approach to an irregular time series on the basis of the fractal theory we present a technique to measure the fractal dimension of the set of points t f t forming the graph of a function f defined on the unit interval first we apply it to a fractional brownian function 1 which has a property of self similarity for all scales and we can get the stable and precise fractal dimension this technique is also applied to the observational data of natural phenomena it does not show self similarity all over the scale but has a different self similarity across the characteristic time scale the present method gives us a stable characteristic time scale as well as the fractal dimension

decorrelation property of discrete wavelet transform under fixed domain asymptotics theoretical aspects of the decorrelation property of the discrete wavelet transform when applied to stochastic processes have been studied exclusively from the increasing domain perspective in which the distance between neighboring observations stays roughly constant as the number of observations increases to understand the underlying data generating process and to obtain good interpolations fixed domain asymptotics in which the number of observations increases in a fixed region is often more appropriate than increasing domain asymptotics in the fixed domain setting we prove that for a general class of inhomogeneous covariance functions with suitable choice of wavelet filters the wavelet transform of a nonstationary process has mostly asymptotically uncorrelated components

optimized wavelet denoising for self similar alpha stable processes we investigate the performance of wavelet shrinkage methods for the denoising of symmetric alpha stable text s alpha text s self similar stochastic processes corrupted by additive white gaussian noise awgn where alpha is tied to the sparsity of the process the wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean square approximation error mmse estimator we derive the corresponding formula for the expected value of the averaged estimation error we show that the predicted mmse is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process using the calculus of variations we then optimize this criterion to find the best performing wavelet within the extended family of meyer wavelets which are bandlimited these are compared with the daubechies wavelets which are compactly supported in time we find that the wavelets that are shorter in time in particular the haar basis are better suited to denoise the sparser processes say alpha while the bandlimited ones including the held and shannon wavelets offer the best performance for alpha 1 6 the limit corresponding to the gaussian case fbm with alpha 2

computer evaluation of kauffman polynomials by using gauss codes with a skein template algorithm homfly polynomials of knots and links are evaluated on computers by using an algebraic version with gauss codes of the skein template algorithm the computer codes are written with the language c whose structure is particularly suitable to such evaluations jones conway and alexander polynomials may then be readily obtained as special cases

diverging moments and parameter estimation heavy tailed distributions are enjoying increased popularity and are becoming more readily applicable as the arsenal of analytical and numerical tools grows they play key roles in modeling approaches in networking finance and hydrology to name but a few areas the tail parameter is of central importance because it governs both the existence of moments of positive order and the thickness of the tails of the distribution some of the best known tail estimators such as those of koutrouvelis and hill are either parametric or show a lack of robustness or accuracy this article develops a shift and scale invariant nonparametric estimator for both upper and lower bounds for orders with finite moments the estimator builds on the equivalence between tail behavior and the regularity of the characteristic function at the origin and achieves its goal by deriving a simplified wavelet analysis that is particularly suited to characteristic functions

computational methods for hidden markov tree models an application to wavelet trees the hidden markov tree models were introduced by crouse et al in 1998 for modeling nonindependent non gaussian wavelet transform coefficients in their paper they developed the equivalent of the forward backward algorithm for hidden markov tree models and called it the upward downward algorithm this algorithm is subject to the same numerical limitations as the forward backward algorithm for hidden markov chains hmcs in this paper adapting the ideas of devijver from 1985 we propose a new upward downward algorithm which is a true smoothing algorithm and is immune to numerical underflow furthermore we propose a viterbi like algorithm for global restoration of the hidden state tree the contribution of those algorithms as diagnosis tools is illustrated through the modeling of statistical dependencies between wavelet coefficients with a special emphasis on local regularity changes

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

rapid construction of equivalent sources using wavelets we have developed a fast algorithm for generating an equivalent source by using fast wavelet transforms based on orthonormal compactly supported wavelets we apply a 2d wavelet transform to each row and column of the coefficient matrix and subsequently threshold the transformed matrix to generate a sparse representation in the wavelet domain the algorithm then uses this sparse matrix to construct the the equivalent source directly in the wavelet domain performing an inverse wavelet transform then yields the equivalent source in the space domain using upward continuation of total field magnetic data between uneven surfaces as examples we have compared this approach with the direct solution using the dense matrix in the space domain we have shown that the wavelet approach can reduce the cpu time by as many as two orders of magnitude

optimized compact support interpolation kernels in this paper we investigate the problem of designing compact support interpolation kernels for a given class of signals by using calculus of variations we simplify the optimization problem from an nonlinear infinite dimensional problem to a linear finite dimensional case and then find the optimum compact support function that best approximates a given filter in the least square sense l2 norm the benefit of compact support interpolants is the low computational complexity in the interpolation process while the optimum compact support interpolant guarantees the highest achievable signal to noise ratio snr our simulation results confirm the superior performance of the proposed kernel compared to other conventional compact support interpolants such as cubic spline

analysis of trapped entanglements in polymer networks this work reviews different standard methods to analyze trapped entanglements of polymer networks and discusses their advantages and drawbacks with respect to simplicity computing time and accuracy of the results since this standard analysis is based on the pairwise test of closed cycles two algorithms for the determination of these cycles are introduced and compared the decomposition into meshes creates unsolved problems regarding the non ambiguity and the completeness of the received results examples are given in order to show their effects and to start further discussion to receive more convenient methods as one possible solution vassiliev s invariants are discussed and an approximation method is introduced to correct the possible multiple counting of entanglements

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

fractal connectivity of long memory networks using the multivariate long memory lm model and taylor expansions we find the conditions for convergence of the wavelet correlations between two lm processes on an asymptotic value at low frequencies these mathematical results and a least squares estimator of lm parameters are validated in simulations and applied to neurophysiological human brain and financial market time series both brain and market systems had multivariate lm properties including a fractal connectivity regime of scales over which wavelet correlations were invariantly close to their asymptotic value this analysis provides efficient and unbiased estimation of long term correlations in diverse dynamic networks

low rank variance approximation in gmrf models single and multiscale approaches we present a versatile framework for tractable computation of approximate variances in large scale gaussian markov random field estimation problems in addition to its efficiency and simplicity it also provides accuracy guarantees our approach relies on the construction of a certain low rank aliasing matrix with respect to the markov graph of the model we first construct this matrix for single scale models with short range correlations and then introduce spliced wavelets and propose a construction for the long range correlation case and also for multiscale models we describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse irregular and noisy measurements and to a gravity inversion problem

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

wavelet based estimations of fractional brownian sheet least squares versus maximum likelihood abstract the scaling behavior or self similarity and long range dependence has been studied in numerous fields such as computer telecommunication hydrology and economics and can be characterized by the hurst parameter h fractional brownian sheet plays an important role in modeling the scaling behavior in multi dimensional signals a wavelet based maximum likelihood estimator for the hurst parameter h h 1 h d of fractional brownian sheet is proposed it is asymptotically normal and compared with the wavelet based least squares estimators introduced in the previous article from theoretical analysis and numerical simulation the theoretical results show that the proposed estimator has the same asymptotic covariance as the least squares estimator of which the weight is equal to the number of available wavelet coefficients and the convergence rates of two estimators are both o 1 i 1 d t i t i is sample length in the i th dimension the simulation results show that the proposed estimator performs better than the least squares estimators in bias and costs less runtime than the generalized least squares estimator realized by the two step method

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

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

analysis and synthesis of 1 f processes via shannon wavelets 1 f processes can be very useful in modeling processes with long term correlation we propose analysis and synthesis procedures to express these processes in terms of the shannon wavelet unlike previous techniques our analysis procedure generates uncorrelated decomposition coefficients for the 1 f process this is done by taking onto account and then removing the residual correlation between the wavelet components the analysis procedure is the major contribution of this work the proposed synthesis algorithm which is a byproduct of the proposed analysis algorithm is competitive with other techniques

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 transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

self similarity part i splines and operators the central theme of this pair of papers parts i and ii in this issue is self similarity which is used as a bridge for connecting splines and fractals the first part of the investigation is deterministic and the context is that of l splines these are defined in the following terms s t is a cardinal l spline iff l s t sigma kisinza k delta t k where l is a suitable pseudodifferential operator our starting point for the construction of self similar splines is the identification of the class of differential operators l that are both translation and scale invariant this results into a two parameter family of generalized fractional derivatives parttau gamma where gamma is the order of the derivative and tau is an additional phase factor we specify the corresponding l splines which yield an extended class of fractional splines the operator parttau gamma is used to define a scale invariant energy measure the squared l2 norm of the gammath derivative of the signal which provides a regularization functional for interpolating or fitting the noisy samples of a signal we prove that the corresponding variational or smoothing spline estimator is a cardinal fractional spline of order 2gamma which admits a stable representation in a b spline basis we characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical butterworth filter of order 2gamma we also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter omega0aplambda 2gamma finally we present an efficient computational solution to the fractional smoothing spline problem it uses the fast fourier transform and takes advantage of the multiresolution properties of the underlying basis functions

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

fast parametric elastic image registration we present an algorithm for fast elastic multidimensional intensity based image registration with a parametric model of the deformation it is fully automatic in its default mode of operation in the case of hard real world problems it is capable of accepting expert hints in the form of soft landmark constraints much fewer landmarks are needed and the results are far superior compared to pure landmark registration particular attention has been paid to the factors influencing the speed of this algorithm the b spline deformation model is shown to be computationally more efficient than other alternatives the algorithm has been successfully used for several two dimensional 2 d and three dimensional 3 d registration tasks in the medical domain involving mri spect ct and ultrasound image modalities we also present experiments in a controlled environment permitting an exact evaluation of the registration accuracy test deformations are generated automatically using a random hierarchical fractional wavelet based generator

wavelet entropy and fractional brownian motion time series abstract we study the functional link between the hurst parameter and the normalized total wavelet entropy when analyzing fractional brownian motion fbm time series these series are synthetically generated both quantifiers are mainly used to identify fractional brownian motion processes l zunino d g perez m garavaglia o a rosso characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform fractals 12 2 2004 223 233 the aim of this work is to understand the differences in the information obtained from them if any

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

local fractal and multifractal features for volumic texture characterization for texture analysis several features such as co occurrence matrices gabor filters and the wavelet transform are used recently fractal geometry appeared to be an effective feature to analyze texture but it is often restricted to 2d images while 3d information can be very important especially in medical image processing moreover applications are limited to the use of fractal dimension this study focuses on the benefits of fractal geometry in a classification method based on volumic texture analysis the proposed methods make use of fractal and multifractal features for a 3d texture analysis of a voxel neighborhood they are validated with synthetic data before being applied on real images their efficiencies are proved by comparison to some other texture features in supervised classification processes adaboost and support vector machine classifiers the results showed that features based on fractal geometry by combining fractal and multifractal features contributed to new texture characterization information on new features was useful and complementary for a classification method this study suggests that fractal geometry can provide a new useful information in 3d texture analysis especially in medical imaging

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

new fast smoothers for multiscale systems we consider the smoothing problem for multiscale stochastic systems based on the wavelet transform these models involve processes indexed by the nodes of a dyadic tree each level of the dyadic tree represents one scale or resolution of the process therefore moving upward on the tree divides the resolution by 2 whereas moving downward multiplies it by 2 the processes are built according to a recursion in scale from coarse to fine to which random details are added to operate the change in scale one must perform an interpolation this is achieved using the qmf pair of operators attached to a wavelet transform these models have proved to be of great value to capture textures or fractal like processes as well as to perform multiresolution sensor fusion an example of which is given here up to now however only subclasses of multiscale systems were amenable to fast algorithms and through different formalisms those relying on haar s wavelet and those involving only one of the two wavelet interpolators we provide here a unifying framework that handles any system based on orthogonal wavelets a smoothing theory is presented to define the field of fast algorithms for markov random fields and give intuition on how to design them this theory reveals the difficulties arising with general multiscale systems we then prove that orthogonality properties of wavelets are the gate to fastness

a note on robust variance estimation for cluster correlated data there is a simple robust variance estimator for cluster correlated data while this estimator is well known it is poorly documented and its wide range of applicability is often not understood the estimator is widely used in sample survey research but the results in the sample survey literature are not easily applied because of complications due to unequal probability sampling this brief note presents a general proof that the estimator is unbiased for cluster correlated data regardless of the setting the result is not new but a simple and general reference is not readily available the use of the method will benefit from a general explanation of its wide applicability

wavelet transform inversion with prior scale information the concept of scale is of central importance in all problems related to the interpretation of earth data inversion of geophysical data does not at present however allow the incorporation of scale dependent information directly this article presents an application of the wavelet transform to traveltime inversion where we are specifically concerned with the selection and application of a priori scale information in the wavelet domain to reconstruct tomographic models because of the nature of the problem the wavelet basis chosen for this example is the haar basis the results are compared with the classic damped least squares dls method which is a special case of the wavelet transform approach when all the scales are treated equally we show that with scale dependent prior information most of the artifacts created by the dls technique are diminished considerably

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

a statistical test for the time constancy of scaling exponents a statistical test is described for determining if scaling exponents vary over time it is applicable to diverse scaling phenomena including long range dependence and exactly self similar processes in a uniform framework without the need for prior knowledge of the type in question it is based on the special properties of wavelet based estimates of the scaling exponent strongly motivating an idealized inference problem the equality or otherwise of means of independent gaussian variables with known variances a uniformly most powerful invariant test exists for this problem and is described a separate uniformly most powerful invariant test is also given for when the scaling exponent undergoes a level change the power functions of both tests are given explicitly and compared using simulation the effect in practice of deviations from the idealizations made of the statistical properties of the wavelet detail coefficients are analyzed and found to be small the tests inherit the significant robustness and computational advantages of the underlying wavelet based estimator a detailed methodology is given describing its use in practical situations the use and benefits of the test are illustrated on the bellcore ethernet data sets

a multiscale guide to brownian motion we revise the levy s construction of brownian motion as a simple though rigorous approach to operate with various gaussian processes a brownian path is explicitly constructed as a linear combination of wavelet based geometrical features at multiple length scales with random weights such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of brownian paths this formula elucidates many classical results about brownian motion e g non differentiability of its path providing intuitive feeling for non mathematicians the illustrative character of the wavelet representation along with the simple structure of the underlying probability space is different from the usual presentation of most classical textbooks similar concepts are discussed for brownian bridge fractional brownian motion ornstein uhlenbeck process gaussian free field and fractional gaussian fields wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate gaussian processes and fields including brownian motion and other diffusive processes in confining domains

evaluation of fractal dimension effectiveness for damage detection in retinal background abstract this work investigates the characterization of bright lesions in retinal fundus images using texture analysis techniques exudates and drusen are evidences of retinal damage in diabetic retinopathy dr and age related macular degeneration amd respectively an automatic detection of pathological tissues could make possible an early detection of these diseases in this work fractal analysis is explored in order to discriminate between pathological and healthy retinal texture after a deep preprocessing step in which spatial and colour normalization are performed the fractal dimension is extracted locally by computing the hurst exponent h along different directions the greyscale image is described by the increments of the fractional brownian motion model and the h parameter is computed by linear regression in the frequency domain the ability of fractal dimension to detect pathological tissues is demonstrated using a home made system based on fractal analysis and support vector machine able to achieve around a 70 and 83 of accuracy in e ophtha and diaretdb1 public databases respectively in a second experiment the fractal descriptor is combined with texture information extracted by the local binary patterns improving the bright lesion detection accuracy sensitivity and specificity values higher than 89 80 and 90 respectively suggest that the method presented in this paper is a robust algorithm for describing retina texture and can be useful in the automatic detection of dr and amd

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

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

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

testing for the source of multifractality in water level records abstract the multifractality in hydrologic data has been studied in many literatures but few literatures focus on its source which is helpful for understanding the underlying mechanisms that generate hydrologic data we propose a hypothesis testing procedure for the source of multifractality in water level records based on the empirical distributions of generalized hurst exponent estimated from a set of shuffled or surrogate series the proposed hypothesis testing procedure can show more details about the source of multifractality than previous methods with some statistics especially about the effects of large and small fluctuations on multifractality the generalized hurst exponents are estimated via multifractal detrended fluctuation analysis the data set contains about two million high frequency water level records of a northern china river at its 10 observation stations the testing results show that the multifractality in water levels is mainly caused by nonlinear correlations in small fluctuations and linear correlations in small and large fluctuations and is also related to the probability distribution of small fluctuations this conclusion is validated via some constructed semi surrogate series and values of statistics for source testing

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

statistical evidence for glm regression parameters a robust likelihood approach when a likelihood ratio is used to measure the strength of evidence for one hypothesis over another its reliability i e how often it produces misleading evidence depends on the specification of the working model when the working model happens to be the true or correct model the probability of observing strong misleading evidence is low and controllable but this is not necessarily the case when the working model is misspecified royall and tsou j r stat soc ser b 2003 65 391 404 show how to adjust working models to make them robust to misspecification likelihood ratios derived from their robust adjusted likelihood are just as reliable asymptotically as if the working model were correctly specified in the first place in this paper we apply and extend these ideas to the generalized linear model glm regression setting we provide several illustrations both from simulated data and real data concerning rates of parasitic infection in philippine adolescents show how the required adjustment factor can be obtained from standard statistical software and draw some connections between this approach and the sandwich estimator for robust standard errors of regression parameters this substantially broadens the availability and the viability of likelihood methods for measuring statistical evidence in regression settings copyright 2007 john wiley sons ltd

on the correlation structure of the wavelet coefficients of fractional brownian motion shows that the interdependence of the discrete wavelet coefficients of fractional brownian motion defined by normalized correlation decays exponentially fast across scales and hyperbolically fast along time

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

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

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

cluster sampling techniques in quantal response teratology and developmental toxicity studies summary this paper presents a model free approach for evaluating teratology and developmental toxicity data involving clustered binary responses in teratology studies a major statistical problem arises from the effect of intralitter correlation or the potential for littermates to respond similarly some statistical methods impose strict distributional assumptions to account for extra binomial variation while others rely on nonparametric resampling and empirical variance estimation techniques quasi likelihood methods and generalized estimating equations gee which model the marginal mean variance relationship also avoid strict distributional assumptions the proposed approach often used to analyze complex sample survey data is based on a first order taylor series approximation and a between cluster variance estimation procedure yielding consistent variance estimates for binomial based proportions and regression coefficients from dose response models the cluster sample technique presented here in the context of a logistic dose response model incorporates many of the advantages of quasi likelihood methods are valid for any underlying nested correlation structure and are adaptable to a variety of analytical settings the results of a simulation study show the cluster sample technique to be a viable competitor to gee methods currently receiving attention

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

statistical study of the wavelet analysis of fractional brownian motion we present a statistical study of wavelet coefficients of a fractional brownian motion a central limit theorem for empirical variances of exact wavelet coefficients is given under conditions on the mother wavelet and the choice of scales a limit theorem is given for fitted wavelet coefficients computed from a time series it provides an estimator for the self similarity parameter of gaussian time series

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

wavelet based estimation of a semiparametric generalized linear model of fmri time series addresses the problem of detecting significant changes in fmri time series that are correlated to a stimulus time course this paper provides a new approach to estimate the parameters of a semiparametric generalized linear model of the fmri time series the fmri signal is described as the sum of two effects a smooth trend and the response to the stimulus the trend belongs to a subspace spanned by large scale wavelets the wavelet transform provides an approximation to the karhunen loeve transform for the long memory noise and we have developed a scale space regression that permits one to carry out the regression in the wavelet domain while omitting the scales that are contaminated by the trend in order to demonstrate that our approach outperforms the state of the art detrending technique we evaluated our method against a smoothing spline approach experiments with simulated data and experimental fmri data demonstrate that our approach can infer and remove drifts that cannot be adequately represented with splines

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated

connecting complexity with spectral entropy using the laplace transformed solution to the fractional diffusion equation analytical solutions to the fractional diffusion equation are often obtained by using laplace and fourier transforms which conveniently encode the order of the time and the space derivatives and as non integer powers of the conjugate transform variables s and k for the spectral and the spatial frequencies respectively this study presents a new solution to the fractional diffusion equation obtained using the laplace transform and expressed as a fox s h function this result clearly illustrates the kinetics of the underlying stochastic process in terms of the laplace spectral frequency and entropy the spectral entropy is numerically calculated by using the direct integration method and the adaptive gauss kronrod quadrature algorithm here the properties of spectral entropy are investigated for the cases of sub diffusion and super diffusion we find that the overall spectral entropy decreases with the increasing and and that the normal or gaussian case with 1 and 2 has the lowest spectral entropy i e less information is needed to describe the state of a gaussian process in addition as the neighborhood over which the entropy is calculated increases the spectral entropy decreases which implies a spatial averaging or coarse graining of the material properties consequently the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion future studies should be designed to examine changes of spectral entropy in physical chemical and biological systems undergoing phase changes chemical reactions and tissue regeneration