the hilbert spectrum via wavelet projections non stationary signals are increasingly analysed in the time frequency domain to determine the variation of frequency components with time it was recently proposed in this journal that such signals could be analysed by projections onto the time frequency plane giving a set of monocomponent signals these could then be converted to analytic signals using the hilbert transform and their instantaneous frequency calculated which when weighted by the energy yields the hilbert energy spectrum for that projection agglomeration over projections yields the complete hilbert spectrum we show that superior results can be obtained using waveletbased projections the maximal overlap undecimated stationary translation invariant discrete wavelet transform and wavelet packet transforms are used with

studies of spectral properties of short genes using the wavelet subspace hilbert huang transform wshht abstract this paper presents a new algorithm for the analysis of spectral properties of short genes using the wavelet transform and the hilbert huang transform hht a wavelet subspace algorithm combined with the empirical mode decomposition emd is introduced to create subdivided intrinsic mode functions imfs and a cross correlation analysis is applied to remove pseudo spectral components experiments are carried out on dna sequences with the double base db curve representation and the results show that the signal to noise ratio of buried signals can be enhanced using the proposed method yielding significant patterns that are rarely observed with conventional methods the wavelet subspace hilbert huang transform wshht algorithm is able to correctly identify spectral patterns of very short genes below 70 bp in dna sequences

phase correlation of foreign exchange time series abstract correlation of foreign exchange rates in currency markets is investigated based on the empirical data of usd dem and usd jpy exchange rates for a period from february 1 1986 to december 31 1996 the return of exchange time series is first decomposed into a number of intrinsic mode functions imfs by the empirical mode decomposition method the instantaneous phases of the resultant imfs calculated by the hilbert transform are then used to characterize the behaviors of pricing transmissions and the correlation is probed by measuring the phase differences between two imfs in the same order from the distribution of phase differences our results show explicitly that the correlations are stronger in daily time scale than in longer time scales the demonstration for the correlations in periods of 1986 1989 and 1990 1993 indicates two exchange rates in the former period were more correlated than in the latter period the result is consistent with the observations from the cross correlation calculation

spectral and convergence analysis of the discrete alif method abstract the adaptive local iterative filtering alif method is a recently proposed iterative procedure to decompose a signal into a finite number of simple components called intrinsic mode functions it is an alternative to the well known and widely used empirical mode decomposition method and it has proved to be a powerful and promising algorithm however so far no convergence analysis is available for the alif algorithm both in the continuous and discrete settings in the present paper we focus on the discrete version of the alif method and we tackle the problem of studying its convergence properties using recent results about sampling matrices and the theory of generalized locally toeplitz sequences which we extend in this paper we perform a spectral analysis of the alif iteration matrices with a special attention to the eigenvalue clustering and the eigenvalue distribution based on the eigenvalue distribution we formulate a necessary condition for the convergence of the discrete alif method moreover we provide a simple criterion to construct appropriate filters satisfying this condition we also present several numerical examples in support of the theoretical study our contribution represents a first significant step toward a complete convergence analysis of the alif method an analysis which appears to be rather difficult from a mathematical viewpoint as the alif iteration matrices possess a peculiar structure that to the best of the authors knowledge has never been investigated in the literature

empirical mode decomposition and correlation properties of long daily ozone records sreceived 16 november 2004 revised manuscript received 17 march 2005 published 27 may 2005d correlations for daily data of total ozone column are investigated by detrended fluctuation analysis sdfad the removal of annual periodicity does not result in a background free signal for the tropical station mauna loa in order to identify the remaining quasiperiodic constituent the relatively new method of empirical mode decomposition semdd is tested we found that the so called intrinsic mode functions do not represent real signal components of the ozone time series their amplitude modulation is very sensitive to local changes such as random data removal or smoothing tests on synthetic data further corroborate the limitations of decomposing quasiperiodic signals from noise with emd nevertheless the emd algorithm helps to identify dominating frequencies in the time series which allows to separate fluctuations from the remaining background we demonstrate that dfa analysis for the cleaned mauna loa record yields scaling comparable to a mid latitude station

physics based foundation for empirical mode decomposition we study the correspondence between analytical and empirical slow flow analyses given a sufficiently dense set of sensors measured time series recorded throughout a mechanical or structural system contains all information regarding the dynamics of that system empirical mode decomposition is a useful tool for decomposing the measured time series in terms of intrinsic mode functions which are oscillatory modes embedded in the data that fully reproduce the time series the equivalence of responses of the analytical slow flow models and the dominant intrinsic mode functions derived from empirical mode decomposition provides a physics based theoretical foundation for empirical mode decomposition which currently is performed formally in an ad hoc fashion to demonstrate correspondence between analytical and empirical slow flows we derive appropriate mathematical expressions governing the empirical slow flows and based on analyticity conditions several nonlinear dynamical systems are considered to demonstrate this correspondence and the agreement between the analytical and empirical slow dynamics proves the assertion

emd based refined composite multiscale entropy analysis of complex signals abstract multiscale entropy mse is an effective method to measure the complexity of signals from complex systems which has been applied to various fields successfully however mse may yield an inaccurate estimate of entropy and induce undefined entropy as the coarse graining procedure reduces the length of data considerably at large scales refined composite multiscale entropy rcmse is then developed to solve this problem however trends superimposed in signals may significantly affect the complexity measurement thus we introduce an empirical mode decomposition based rcmse called emd rcmse to first eliminate such effects of trends and then measure the complexity of signals it is validated from simulated signals and has good estimation in addition this method is also applied to study the complexity of traffic signals and we obtain some interesting results 1 traffic signals are more complex than the results showing from rcmse mse 2 weekday and weekend patterns different combination of trends greatly affect the results 3 complexity indices change with time at each day due to the degree of human activities

b splines of blaschke product type in this paper we construct a class of new splines related to a blaschke product they emerge naturally when studying the filter functions of a class of linear time invariant systems which are related to the boundary values of a blaschke product for the purpose of sampling non bandlimited signals using nonlinear fourier atoms the new splines generalize the well known symmetric b splines we establish their properties such as integral representation property a partition of unity property a recurrence relation and difference property we also investigate their random behaviour finally our numerical experiments confirm our theories

reversible joint hilbert and linear canonical transform without distortion generalized analytic signal associated with the linear canonical transform lct was proposed recently by fu and li generalized analytic signal associated with linear canonical transform opt commun vol 281 pp 1468 1472 2008 however most real signals especially for baseband real signals cannot be perfectly recovered from their generalized analytic signals therefore in this paper the conventional hilbert transform ht and analytic signal associated with the lct are concerned to transform a real signal into the lct of its ht two integral transforms i e the ht and lct are required the goal of this paper is to simplify cascades of multiple integral transforms which may be the ht analytic signal lct or inverse lct the proposed transforms can reduce the complexity when realizing the relationships among the following six kinds of signals a real signal its ht and analytic signal and the lct of these three signals most importantly all the proposed transforms are reversible and undistorted using the proposed transforms several signal processing applications are discussed and show the advantages and flexibility over simply using the analytic signal or the lct

the application of the hilbert spectrum to the analysis of electromyographic signals this paper investigates the application of the hilbert spectrum hs which is a recent tool for the analysis of nonlinear and nonstationary time series to the study of electromyographic emg signals the hs allows for the visualization of the energy of signals through a joint time frequency representation in this work we illustrate the use of the hs in two distinct applications the first is for feature extraction from emg signals our results showed that the instantaneous mean frequency imnf estimated from the hs is a relevant feature to clinical practice we found that the median of the imnf reduces when the force level of the muscle contraction increases in the second application we investigated the use of the hs for detection of motor unit action potentials muaps the detection of muaps is a basic step in emg decomposition tools which provide relevant information about the neuromuscular system through the morphology and firing time of muaps we compared visually how muap activity is perceived on the hs with visualizations provided by some traditional e g scalogram spectrogram wigner ville time frequency distributions furthermore an alternative visualization to the hs for detection of muaps is proposed and compared to a similar approach based on the continuous wavelet transform cwt our results showed that both the proposed technique and the cwt allowed for a clear visualization of muap activity on the time frequency distributions whereas results obtained with the hs were the most difficult to interpret as they were extremely affected by spurious energy activity

singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series we distinguish between two dimensions of a dynamical system given by experimental time series statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data taking into account sampling and noise problems the dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data singular spectrum analysis ssa provides estimates of the statistical dimension ssa also describes the main physical phenomena reflected by the data it gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data we apply ssa to four paleoclimatic records the principal climatic oscillations and the regime changes in their amplitude are detected about 10 degrees of freedom are statistically significant in the data large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension

asymmetric multiscale detrended fluctuation analysis of california electricity spot price abstract in this paper we develop a new method called asymmetric multiscale detrended fluctuation analysis which is an extension of asymmetric detrended fluctuation analysis a dfa and can assess the asymmetry correlation properties of series with a variable scale range we investigate the asymmetric correlations in california 1999 2000 power market after filtering some periodic trends by empirical mode decomposition emd our findings show the coexistence of symmetric and asymmetric correlations in the price series of 1999 and strong asymmetric correlations in 2000 what is more we detect subtle correlation properties of the upward and downward price series for most larger scale intervals in 2000 meanwhile the fluctuations of s asymmetry and s absolute asymmetry are more significant in 2000 than that in 1999 for larger scale intervals and they have similar characteristics for smaller scale intervals we conclude that the strong asymmetry property and different correlation properties of upward and downward price series for larger scale intervals in 2000 have important implications on the collapse of california power market and our findings shed a new light on the underlying mechanisms of power price

improved consistent estimation of the order of stochastic control systems a more consistent method of estimating the order of feedback control systems is developed the algorithm is of the recursive type strong consistency is derived under less stringent conditions than those of h f chen and l guo ibid vol ac 32 p 531 5 1987

instantaneous vortex shedding behaviour in periodically varying flow vortex shedding behind a stationary t shaped cylinder in a circular pipe subjected to periodically varying flow was studied at reynolds numbers between 6 17 103 and 2 46 104 whereas the frequency ratio fs fo ranged from 0 29 to 14 64 fs denotes the natural vortex shedding frequency referred to the mean flow and fo denotes the frequency of periodically varying flow by adopting the hilbert transform to analyse the velocity signals measured the instantaneous vortex shedding frequency was obtained based on this quantity one could categorize the vortex shedding phenomenon observed into three regimes namely quasi steady vortex shedding for fs fo 4 37 hysteresis vortex shedding for fs fo 1 56 4 37 and non interactive vortex shedding for 0 29 fs fo 1 56 in the regime of quasi steady vortex shedding the instantaneous vortex shedding frequency follows the periodically varying flow without phase lag in the regime of hysteresis vortex shedding the instantaneous vortex shedding frequency lags behind the periodically varying flow phase lag roughly varies linearly with fs fo further the variations of non dimensionalized instantaneous vortex shedding frequencies obtained in the accelerating and decelerating portions of the periodically varying flow are found to depend on fs fo and u in the regime of non interactive vortex shedding the vortex shedding frequency tends to vary with the mean velocity of periodically varying flow excluding the occurrences of primary and secondary lock on at fs fo 0 97 1 03 and 0 495 0 514 respectively

derivative optimized empirical mode decomposition for the hilbert huang transform in the empirical mode decomposition emd for the hilbert huang transform hht a nonlinear and non stationary signal is adaptively decomposed by an hht into a series of intrinsic mode functions imfs with the lowest one as the trend at each step of the emd the low frequency component is mainly determined by the average of the upper envelope consisting of local maxima and the lower envelope consisting of local minima the high frequency component is the deviation of the signal relative to the low frequency component the fact that no local maximum and minimum can be determined at the two end points leads to detrend uncertainty and in turn causes uncertainty in the hht to reduce such uncertainty hermitian polynomials are used to obtain the upper and lower envelopes with the first derivatives at the two end points q l q r as parameters which are optimally determined on the base of minimum temporal variability of the low frequency component in the each step of the decomposition this well posed mathematical system is called the derivative optimized emd demd with the demd the end effect and detrend uncertainty are drastically reduced and scales are separated naturally without any a priori subjective selection criterion

hilbert transform and its engineering applications the hilbert transform is useful in calculating instantaneous attributes of a time series especially the envelope amplitude and instantaneous frequency the instantaneous envelope is the amplitude of the complex hilbert transform the instantaneous frequency is the time rate of change of the instantaneous phase angle these properties can be applied to identify dynamic characteristics of a linear as well as a nonlinear system however the conventional discrete hilbert transform which is based on fast fourier transform and inverse fourier transform has shown the lack of accuracy for time derivative calculations in this paper we first introduce the hilbert transform and its applications to the nonlinear system parameter identification then we address the practical issues in applying the hilbert transform to engineering applications to increase the accuracy of the envelope detection we propose a hilbert transform technique based on local maxima interpolation analyses and simulations are carried out to demonstrate the advantages of the proposed technique finally we employ the proposed local maxima interpolation technique in identifying the nonlinear dynamic characteristics of industrial examples

nonlinear stochastic exclusion financial dynamics modeling and time dependent intrinsic detrended cross correlation abstract in attempt to reproduce price dynamics of financial markets a stochastic agent based financial price model is proposed and investigated by stochastic exclusion process the exclusion process one of interacting particle systems is usually thought of as modeling particle motion with the conserved number of particles in a continuous time markov process in this work the process is utilized to imitate the trading interactions among the investing agents in order to explain some stylized facts found in financial time series dynamics to better understand the correlation behaviors of the proposed model a new time dependent intrinsic detrended cross correlation tdi dcc is introduced and performed also the autocorrelation analyses are applied in the empirical research furthermore to verify the rationality of the financial price model the actual return series are also considered to be comparatively studied with the simulation ones the comparison results of return behaviors reveal that this financial price dynamics model can reproduce some correlation features of actual stock markets

asymptotic inversion of the incomplete beta function abstract the normalized incomplete beta function ix a b is inverted for large values of the parameters a and b that is x solutions of the equation ix a b p p 0 1 are considered especially for large values of a and b the approximations are obtained by using uniform asymptotic expansions of the incomplete beta function in which an error function or an incomplete gamma function is the dominant term the inversion problem is started by inverting this dominant term further terms in the expansion are obtained by using standard perturbation methods which were recently introduced in a paper describing a method for asymptotic inversion of the incomplete gamma functions numerical results indicate that for obtaining an accuracy of four correct digits the asymptotic method can already be used for a b 5

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

adaptive ecg denoising using genetic algorithm based thresholding and ensemble empirical mode decomposition this paper proposes an adaptive denoising methodology for electrocardiogram ecg signals that employs ensemble empirical mode decomposition eemd and a genetic algorithm ga based thresholding technique in this method a noisy ecg signal is first decomposed by means of eemd into a series of intrinsic mode functions imfs which are then separated into signal and noise dominant groups using a similarity measure based on kullback leibler divergence and a probability density function a ga based thresholding technique is then used adaptively to remove the noise inherent in noise dominant imfs finally a denoised signal is reconstructed by combining the signal dominant imfs and the denoised noise dominant imfs the performance of the proposed denoising methodology is evaluated in the present work by using the mit bih arrhythmia ecg database and results from the proposed method are compared with those of other conventional approaches in various noisy environments experimental results indicate that the proposed denoising method outperforms other denoising methodologies in terms of signal to noise ratio mean square error and percent root mean square difference

rbf based technique for statistical demodulation of pathological tremor this paper presents an innovative technique based on the joint approximation capabilities of radial basis function rbf networks and the estimation capability of the multivariate iterated hilbert transform iht for the statistical demodulation of pathological tremor from electromyography emg signals in patients with parkinson s disease we define a stochastic model of the multichannel high density surface emg by means of the rbf networks applied to the reconstruction of the stochastic process characterizing the disease modeled by the multivariate relationships generated by the karhunen loeve transform in hilbert spaces next we perform a demodulation of the entire random field by means of the estimation capability of the multivariate iht in a statistical setting the proposed method is applied to both simulated signals and data recorded from three parkinsonian patients and the results show that the amplitude modulation components of the tremor oscillation can be estimated with signal to noise ratio close to 30 db with root mean square error for the estimates of the tremor instantaneous frequency additionally the comparisons with a large number of techniques based on all the combinations of the rbf extreme learning machine backpropagation support vector machine used in the first step of the algorithm and iht empirical mode decomposition multiband energy separation algorithm periodic algebraic separation and energy demodulation used in the second step of the algorithm clearly show the effectiveness of our technique these results show that the proposed approach is a potential useful tool for advanced neurorehabilitation technologies that aim at tremor characterization and suppression

methodology for nonlinear quantification of a flexible beam with a local strong nonlinearity abstract this study presents a methodology for nonlinear quantification i e the identification of the linear and nonlinear regimes and estimation of the degree of nonlinearity for a cantilever beam with a local strongly nonlinear stiffness element the interesting feature of this system is that it behaves linearly in the limits of extreme values of the nonlinear stiffness an euler bernoulli cantilever beam with two nonlinear configurations is used to develop and demonstrate the methodology one configuration considers a cubic spring attached at a distance from the beam root to achieve a smooth nonlinear effect the other configuration considers a vibro impact element that generates non smooth effects both systems have the property that in the limit of small and large values of a configuration parameter the system is almost linear and can be modeled as such with negligible error for the beam with a cubic spring attachment the forcing amplitude is the varied parameter while for the vibro impact beam this parameter is the clearance between the very stiff stops and the beam at static equilibrium proper orthogonal decomposition is employed to obtain an optimal orthogonal basis used to describe the nonlinear system dynamics for varying parameter values the frequencies of the modes that compose the basis are then estimated using the rayleigh quotient the variations of these frequencies are studied to identify parameter values for which the system behaves approximately linearly and those for which the dynamical response is highly nonlinear moreover a criterion based on the betti maxwell reciprocity theorem is used to verify the existence of nonlinear behavior for the set of parameter values suggested by the described methodology the developed methodology is general and applicable to discrete or continuous systems with smooth or nonsmooth nonlinearities

identification of linear time varying mdof dynamic systems from forced excitation using hilbert transform and emd method an algorithm is developed for the identification of linear time varying ltv multiple degrees of freedom systems it is based on the hilbert transform and the empirical mode decomposition with forced vibration response data the proposed identification algorithm is applied to single degree of freedom and multi degrees of freedom dynamic systems three ideal cases of ltv systems with smoothly varying abruptly varying and periodically varying stiffness and damping are studied to illustrate the capability of the algorithm to track the variations of the system simulation results demonstrate the effectiveness and the robustness of the proposed identification algorithm and the lack of complete orthogonality for any two intrinsic mode functions is one of the sources of error in the identification

machine fault diagnosis through an effective exact wavelet analysis continuous wavelet transforms cwts are widely recognized as effective tools for vibration based machine fault diagnosis as cwts can detect both stationary and transitory signals however due to the problem of overlapping a large amount of redundant information exists in the results that are generated by cwts the appearance of overlapping can smear the spectral features and make the results very difficult to interpret for machine operators misinterpretation of results may lead to false alarms or failures to detect anomalous signals moreover as conventional cwts only use a single mother wavelet to generate daughter wavelets the distortion of the original signal in the resultant coefficients is inevitable obviously this will significantly affect the accuracy in anomalous signal detection to minimize the effect of overlapping and to enhance the accuracy of fault detection a novel wavelet transform which is named as exact wavelet analysis has been designed for use in vibration based machine fault diagnosis the design of exact wavelet analysis is based on genetic algorithms at each selected time frame the algorithms will generate an adaptive daughter wavelet to match the inspected signal as exactly as possible the optimization process of exact wavelet analysis is different from other adaptive wavelets as it considers both the optimization of wavelet coefficients and the satisfaction of the admissibility conditions of wavelets the results obtained from simulated and practical experiments prove that exact wavelet analysis not only minimizes the undesirable effect of overlapping but also helps operators to detect faults and distinguish the causes of faults with the help from exact wavelet analysis sudden shutdowns of production and services due to the fatal breakdown of machines could be avoided

analytic signals and the transition to chaos in deterministic flows the transition from regular to chaotic motions in deterministic flows is characterized by a change from a discrete fourier spectrum to a broadband one the onset of chaos is thus associated with the creation of an infinite number of new fourier modes given a system that generates a time series x t we study the transition to chaos from the perspective of analytic signals which are defined via the hilbert transform in order to identify distinct analytic signals we decompose the original time series x t into a finite number of modes that correspond to proper rotations in the complex plane of their analytic signals we provide numerical evidence that at the transition there is no substantial change in the number of analytic signals characterizing x t furthermore the distributions of the instantaneous frequencies of the analytic signals in the chaotic regime are well localized and exhibit no broadband feature these results suggest a simple organization of chaos in terms of analytic signals

multiscale multifractal detrended fluctuation analysis of earthquake magnitude series of southern california abstract the multifractal characteristics of magnitude time series of earthquakes that occurred in southern california from 1990 to 2010 are studied in this work a method for the scale division of the magnitude of these earthquakes based on empirical mode decomposition emd and multifractal analysis is proposed this method gains a new insight into measuring multifractal properties of the magnitude time series at multiple scales and it reveals further information about the dynamic seismic behavior by using emd a time series can be decomposed into mode time series that represent different time frequency components we find that time frequency components show long range correlation with different hurst exponents by using r s analysis based on the different fractal structures of components we consider three different scale series micro mid and macro scale subsequences which are superposed and reconstructed by the components the multifractal properties of the three scale subsequences are analyzed by using multifractal detrended fluctuation analysis mf dfa the results show that the three different scale subsequences have various shapes of multifractal spectra and corresponding distinct properties the micro scale subsequence singularity spectrum shows left skewed indicating a relative dominance of the lower hurst exponent the mid scale subsequence has a right skewed singularity spectrum the macro scale subsequence exhibits the most significant persistence and shows the strongest multifractality

automatic noise removal signal removal based on general cross validation thresholding in synchrosqueezed domain and its application on earthquake data abstractrecorded seismic signals are often corrupted by noise we have developed an automatic noise attenuation method for single channel seismic data based upon high resolution time frequency analysis synchrosqueezing is a time frequency reassignment method aimed at sharpening a time frequency picture noise can be distinguished from the signal and attenuated more easily in this reassigned domain the threshold level is estimated using a general cross validation approach that does not rely on any prior knowledge about the noise level the efficiency of the thresholding has been improved by adding a preprocessing step based on kurtosis measurement and a postprocessing step based on adaptive hard thresholding the proposed algorithm can either attenuate the noise either white or colored and keep the signal or remove the signal and keep the noise hence it can be used in either normal denoising applications or preprocessing in ambient noise studies we tested the performance of the proposed method on s

scaling analysis of phase fluctuations in experimental three phase flows abstract the characterization of complex patterns arising from three phase e g oil gas water flows is an important problem with significant engineering and industrial applications based solely on measured conductance fluctuation signals from experimental three phase flows we propose a method to characterize and distinguish three commonly observed flow patterns using the phase characterization method we first calculate the instantaneous phase from the signals then through performing a scaling analysis detrended fluctuation analysis dfa we extract scaling behaviors associated with the phase fluctuations and find that the dfa scaling exponent is sensitive to the transition among different flow patterns which can be used to characterize nonlinear dynamics of the three phase flow from a novel perspective we investigate the three phase flow in terms of phase characterization and scaling analysis the results indicate that our method can provide new insights into the exploration of complex mechanism in flow pattern transition the effectiveness of the method is demonstrated and its broader applicability is articulated

study of boundary conditions in the iterative filtering method for the decomposition of nonstationary signals abstract nonstationary and non linear signals are ubiquitous in real life their decomposition and analysis is an important research topic of signal processing recently a new technique called iterative filtering has been developed with the goal of decomposing such signals into simple oscillatory components several papers have been devoted to the investigation of this technique from a mathematical point of view all these works start with the assumption that each compactly supported signal is extended periodically outside the boundaries in this work we tackle the problem of studying the influence of different boundary conditions on the decompositions produced by the iterative filtering method in particular the choice of boundary conditions gives rise to different types of structured matrices thus we describe their spectral properties and the convergence properties of iterative filtering algorithm when such matrices are involved numerical results on artificial and real life signals provide interesting insight on important aspects such as accuracy and error propagation of the proposed technique and pave the way for further promising developments

efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios algorithms for the numerical evaluation of the incomplete gamma function ratios p a x gamma a x gamma a and q a x gamma a x gamma a are described for positive values of a and x also inversion methods are given for solving the equations p a x p q a x q with 0 p q 1 both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability the analytical approach from earlier literature is summarized and new initial estimates are derived for starting the inversion algorithms the performance of the associated software to our algorithms the fortran 90 module incgamfi is analyzed and compared with earlier published algorithms

volatility behavior of visibility graph emd financial time series from ising interacting system abstract a financial market dynamics model is developed and investigated by stochastic ising system where the ising model is the most popular ferromagnetic model in statistical physics systems applying two graph based analysis and multiscale entropy method we investigate and compare the statistical volatility behavior of return time series and the corresponding imf series derived from the empirical mode decomposition emd method and the real stock market indices are considered to be comparatively studied with the simulation data of the proposed model further we find that the degree distribution of visibility graph for the simulation series has the power law tails and the assortative network exhibits the mixing pattern property all these features are in agreement with the real market data the research confirms that the financial model established by the ising system is reasonable

vibration signal component separation by iteratively using basis pursuit and its application in mechanical fault detection abstract using different transform basis the vibration signal components such as impulses harmonics and modulated components can be effectively separated by basis pursuit for multicomponent vibration signals this paper proposes a new vibration signal component separation approach by iteratively using basis pursuit the signal is firstly denoised by basis pursuit denoising and the impulsive component can be separated by using identity matrix and redundant fourier basis then signal components with different frequency ranges can be separated by using short time fourier transform basis with different window lengths and the components with high frequency are separated after each iteration to choose the best basis from the dictionary the separation coefficient is proposed for evaluating the separating performance of the transform basis the proposed approach empirical mode decomposition emd and matching pursuit are respectively applied to analyzing a simulated faulty vibration and a simulated faulty gear vibration signal the comparison results demonstrate its superiority to emd and matching pursuit in separation accuracy finally the proposed approach is used for detecting the faults of a rotor and rolling bearings the results further show that this proposed method can be effectively applied to mechanical fault diagnosis

an improved hilbert huang transform and its application in vibration signal analysis abstract the vibration generated by industrial machines always contains nonlinear and non stationary signals recently a number of new methods have been proposed to analyse these signals one of the promising methods is the hilbert huang transform hht the hht is derived from the principals of empirical mode decomposition emd and the hilbert transform when applying the hht first the emd will decompose the acquired signal into a collection of intrinsic mode functions imf the imf is a kind of complete adaptive and almost orthogonal representation for the analysed signal since the imf is almost monocomponent it can determine all the instantaneous frequencies from the nonlinear or non stationary signal second the local energy of each instantaneous frequency can be derived through the hilbert transform hence the result is an energy frequency time distribution of the signal since applying the process of hht is not computational intensive the hht becomes a promising method to extract the properties of nonlinear and non stationary signal however after the completion of a thorough experiment the result generated by the hht has its deficiency first the emd will generate undesirable imfs at the low frequency region that may cause misinterpretation to the result second depends on the analysed signal the first obtained imf may cover too wide a frequency range such that the property of monocomponent cannot be achieved third the emd operation cannot separate signals that contain low energy components in this study new techniques have been applied to improve the result of hht in the improved version of hht the wavelet packet transform wpt is used as preprocessing to decompose the signal into a set of narrow band signals prior to the application of emd with the help from wpt each imf derived from the emd can truly become monocomponent then a screening process is conducted to remove unrelated imfs from the result both simulated and experimental vibration signals of having a rotary system with the fault of rubbing occurred have proven that the improved hht does show the rubbing symptoms more clear and accurate than the original hht hence the improved hht is a precise method for nonlinear and non stationary signal analysis

a sampling theorem for non bandlimited signals using generalized sinc functions a ladder shaped filter of two real parameters a 1 a 2 1 1 is introduced in this note the impulse response of the corresponding linear time invariant lti system is a generalized sinc function of two parameters consequently a generalized shannon type sampling theorem is established for a class of non bandlimited signals with special spectrum properties associated with a ladder shaped filter of two parameters finally a mathematical characterization for the class of non bandlimited signals satisfying the generalized sampling theorem is offered these signals are restrictions to the real line of certain analytic functions in stripped domains symmetric about the real axis in the complex plane for these signals their spectra in higher frequency bands are measured by the spectrum of their base bands

asymptotic inversion of the erlang b formula the erlang b formula represents the steady state blocking probability in the erlang loss model or m m s s queue we derive asymptotic expansions for the offered load that matches for a given number of servers a certain blocking probability in addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function a similar inversion problem is investigated for the erlang c formula

a green s function based bi dimensional empirical mode decomposition a new git bemd algorithm is developed based on green s function with tension much faster than canonical bemd algorithms easy to implement and very robust superior performance tested on artificial and real images bidimensional empirical mode decomposition bemd interprets an image as a superposition of bidimensional intrinsic mode functions bimfs they are extracted by a process called sifting which encompasses two dimensional surface interpolations connecting a set of local maxima or minima to form corresponding envelope surfaces existing surface interpolation schemes are computationally very demanding and often induce artifacts in the extracted modes this paper suggests a novel method of envelope surface interpolation based on green s functions including surface tension greatly improves the stability of the new method which we call green s function in tension based bemd git bemd simulation results using toy images with various textures facial images and functional neuroimages demonstrate the superior performance of the new method when compared to its canonical bemd counterpart git bemd strongly speeds up computations and achieves a higher quality of the extracted bimfs furthermore git bemd can be extended simply to an ensemble based variant git beemd if needed in summary the study suggests the new variant git bemd as a highly competitive fast and stable alternative to existing bemd techniques for image analysis

on fourier interpolation error for band limited signals fourier interpolation is a useful tool to reconstruct continuous signals from discrete samples this correspondence derives fourier interpolation expression from fourier series for periodic signals and investigates its error expression and upper bound for general band limited signals two numerical examples are given to interpret the meaning of the formulas

instantaneous frequency time analysis of physiology signals the application of pregnant women s radial artery pulse signals abstract this study used the hilbert huang transform a recently developed instantaneous frequency time analysis to analyze radial artery pulse signals taken from women in their 36th week of pregnancy and after pregnancy the acquired instantaneous frequency time spectrum hilbert spectrum is further compared with the morlet wavelet spectrum results indicate that the hilbert spectrum is especially suitable for analyzing the time series of non stationary radial artery pulse signals since in the hilbert huang transform signals are decomposed into different mode functions in accordance with signal s local time scale therefore the hilbert spectrum contains more detailed information than the morlet wavelet spectrum from the hilbert spectrum we can see that radial artery pulse signals taken from women in their 36th week of pregnancy and after pregnancy have different patterns this approach could be applied to facilitate non invasive diagnosis of fetus physiological signals in the future

multiscale image fusion using complex extensions of emd empirical mode decomposition emd is a fully data driven technique for decomposing signals into their natural scale components however the problem of uniqueness caused by the empirical nature of the algorithm and its sensitivity to changes in parameters makes it difficult to perform fusion of data from multiple and heterogeneous sources a solution to this problem is proposed using recent complex extensions of emd which guarantees the same number of decomposition levels that is the uniqueness of the scales the methodology is used to address multifocus image fusion whereby two or more partially defocused images are combined in automatic fashion so as to create an all in focus image

empirical mode decomposition using rational splines an application to rainfall time series empirical mode decomposition emd a relatively new form of time series decomposition has the feature of not assuming that a time series is linear or stationary as is implicitly done in fourier analysis in natural time series such as records of rainfall streamflow temperature etc where most variables exhibit nonlinear and non stationary behaviour this feature is particularly useful allowing more meaningful quantification of the proportion of variance in a time series due to fluctuations at different time scales than previous spectral analysis techniques however in its original form the emd algorithm relies on cubic spline interpolation of the extrema which often inflate the variance of the resultant components intrinsic mode functions and residual in this paper a suggested improvement to the emd algorithm using rational splines and flexible treatment of the end conditions is outlined and the consequent effect on three exemplary annual rainfall time series is assessed as a proof of concept it is recognized that many more examples are needed for the fine tuning of the ideas before it can be widely recommended and this work is currently underway

analytical signal analysis of strange nonchaotic dynamics we apply an analytical signal analysis to strange nonchaotic dynamics through this technique it is possible to obtain the spectrum of instantaneous intrinsic mode frequencies that are present in a given signal we find that the second mode frequency and its variance are good order parameters for dynamical transitions from quasiperiodic tori to strange nonchaotic attractors snas and from snas to chaotic attractors phase fluctuation analysis shows that snas and chaotic attractors behave identically within short time windows as a consequence of local instabilities in the dynamics in longer time windows however the globally stable character of snas becomes apparent this methodology can be of great utility in the analysis of experimental time series and representative applications are made to signals obtained from rossler and duffing oscillators

dynamic fluctuations of cross correlations in multi time scale abstract with the empirical mode decomposition we investigate the dynamic fluctuations of cross correlations in multi time scale in this approach the time series of price returns in financial markets is decomposed into a small number of intrinsic mode functions and the corresponding time series of phases can be obtained with the hilbert transform through the empirical mode decomposition we uncover that the cross correlations between stocks change significantly for different time scales and exhibit singular behaviors during a financial crisis with the phase correlation analysis we observe that the cross correlations between market indices vary also for different time scales dividing the total time interval into two sub intervals by the year 2007 we detect that the market indices are much more correlated after 2007

lagrangian single particle turbulent statistics through the hilbert huang transform the hilbert huang transform is applied to analyze single particle lagrangian velocity data from numerical simulations of hydrodynamic turbulence the velocity trajectory is described in terms of a set of intrinsic mode functions c i t and of their instantaneous frequency i t on the basis of this decomposition we define the conditioned statistical moments of the c i modes named q order hilbert spectra hs we show that such quantities have enhanced scaling properties as compared to traditional fourier transform or correlation based structure functions statistical indicators thus providing better insights into the turbulent energy transfer process we present clear empirical evidence that the energylike quantity i e the second order hs displays a linear scaling in time in the inertial range as expected from a dimensional analysis we also measure high order moment scaling exponents in a direct way without resorting to the extended self similarity procedure this leads to an estimate of the lagrangian structure function exponents which are consistent with the multifractal prediction in the lagrangian frame as proposed by biferale et al

the spillover effects between natural gas and crude oil markets the correlation network analysis based on multi scale approach abstract this paper proposes a new compound model to investigate the dynamic linkage mechanism between the natural gas and crude oil markets from the multi scale perspective in the proposed model two main steps are involved multi scale analysis and network research based on the bivariate empirical mode decomposition bemd and fine to coarse algorithm the multi scale analysis ensures the relationship between price fluctuations can be studied under three different time scales which corresponding to market disequilibrium significant events and long term trend by integrating the grey correlation degree and coarse gaining algorithm the network research reveals the dynamic spillover effects of the two markets at different time scales to capture the different linked characteristic in different periods the sample data of henry hub and wti spot prices from 1997 to 2017 is divided into three periods we transfer the daily correlations between the price fluctuations with specific time scale into the correlation patterns and establish the network based on their transmission relations the crucial correlation patterns and the significant transmission relations are revealed by some network indicators and the proposed approach in this paper can be applied to other field of research

hilbert huang transform stability spectral analysis applied to flutter flight test data a new method the hilbert huang transform hht has been applied to analyze the aerostructures test wing atw flight flutter data from nasa dryden flight research center the analysis shows the yielding of the wing after the onset of flutter but just before breaking off of the wingtip based on hht a new stability spectral analysis is also proposed that shows both positive stable and negative unstable damping the stability spectral analysis further shows that the flutter of atw bending occurs at 2 5 hz in addition to 18 hz as determined by modal analysis and identification both hht and the teager energy operator based nonlinearity indicator show that the vibrations of the atw are nonlinear throughout the flight test flutter maneuver

consistent estimation of the order of stochastic control systems a consistent estimate of the order of feedback control systems with unknown matrix coefficients estimated by the least squares method is derived by minimizing a modified version of the bayesian information criterion

time varying lapped transforms and wavelet packets the perfect reconstruction conditions for a time varying lapped transform paraunitary filter bank are developed through the factorization of the transform matrix into sparse factors a general formulation is presented allowing one to switch between two paraunitary filter banks however the extended lapped transform elt is often used as an example furthermore an adaptive wavelet packet is developed employing a time varying tree association of elts in all cases perfect reconstruction is inherently assured

asymptotic inversion of incomplete gamma functions the normalized incomplete gamma functions p a x and q a x are inverted for large values of the parameter a that is x solutions of the equations p a x p q a x q p e o 1 q i p are considered especially for large values of a the approximations are obtained by using uniform asymptotic expansions of the incomplete gamma functions in which an error function is the dominant term the inversion problem is started by inverting this error function term numerical results indicate that for obtaining an accuracy of four correct digits the method can already be used for a 2 although a is a large parameter it is indicated that the method can be applied to other cumulative distribution functions

arbitrary order hilbert spectral analysis for time series possessing scaling statistics comparison study with detrended fluctuation analysis and wavelet leaders in this paper we present an extended version of hilbert huang transform namely arbitrary order hilbert spectral analysis to characterize the scale invariant properties of a time series directly in an amplitude frequency space we first show numerically that due to a nonlinear distortion traditional methods require high order harmonic components to represent nonlinear processes except for the hilbert based method this will lead to an artificial energy flux from the low frequency large scale to the high frequency small scale part thus the power law if it exists is contaminated we then compare the hilbert method with structure functions sf detrended fluctuation analysis dfa and wavelet leader wl by analyzing fractional brownian motion and synthesized multifractal time series for the former simulation we find that all methods provide comparable results for the latter simulation we perform simulations with an intermittent parameter 0 15 we find that the sf underestimates scaling exponent when q 3 the hilbert method provides a slight underestimation when q 5 however both dfa and wl overestimate the scaling exponents when q 5 it seems that hilbert and dfa methods provide better singularity spectra than sf and wl we finally apply all methods to a passive scalar temperature data obtained from a jet experiment with a taylor s microscale reynolds number re 250 due to the presence of strong ramp cliff structures the sf fails to detect the power law behavior for the traditional method the ramp cliff structure causes a serious artificial energy flux from the low frequency large scale to the high frequency small scale part thus dfa and wl underestimate the scaling exponents however the hilbert method provides scaling exponents q quite close to the one for longitudinal velocity indicating a less intermittent passive scalar field than what was believed before 2011 american physical society

detecting intrinsic dynamics of traffic flow with recurrence analysis and empirical mode decomposition abstract in this paper we apply the empirical mode decomposition emd method to the recurrence plot rp and recurrence quantification analysis rqa to evaluate the frequency and time evolving dynamics of the traffic flow based on the cumulative intrinsic mode functions extracted by the emd the frequency evolving rp regarding different oscillation of modes suggests that apparent dynamics of the data considered are mainly dominated by its components of medium and low frequencies while severely affected by fast oscillated noises contained in the signal noises are then eliminated to analyze the intrinsic dynamics and consequently the denoised time evolving rqa diversely characterizes the properties of the signal and marks crucial points more accurately where white bands in the rp occur whereas a strongly qualitative agreement exists between all the non denoised rqa measures generally the emd combining with the recurrence analysis sheds more reliable abundant and inherent lights into the traffic flow which is meaningful to the empirical analysis of complex systems

fault identification of rotor bearing system based on ensemble empirical mode decomposition and self zero space projection analysis abstract accurately identifying faults in rotor bearing systems by analyzing vibration signals which are nonlinear and nonstationary is challenging to address this issue a new approach based on ensemble empirical mode decomposition eemd and self zero space projection analysis is proposed in this paper this method seeks to identify faults appearing in a rotor bearing system using simple algebraic calculations and projection analyses first eemd is applied to decompose the collected vibration signals into a set of intrinsic mode functions imfs for features second these extracted features under various mechanical health conditions are used to design a self zero space matrix according to space projection analysis finally the so called projection indicators are calculated to identify the rotor bearing system s faults with simple decision logic experiments are implemented to test the reliability and effectiveness of the proposed approach the results show that this approach can accurately identify faults in rotor bearing systems

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

order determination theorems and system identification theorems for order determination without a priori knowledge of upper bounds on the order in mimo dynamic systems are developed also deterministic procedures are introduced to determine orders and estimate parameters simultaneously by recursively computing the order determining quantity s sub n a b k which plays a crucial role in order determination procedures and the least squares estimate theta sub n a b of theta p q with p and q denoting the true orders

cross correlation analysis of stock markets using emd and eemd abstract empirical mode decomposition emd is a data driven signal analysis method for nonlinear and nonstationary data since it is intuitive direct posterior and adaptive emd is widely applied to various fields of study in this paper emd and ensemble empirical mode decomposition eemd a modified method of emd are applied to financial time series through analyzing the intrinsic mode functions imfs of emd and eemd we find eemd method performs better on the orthogonality of imfs than emd with clustering the ordered frequencies of imfs the imfs obtained from eemd method are grouped into high medium and low frequency components representing the short medium and long term volatilities of the index sequences respectively with the cross correlation analysis of dcca cross correlation coefficient our findings allow us to gain further and detailed insight into the cross correlations of stock markets

a hybrid detrending method for fractional gaussian noise abstract determining trend and implementing detrending operations are important steps in data analysis yet there is neither precise definition of trend nor any logical algorithm for extracting it in this paper we propose a hybrid detrending method hdm which is based on the empirical mode decomposition emd and the detrended fluctuation analysis dfa our method can remove the polynomial and sinusoidal trends from the fractional gaussian noise fgn background we illustrate the method by selected examples from artificial time series and climate data in comparison with existing frequency domain based detrending methods our method is a posteriori the trend defined by our method is only derived from the data further our method also preserves the scaling behavior of the original signals

an improved empirical mode decomposition method based on the cubic trigonometric b spline interpolation algorithm abstract empirical mode decomposition emd is a new method presented recently for analyzing nonlinear and non stationary signals its basic idea is to decompose the signal into a series of complete orthogonal intrinsic mode functions imfs based on the local characteristics of the signal in time domain the key step of emd is to use the cubic spline interpolation to connect the maximum and minimum values of the signals into upper and lower envelopes respectively and then calculate the mean values of upper and lower envelopes based on the cubic trigonometric b spline interpolation algorithm a new improved method for emd is proposed named ctb emd in this paper in this method the interpolation curve is more flexible because of the adjustability of shape of the cubic trigonometric b splines curve thus the overshoot and undershoot problems in the cubic spline interpolation curve can be avoided and then the decomposition of the signal is more accurate and effect through numerical experiments we compare the effect of this method with other methods on decomposing simulation signals and real signals experimental results show that this method can decompose signals more effectively and accurately

a class of multivariate denoising algorithms based on synchrosqueezing univariate thresholding techniques based on high resolution time frequency algorithms such as the synchrosqueezing transform have emerged as important tools in removing noise from real world data low cost multichannel sensor technology has highlighted the need for direct multivariate denoising and to this end we introduce a class of multivariate denoising techniques based on the synchrosqueezing transform this is achieved by partitioning the time frequency domain so as to identify a set of modulated oscillations common to the constituent data channels within multivariate data and by employing a modified universal threshold in order to remove noise components while retaining signal components of interest this principle is used to introduce both the wavelet and fourier based multivariate synchrosqueezing denoising algorithms the performance of the proposed multivariate denoising algorithm is illustrated on both synthetic and real world data

2d prony huang transform a new tool for 2d spectral analysis this paper provides an extension of the 1d hilbert huang transform for the analysis of images using recent optimization techniques the proposed method consists of 1 adaptively decomposing an image into oscillating parts called intrinsic mode functions imfs using a mode decomposition procedure and 2 providing a local spectral analysis of the obtained imfs in order to get the local amplitudes frequencies and orientations for the decomposition step we propose two robust 2d mode decompositions based on nonsmooth convex optimization 1 a genuine 2d approach which constrains the local extrema of the imfs and 2 a pseudo 2d approach which separately constrains the extrema of lines columns and diagonals the spectral analysis step is an optimization strategy based on prony annihilation property and applied on small square patches of the imfs the resulting 2d prony huang transform is validated on simulated and real data

bifurcation and chaos in a rub impact jeffcott rotor system non linear vibration characteristics of a rub impact jeffcott rotor are investigated the system is two dimensional non linear and periodic fourier series analysis and the floquet theory are used to perform qualitative global analysis on bifurcation and stability the governing ordinary differential equations are also integrated using a numerical method to give the quantitative result this preliminary study reveals the chaotic feature of the system after the rub impact as the rotating speed is increased three kinds of routes to chaos are found that is from a stable periodic motion through period doubling bifurcation grazing bifurcation and a sudden transition from periodic motion to chaos quasi periodic motions are also found

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