direct approach to the study of soliton perturbations a direct approach in studying the soliton perturbations of nonlinear evolution equations has been developed it is based on the method of the derivative expansions for linearization of the perturbed equations and the separation of variables for solution of the linearized equations if differs substantially from the past direct methods the apparent advantage of this approach is that it relies on no knowledge of the inverse scattering transform besides it is very concise and easy to understand as an example we use it to study the perturbed korteweg de vries equation the results we obtained agree with what other authors have found

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

mixed effects nonlinear regression for unbalanced repeated measures repeated measures data such as clinical pharmacokinetic data growth data and dose response data are often inherently nonlinear with respect to a given response function and are frequently incomplete and or unbalanced nonlinear random effects models together with a variety of estimation procedures have been proposed for the analysis of such data this paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed effects nonlinear regression model the asymptotic properties of the proposed estimators are given and large sample tests of hypothesis provided the results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure

use of ridge regression for improved estimation of kinetic constants from pet data the estimation of parameters in radio tracer models from positron emission tomography pet data by nonlinear least squares nls often leads to results with unacceptable mean square error mse characteristics the introduction of constraints on parameters has the potential to address this problem we examine a ridge regression technique that augments the standard nls criterion by the addition of a term which penalizes estimates which deviate from physiologically reasonable values a variation on a plug in methodology of hoerl et al 1975 is examined for data dependent selection of the degree of reliance to place on the penalizing term a simulation study is carried out to evaluate the performance of this approach in the context of estimation of kinetic constants in the three compartment model used to analyze data from pet studies with fluoro deoxyglucose fdg the results show that over a range of realistic noise levels the ridge regression procedure can be expected to reduce the root mse of parameter estimates by 60 this result is not found to be substantially dependent on the precise formulation of the penalty function used thus the use of ridge regression for the estimation of kinetic parameters in pet studies is thus considered to be a promising tool

parametric imaging with bayesian priors a validation study with 11c altropane pet abstract it has been suggested that bayesian estimation methods may be used to improve the signal to noise ratio of parametric images however there is little experience with the method and some of the underlying assumptions and performance properties of bayesian estimation remain to be investigated we used a sample population of 54 subjects studied previously with 11 c altropane to empirically evaluate the assumptions performance and some practical issues in forming parametric images by using normality tests we showed that the underpinning normality assumptions of data and parametric distribution apply to more than 80 of voxels the standard deviation of the binding potential can be reduced 30 50 using bayesian estimation without introducing substantial bias the sample size required to form the a priori information was found to be modest as little as ten subjects may be sufficient and the choice of specific subjects has little effect on bayesian estimation a realistic simulation study showed that detection of localized differences in parametric images e g by statistical parametric mapping spm could be made more reliable and or conducted with smaller sample size using bayesian estimation in conclusion bayesian estimation can improve the snr of parametric images and better detect localized changes in cohorts of subjects

wavelet analysis of dynamic pet data application to the parametric imaging of benzodiazepine receptor concentration abstract receptor density and ligand affinity can be assessed using positron emission tomography pet biological parameters b max k 1 k 2 k on v r k off are estimated using a compartmental model and a multi injection protocol parametric imaging of the ligand receptor model has been shown to be of special interest to study certain brain disorders however the low signal to noise ratio in kinetic curves at the pixel level hampers an adequate estimation of model parameters during the optimization procedure for this reason mapping requires a spatial filter resulting in a loss of resolution filtering the kinetic curves in the frequency domain using the fourier transform is not appropriate because of difficulties in choosing a correct and efficient cutoff frequency a wavelet based filter is more appropriate to such tracer kinetics the purpose of this study is to build up parametric images at the pixel level while conserving the original spatial resolution using wavelet based filtering data from 11 c flumazenil studies mapping the benzodiazepine receptor density were used an invertible discrete wavelet transform was used to calculate the time frequency signals of the time concentration pet curves on a pixel by pixel basis kinetic curves observed from large regions of interest in high and low receptor density regions were used to calibrate the threshold of wavelet coefficients the shrunken wavelet coefficients were then transformed back to the original domain in order to obtain the filtered pet signal maps of all binding parameters were obtained at the pixel level with acceptable coefficients of variation of less than 30 for the b max parameter in most of the gray matter a strong correlation between model parameter estimates using the usual regions of interest and parametric imaging was observed for all model parameters r 0 949 for the parameter b max we conclude that wavelet based filters are useful for building binding parameter maps without loss of the original spatial resolution of the pet scanner the use of the wavelet based filtering method can be extended far beyond the multi injection protocol it is likely to be also effective for other dynamic pet studies

exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory abstract we give a short review of the techniques that have been used to study soliton perturbations for nonlinear integrable systems in 1 1 dimensions in particular we discuss the connection between direct perturbation methods using squared eigenfunction expansions and quasistationary perturbation theory we show for the kdv korteweg de vries how the results for quasistationary approximations are related to those for the standard adiabatic perturbation theory

sparsity constrained mixture modeling for the estimation of kinetic parameters in dynamic pet the estimation and analysis of kinetic parameters in dynamic positron emission tomography pet is frequently confounded by tissue heterogeneity and partial volume effects we propose a new constrained model of dynamic pet to address these limitations the proposed formulation incorporates an explicit mixture model in which each image voxel is represented as a mixture of different pure tissue types with distinct temporal dynamics we use cramer rao lower bounds to demonstrate that the use of prior information is important to stabilize parameter estimation with this model as a result we propose a constrained formulation of the estimation problem that we solve using a two stage algorithm in the first stage a sparse signal processing method is applied to estimate the rate parameters for the different tissue compartments from the noisy pet time series in the second stage tissue fractions and the linear parameters of different time activity curves are estimated using a combination of spatial regularity and fractional mixture constraints a block coordinate descent algorithm is combined with a manifold search to robustly estimate these parameters the method is evaluated with both simulated and experimental dynamic pet data

resolution and noise properties of map reconstruction for fully 3 d pet derives approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three dimensional 3 d positron emission tomography pet data using maximum a posteriori map estimation these expressions explicitly account for the spatially variant detector response and sensitivity of a 3 d tomograph the resulting spatially variant impulse response and covariance are computed using 3 d fourier transforms a truncated gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in map reconstruction using monte carlo simulations and phantom data from the micropet small animal scanner the authors show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of map reconstruction for priors with quadratic energy functions they also describe how these analytical results can be used to achieve near uniform contrast recovery throughout the reconstructed volume

comparing methods of analyzing fmri statistical parametric maps approaches for the analysis of statistical parametric maps spms can be crudely grouped into three main categories in which different philosophies are applied to delineate activated regions these being type i error control thresholding false discovery rate fdr control thresholding and posterior probability thresholding to better understand the properties of these main approaches we carried out a simulation study to compare the approaches as they would be used on real data sets using default settings we find that posterior probability thresholding is the most powerful approach and type i error control thresholding provides the lowest levels of type i error false discovery rate control thresholding performs in between the other approaches for both these criteria although for some parameter settings this approach can approximate the performance of posterior probability thresholding based on these results we discuss the relative merits of the three approaches in an attempt to decide upon an optimal approach we conclude that viewing the problem of delineating areas of activation as a classification problem provides a highly interpretable framework for comparing the methods within this framework we highlight the role of the loss function which explicitly penalizes the types of errors that may occur in a given analysis

an unbiased parametric imaging algorithm for nonuniformly sampled biomedical system parameter estimation an unbiased algorithm of generalized linear least squares glls for parameter estimation of nonuniformly sampled biomedical systems is proposed the basic theory and detailed derivation of the algorithm are given this algorithm removes the initial values required and computational burden of nonlinear least regression and achieves a comparable estimation quality in terms of the estimates bias and standard deviation therefore this algorithm is particular useful in image wide pixel by pixel based parameter estimation e g to generate parametric images from tracer dynamic studies with positron emission tomography an example is presented to demonstrate the performance of this new technique this algorithm is also generally applicable to other continuous system parameter estimation

an evaluation of bayesian regression for estimating cerebral oxygen utilization with oxygen 15 and dynamic pet the use of bayesian regression for improving the estimation of cerebral oxygen utilization rate cmro was investigated based on plasma time activity curves of sup 15 o oxygen and sup 15 o water that were obtained previously in a normal subject realistic tissue time activity curves of sup 15 o radioactivity in gray and white matter as obtained in dynamic pet positron emission tomography sup 15 o oxygen studies were computer simulated for various noise conditions and blood flow variations bayesian regression and regular least square regressions were applied to estimate cmro from the kinetics the accuracy and precision of the estimates were evaluated the improvement in precision was as high as 50 for gray matter of 1 cm sup 3 size in a study of 0 5 million counts slice over 3 min similar improvement was achieved for white matter in 1 million count studies the amount of improvement is larger for higher noise cases

a general method of bayesian estimation for parametric imaging of the brain abstract we report a general method of bayesian estimation that uses prior measurements to improve the signal to noise ratio of parametric images computed from dynamic pet scanning in our method the ordinary weighted least squares cost function is augmented by a penalty term to yield k s min k c f k t c 1 c f k s k s 0 k k t k 1 k k where c is a pet concentration history and c is its variance f is the model of the concentration history k k 1 k 2 k m t is the parameter vector k is the vector of population means for the model parameters k is its covariance k k s 0 is the conventional weighted sum of squares s 0 is chosen to control the balance between the prior and new data data from a prior population of subjects are analyzed with standard methods to provide maps of the mean parameter values and their variances as an example of this approach we used the dynamic image data of 10 normal subjects who had previously been studied with 11 c raclopride to estimate the prior distribution the dynamic data were transformed to stereotactic coordinates and analyzed by standard methods the resulting parametric maps were used to compute the voxel wise sample statistics then the cohort of priors was analyzed as a function of s using nonlinear least squares estimation and the cost function shown above as s is increased the standard error in estimating bp in single subjects was substantially reduced allowing measurement in bp in thalamus cortex brain stem etc additional studies demonstrate that a range of s values exist for which the bias is not excessive even when parameter values differ markedly from the sample mean this method can be used with any kinetic model so long as it is possible to compute a map of a priori mean parameters and their variances

the perturbed korteweg de vries equation considered anew the perturbed korteweg de vries equation is studied in a new way by a green s function formalism without use of inverse scattering methods the green s function is determined by employing the backlund transformation and green s theorem after a thorough analysis of the exact first order solution with regard to secular terms a two time scale expansion leads to the adiabatic approximation and the first order correction in accordance with the results of karpman and maslov contrary to statements in the literature the term tanh2 z in the expression for the modified phase of the perturbed soliton arises as a consequence of the systematically conducted first order perturbation theory

quadratic forms and systems of forms in many variables let f 1 ldots f r be quadratic forms with integer coefficients in n variables when n ge 9r and the variety v f 1 ldots f r is a smooth complete intersection we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish which in particular implies the hasse principle for v f 1 ldots f r previous work in this direction required n to grow at least quadratically with r we give a similar result for r forms of degree d conditional on an upper bound for the number of solutions to an auxiliary inequality in principle this result may apply as soon as n d2 dr in the case that d ge 3 several strategies are available to prove the necessary upper bound for the auxiliary inequality in a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients

conservation laws and the perturbed kdv equation examines the integral for the correction to the soliton position in the adiabatic approximation using the modified conservation laws for the mass and the energy the general correction to the soliton position is obtained it is easily seen that this correction is due to the formation of the shelf

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

wavelet domain filtering for photon imaging systems many imaging systems rely on photon detection as the basis of image formation one of the major sources of error in these systems is poisson noise due to the quantum nature of the photon detection process unlike additive gaussian white noise the variance of poisson noise is proportional to the underlying signal intensity and consequently separating signal from noise is a very difficult task in this paper we perform a novel gedankenexperiment to devise a new wavelet domain filtering procedure for noise removal in photon imaging systems the filter adapts to both the signal and the noise and balances the trade off between noise removal and excessive smoothing of image details designed using the statistical method of cross validation the filter is simultaneously optimal in a small sample predictive sum of squares sense and asymptotically optimal in the mean square error sense the filtering procedure has a simple interpretation as a joint edge detection estimation process moreover we derive an efficient algorithm for performing the filtering that has the same order of complexity as the fast wavelet transform itself the performance of the new filter is assessed with simulated data experiments and tested with actual nuclear medicine imagery

renormalization group theory for perturbed evolution equations we show that proto rg renormaliation group theory can be used to give a systematic description of the evolution of soltion of perturbed equations the equations describing the deformation of its shape as the effect of perturbation are proto rg equations the rg approach may be simpler than inverse scattering theory ist and another approaches because it dose not rely on any knowledge of ist it is very concise and easy to understand

quasi stationary perturbations of the kdv soliton in this paper we show how the results of quasi stationary perturbation theory relate to those based upon squared eigenfunction expansions we show that the two give the same results for a quasi stationary approximation and also show how standard adiabatic perturbation theory connects to the quasi stationary theory

estimating neurotransmitter kinetics with ntpet a simulation study of temporal precision and effects of biased data abstract we recently introduced neurotransmitter pet ntpet an analysis technique that estimates the kinetics of stimulus induced neurotransmitter nt release here we evaluate two formulations of ntpet the arterial art approach measures the tracer input function tif directly the reference ref approach derives the tif from reference region data arterial sampling is considered the gold standard in pet modeling but reference region approaches are preferred for reduced cost and complexity if simulated pet data with unbiased tifs were analyzed using art or ref temporal precision was better than 3 min provided nt concentration peaked less than 30 min into the scanning session the consequences of biased tifs or stimulus induced changes in tracer delivery were also evaluated art tifs were biased by the presence of uncorrected radiometabolites in the plasma whereas ref tifs were biased by specific binding in the reference region simulated changes in tracer delivery emulated ethanol induced blood flow alterations observed previously with pet art performance deteriorated significantly if metabolites amounted to 50 of plasma radioactivity by 60 min the accuracy and precision of ref were preserved even if the reference region contained 40 of the receptor density of the target region both methods were insensitive to blood flow alterations proportional changes in k1 and k2 our results suggest that pet data contain information heretofore not extracted about the timing of nt release the ref formulation of ntpet proved to be robust to many plausible model violations and under most circumstances is an appropriate alternative to art

the stochastic damped kdv equation the author applies singular perturbation theory to the study of a damped kdv soliton under the influence of space and time dependent external noise he finds that asymptotically the shape of the averaged solution approaches that of a gaussian packet whose amplitude decays and width grows in the same way as in the case of purely time dependent noise

complete eigenfunctions of linearized integrable equations expanded around a soliton solution complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory in this article we explicitly construct such eigenfunctions for a large class of integrable equations including the kdv nls and mkdv hierarchies we establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions furthermore these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem the key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro differential operator which generates the integrable equation and the linearization operator of the lowest order integrable equation in the same hierarchy we also obtain similar results for the adjoint linearization operator of an integrable equation even though our analysis is conducted only for the kdv nls and mkdv hierarchies similar results are expected for other integrable hierarchies as well we further explicitly present the complete eigenfunctions for the kdv nls and mkdv hierarchy equations and give their inner products thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations

linear regression with spatial constraint to generate parametric images of ligand receptor dynamic pet studies with a simplified reference tissue model abstract for the quantitative analysis of ligand receptor dynamic positron emission tomography pet studies it is often desirable to apply reference tissue methods that eliminate the need for arterial blood sampling a common technique is to apply a simplified reference tissue model srtm applications of this method are generally based on an analytical solution of the srtm equation with parameters estimated by nonlinear regression in this study we derive based on the same assumptions used to derive the srtm a new set of operational equations of integral form with parameters directly estimated by conventional weighted linear regression wlr in addition a linear regression with spatial constraint lrsc algorithm is developed for parametric imaging to reduce the effects of high noise levels in pixel time activity curves that are typical of pet dynamic data for comparison conventional weighted nonlinear regression with the marquardt algorithm wnlrm and nonlinear ridge regression with spatial constraint nlrrsc were also implemented using the nonlinear analytical solution of the srtm equation in contrast to the other three methods lrsc reduces the percent root mean square error of the estimated parameters especially at higher noise levels for estimation of binding potential bp wlr and lrsc show similar variance even at high noise levels but lrsc yields a smaller bias results from human studies demonstrate that lrsc produces high quality parametric images the variance of r1 and k2 images generated by wlr wnlrm and nlrrsc can be decreased 30 60 by using lrsc the quality of the bp images generated by wlr and lrsc is visually comparable and the variance of bp images generated by wnlrm can be reduced 10 40 by wlr or lrsc the bp estimates obtained using wlr are 3 5 lower than those estimated by lrsc we conclude that the new linear equations yield a reliable computationally efficient and robust lrsc algorithm to generate parametric images of ligand receptor dynamic pet studies

generalized linear least squares method for fast generation of myocardial blood flow parametric images with n 13 ammonia pet in this paper the authors developed and tested strategies for estimating myocardial blood flow mbf and generating mbf parametric images using positron emission tomography pet n 13 ammonia and the generalized linear least square glls method glls was generalized to the general linear compartment model modified for the correction of spillover validated using simulated n 13 ammonia data and examined using pet data from several patient studies in comparison to the standard model fitting procedure the glls method provided similar accuracy and superior computational speed

a direct perturbation theory for dark solitons based on a complete set of the squared jost solutions because of the essential difficulty caused by the non vanishing boundary condition a systematic perturbation approach for dark solitons has not yet been found based on a rigorous proof of the completeness of the squared jost solution with explicit expressions for one soliton case a direct perturbation approach for dark solitons is developed in this paper difficulties caused by the background are overcome as an example of the approach the problem of damping is treated as a perturbation

a non linear mixed effect modelling approach for metabolite correction of the arterial input function in pet studies abstract quantitative pet studies with arterial blood sampling usually require the correction of the measured total plasma activity for the presence of metabolites in particular if labelled metabolites are found in the plasma in significant amounts their presence has to be accounted for because it is the concentration of the parent tracer which is required for data quantification this is achieved by fitting a parent plasma fraction ppf model to discrete metabolite measurements the commonly used method is based on an individual approach i e for each subject the ppf model parameters are estimated from its own metabolite samples which are in general sparse and noisy this fact can compromise the quality of the reconstructed arterial input functions and consequently affect the quantification of tissue kinetic parameters in this study we proposed a non linear mixed effect modelling nlmem approach to describe metabolite kinetics since nlmem has been developed to provide robust parameter estimates in the case of sparse and or noisy data it has the potential to be a reliable method for plasma metabolite correction three different pet datasets were considered 11c phno 54 scans 11c pib 22 scans and 11c dasb 30 scans for each tracer both simulated and measured data were considered and nlmem performance was compared with that provided by individual analysis results showed that nlmem provided improved estimates of the plasma parent input function over the individual approach when the metabolite data were sparse or contained outliers

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

bayesian hierarchical multi subject multiscale analysis of functional mri data we develop a methodology for bayesian hierarchical multi subject multiscale analysis of functional magnetic resonance imaging fmri data we begin by modeling the brain images temporally with a standard general linear model after that we transform the resulting estimated standardized regression coefficient maps through a discrete wavelet transformation to obtain a sparse representation in the wavelet space subsequently we assign to the wavelet coefficients a prior that is a mixture of a point mass at zero and a gaussian white noise in this mixture prior for the wavelet coefficients the mixture probabilities are related to the pattern of brain activity across different resolutions to incorporate this information we assume that the mixture probabilities for wavelet coefficients at the same location and level are common across subjects furthermore we assign for the mixture probabilities a prior that depends on a few hyperparameters we develop an empirical bayes methodology to estimate the hyperparameters and as these hyperparameters are shared by all subjects we obtain precise estimated values then we carry out inference in the wavelet space and obtain smoothed images of the regression coefficients by applying the inverse wavelet transform to the posterior means of the wavelet coefficients an application to computer simulated synthetic data has shown that when compared to single subject analysis our multi subject methodology performs better in terms of mean squared error finally we illustrate the utility and flexibility of our multi subject methodology with an application to an event related fmri dataset generated by postle 2005 through a multi subject fmri study of working memory related brain activation

resampling methods for improved wavelet based multiple hypothesis testing of parametric maps in functional mri abstract two or three dimensional wavelet transforms have been considered as a basis for multiple hypothesis testing of parametric maps derived from functional magnetic resonance imaging fmri experiments most of the previous approaches have assumed that the noise variance is equally distributed across levels of the transform here we show that this assumption is unrealistic fmri parameter maps typically have more similarity to a 1 f type spatial covariance with greater variance in 2d wavelet coefficients representing lower spatial frequencies or coarser spatial features in the maps to address this issue we resample the fmri time series data in the wavelet domain using a 1d discrete wavelet transform dwt to produce a set of permuted parametric maps that are decomposed using a 2d dwt to estimate level specific variances of the 2d wavelet coefficients under the null hypothesis these resampling based estimates of the wavelet variance spectrum are substituted in a bayesian bivariate shrinkage operator to denoise the observed 2d wavelet coefficients which are then inverted to reconstitute the observed denoised map in the spatial domain multiple hypothesis testing controlling the false discovery rate in the observed denoised maps then proceeds in the spatial domain using thresholds derived from an independent set of permuted denoised maps we show empirically that this more realistic resampling based algorithm for wavelet based denoising and multiple hypothesis testing has good type i error control and can detect experimentally engendered signals in data acquired during auditory linguistic processing

a maximum likelihood estimation method for random coefficient regression models summary a method for estimating the distribution of the parameters of a random coefficient regression model is proposed this distribution accounting for interindividual variability is assumed to lie in a wide class of probability distributions rather than in a given parametric class estimation is based on observations from a sample of individuals and likelihood is the estimation criterion experimental designs may be different among individuals allowing the method to apply to routinely collected data the problem has strong connections with the theory of optimum design of experiments conditions are given under which the problem has a unique solution which then corresponds to a discrete distribution a simple pharmacokinetic model involving two parameters is used as an example these parameters have a bimodal distribution as statistical specification moreover only one observation is available per individual thus the method applies even when the model is not identifiable

wavelets and functional magnetic resonance imaging of the human brain the discrete wavelet transform dwt is widely used for multiresolution analysis and decorrelation or whitening of nonstationary time series and spatial processes wavelets are naturally appropriate for analysis of biological data such as functional magnetic resonance images of the human brain which often demonstrate scale invariant or fractal properties we provide a brief formal introduction to key properties of the dwt and review the growing literature on its application to fmri we focus on three applications in particular i wavelet coefficient resampling or wavestrapping of 1 d time series 2 to 3 d spatial maps and 4 d spatiotemporal processes ii wavelet based estimators for signal and noise parameters of time series regression models assuming the errors are fractional gaussian noise fgn and iii wavelet shrinkage in frequentist and bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps we conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fmri data

map based kinetic analysis for voxel by voxel compartment model estimation detailed imaging of the cerebral glucose metabolism using fdg we propose a novel algorithm for voxel by voxel compartment model analysis based on a maximum a posteriori map algorithm voxel by voxel compartment model analysis can derive functional images of living tissues but it suffers from high noise statistics in voxel based pet data and extended calculation times we initially set up a feature space of the target radiopharmaceutical composed of a measured plasma time activity curve and a set of compartment model parameters and measured the noise distribution of the pet data the dynamic pet data were projected onto the feature space and then clustered using the mahalanobis distance our method was validated using simulation studies and compared with roi based ordinary kinetic analysis for fdg the parametric images exhibited an acceptable linear relation with the simulations and the roi based results and the calculation time took about 10 min we therefore concluded that our proposed map based algorithm is practical

a direct approach to studying soliton perturbations starting with an integrable nonlinear evolution equation the author investigates perturbations about a one soliton solution through the inversion of a linear equation for the first order correction to the soliton solution this inversion differs from past methods as the proposed method takes place in coordinate space not spectral space while it employs some of the tools of inverse scattering theory the method is applied to the korteweg de vries nonlinear schrodinger and sine gordon equations the first order corrections are then obtained

on the number of linear spaces on hypersurfaces with a prescribed discriminant for a given form we apply the circle method in order to give an asymptotic estimate of the number of m tuples spanning a linear space on the hypersurface with the property that this allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive

penalized discriminant analysis of sup 15 o water pet brain images with prediction error selection of smoothness and regularization hyperparameters the authors propose a flexible comprehensive approach for analysis of sup 15 o water positron emission tomography pet brain images using a penalized version of linear discriminant analysis pda they applied it to scans from 20 subjects eight scans subject performing a finger movement task and analyzed 1 two classes to obtain a covariance normalized baseline activation image and 2 eight classes for the mean within subject temporal structure which contained baseline activation and time dependent changes in a two dimensional canonical subspace the authors imposed spatial smoothness on the resulting image s by expanding it in five tensor product b spline tps bases of varying smoothness and further regularized with a ridge type penalty on the noise covariance matrix the discrimination approach of pda provides a probabilistic framework within which prediction error pe estimates are derived the authors used these to optimize over tps bases and a ridge hyperparameter expressed as equivalent degrees of freedom edf they obtained unbiased low variance pe estimates using modern resampling tools 632 bootstrap and cross validation and compared pda of 1 tps projected mean normalized and unnormalized scans and 2 mean normalized scans with and without additional presmoothing by examining the tradeoffs between pe and edf as a function of basis selection and image smoothing the authors demonstrate the utility of pda the pe framework and the relationship between singular value decomposition and smooth tps bases in the analysis of functional neuroimages

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

green s function method for perturbed dnls equation with the squared spectrum parameter abstract a general direct perturbation theory for the perturbed dnls equation is developed the completeness relation is given by the green s function method similar to that used for the linear equations the effect of damping on a single soliton is calculated as an example and the result is good with the squared spectrum parameter

fast and reliable estimation of multiple parametric images using an integrated method for dynamic spect dynamic single photon emission computed tomography spect has demonstrated the potential to quantitatively estimate physiological parameters in the brain and the heart the generalized linear least square glls method is a well established method for solving linear compartment models with fast computational speed however the high level of noise intrinsic in the spect data leads to reliability and instability problems of glls for generating parametric images an integrated method is proposed to restrict the noise in both the temporal and spatial domains to estimate multiple parametric images for dynamic spect this method comprises three steps which are optimum image sampling schedule in the projection space cluster analysis applied postreconstruction and parametric image generation with glls the simulation and experimental studies for the neuronal nicotine acetylcholine receptor tracer of 5 123i iodo a 85380 were employed to evaluate the performance of the proposed method the results of influx rate of k1 and volume of distribution of vd demonstrated that the integrated method was successful in generating low noise parametric images for high noise spect data without enhancing the partial volume effect furthermore the integrated method is computationally efficient for potential clinical applications

wavelet variance components in image space for spatiotemporal neuroimaging data neuroimaging studies place great emphasis on not only the estimation but also the standard error estimates of underlying parameters derived from a temporal model this allows inferences to be made about the signal estimates and resulting conclusions to be drawn about the underlying data it can often be advantageous to interrogate temporal models after spatial transformation of the data into the wavelet domain wavelet bases provide a multiresolution decomposition of the spatial data dimension and an ensuing reduction in spatial correlation however widespread acceptance of these wavelet techniques has been hampered by the limited ability to reconstruct both parametric and error estimates into the image domain after analysis of temporal models in the wavelet domain this paper introduces a derivation and a fast implementation of a method for the calculation of the variance of the parametric images obtained from wavelet filters the technique is proposed for a class of estimators that have been shown to be useful in neuroimaging studies the techniques are demonstrated for both functional magnetic resonance imaging fmri and positron emission tomography pet data sets

a linear wavelet filter for parametric imaging with dynamic pet describes a new filter for parametric images obtained from dynamic positron emission tomography pet studies the filter is based on the wavelet transform following the heuristics of a previously published method that are here developed into a rigorous theoretical framework it is shown that the space time problem of modeling a dynamic pet sequence reduces to the classical one of estimation of a normal multivariate vector of independent wavelet coefficients that under least squares risk can be solved by straightforward application of well established theory from the study of the distribution of wavelet coefficients of pet images it is inferred that a james stein linear estimator is more suitable for the problem than traditional nonlinear procedures that are incorporated in standard wavelet filters this is confirmed by the superior performance of the james stein filter in simulation studies compared to a state of the art nonlinear wavelet filter and a nonstationary filter selected from literature finally the formal framework is interpreted for the practitioner s point of view and advantages and limitations of the method are discussed

a study on statistically reliable and computationally efficient algorithms for generating local cerebral blood flow parametric images with positron emission tomography with the advent of positron emission tomography pet a variety of techniques have been developed to measure local cerebral blood flow lcbf noninvasively in humans a potential class of techniques which includes linear least squares ls linear weighted least squares wls linear generalized least squares gls and linear generalized weighted least squares gwls is proposed the statistical characteristics of these methods are examined by computer simulation the authors present a comparison of these four methods with two other rapid estimation techniques developed by huang et al 1982 and alpert 1984 and two classical methods the unweighted and weighted nonlinear least squares regression the results show that these methods can take full advantage of the contribution from the fine temporal sampling data of modern tomographs and thus provide statistically reliable estimates that are comparable to those obtained from nonlinear ls regression these methods also have high computational efficiency and the parameters can be estimated directly from operational equations in one single step therefore they can potentially be used in image wide estimation of local cerebral blood flow and distribution volume with pet

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

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

an evaluation of the algorithms for determining local cerebral metabolic rates of glucose using positron emission tomography dynamic data measurement of the local cerebral metabolic rate of glucose lcmrglc and the individual rate constant parameters of the sup 18 f 2 fluoro 2 deoxy d glucose fdg model can provide a clearer understanding and insight to the physiological processes in the human brain and a quicker and more accurate means of diagnosis in clinical applications a systematic study using simulated and clinical tissue time activity data is presented to evaluate several existing and newly developed major algorithms used for determining lcmrglc and the individual rate constants from positron emission tomography dynamic data the computational and statistical properties of the autoradiographic approach weighted and unweighted nonlinear least squares methods patlak graphic approach weighted integration method linear least squares and generalized linear least squares methods are investigated and discussed in this paper

a perturbation method for dark solitons based on a complete set of the squared jost solutions a perturbation method for dark solitons is developed which is based on the construction and the rigorous proof of the complete set of squared jost solutions the general procedure solving the adiabatic solution of perturbed nonlinear schrodinger equation the time evolution equation of dark soliton parameters and a formula for calculating the first order correction are given the method can also overcome the difficulties resulting from the non vanishing boundary condition

improved signal to noise ratio in parametric images by cluster analysis abstract parametric images are formed by analyzing the concentration history of every voxel in pet data sets because pet concentration data at the voxel level are rather noisy noise propagation into the parametric image is often quite noticeable to address this problem a model based clustering method has been developed to generate parametric images the basic idea of the clustering method is to average over voxels whose concentration histories have the same shape we applied the method to a two parameter k 1 k 2 compartment model of local cerebral blood flow the statistic r tc t dt c t dt te k 2 t c a t dt e k 2 t c a t dt classifies curves in terms of k 2 where c t and c a t denote the tissue and blood concentration histories respectively and is the convolution operator simulation studies of noise propagation in the clustering statistic showed that 30 voxel noise yielded a 2 standard deviation in r parametric images of blood flow and partition coefficient were computed for an o 15 study with and without clustering cluster size affected bias statistical precision and computation time with clusters of 400 voxels the variance of the flow parameter was around 1 50 smaller with clustering with negligible bias and a computation time of 30 s on a 64 mhz workstation for 15 128 128 images with matlab 5 1

global two stage filtering of clinical pet parametric maps application to 11 c r pk11195 abstract introduction in positron emission tomography pet quantification of physiological parameters at the voxel level may result in unreliable estimates due to the high noise of voxel time activity curves global two stage gts an estimation technique belonging to the group of population approaches can be used to tackle this problem gts was previously tested on simulated pet data and yielded substantial improvements when compared to standard estimation approaches such as weighted nonlinear least squares wnlls and basis function method bfm in this work gts performance is assessed in a clinical context using the neuroinflammation marker 11 c r pk11195 applied to a cohort of huntington s disease hd patients with and without symptoms materials and methods parametric maps of binding potential bp nd of 12 normal controls nc 9 symptomatic and 9 presymptomatic hd patients were generated by applying a modified reference tissue model that accounts for tracer vascular activity in both reference and target tissues srtmv gts was then applied to srtmv maps and its performance compared with that of srtmv three smoothed versions of srtmv obtained by filtering the original srtmv maps with gaussian filters of 3 mm 5 mm and 7 mm full width half maximum fwhm were also included in the comparison since striatal degeneration is the hallmark of hd sensitivity was assessed for all methods by computing the mean of z scores in caudate putamen and globus pallidus in the voxel by voxel statistical comparison of bp nd between hd and nc results application of gts to parametric maps brought a substantial qualitative improvement to srtmv maps to the extent that anatomical structures often became visible in addition most parameter estimates that were outside the physiological range with srtmv were corrected by gts gts yielded a 2 3 fold increase in sensitivity with respect to srtmv for the symptomatic cohort mean of striatal z scores of 0 76 for srtmv and 1 79 for gts and an even more substantial increase for the presymptomatic cohort mean of striatal z scores of 0 34 for srtmv and 0 96 for gts the sensitivity of gts was similar to the one obtained with a filter of 7 mm fwhm applied to the initial srtmv maps but gts images were not characterized by the notable loss of resolution typical of smoothed maps gts additionally does not require to change define settings according to the tracer and level of noise whereas the choice of the fwhm value of the gaussian filter normally employed in the smoothing procedure is typically arbitrary conclusions gts is a powerful and robust tool for improving the quality of parametric maps in pet the method is particularly appealing in that it can be applied to any tracer and estimation method provided that initial estimates of the parameter vector and of its covariance are available although the benefits of gts are far from being exhaustively assessed the significant improvements obtained both on real and simulated data suggest that it could become an important tool for dynamic pet in the future

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

kinetic modelling using basis functions derived from two tissue compartmental models with a plasma input function general principle and application to 18f fluorodeoxyglucose positron emission tomography abstract a kinetic modelling method for the determination of influx constant k i is given that utilises basis functions derived from plasma input two tissue compartmental models bafpic two forms of the basis functions are given bafpic i with k 4 0 no product loss and bafpic r with k 4 non zero simulations were performed using literature rate constant values for 18 f fluorodeoxyglucose fdg in both normal and abnormal brain pathology both homogeneous and heterogeneous tissues were simulated and this data was also used as input for other methods commonly used to determine k i non linear least squares compartmental modelling nlls autoradiographic method and patlak gjedde graphical analysis pga the four methods were also compared for real fdg positron emission tomography pet data for both k 4 0 and k 4 non zero simulated data bafpic had the best bias properties of the four methods the autoradiographic method was always the best for variability but bafpic i had lower variability than pga and nlls for non zero k 4 data the variance of bafpic r was inferior to pga but still significantly superior to nlls k i maps calculated from real data substantiate the simulation results with bafpic i having lower noise than pga voxel k i values from bafpic i correlated well with those from pga r 2 0 989 bafpic is easy to implement and combines low bias with good noise properties for voxel wise determination of k i for fdg bafpic is suitable for determining k i for other tracers well characterised by a serial two tissue compartment model and has the advantage of also producing values for individual kinetic constants and blood volume