Blind source separation using second-order cyclostationary statistics

This paper studies the blind source separation (BSS) problem with the assumption that the source signals are cyclostationary. Identifiability and separability criteria based on second-order cyclostationary statistics (SOCS) alone are derived. The identifiability condition is used to define an appropriate contrast function. An iterative algorithm (ATH2) is derived to minimize this contrast function. This algorithm separates the sources even when they do not have distinct cycle frequencies .

Application of higher order statistics/spectra in biomedical signals : a review

For many decades correlation and power spectrum have been primary tools for digital signal processing applications in the biomedical area. The information contained in the power spectrum is essentially that of the autocorrelation sequence; which is sufficient for complete statistical descriptions of Gaussian signals of known means. However, there are practical situations where one needs to look beyond autocorrelation of a signal to extract information regarding deviation from Gaussianity and the presence of phase relations. Higher order spectra, also known as polyspectra, are spectral representations of higher order statistics, i.e. moments and cumulants of third order and beyond. HOS (higher order statistics or higher order spectra) can detect deviations from linearity, stationarity or Gaussianity in the signal. Most of the biomedical signals are non-linear, non-stationary and non-Gaussian in nature and therefore it can be more advantageous to analyze them with HOS compared to the use of second order correlations and power spectra. In this paper we have discussed the application of HOS for different bio-signals. HOS methods of analysis are explained using a typical heart rate variability (HRV) signal and applications to other signals are reviewed.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Cell image classification using histograms, higher order statistics and adaboost

A cell classification algorithm that uses first, second and third order statistics of pixel intensity distributions over pre-defined regions is implemented and evaluated. A cell image is segmented into 6 regions extending from a boundary layer to an inner circle. First, second and third order statistical features are extracted from histograms of pixel intensities in these regions. Third order statistical features used are one-dimensional bispectral invariants. 108 features were considered as candidates for Adaboost based fusion. The best 10 stage fused classifier was selected for each class and a decision tree constructed for the 6-class problem. The classifier is robust, accurate and fast by design.

Blind extraction using fractional lower-order statistics

In traditional method to blindly extract interesting source signals sequentially, the second-order or higher-order statistics of signals are often utilized. However, for impulsive sources, both of the second-order and higher-order statistics may degenerate. Therefore, it is necessary to exploit new method for the blind extraction of impulsive sources. Based on the best compression-reconstruction principle, a novel model is proposed in this work, together with the corresponding algorithm. The proposed method can be used for blind extraction of sources which are distributed from alpha stable process. Simulations are given to illustrate availability and robustness of our algorithm.

On a Second Order Condition for Max-Semistable Laws

2000 Mathematics Subject Classification: 62G32, 62G20.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

Supplement : Efficient weak second-order stochastic Runge-Kutta methods for non-commutative Stratonvich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Testing second order cyclostationarity in the squared envelope spectrum of non-white vibration signals

Cyclostationary models for the diagnostic signals measured on faulty rotating machineries have proved to be successful in many laboratory tests and industrial applications. The squared envelope spectrum has been pointed out as the most efficient indicator for the assessment of second order cyclostationary symptoms of damages, which are typical, for instance, of rolling element bearing faults. In an attempt to foster the spread of rotating machinery diagnostics, the current trend in the field is to reach higher levels of automation of the condition monitoring systems. For this purpose, statistical tests for the presence of cyclostationarity have been proposed during the last years. The statistical thresholds proposed in the past for the identification of cyclostationary components have been obtained under the hypothesis of having a white noise signal when the component is healthy. This need, coupled with the non-white nature of the real signals implies the necessity of pre-whitening or filtering the signal in optimal narrow-bands, increasing the complexity of the algorithm and the risk of losing diagnostic information or introducing biases on the result. In this paper, the authors introduce an original analytical derivation of the statistical tests for cyclostationarity in the squared envelope spectrum, dropping the hypothesis of white noise from the beginning. The effect of first order and second order cyclostationary components on the distribution of the squared envelope spectrum will be quantified and the effectiveness of the newly proposed threshold verified, providing a sound theoretical basis and a practical starting point for efficient automated diagnostics of machine components such as rolling element bearings. The analytical results will be verified by means of numerical simulations and by using experimental vibration data of rolling element bearings.

Second-order inelastic analysis of composite framed structures based on the refined plastic hinge method

Composite steel-concrete structures experience non-linear effects which arise from both instability-related geometric non-linearity and from material non-linearity in all of their component members. Because of this, conventional design procedures cannot capture the true behaviour of a composite frame throughout its full loading range, and so a procedure to account for those non-linearities is much needed. This paper therefore presents a numerical procedure capable of addressing geometric and material non-linearities at the strength limit state based on the refined plastic hinge method. Different material non-linearity for different composite structural components such as T-beams, concrete-filled tubular (CFT) and steel-encased reinforced concrete (SRC) sections can be treated using a routine numerical procedure for their section properties in this plastic hinge approach. Simple and conservative initial and full yield surfaces for general composite sections are proposed in this paper. The refined plastic hinge approach models springs at the ends of the element which are activated when the surface defining the interaction of bending and axial force at first yield is reached; a transition from the first yield interaction surface to the fully plastic interaction surface is postulated based on a proposed refined spring stiffness, which formulates the load-displacement relation for material non-linearity under the interaction of bending and axial actions. This produces a benign method for a beam-column composite element under general loading cases. Another main feature of this paper is that, for members containing a point of contraflexure, its location is determined with a simple application of the method herein and a node is then located at this position to reproduce the real flexural behaviour and associated material non-linearity of the member. Recourse is made to an updated Lagrangian formulation to consider geometric non-linear behaviour and to develop a non-linear solution strategy. The formulation with the refined plastic hinge approach is efficacious and robust, and so a full frame analysis incorporating geometric and material non-linearity is tractable. By way of contrast, the plastic zone approach possesses the drawback of strain-based procedures which rely on determining plastic zones within a cross-section and which require lengthwise integration. Following development of the theory, its application is illustrated with a number of varied examples.

Higher-order non-linear analysis of steel structures. Part I : elastic second-order formulation

This paper presents a higher-order beam-column formulation that can capture the geometrically non-linear behaviour of steel framed structures which contain a multiplicity of slender members. Despite advances in computational frame software, analyses of large frames can still be problematic from a numerical standpoint and so the intent of the paper is to fulfil a need for versatile, reliable and efficient non-linear analysis of general steel framed structures with very many members. Following a comprehensive review of numerical frame analysis techniques, a fourth-order element is derived and implemented in an updated Lagrangian formulation, and it is able to predict flexural buckling, snap-through buckling and large displacement post-buckling behaviour of typical structures whose responses have been reported by independent researchers. The solutions are shown to be efficacious in terms of a balance of accuracy and computational expediency. The higher-order element forms a basis for augmenting the geometrically non-linear approach with material non-linearity through the refined plastic hinge methodology described in the companion paper.