertheless, as it requires the measurement of the complete interelements response matrix and time-consuming computation, the separation of multiple targets may not be achieved in real time. In this study, a new real time technique is proposed for multitarget selective focusing that does not require the experimental acquisition of the time reversal operator. This technique achieves the operator decomposition using a particular sequence of filtered waves propagation instead of computational power. Due to its simplicity of implementation, this iterative process can be achieved in real time. This high speed selective focusing is experimentally demonstrated by detecting targets through a heterogeneous medium and in a speckle environment. A theoretical analysis compares this technique to the DORT formalism.

Calculation of imaging errors of awg

This paper reports on the modeling of arrayed-waveguide gratings (AWGs) by means of Fourier optics where the coupling between the grating waveguides and defocus of the star couplers is taken into account as well. Special interest is drawn to the geometrical modeling of AWGs. The paraxial approximation for the free-space propagation within the star couplers, the exact calculation of the coupling coefficients for the star couplers by an orthonormal basis (ONB) expansion, and the drawbacks due to the nonunitary transfer matrix provided by the Green's functions approach for the coupling between the grating waveguides, as well as possible improvements thereof, are identified as crucial points. An efficient orthonormalization procedure by means of an eigenvector decomposition is presented. A unitary approximation of the transfer matrix, which is in itself approximate but in general not unitary, introduced within the impulse response method, is obtained. The presented simulation technique allows for ab initio simulations, which do not fall back on any experimental input, for the spectral response of AWGs, which are found in good agreement with experimental results.