multivariate empirical mode decomposition despite empirical mode decomposition emd becoming a de facto standard for time frequency analysis of nonlinear and non stationary signals its multivariate extensions are only emerging yet they are a prerequisite for direct multichannel data analysis an important step in this direction is the computation of the local mean as the concept of local extrema is not well defined for multivariate signals to this end we propose to use real valued projections along multiple directions on hyperspheres n spheres in order to calculate the envelopes and the local mean of multivariate signals leading to multivariate extension of emd to generate a suitable set of direction vectors unit hyperspheres n spheres are sampled based on both uniform angular sampling methods and quasi monte carlo based low discrepancy sequences the potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real world human motion signals

adaptive signal decomposition based on local narrow band signals we propose an operator based method of adaptive signal decomposition whereby a local narrow band signal is defined in the null space of a singular local linear operator based on the definition and the algorithm we propose two types of local narrow band signals and two singular operator estimation methods for adaptive signal decomposition we show that our approach can solve a special case of huang s empirical mode decomposition algorithm for signals that cannot be resolved by our method or the empirical mode decomposition algorithm we propose a hybrid approach conceptually the approach applies the empirical mode decomposition algorithm followed by our algorithms our experiments show that the proposed hybrid approach can solve a wide range of complex signals effectively

some analysis methods for rotating systems with periodic coefficients two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed the use of floquet methods and the use of multiblade coordinate and harmonic balance methods the analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems previously announced in star as n82 23702

fractal boundaries of basin of attraction of newton raphson method in helicopter trim a key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim particularly when free wake models are used trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero an appropriate initial estimate of the trim state is needed for successful helicopter trim this study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations set of initial guess points from which the nonlinear equations converge it is illustrated that the three main rotor pitch controls of collective pitch longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution trajectories of the newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed it is found that the basins of attraction can have fractal boundaries

numerical methods for determining the stability and response of periodic systems with applications to helicopter rotor dynamics and aeroelasticity abstract this paper reviews and extends some recently developed numerical techniques for analyzing the stability linear response and nonlinear response of periodic systems which are governed by ordinary differential equations with periodic coefficients the main emphasis is on applications aimed at rotor blade aeroelastic stability and response calculations in forward flight first floquer theory is br4iefly reviewed and numerically efficient methods for evaluating the transition matrix at the end of one period are described next the numerical treatment of the linear response problem is discussed finally the numerical solution of the nonlinear response problem is treated using quasilinearization and periodic shooting applications illustrating numerical properties of the methods are described

one or two frequencies the empirical mode decomposition answers this paper investigates how the empirical mode decomposition emd a fully data driven technique recently introduced for decomposing any oscillatory waveform into zero mean components behaves in the case of a composite two tones signal essentially two regimes are shown to exist depending on whether the amplitude ratio of the tones is greater or smaller than unity and the corresponding resolution properties of the emd turn out to be in good agreement with intuition and physical interpretation a refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments the analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the emd are assessed

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

matrix discriminant analysis with application to colorimetric sensor array data with the rapid development of nano technology a colorimetric sensor array csa that is referred to as an optical electronic nose has been developed for the identification of toxicants unlike traditional sensors that rely on a single chemical interaction csa can measure multiple chemical interactions by using chemo responsive dyes the color changes of the chemo responsive dyes are recorded before and after exposure to toxicants and serve as a template for classification the color changes are digitalized in the form of a matrix with rows representing dye effects and columns representing the spectrum of colors thus matrix classification methods are highly desirable in this article we develop a novel classification method matrix discriminant analysis mda which is a generalization of linear discriminant analysis lda for the data in matrix form by incorporating the intrinsic matrix structure of the data in discriminant analysis the proposed method can improve csa s sensitivity and more impor

nonlinear chirp mode decomposition a variational method variational mode decomposition vmd a recently introduced method for adaptive data analysis has aroused much attention in various fields however the vmd is formulated based on the assumption of narrow band property of the signal model to analyze wide band nonlinear chirp signals ncss we present an alternative method called variational nonlinear chirp mode decomposition vncmd the vncmd is developed from the fact that a wideband ncs can be transformed to a narrow band signal by using demodulation techniques our decomposition problem is thus formulated as an optimal demodulation problem which is efficiently solved by the alternating direction method of multipliers our method can be viewed as a time frequency filter bank which concurrently extracts all the signal modes some simulated and real data examples are provided showing the effectiveness of the vncmd in analyzing ncss containing close or even crossed modes

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

emd revisited a new understanding of the envelope and resolving the mode mixing problem in am fm signals empirical mode decomposition emd is an adaptive and data driven approach for analyzing multicomponent nonlinear and nonstationary signals the stop criterion envelope technique and mode mixing problem are the most important topics that need to be addressed in order to improve the emd algorithm in this paper we study the envelope technique and the mode mixing problem caused by separating multicomponent am fm signals with the emd algorithm we present a new necessary condition on the envelope that questions the current assumption that the envelope passes through the extreme points of an intrinsic mode function imf then we present a solution to the mode mixing problem that occurs when multicomponent am fm signals are separated we experiment on several signals including simulated signals and real life signals to demonstrate the efficacy of the proposed method in resolving the mode mixing problem

the inner structure of empirical mode decomposition abstract the empirical mode decomposition emd is a nonlinear method that is truly adaptive with good localization property in the time domain for analyzing non stationary complex data the emd has been proven useful in a wide range of applications however due to the nonlinear and complex nature of the sifting process the most essential step of the emd a firm mathematical foundation or a transparent physical description are still lacked for emd here we embark on constructing a mathematical theory of the sifting operator we first show that the sifting operator can be expressed as the data plus the sum of the responses to the impulses multiplied by the data value at the extrema such an expression of the sifting operator is then used to investigate the adaptive nature and the localizing effect of the emd alternatively the sifting operator can also be represented by a sifting matrix which depends nonlinearly on the extrema distribution based on the eigen decomposition of the sifting matrix the transfer function of the sifting process is analyzed finally we answer what an intrinsic mode function imf is from the wave perspective by exploring the physical basis of the imfs

multifractal description of wind power fluctuations using arbitrary order hilbert spectral analysis a b s t r a c t the objectives are to study and model the aggregate wind power fluctuations dynamics in the multifractal framework we present here the analysis of aggregate power output sampled at 1 hz during three years we decompose the data into several intrinsic mode functions imfs using empirical mode decomposition emd we use a new approach ar bitrary order hilbert spectral analysis a combination of the emd approach with hilbert spectral analysis or hilbert huang transform and the classical structure function anal ysis to extract the scaling exponents or multifractal spectrum q this function provides a full characterization of a process at all intensities and all scales the application of both methods i e structure function and arbitrary order hilbert spectral analyses gives similar results indicating that the aggregate power output from a wind farm possesses intermit tent and multifractal properties in order to check this result we generate stochastic simu lations of a multifractal random walk mrw using a log normal stochastic equation we show that the simulation results are fully compatible with the experimental results

1d lda vs 2d lda when is vector based linear discriminant analysis better than matrix based recent advances have shown that algorithms with 2d matrix based representation perform better than the traditional 1d vector based ones in particular 2d lda has been widely reported to outperform 1d lda however would the matrix based linear discriminant analysis be always superior and when would 1d lda be better in this paper we investigate into these questions and have a comprehensive comparison between 1d lda and 2d lda in theory and in experiments we analyze the heteroscedastic problem in 2d lda and formulate mathematical equalities to explore the relationship between 1d lda and 2d lda then we point out potential problems in 2d lda it is shown that 2d lda has eliminated the information contained in the covariance information between different local geometric structures such as the rows or the columns which is useful for discriminant feature extraction whereas 1d lda could preserve such information interestingly this new finding indicates that 1d lda is able to gain higher fisher score than 2d lda in some extreme case furthermore sufficient conditions on which 2d lda would be bayes optimal for two class classification problem are derived and comparison with 1d lda in this aspect is also analyzed this could help understand how 2d lda is expected to achieve at its best further discover its relationship with 1d lda and well support other findings after the theoretical analysis comprehensive experimental results are reported by fairly and extensively comparing 1d lda with 2d lda in contrast to the existing view that some 2d lda based algorithms would perform better than 1d lda when the number of training samples for each class is small or when the number of discriminant features used is small we show that it is not always true and show that some standard 1d lda based algorithms could perform better in those cases on some challenging data sets

investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach empirical mode decomposition emd is a relatively new data driven adaptive technique for analyzing multicomponent signals although it has many interesting features and often exhibits an ability to decompose nonlinear and nonstationary signals it lacks a strong theoretical basis which would allow a performance analysis and hence the enhancement and optimization of the method in a systematic way in this paper the optimization of emd is attempted in an alternative manner using specially defined multicomponent signals the optimum outputs can be known in advance and used in the optimization of the emd free parameters within a genetic algorithm framework the contributions of this paper are two fold first the optimization of both the interpolation points and the piecewise interpolating polynomials for the formation of the upper and lower envelopes of the signal reveal important characteristics of the method which where previously hidden second basic directions for the estimates of the optimized parameters are developed leading to significant performance improvements

variational mode decomposition during the late 1990s huang introduced the algorithm called empirical mode decomposition which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands emd is known for limitations like sensitivity to noise and sampling these limitations could only partially be addressed by more mathematical attempts to this decomposition problem like synchrosqueezing empirical wavelets or recursive variational decomposition here we propose an entirely non recursive variational mode decomposition model where the modes are extracted concurrently the model looks for an ensemble of modes and their respective center frequencies such that the modes collectively reproduce the input signal while each being smooth after demodulation into baseband in fourier domain this corresponds to a narrow band prior we show important relations to wiener filter denoising indeed the proposed method is a generalization of the classic wiener filter into multiple adaptive bands our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand the variational model is efficiently optimized using an alternating direction method of multipliers approach preliminary results show attractive performance with respect to existing mode decomposition models in particular our proposed model is much more robust to sampling and noise finally we show promising practical decomposition results on a series of artificial and real data

second order structure function in fully developed turbulence we relate the second order structure function of a time series with the power spectrum of the original variable taking an assumption of statistical stationarity with this approach we find that the structure function is strongly influenced by the large scales the large scale contribution and the contribution range are respectively 79 and 1 4 decades for a kolmogorov ensuremath 5 3 power spectrum we show numerically that a single scale influence range over smaller scales is about 2 decades we argue that the structure function is not a good method to extract the scaling exponents when the data possess large energetic scales an alternative methodology the arbitrary order hilbert spectral analysis which may constrain this influence within 0 3 decade is proposed to characterize the scaling property directly in an amplitude frequency space an analysis of passive scalar temperature turbulence time series is presented to show the influence of large scale structures in real turbulence and the efficiency of the hilbert based methodology the corresponding scaling exponents ensuremath zeta ensuremath theta q provided by the hilbert based approach indicate that the passive scalar turbulence field may be less intermittent than what was previously believed

response of linear slowly varying systems under external excitations abstract an approximate closed form solution for a linear slowly varying system under external excitation is derived the derivation of the solution is based on the technique of freezing slowly varying parameters a technique used in establishing the stability of slowly varying homogeneous systems the error between the approximate and exact solutions is given and an upper bound on the norm of the error in terms of a measure of the rate of change of the system matrix and the norm of the exact solution of the system is derived approximate responses of two particular linear slowly varying systems under external excitation are obtained by the proposed technique good agreement between the approximate and numerically obtained solutions is observed

null space pursuit an operator based approach to adaptive signal separation the operator based signal separation approach uses an adaptive operator to separate a signal into additive subcomponents the approach can be formulated as an optimization problem whose optimal solution can be derived analytically however the following issues must still be resolved estimating the robustness of the operator s parameters and the lagrangian multipliers and determining how much of the information in the null space of the operator should be retained in the residual signal to address these problems we propose a novel optimization formula for operator based signal separation and show that the parameters of the problem can be estimated adaptively we demonstrate the effectiveness of the proposed method by processing several signals including real life signals

development of emd based denoising methods inspired by wavelet thresholding one of the tasks for which empirical mode decomposition emd is potentially useful is nonparametric signal denoising an area for which wavelet thresholding has been the dominant technique for many years in this paper the wavelet thresholding principle is used in the decomposition modes resulting from applying emd to a signal we show that although a direct application of this principle is not feasible in the emd case it can be appropriately adapted by exploiting the special characteristics of the emd decomposition modes in the same manner inspired by the translation invariant wavelet thresholding a similar technique adapted to emd is developed leading to enhanced denoising performance

an efficient computational scheme for the analysis of periodic systems abstract in this paper a new efficient numerical scheme for the stability analysis of linear systems with periodic parameters is suggested the approach is based on the idea that the state vector and the periodic matrix of the system can be expanded in terms of chebyshev polynomials over the principal period such an expansion reduces the original problem to a set of linear algebraic equations from which the solution in the interval of one period can be obtained furthermore the technique is combined with floquet theory to yield the transition matrix at the end of one period and provide the stability conditions via an eigen analysis procedure two formulations are presented the first formulation is suitable for a set of equations written in state space form while the second can be applied directly to a set of second order equations the application is demonstrated through practical illustrative examples including the parametric excitation problem of a fixed fixed column the numerical results thus obtained are compared with those obtained from dverk a runge kutta code available in the imsl library an error bound analysis is also included it is concluded that the suggested schemes not only provide accurate results but are also computationally very efficient in particular the second formulation is found to be several times faster than the standard runge kutta type codes

a study of the characteristics of white noise using the empirical mode decomposition method based on numerical experiments on white noise using the empirical mode decomposition emd method we find empirically that the emd is effectively a dyadic filter the intrinsic mode function imf components are all normally distributed and the fourier spectra of the imf components are all identical and cover the same area on a semi logarithmic period scale expanding from these empirical findings we further deduce that the product of the energy density of imf and its corresponding averaged period is a constant and that the energy density function is chi squared distributed furthermore we derive the energy density spread function of the imf components through these results we establish a method of assigning statistical significance of information content for imf components from any noisy data southern oscillation index data are used to illustrate the methodology developed here

bilinear formulation applied to the response and stability of helicopter rotor blade a time finite element method based on hamilton s law of varying action is applied to the analysis of helicopter blade dynamics the bilinear formulation of hamilton s law of varying action is used to discretize the temporal dependence of the equations of motion variable order shape and test functions based on legendre polynomials are used two approaches for the numerical implementation of the bilinear formulation are used one approach obtains the response of a general dynamic system using marching in time the other approach is based on the assumption that the solution is identical at the beginning and end of one period therefore this approach is suitable for periodic systems the bilinear formulation in imposed periodicity mode is used to compute the response of a rigid blade with flap and lag degrees of freedom in forward flight the bilinear formulation in marching mode is applied to find the floquet transition matrix necessary to test the stability of blade motion

sparse discriminant analysis we consider the problem of performing interpretable classification in the high dimensional setting in which the number of features is very large and the number of observations is limited this setting has been studied extensively in the chemometrics literature and more recently has become commonplace in biological and medical applications in this setting a traditional approach involves performing feature selection before classification we propose sparse discriminant analysis a method for performing linear discriminant analysis with a sparseness criterion imposed such that classification and feature selection are performed simultaneously sparse discriminant analysis is based on the optimal scoring interpretation of linear discriminant analysis and can be extended to perform sparse discrimination via mixtures of gaussians if boundaries between classes are nonlinear or if subgroups are present within each class our proposal also provides low dimensional views of the discriminative directions

on analysis of bi dimensional component decomposition via bemd this paper investigates how the bi dimensional empirical mode decomposition bemd behaves in digital images via two components first some definitions of components are given then the theoretical analysis is presented to show that in digital images there might be sparse extrema sets and the sifting process of bemd converges to the true component provided that the sparse extrema sets do exist also the feature of 2d component in digital images is explored via composite two component signals the three dimensional cubes disclosing the performance of bemd are presented which turn out to be in good agreement with intuition and physical interpretation this paper has also shown that the sampling period the cross angle the amplitude ratio and the frequency ratio between the components will affect the separation of them and some instructive conclusions are achieved as well the theoretical analysis is provided for analyzing the observed behaviors and supported by numerical experiments the main aim of this study is primarily to contribute to a better understanding of the possibilities and limitations offered by bemd in digital images

an optimization based empirical mode decomposition scheme the empirical mode decomposition emd has been developed by n e huang et al in 1998 as an iterative method to decompose a nonlinear and nonstationary univariate function additively into multiscale components these components called intrinsic mode functions imfs are constructed such that they are approximately orthogonal to each other with respect to the l 2 inner product moreover the components allow for a definition of instantaneous frequencies through complexifying each component by means of the application of the hilbert transform this approach via analytic signals however does not guarantee that the resulting frequencies of the components are always non negative and thus physically meaningful and that the amplitudes can be interpreted as envelopes in this paper we formulate an optimization problem which takes into account important features desired of the resulting emd specifically we propose a data adapted iterative method which minimizes in each iteration step a smoothness functional subject to inequality constraints involving the extrema in this way our method constructs a sparse data adapted basis for the input function as well as a mathematically stringent envelope for the function moreover we present an optimization based normalization to extract instantaneous frequencies from the analytic function approach we present corresponding algorithms together with several examples