characterization of low degree a stable symmetric rk collocation methods abstract a necessary and sufficient condition for symmetric collocation nodes to give an a stable rk collocation method is presented for low degree methods n 7 the results are obtained by means of the routh hurwitz algorithm some order stars movies are shown to illustrate their geometry in the space of parameters

analysis of moments based methods for fractional gaussian noise estimation fractional gaussian noise given as the increment of fractional brownian motion is a stationary gaussian process characterized by the hurst parameter in the paper moments based estimators of the hurst parameter are presented and analyzed with respect to asymptotic variance

on the a stability of runge kutta collocation methods based on orthogonal polynomials some results about the a stability of runge kutta collocation methods based on the zeros and extrema of the ultraspherical or gegenbauer polynomials are presented these methods are implicit and the a stability is an important property usually demanded the routh hurwitz algorithm is used to establish the main results about the a stability of the methods depending on the parameter lambda of the polynomials for low degree methods some analytical results are presented while for higher degree an open problem based on numerical tests is proposed

algorithms for the integration and derivation of chebyshev series general formulas for the mth integral and derivative of a chebyshev polynomial of the first or second kind are presented the result is expressed as a finite series of the same kind of chebyshev polynomials these formulas permit to accelerate the determination of such integrals or derivatives besides it is presented formulas for the mth integral and derivative of finite chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series

chebyshev collocation methods for fast orbit determination we apply some chebyshev collocation methods for fast and accurate determination of ephemeris of artificial satellites this formulation is based on the approximation of the differential system by means of chebyshev polynomials in order to assess the convergence of the iterative process and to reduce the number of operations the special formulation of dziobek brouwer is used from several numerical tests we conclude that the method presents very good behaviour for geostationary and low eccentric orbits the most frequent cases the number of operations with this method in some cases is 20 times less than that with a standard integrator besides a typical dense output in a polynomial form is obtained

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

estimation of fractional brownian motion embedded in a noisy environment using nonorthogonal wavelets we show that nonorthogonal wavelets can characterize the fractional brownian motion fbm that is in white noise we demonstrate the point that discriminating the parameter of fbm from that of noise is equivalent to discriminating the composite singularity formed by superimposing a peak singularity on a dirac singularity we characterize the composite singularity by formalizing this problem as a nonlinear optimization problem this yields our parameter estimation algorithm for fractal signal estimation wiener filtering is explicitly formulated as a function of the signal and noise parameters and the wavelets we show that the estimated signal is a 1 f process comparative studies through numerical simulations of our methods with those of wornell and oppenheim 1992 are presented

an improved method for 2 d self similar image synthesis we propose a new method called incremental fourier synthesis to generate 2 d self similar images based on a 2d fractional brownian motion fbm model with this method the stationary increments of fbm are created by a fourier synthesis method and the increments are added up to generate the nonstationary 2d fbm process since the new method takes advantage of the fft its computational complexity is only o n sup 2 log sub 2 n and its memory requirement is only o n sup 2 for a self similar image of size n spl times n

comparison of whole body vertical stiffness and leg stiffness during single leg hopping in place in children and adults in the hopping literature whole body vertical stiffness and leg stiffness are used interchangeably due to most of the movement occurring in the vertical direction however there is some anterior posterior movement of the center of mass and displacements of the foot during hopping in place in both children and adults further it is not understood if leg stiffness show a similar pattern as whole body vertical stiffness when increasing hopping frequency the purpose of this study was to test if whole body vertical stiffness and leg stiffness are different during single leg hopping in place in children and adults across a range of frequencies seventeen children aged 5 11years and 16 young adults participated in this study the subjects hopped at their preferred frequency as well as 20 below 20 above and 40 above preferred frequency our results demonstrate that both whole body vertical stiffness and leg stiffness increase when increasing hopping frequency for children and adults however whole body vertical stiffness consistently overestimates leg stiffness due to a similar peak force but a greater leg length change compared to vertical com displacement this suggests a considerable horizontal com movement from landing to mid stance during hopping children aged 5 11years old showed lower absolute values but higher normalized values of two stiffness measures than adults this suggests somewhat adult like stiffness control in children but a reduced ability to manipulate the horizontal movement during single leg hopping in place when compared to adults

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

on the wavelet transform of fractional brownian motion a theorem characterizing fractional brownian motion by the covariance structure of its wavelet transform is established the authors examine whether there are alternate gaussian processes whose wavelet transforms have a natural covariance structure in addition the authors examine if there are any gaussian processes whose wavelet transform is stationary with respect to the affine group i e the statistics of the wavelet transform do not depend on translations and dilations of the process

a family of implicit chebyshev methods for the numerical integration of second order differential equations abstract a family of implicit methods based on intra step chebyshev interpolation is developed for the solution of initial value problems whose differential equations are of the special second order form y f y x x the general procedure allows stepsizes which are considerably larger than commonly used in conventional methods computation overhead is comparable to that required by high order single or multistep procedures in addition the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

wavelet packets of fractional brownian motion asymptotic analysis and spectrum estimation this paper provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional brownian motion it also discusses the convergence speed to the limit autocorrelation function when the input random process is either a fractional brownian motion or a wide sense stationary second order random process the analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the shannon paraunitary filters from this analysis we derive wavelet packet based spectrum estimation for fractional brownian motions and wide sense stationary random processes experimental tests show good results for estimating the spectrum of 1 f processes

fractal estimation using models on multiscale trees we estimate the hurst parameter h of fractional brownian motion or by extension the fractal exponent spl phi of stochastic processes having 1 f sup spl phi like spectra by applying a multiresolution framework this framework admits an efficient likelihood function evaluation allowing us to compute the maximum likelihood estimate of this fractal parameter with relative ease in addition to yielding results that compare well with other proposed methods and in contrast with other approaches our method is directly applicable with at most very simple modification in a variety of other contexts including fractal estimation given irregularly sampled data or nonstationary measurement noise and the estimation of fractal parameters for 2 d random fields

a multifractal wavelet model with application to network traffic we develop a new multiscale modeling framework for characterizing positive valued data with long range dependent correlations 1 f noise using the haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results the model provides a rapid o n cascade algorithm for synthesizing n point data sets we study both the second order and multifractal properties of the model the latter after a tutorial overview of multifractal analysis we derive a scheme for matching the model to real data observations and to demonstrate its effectiveness apply the model to network traffic synthesis the flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics variance time plots and moment scaling and queuing behavior although for illustrative purposes we focus on applications in network traffic modeling the multifractal wavelet model could be useful in a number of other areas involving positive data including image processing finance and geophysics

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

modifications of the method of variation of parameters in spite of being a classical method for solving differential equations the method of variation of parameters continues having a great interest in theoretical and practical applications as in astrodynamics in this paper we analyse this method providing some modifications and generalised theoretical results finally we present an application to the determination of the ephemeris of an artificial satellite showing the benefits of the method of variation of parameters for this kind of problems

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

fractional brownian motion a maximum likelihood estimator and its application to image texture fractals have been shown to be useful in characterizing texture in a variety of contexts use of this methodology normally involves measurement of a parameter h which is directly related to fractal dimension in this work the basic theory of fractional brownian motion is extended to the discrete case it is shown that the power spectral density of such a discrete process is only approximately proportional to f a instead of in direct proportion as in the continuous case an asymptotic cramer rao bound is derived for the variance of an estimate of h subsequently a maximum likelihood estimator mle is developed to estimate h it is shown that the variance of this estimator nearly achieves the minimum bound a generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the mle when the discrete fractional brownian process is contaminated with additive gaussian noise the results show that even at signal to noise ratios of 30 db significant errors in estimation of h can result when noise is present the mle is then applied to x ray images of the human calcaneus to demonstrate how the line to line formulation can be applied to the two dimensional case these results indicate that it has strong potential for quantifying texture

estimation of 1 f signals on the basis of curve fitting an algorithm is proposed for the identification of fractal signals with wavelet based models from corrupted data this algorithm is based on curve fitting and wavelet transformation it is proved that whenever spl gamma is greater than 0 the algorithm provides an almost consistent estimation moreover the estimated parameters are asymptotically gaussian distributed a mean square asymptotic convergence rate of the estimated parameters has also been established simulation results verify the efficiency of the proposed algorithm

the modeling and estimation of statistically self similar processes in a multiresolution framework statistically self similar sss processes can be used to describe a variety of physical phenomena yet modeling these phenomena has proved challenging most of the proposed models for sss and approximately sss processes have power spectra that behave as 1 f sup spl gamma such as fractional brownian motion fbm fractionally differenced noise and wavelet based syntheses the most flexible framework is perhaps that based on wavelets which provides a powerful tool for the synthesis and estimation of 1 f processes but assumes a particular distribution of the measurements an alternative framework is the class of multiresolution processes proposed by chou et al 1994 which has already been shown to be useful for the identification of the parameters of fbm these multiresolution processes are defined by an autoregression in scale that makes them naturally suited to the representation of sss and approximately sss phenomena both stationary and nonstationary also this multiresolution framework is accompanied by an efficient estimator likelihood calculator and conditional simulator that make no assumptions about the distribution of the measurements we show how to use the multiscale framework to represent sss or approximately sss processes such as fbm and fractionally differenced gaussian noise the multiscale models are realized by using canonical correlations cc and by exploiting the self similarity and possible stationarity or stationary increments of the desired process a number of examples are provided to demonstrate the utility of the multiscale framework in simulating and estimating sss processes

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

extending self similarity for fractional brownian motion the fractional brownian motion fbm model has proven to be valuable in modeling many natural processes because of its persistence for large time lags however the model is characterized by one single parameter that cannot distinguish between short and long term correlation effects this article investigates the idea of extending self similarity to create a correlation model that generalizes discrete fbm referred to as asymptotic fbm afbm namely afbm is parameterized by variables controlling short and long term correlation effects it proposes a fast parameter estimation algorithm for afbm based on the haar transform and demonstrates the performance of this parameter estimation algorithm with numerical simulations

vsvo formulation of the taylor method for the numerical solution of odes this paper presents an analysis of the taylor method for the numerical solution of odes when a very high precision of the solution is required some theoretical properties of the taylor method are considered from the practical point of view a variable stepsize variable order vsvo scheme is presented and its utility is discussed with several examples to reach the goal of high precision the use of multiprecision libraries is considered finally some numerical tests based on the test problems given in 1 and on a set of important problems in dynamical systems and astrodynamics are presented showing the benefits of the vsvo formulation especially for high precision demands compared with a well established runge kutta code

fast exact synthesis of gaussian and nongaussian long range dependent processes 1 f noise and statistically self similar random processes such as fractional brownian motion fbm and fractional gaussian noise fgn are fundamental models for a host of real world phenomena from network traffic to dna to the stock market synthesis algorithms play a key role by providing the feedstock of data necessary for running complex simulations and accurately evaluating analysis techniques unfortunately current algorithms to correctly synthesize these long range dependent lrd processes are either abstruse or prohibitively costly which has spurred the wide use of inexact approximations to fill the gap we develop a simple fast o n log n operations for a length n signal framework for exactly synthesizing a range of gaussian and nongaussian lrd processes as a bonus we introduce and study a new bi scaling fbm process featuring a kinked correlation function that exhibits distinct scaling laws at coarse and fine scales

wavelet estimation of fractional brownian motion embedded in a noisy environment this correspondence proposes a wavelet based fractional brownian motion fbm signal estimation scheme despite the fact that wavelet transform approximately whitens the fbm processes it is observed that statistical dependencies still exist across adjacent wavelet scales and between neighboring wavelet coefficients these dependencies can be exploited to improve the estimation of fbm signals embedded into noise the idea is to reorganize the wavelet coefficients into a scale time mixture model and then carry out the minimum mean square error estimation mmse using the model experiments show that the proposed scheme obtains better estimates than wornell and oppenheim s algorithm in which the wavelet dependencies are not utilized

statistical analysis of dwt coefficients of fgn processes using arfima p d q models abstract fractional gaussian noise fgn provides an important parametric representation for the data recorded from long memory processes also it has been well established in literature that the orthogonal wavelet transforms prove to be the optimal bases to represent the data as fgn or fbm fractional brownian motion models this paper highlights the statistical properties of discrete wavelet transform dwt coefficients in the wavelet expansion of fgn statistical analysis was carried out by analyzing the inter scale and intra scale correlations of the dwt coefficients for wavelets with varying vanishing moments two types of auto regressive moving average arma models were fit to the wavelet coefficients of fgn namely i arma p q and ii arfima p d q models the latter represents the arma models with fractional differencing using the akaike information criteria aic and the bayesian information criteria bic it has been shown that arfima models best represent the wavelet coefficients of fgn the above observation holds good when wavelets with increasing number of vanishing moments are used for obtaining dwt coefficients after estimating the optimal model and its parameters different properties pertaining to the inter scale and intra scale correlations were verified using these models

wavelet based estimation of 1 f type signal parameters confidence intervals using the bootstrap we propose to construct confidence intervals of parameters of 1 f type signals using a nonparametric wavelet based bootstrap method bootstrap based confidence intervals of maximum likelihood parameter estimates are compared to the confidence intervals derived from the cramer rao lower bound crlb for moderately large data sample sizes the bootstrap approach achieves the nominal coverage and may perform better than the crlb based parametric approach

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

wavelet packet computation of the hurst exponent wavelet packet analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis a new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power law to estimate the hurst exponent profile irregularity and long range correlations in self affine systems can be identified and indexed with the hurst exponent and synthetic one dimensional fractional brownian motion fbm type profiles are used to illustrate and test the proposed wavelet packet expansion we also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self affine trace or an enzyme walk the robustness of the wavelet approach applied to this stochastic system is presented and comparison is made between the wavelet packet method and the root mean square roughness and second moment approaches for both examples the wavelet packet method to estimate the global hurst exponent appears to have similar accuracy compared with other methods but its main advantage is the extensive choice of available analysing wavelet filter functions for characterizing periodic and oscillatory signals

a new method for fractional brownian motion interpolation this article presents a new method for interpolating long fractional brownian motion fbm sequences called incremental fourier interpolation ifi instead of computing the interpolated samples directly as is the case with existing algorithms ifi computes the first order increments between the original and interpolated samples for long sequences these increments can be computed using the computationally efficient fast fourier transform estimators for the fbm parameters are also incorporated into the algorithm simulations are presented for both known and unknown parameter cases that demonstrate the accuracy of ifi even for relatively short length sequences

estimation of 2 d noisy fractional brownian motion and its applications using wavelets the two dimensional 2 d fractional brownian motion fbm model is useful in describing natural scenes and textures most fractal estimation algorithms for 2 d isotropic fbm images are simple extensions of the one dimensional 1 d fbm estimation method this method does not perform well when the image size is small say 32 spl times 32 we propose a new algorithm that estimates the fractal parameter from the decay of the variance of the wavelet coefficients across scales our method places no restriction on the wavelets also it provides a robust parameter estimation for small noisy fractal images for image denoising a wiener filter is constructed by our algorithm using the estimated parameters and is then applied to the noisy wavelet coefficients at each scale we show that the averaged power spectrum of the denoised image is isotropic and is a nearly 1 f process the performance of our algorithm is shown by numerical simulation for both the fractal parameter and the image estimation applications to coastline detection and texture segmentation in a noisy environment are also demonstrated

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed