exact confidence intervals for a variance ratio or heritability in a mixed linear model a family of procedures is given to construct confidence intervals for the heritability coefficient in a mixed linear model the procedures are applicable for constructing confidence intervals for a ratio of variance components in a mixed linear model having two sources of variation if the random effects are correlated the procedure is valid even when there are zero degrees of freedom for error the resulting intervals are evaluated in terms of bias and expected length a sufficient condition for local unbiasedness is given and a numerical procedure is discussed for computing expected lengths the investigator may select the best confidence interval procedure from the family of procedures based on these criteria computer software for obtaining the best interval is available from the authors

closed form approximations to the reml estimator of a variance ratio or heritability in a mixed linear model in this article we estimate heritability or intraclass correlation in a mixed linear model having two sources of variation in most applications the commonly used restricted maximum likelihood reml estimator can only be obtained via an iterative approach in some cases the algorithm used to compute reml estimates may be slow or may even fail to converge we develop a set of closed form approximations to the reml estimator and the performance of these estimators is compared with that of the reml estimator we provide guidelines regarding how to choose the estimator that best approximates the reml estimator examples presented in the article suggest that the closed form estimators compete with and in some cases outperform the reml estimator

consistent estimation of linear panel data models with measurement error abstract measurement error causes a bias towards zero when estimating a panel data linear regression model the panel data context offers various opportunities to derive instrumental variables allowing for consistent estimation we consider three sources of moment conditions i restrictions on the covariance matrix of the errors in the equations ii nonzero third moments of the regressors and iii heteroskedasticity and nonlinearity in the relation between the error ridden regressor and another error free regressor in simulations these approaches appear to work well

confidence intervals for a variance ratio or for heritability in an unbalanced mixed linear model a procedure is presented for constructing an exact confidence interval for the ratio of the two variance components in a possibly unbalanced mixed linear model that contains a single set of m random effects this procedure can be used in animal and plant breeding problems to obtain an exact confidence interval for a heritability the confidence interval can be defined in terms of the output of a least squares analysis it can be computed by a graphical or iterative technique requiring the diagonalization of an m x m matrix or alternatively the inversion of a number of m x m matrices confidence intervals that are approximate can be obtained with much less computational burden using either of two approaches the various confidence interval procedures can be extended to some problems in which the mixed linear model contains more than one set of random effects corresponding to each interval procedure is a significance test and one or more estimators

a comparison of confidence interval methods for the intraclass correlation coefficient summary different methods of obtaining confidence intervals for the intraclass correlation coefficient p in the unbalanced one way random effects model are investigated focusing on applications to family studies methods based on simple modifications of formulas for the case of equal group sizes are found to provide adequate coverage at small to moderate values of p a method based on the large sample standard error of the sample intraclass correlation as derived by smith 1956 annals of human genetics 21 363 373 is shown to provide consistently good coverage at all values of p a method proposed by thomas and hultquist 1978 annals of statistics 6 582 587 also provides consistently good coverage but generates mean interval widths substantially greater than those generated by smith s method at values of p likely to arise in practice

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

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

parameter estimation in regression models with errors in the variables and autocorrelated disturbances abstract for regression models with first order autocorrelated disturbances the traditional prescription of econometricians is to correct for serial correlation by using appropriate estimation techniques such as the cochrane orcutt hildreth lu or prais winsten procedures this suggestion however does not take into account the fact that in most economic applications the independent variables contain measurement errors when there are errors in the variables ordinary least squares without any attempt to correct for serial correlation may in several cases yield less unreliable and misleading results in terms of biases and mean squared errors of the estimators this paradoxical finding is demonstrated analytically for the asymptotic case and illustrated for finite samples by monte carlo experiments for the regression model with a single stationary explanatory variable the monte carlo experiments also suggest that the same conclusion holds if the independent variable follows a random walk at least for medium size samples

regularized liml for many instruments the use of many moment conditions improves the asymptotic efficiency of the instrumental variables estimators however in finite samples the inclusion of an excessive number of moments increases the bias to solve this problem we propose regularized versions of the limited information maximum likelihood liml based on three different regularizations tikhonov landweber fridman and principal components our estimators are consistent and asymptotically normal under heteroskedastic error moreover they reach the semiparametric efficiency bound assuming homoskedastic error we show that the regularized liml estimators possess finite moments when the sample size is large enough the higher order expansion of the mean square error mse shows the dominance of regularized liml over regularized two staged least squares estimators we devise a data driven selection of the regularization parameter based on the approximate mse a monte carlo study and two empirical applications illustrate the relevance of our estimators

matrix algebra for higher order moments abstract a large part of statistics is devoted to the estimation of models from the sample covariance matrix the development of the statistical theory and estimators has been greatly facilitated by the introduction of special matrices such as the commutation matrix and the duplication matrix and the corresponding matrix algebra some more extensive models require however estimation based on higher order moments typically third and fourth order moments an example is the popular kenny judd model that includes interactions between latent variables this paper introduces some special matrices that can be used to simplify the model expressions for third fourth and higher order moments gives some relationships between these matrices and related matrices and gives some formulas for kronecker products of three and four matrices the theory is applied to derive convenient expressions for third and fourth order moments of some structural equation models

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 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

nonparametric identification and semiparametric estimation of classical measurement error models without side information virtually all methods aimed at correcting for covariate measurement error in regressions rely on some form of additional information e g validation data known error distributions repeated measurements or instruments in contrast we establish that the fully nonparametric classical errors in variables model is identifiable from data on the regressor and the dependent variable alone unless the model takes a very specific parametric form this parametric family includes but is not limited to the linear specification with normally distributed variables as a well known special case this result relies on standard primitive regularity conditions taking the form of smoothness constraints and nonvanishing characteristic functions assumptions our approach can handle both monotone and nonmonotone specifications provided the latter oscillate a finite number of times given that the very specific unidentified parametric functional form is arguably the exception rather than the rule this identification r

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

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

simulated minimum distance estimation of dynamic models with errors in variables abstract empirical analysis often involves using inexact measures of the predictors suggested by economic theory the bias created by the correlation between the mismeasured regressors and the error term motivates the need for instrumental variable estimation this paper considers a class of estimators that can be used in dynamic models with measurement errors when external instruments may not be available or are weak the idea is to exploit the relation between the parameters of the model and the least squares biases in cases when the latter are not analytically tractable a special algorithm is designed to simulate the model without completely specifying the processes that generate the latent predictors the proposed estimators perform well in simulations of the autoregressive distributed lag model the methodology is used to estimate the long run risks model

higher moment estimators for linear regression models with errors in the variables abstract this paper proposes consistent instrumental variable estimators for linear regression models with errors in the variables that require no extraneous information these estimators are based on sample moments of order higher than two while similar estimators proposed previously in the literature seem to be quite erratic our experimental f findings suggest that our estimators perform better than ordinary least squares estimators in terms of root mean squared errors and also in terms of size of type i errors of standard tests in many typical situations of economic analyses tests for the presence of errors in the variables are also described

mra of processes synthesized by fractional integration in this correspondence we show that the multiresolution analysis mra of the reproducing kernel hilbert space associated with processes synthesized by fractional integration of white processes leads to an orthogonal multiresolution decomposition of the process we show that the decomposition converges almost surely and uniformly on finite intervals we also show that the random coefficients in the decomposition can be evaluated almost surely

fiducial intervals for variance components in an unbalanced two component normal mixed linear model in this article we propose a new method for constructing confidence intervals for 2 2 and the intraclass correlation 2 2 e2 in a two component mixed effects linear model this method is based on an extension of r a fisher s fiducial argument we conducted a simulation study to compare the resulting interval estimates with other competing confidence interval procedures from the literature our results demonstrate that the proposed fiducial intervals have satisfactory performance in terms of coverage probability as well as shorter average confidence interval lengths overall we also prove that these fiducial intervals have asymptotically exact frequentist coverage probability the computations for the proposed procedures are illustrated using real data examples

multiresolution sequence detection in rapidly fading channels based on focused wavelet decompositions the problem of detecting a sequence using the maximum likelihood strategy when the time varying multipath channel is unknown can be in principle solved using the generalized likelihood detector gld the gld metric involves finding the orthogonal projection of the received signal onto the subspace of the transmitted signal distorted by the multipath channel the faded signal corresponding to one of the transmitted sequences is in fact known to lie in a subspace but its exact location is not known because the channel parameters are unknown this detector which in the case of a static channel collapses to the per survivor processing method is also called a matched subspace detector because its statistic is matched to the a priori known signal subspace unfortunately the computation of the perfectly matched orthogonal projection of the received signal onto the multipath faded signal subspace is in the general time varying case impossible we introduce in this work the idea of using new wavelet based subspaces that approximate the original signal subspace the nested sequence of linear vector spaces defined by a wavelet based multiresolution decomposition of the fading channel time variations provides a set of subspaces that at an increasingly high level of detail are efficient representations of the original signal subspace for each of these representations sequence detection at different levels of resolution can be performed

errors in survey reports of earnings hours worked and hourly wages abstract using administrative records on earnings hours worked and hourly wages for hourly employees of a single manufacturing firm we found standard assumptions about errors in survey measures to be violated to varying extents errors are correlated with true scores and errors in reports of earnings and hours in different periods are generally positively correlated with one another particularly for reports given in a single interview although a high proportion of reporting errors are from an approximately normal distribution a small proportion comes from a distribution with a much greater variance and these cases often have considerable influence on estimates of relationships between variables

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

a regularization approach to the many instruments problem abstract this paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite in order to improve the small sample properties of standard instrumental variable iv estimators we propose three modified iv estimators based on three different ways of inverting the covariance matrix of the instruments these inverses involve a regularization or smoothing parameter it should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation we show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound higher order analysis of the mse reveals that the bias of the modified estimators does not depend on the number of instruments finally we suggest a data driven method for selecting the regularization parameter interestingly our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

inference for a random wavelet packet model of single channel event related potentials event related potentials erps are brain electrical potentials associated with sensory and cognitive processing erp researchers typically wish to separate a recorded time series into functionally distinct component waveforms and to estimate the effects of experimental conditions on each component we present an integrated statistical approach to the decomposition of single channel erps and to inference concerning the component waveforms and the effects of experimental conditions on the amplitude and latency lag from stimulus presentation of each component a wavelet packet model of a single individual s data defines a unique decomposition based on prior time frequency information and variation among experimental conditions a particular orthogonal wavelet packet basis is selected using the best basis algorithm with a special cost function that incorporates prior information our statistical model allows individual specific parameters to vary randomly among individuals because the number of observati

matching wavelet packets to gaussian random processes we consider the problem of approximating a set of arbitrary discrete time gaussian random processes by a single representative wavelet based gaussian process we measure the similarity between the original processes and the wavelet based process with the bhattacharyya 1943 coefficient by manipulating the bhattacharyya coefficient we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet based eigenvectors an associated diagonal matrix of eigenvalues and a mean vector the matching algorithm we derive maximizes the nonadditive bhattacharyya coefficient in three steps a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet based eigenvectors and two separate fixed point algorithms that derive an appropriate set of eigenvalues and a mean vector we illustrate the method with two different classes of processes first order markov and bandlimited the technique is also applied to the problem of robust terrain classification in polarimetric sar images

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

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

some alternatives to wald s confidence interval and test abstract the problem considered is that of inference for the variance ratio 2 s 2 e in mixed linear models of the form y x zs e where is a column vector of unknown parameters and s and e are statistically independent multivariate normal random vectors with e s 0 var s 2 s i e e 0 and var e 2 e i the case where there is a known upper bound on is emphasized the 100 1 confidence sets corresponding to the following two sided tests of the null hypothesis h 0 0 are discussed and compared 1 a size wald s test and 2 the test that rejects h 0 whenever h 0 is rejected by the most powerful size 1 invariant test of h 0 versus the alternative u or by the most powerful size 2 invariant test of h 0 versus the alternative l l 2 l 0 u if u and l are close to 0 the latter test is essentially equivalent to a two sided version of the locally most powerful invariant test

case deletion diagnostics for mixed models mixed linear models arise in many areas of application standard estimation methods for mixed models are sensitive to bizarre observations such influential observations can completely distort an analysis and lead to inappropriate actions and conclusions we develop case deletion diagnostics for detecting influential observations in mixed linear models diagnostics for both fixed effects and variance components are proposed computational formulas are given that make the procedures feasible the methods are illustrated using examples

exact confidence sets for variance components in unbalanced mixed linear models we present a general procedure for obtaining an exact confidence set for the variance components in a mixed linear model the procedure can be viewed as a generalization of the anova method used with balanced models our procedure uses as pivotal quantities quadratic forms that are distributed independently as chi squared variables these quadratic forms are constructed with reference to spaces that are orthogonal with respect to the covariance matrix of the observation vector which is a function of the variance components for balanced models these pivotal quantities simplify to multiples of the sums of squares used in the anova method an exact confidence set for the vector of ratios of the effect variances to the error variance is also presented based on the same collection of quadratic forms computing formulas for calculating approximations to these confidence sets are presented and the results of their application to several two way data sets are given

some further results on the exact small sample properties of the instrumental variable estimator we present new results on the exact small sample distribution of the instrumental variable estimator in particular we compare the small sample distribution to the asymptotic distribution

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

some bayesian and non bayesian procedures for the analysis of comparative experiments and for small area estimation computational aspects frequentist properties and relationships abstract the estimation of a treatment contrast from experimental data and the estimation of a small area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two part mixed linear model in this article a bayesian approach to point and interval prediction is presented and its computational requirements are examined differences between the bayesian approach and the traditional classical approach are discussed in general terms and in addition in terms of two examples taken from the literature 1 the comparison of drug formulations in a bioavailability trial westlake and 2 the estimation of corn crop areas using satellite data battese harter and fuller some deficiencies in the classical approach are pointed out and the bayesian approach is considered from a frequentist perspective it is shown via a monte carlo study that for certain noninformative choices of the prior distribution the frequentist prop

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

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

fast positive definite linear system solvers the authors show that the m band wavelet transforms of a wide class of covariance matrices consist of subblocks that are essentially banded furthermore they prove that the cholesky factors of the transformed covariance matrices also consist of subblocks that are essentially banded they combine these two observations to construct a fast o n sup 2 algorithm for solving the n spl times n linear positive definite systems of equations that arise in statistical signal processing finally they provide an error analysis of the proposed linear positive definite system solver

estimating a class of triangular simultaneous equations models without exclusion restrictions this paper provides a control function estimator to adjust for endogeneity in the triangular simultaneous equations model where there are no available exclusion restrictions to generate suitable instruments our approach is to exploit the dependence of the errors on exogenous variables e g heteroscedasticity to adjust the conventional control function estimator the form of the error dependence on the exogenous variables is subject to restrictions but is not parametrically specified in addition to providing the estimator and deriving its large sample properties we present simulation evidence which indicates the estimator works well

exact tests for the main effects variance components in an unbalanced random two way model abstract exact tests are derived for testing hypotheses concerning the variance components of the main effects in an unbalanced random two way crossed classification with interaction model the tests are based on four sums of squares which are distributed independently as scalar multiples of chi squared variates these sums of squares can also be used to find an exact test concerning the interaction variance component and to obtain simultaneous confidence intervals on all continuous functions of the model s variance components a study is made concerning the power of the proposed tests including a comparison with other approximate tests author

consistent noisy independent component analysis we study linear factor models under the assumptions that factors are mutually independent and independent of errors and errors can be correlated to some extent under the factor non gaussianity second to fourth order moments are shown to yield full identification of the matrix of factor loadings we develop a simple algorithm to estimate the matrix of factor loadings from these moments we run monte carlo simulations and apply our methodology to data on cognitive test scores and financial data on stock returns

minimum distance estimation of the errors in variables model using linear cumulant equations abstract we consider a multiple mismeasured regressor errors in variables model we develop closed form minimum distance estimators from any number of estimating equations which are linear in the third and higher cumulants of the observable variables using the cumulant estimators alters qualitative inference relative to ordinary least squares in two applications related to investment and leverage regressions the estimators perform well in monte carlos calibrated to resemble the data from our applications although the cumulant estimators are asymptotically equivalent to the moment estimators from erickson and whited 2002 the finite sample performance of the cumulant estimators exceeds that of the moment estimators

reciprocal processes on a tree modeling and estimation issues motivated by multiresolution decomposition methods such as the discrete wavelet transformation the authors introduce reciprocal processes on truncated n ary trees the authors discuss the relationship between such processes and nearest neighbor models the authors show that they can derive a recursive description of the process and that all reciprocal processes on n ary trees reduce to autoregressive processes in the case of zero valued boundary values at the bottom of the tree corresponding to truncation of the tree the authors then study the smoothing equations associated with such models

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

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