exact solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far field or intermediate regions exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain a stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field it is found that the degree of ill posedness of the inverse problem is significantly reduced when considering a hemi circular observation arc with respect to a linear array of sensors by a factor scaling on the small angular aperture subtended by the observation line estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

analytic solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

a semiparametric approach to density estimation abstract one method of fitting a parametric density function f x is first to estimate by maximum likelihood say and then to estimate f x by on the other hand when the parametric model does not hold the true density f x may be estimated nonparametrically as in the case of a kernel estimate the key idea proposed is to fit a combination of the parametric and nonparametric estimates namely where 0 1 is unknown the parameter is estimated from the data and its estimate is then used in as the proposed density estimate the main point is that we expect to be close to unity when the parametric model prevails and close to zero when it does not hold we show that under certain conditions converges to the parametric density when the parametric model holds and approaches the true f x when the parametric model does not hold the procedure was applied to a number of actual data sets in each case the maximum likelihood estimate was readily obtained and the semiparametric density es

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

analytic solutions of the radiation modes problem and the active control of sound power this paper explores the common mathematical foundation of two different problems the first one arises in electrical engineering for the detection and the spectral estimation of signals in noise and the second one appears in acoustics for the calculation of the acoustic radiation modes of rectangular structures although apparently unrelated it is found that both applications draw on the so called concentration problem of determining which functions that are band limited in one domain have maximal energy concentration within a region of the transform domain the analytic solutions to problems of this form are seen to involve prolate spheroidal wave functions in particular exact expressions are given for the radiation efficiencies and shapes of the radiation modes of a baffled beam as well as their asymptotics it is shown that a generalization of the concentration problem to the two dimensional case provides analytic solutions that solve with a good accuracy although approximately the radiation problem the properties of these special functions provide a rigorous basis of understanding some previously observed features of these applications namely the grouping property of the radiation modes of a baffled panel and the physical limitations for the active control of sound from a panel

fast fourier transform and singular value decomposition formulations for patch nearfield acoustical holography nearfield acoustical holography nah requires the measurement of the pressure field over a complete surface in order to recover the normal velocity on a nearby concentric surface the latter generally coincident with a vibrator patch nah provides a major simplification by eliminating the need for complete surface pressure scans only a small area needs to be scanned to determine the normal velocity on the corresponding small area concentric patch on the vibrator the theory of patch nah is based on 1 an analytic continuation of the patch pressure which provides a spatially tapered aperture extension of the field and 2 a decomposition of the transfer function pressure to velocity and or pressure to pressure between the two surfaces using the singular value decomposition svd for general shapes and the fast fourier transform fft for planar surfaces inversion of the transfer function is stabilized using tikhonov regularization and the morozov discrepancy principle experimental results show that root mean square errors of the normal velocity reconstruction for a point driven vibrator over 200 2700 hz average less than 20 for two small concentric patch surfaces 0 4 cm apart reconstruction of the active normal acoustic intensity was also successful with less than 30 error over the frequency band

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

inverse aeroacoustic problem for a streamlined body i basic formulation the inverse aeroacoustic problem associated with a flat plate airfoil in unsteady compressible flow is formulated in terms of a fredholm integral equation of the first kind the feasibility of determining the unsteady pressure along the airfoil surface from the radiated acoustic signal is demonstrated it is shown that the hadamard conditions of existence and uniqueness of the inverse solution can be satisfied the third hadamard condition for the continuous dependence of the solution on the input acoustic data shows the problem to be ill posed the singular value decomposition method with regularization is used to treat the associated ill conditioned algebraic problem discretization and collocation techniques are used to represent the unsteady pressure on the airfoil both methods give very accurate reconstructions when perfect input data are used the collocation method requires only a small amount of input data either from the far field or the near field thus making it more suitable for applications

reconstruction of vibro acoustic fields using hybrid nearfield acoustic holography hybrid nearfield acoustical holography is utilized to reconstruct the vibro acoustic fields generated by an arbitrary structure vibrating in an unbounded fluid medium this hybrid nah is based on the modified helmholtz equation least squares method that expresses the field acoustic pressure in terms of expansions of outgoing and incoming waves the expansion coefficients are determined by solving an over determined linear system of equations obtained by matching the assumed form solution to a finite number of acoustic pressures measured in the field via least squares the measurements are taken over a conformal surface around the source at close range so that the evanescent waves can be captured once the expansion coefficients are specified enough field acoustic pressures are regenerated on a conformal surface around the source and are taken as input to the helmholtz integral formulation implemented numerically by the boundary element method the acoustic pressure and normal velocity on the source surface are reconstructed using singular value decomposition modified tikhonov regularization and generalized cross validation method this hybrid nah enables one to obtain a satisfactory reconstruction cost effectively because the majority of the input data are calculated but not measured numerical examples of reconstructing the vibro acoustic quantities on the surfaces of an engine block and a highly elongated cylinder are demonstrated

sound power radiation from a vibrating structure in terms of structure dependent radiation modes abstract as a good supplement of conventional acoustic radiation modes a modes a set of so called structure dependent radiation modes s modes is introduced to describe the sound power radiation from a vibrating structure differing from a modes s modes are determined by not only the acoustic resistance matrix of the structure but also the frequency independent normal modes of the structure such a new definition has the following main advantages over the conventional one 1 it can reflect directly the influences of dynamic properties e g boundary conditions of the structures on its sound power radiation 2 the number of s modes generated is generally less than that of a modes since the former depends on the number of structural modes involved in the vibration while the latter depends on the number of segmented elemental radiators of the structure and consequently the demand for large data storage can be greatly alleviated especially for large structures and or higher frequency vibrations 3 the set of s modes possesses a better convergence than that of a modes because the higher ordered s modes can decay more rapidly than the same ordered a modes two baffled finite models i e a simple beam and a thin plate are employed to investigate numerically the acoustic properties of s modes and then compared with those of a modes it has been shown that the two sets of radiation modes share a very similar frequency dependent behavior in that the radiation efficiency falls off very rapidly with increasing mode order at low frequency range typically with kl

using specially designed exponential families for density estimation i this paper have y being portions of the real line or of the plane but the methodology applies just as well to higher dimensionalities and to more complicated spaces estimates of g y are traditionally constructed in two quite different ways by maximum likelihood fitting within some parametric family such as the normal or by nonparametric methods such as kernel density estimation these two methods can be combined by putting an exponential family through a nonparametric estimator the resulting hybrid estimators are the specially designed exponential families of the title figure 1 shows a simple example of this methodology the y are pain i scores for n s 67 women each obtained by averaging the results from a questionnaire administered after an operation the scale runs from y s 0 s w x

nonparametric density estimation with a parametric start the traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes the present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities for example the normal while not losing much in precision when the true density is far from the parametric class the idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor this works well in cases where the correction factor function is less rough than the original density itself extensive comparisons with the kernel estimator are carried out including exact analysis for the class of all normal mixtures the new method with a normal start wins quite often even in many cases where the true density is far from normal procedures for choosing the smoothing parameter of the estimator are also discussed the new estimator should be particularly useful in higher dimensions where the usual nonparametric methods have problems the idea is also spelled out for nonparametric regression

regularization methods for near field acoustical holography the reconstruction of the pressure and normal surface velocity provided by near field acoustical holography nah from pressure measurements made near a vibrating structure is a linear ill posed inverse problem due to the existence of strongly decaying evanescentlike waves regularization provides a technique of overcoming the ill posedness and generates a solution to the linear problem in an automated way we present four robust methods for regularization the standard tikhonov procedure along with a novel improved version landweber iteration and the conjugate gradient approach each of these approaches can be applied to all forms of interior or exterior nah problems planar cylindrical spherical and conformal we also study two parameter selection procedures the morozov discrepancy principle and the generalized cross validation which are crucial to any regularization theory in particular we concentrate here on planar and cylindrical holography these forms of nah which rely on the discrete fourier transform are important due to their popularity and to their tremendous computational speed in order to use regularization theory for the separable geometry problems we reformulate the equations of planar cylindrical and spherical nah into an eigenvalue problem the resulting eigenvalues and eigenvectors couple easily to regularization theory which can be incorporated into the nah software with little sacrifice in computational speed the resulting complete automation of the nah algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for nah

the kernel estimate of a regression function in likelihood based models abstract smoothing splines have a penalized likelihood motivation good and gaskins 1971 allowing direct application to nonparametric regression in likelihood based models the notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed generalizing the local likelihood theory of tibshirani and hastie 1987 let the data be of the form xi yi i 1 n where xi 0 1 d are lattice points and the yi are independent random variables from a family of distributions with parameter i g xi with g having continuous partial derivatives of order k 2 the goal is to arrive at a nonparametric estimate o of o g xo for a fixed point xo 0 1 d we consider the estimator o that maximizes the weighted likelihood function w n i 1 w xo xi b log f yi with f the density of yi w a symmetric kernel with compact support and b the bandwidth that controls the degree of smoothing sufficient conditions for consistency and asympt

sound source reconstruction using inverse boundary element calculations whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure the aim in this work is to reverse the process i e to determine vibration from acoustic field data this inverse problem is brought on a form suited for solution by means of an inverse boundary element method since the numerical treatment of the inverse source reconstruction results in a discrete ill posed problem regularization is imposed to avoid unstable solutions dominated by errors in the present work the emphasis is on tikhonov regularization and parameter choice methods not requiring an error norm estimate for choosing the right amount of regularization several parameter choice strategies have been presented lately but it still remains to be seen how well these can handle industrial applications with real measurement data in the present work it is demonstrated that the l curve criterion is robust with respect to the errors in a real measurement situation in particular it is shown that the l curve criterion is superior to the more conventional generalized cross validation gcv approach for the present tire noise studies

spatial resolution limits for the reconstruction of acoustic source strength by inverse methods one method for deducing the strength of an acoustic source distribution from measurement of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution in practice the frequency response function matrix to be inverted may very often be ill conditioned this ill conditioning will also often result in an ill posed problem and thus regularization algorithms are used to produce reasonable solutions for this purpose tikhonov regularization has been applied and generalized cross validation gcv has been introduced as an effective method for determining the proper amount of regularization without prior knowledge of either the source distribution or the contaminating errors in the present work the emphasis is placed on the relationship between the spatial resolution of the reconstructed source distribution and the small singular values of the frequency response function matrix to be inverted however the use of tikhonov regularization often suppresses the effect of small singular values and these are in turn often associated with high spatial frequencies of the source distribution thus the process of regularization produces a useful estimate of the acoustic source strength distribution but with a limited spatial resolution furthermore in the field of fourier acoustics the spatial resolution of the reconstructed source distribution is usually limited by the wavelength of the radiation this paper expresses the relationship between estimation accuracy spatial resolution noise level and source sensor geometry when a range of inverse sound radiation problems are regularised using tikhonov regularization with gcv the results presented form the basis of guidelines that enable the reconstruction of acoustic source strength with a resolution that is finer than the intrinsic half wavelength limit

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

variable bandwidth and local linear regression smoothers in this paper we introduce an appealing nonparametric method for estimating the mean regression function the proposed method combines the ideas of local linear smoothers and variable bandwidth hence it also inherits the advantages of both approaches we give expressions for the conditional mse and mise of the estimator minimization of the mise leads to an explicit formula for an optimal choice of the variable bandwidth moreover the merits of considering a variable bandwidth are discussed in addition we show that the estimator does not have boundary effects and hence does not require modifications at the boundary the performance of a corresponding plug in estimator is investigated simulations illustrate the proposed estimation method

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

local likelihood based on kernel censoring by drawing an analogy with likelihood for censored data a local likelihood function is proposed which gives more weight to observations near a region of interest in the sample space resulting methods can be used for assessing local departures from a parametric model and for semiparametric density estimation some theory and three examples is given

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

local likelihood density estimation local likelihood was introduced by tibshirani and hastie as a method of smoothing by local polynomials in non gaussian regression models in this paper an extension of these methods to density estimation is discussed and comparison with other methods of density estimation presented the local likelihood method has particularly strong advantages over kernel methods when estimating tails of densities and in multivariate settings suppose constraints are incorporated in a simple manner asymptotic properties of the estimate are discussed a method for computing the estimate is outlined c code to implement the estimation procedure described in this paper together with s interfaces for graphical display of results are available at http cm bell labs com stat project locfit index html

helmholtz equation least squares method for reconstructing the acoustic pressure field a method using spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating object is developed the radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the gram schmidt orthonormalization with respect to the particular solutions to the helmholtz equation on the vibrating surface under consideration the coefficients associated with these independent functions are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points the errors incurred in this process are minimized by the least squares method once these coefficients are specified the acoustic pressure at any point including the source surface is completely determined in this paper this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field it is shown that this method can be applied to both separable and nonseparable geometries and the

fir system identification using higher order statistics alone this paper focuses on fir system identification in gaussian arma noise using higher order cumulants alone two new types of cumulant based normal equations are established and two algorithms are proposed for ma parameter estimation simulations verify the high performance of the new developments

versions of kernel type regression estimators abstract we explore the aims of and relationships between various kernel type regression estimators to do so we identify two general types of direct kernel estimators differing in their treatment of the nuisance density function associated with regressor variable design we look at the well known gasser muller nadaraya watson and priestley chao methods in this light in the random design case none of these methods is totally adequate and we mention a novel direct kernel method with appropriate properties disadvantages of even the latter idea are remedied by kernel weighted local linear fitting a well known technique that is currently enjoying renewed popularity we see how to fit this approach into our general framework and hence form a unified understanding of how these kernel type smoothers interrelate though the mission of this article is unificatory and even pedagogical the desire for better understanding of superficially different approaches is motivated by the need to improve prac

source visualization by using statistically optimized near field acoustical holography in cylindrical coordinates nearfield acoustical holography nah is a useful tool for visualizing noise sources however to avoid spatial fourier transform related truncation effects the measurement or hologram surface must extend beyond the source to a region where the sound pressure drops to a level significantly lower than the peak level within the measurement aperture statistically optimized nearfield acoustical holography sonah first derived by steiner and hald in planar geometry is based on a formulation similar to that of nah however in sonah surface to surface projection of the sound field is performed by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy i e no spatial fourier transforms are performed thus the requirement that the measurement surface be extended is eliminated without compromising the accuracy of the procedure in the present work sonah was re formulated in cylindrical coordinates and wa

a generalization of the two dimensional prolate spheroidal wave function method for nonrectilinear mri data acquisition methods the two dimensional 2 d prolate spheroidal wave function 2 d pswf method was previously introduced as an efficient method for trading off between spatial and temporal resolution in magnetic resonance imaging mri with minimal penalty due to truncation and partial volume effects in the 2 d pswf method the k space sampling area and a matching 2 d pswf filter with optimal signal concentration and minimal truncation artifacts are determined by the shape and size of a given convex region of interest roi the spatial information in the reduced k space data is used to calculate the total image intensity over a nonsquare roi instead of producing a low resolution image this method can be used for tracking dynamic signals from non square rois using a reduced k space sampling area while achieving minimal signal leakage however the previous theory is limited to the case of rectilinear sampling in order to make the 2 d pswf method more suitable for dynamic studies this paper presents a generalized version of the 2 d pswf theory that can be applied to nonrectilinear data acquisition methods the method is applied to an fmri study using a spiral trajectory which illustrates the methods efficiency at tracking hemodynamic signals with high temporal resolution

correlation curves measures of association as functions of covariate values for experiments where the strength of association between a response variable y and a covariate x is different over different regions of values for the covariate x we propose local nonparametric dependence functions which measure the strength of association between y and x as a function of x x our dependence functions are extensions of galton s idea of strength of co relation from the bivariate normal case to the nonparametric case in particular a dependence function is obtained by expressing the usual galton pearson correlation coefficient in terms of the regression line slope 3 and the residual variance o 2 and then replacing 3 and 72 by a nonparametric regression slope 13 x and a nonparametric residual variance o 2 x var yix respectively our local dependence functions are standardized nonparametric regression curves which provide universal scale free measures of the strength of the relationship between variables in nonlinear models they share most of the properties of the correlation coefficient and they reduce to the usual correlation coefficient in the bivariate normal case for this reason we call them correlation curves we show that in a certain sense they quantify lehmann s notion of regression dependence finally the correlation curve concept is illustrated using data from a study of the relationship between cholesterol levels x and triglyceride concentrations y of heart patients