measuring time frequency information content using the renyi entropies the generalized entropies of renyi inspire new measures for estimating signal information and complexity in the time frequency plane when applied to a time frequency representation tfr from cohen s class or the affine class the renyi entropies conform closely to the notion of complexity that we use when visually inspecting time frequency images these measures possess several additional interesting and useful properties such as accounting and cross component and transformation invariances that make them natural for time frequency analysis this paper comprises a detailed study of the properties and several potential applications of the renyi entropies with emphasis on the mathematical foundations for quadratic tfrs in particular for the wigner distribution we establish that there exist signals for which the measures are not well defined

temporal autocorrelation in univariate linear modeling of fmri data in functional magnetic resonance imaging statistical analysis there are problems with accounting for temporal autocorrelations when assessing change within voxels techniques to date have utilized temporal filtering strategies to either shape these autocorrelations or remove them shaping or coloring attempts to negate the effects of not accurately knowing the intrinsic autocorrelations by imposing known autocorrelation via temporal filtering removing the autocorrelation or prewhitening gives the best linear unbiased estimator assuming that the autocorrelation is accurately known for single event designs the efficiency of the estimator is considerably higher for prewhitening compared with coloring however it has been suggested that sufficiently accurate estimates of the autocorrelation are currently not available to give prewhitening acceptable bias to overcome this we consider different ways to estimate the autocorrelation for use in prewhitening after high pass filtering is performed a tukey taper set to smooth the spectral density more than would normally be used in spectral density estimation performs best importantly estimation is further improved by using nonlinear spatial filtering to smooth the estimated autocorrelation but only within tissue type using this approach when prewhitening reduced bias to close to zero at probability levels as low as 1 x 10 5

a computational method for solving the branched channel flow problem in the present work a method is suggested to find the solution of poisson s equation with dirichlet boundary conditions in polar coordinates this method is expressed over the unit circle and consequently over the upper half plane from which geometry we find the solution to poisson s equation over the branched channel domain the results seems very reasonable and the derived transformation from unit circle to y shape channel is accurate poisson s equation can be solved similarly over much other geometry the solution looks very encouraging and resembles that is experimentally known

robust identification from band limited data consider the problem of identifying a scalar bounded input bounded output stable transfer function from pointwise measurements at frequencies within a bandwidth we propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity the noise level to zero and asymptotically meeting some gauge constraint outside error bounds are derived and the procedure is illustrated by numerical experiments

unwarping of unidirectionally distorted epi images echo planar imaging epi is a fast nuclear magnetic resonance imaging mri method unfortunately local magnetic field inhomogeneities induced mainly by the subject s presence cause significant geometrical distortion predominantly along the phase encoding direction which must be undone to allow for meaningful further processing so far this aspect has been too often neglected in this paper the authors suggest a new approach using an algorithm specifically developed for the automatic registration of distorted epi images with corresponding anatomically correct mri images they model the deformation field with splines which gives us a great deal of flexibility while comprising the affine transform as a special case the registration criterion is least squares interestingly the complexity of its evaluation does not depend on the resolution of the control grid the spline model gives the authors good accuracy thanks to its high approximation order the short support of splines leads to a fast algorithm a multiresolution approach yields robustness and additional speedup the algorithm was tested on real as well as synthetic data and the results were compared with a manual method a wavelet based sobolev type random deformation generator was developed for testing purposes a blind test indicates that the proposed automatic method is faster more reliable and more precise than the manual one

a spectral collocation method for two dimensional compressible convection abstract a fourier chebyshev spectral simulation of two dimensional compressible convection is presented the fluid is a perfect gas with constant dynamic viscosity and thermal conductivity both slippery and rigid boundary conditions for the velocity are used here the temperature is maintained fixed at the upper and lower boundaries an explicit adams bashforth predictor corrector numerical scheme is used in order to overcome the courant friedriechs lewy condition the nonlinear diffusion terms are handled by an iterative method with finite differences or spectral preconditioning steady state solutions have been obtained for both types of boundary conditions critical exponents are found to be the same as in the incompressible case at least for a weak value of the stratification parameter

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

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

activation detection in fmri using a maximum energy ratio statistic obtained by adaptive spatial filtering an adaptive spatial filtering method is proposed that takes into account contextual information in fmri activation detection this filter replaces the time series of each voxel with a weighted average of time series of a small neighborhood around it the filter coefficients at each voxel are derived so as to maximize a test statistic designed to indicate the presence of activation this statistic is the ratio of the energy of the filtered time series in a signal subspace to the energy of the residuals it is shown that the filter coefficients and the maximum energy ratio can be found through a generalized eigenproblem this approach equates the filter coefficients to the elements of an eigenvector corresponding to the largest eigenvalue of a specific matrix while the largest eigenvalue itself becomes the maximum energy ratio that can be used as a statistic for detecting activation the distribution of this statistic under the null hypothesis is derived by a nonparametric permutation technique in the wavelet domain also in this paper we introduce a new set of basis vectors that define the signal subspace the space spanned by these basis vectors covers a wide range of possible hemodynamic response functions hrf and is applicable to both event related and block design fmri signal analysis this approach circumvents the need for a priori assumptions about the exact shape of the hrf resting state experimental fmri data were used to assess the specificity of the method showing that the actual false alarm rate of the proposed method is equal or less than its expected value analysis of simulated data and motor task fmri datasets from six volunteers using the method proposed here showed an improved sensitivity as compared to a conventional test with a similar statistic applied to spatially smoothed data

the stability of pseudospectral chebyshev methods abstract the stability of pseudospectral chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients the choice of collocation points is discussed numerical examples are given for the case of variable coefficient hyperbolic equations author

towards a statistical test for functional connectivity dynamics abstract sliding window correlation is an emerging method for mapping time resolved resting state functional connectivity to avoid mapping spurious connectivity fluctuations false positives leonardi and van de ville recently recommended choosing a window length exceeding the longest wavelength composing the bold signal usually assumed to be 100 s here we provide further statistical support for this rule of thumb however we demonstrate that non stationary fluctuations in functional connectivity can in theory be detected with much shorter window lengths e g 40 s while maintaining nominal control of false positives we find that statistical power is near maximal for window lengths chosen according to leonardi and van de ville s rule of thumb furthermore we lay some foundations for a parametric test to identify non stationary fluctuations in functional connectivity also noting limitations of the sinusoidal model upon which our work and the work of leonardi and van de ville is based most notably our analytical results pertain to covariances as does our statistical test whereas functional connectivity is more commonly measured using correlations

the multiscale character of evoked cortical activity abstract both the architecture and the dynamics of the brain have characteristic features at different spatial scales however the existence nature and function of dynamical interdependencies between such scales have not been investigated we studied the multiscale properties of functional magnetic resonance imaging fmri data acquired while human subjects viewed a visual image traditional region of interest analysis of this data set revealed evoked activity in primary and extrastriate visual cortex wavelet transform in the spatial domain provides a multiscale representation of this evoked brain activity studying the correlation structure of this representation revealed strong and novel interdependencies in these data within and between different spatial scales we found that such correlations are stronger than those evident in the original data and comparable in magnitude to those obtained after gaussian smoothing however analysis of the data in the wavelet domain revealed additional structure such as positive correlations strong anti correlations and phase lagged interdependencies statistical significance of these effects was inferred through nonparametric bootstrap techniques we conclude that the spatial analysis of functional neuroimaging data in the wavelet domain provides novel information which may reflect complex spatiotemporal neuronal activity and information encoding it also affords a quantitative means of testing hierarchical and multiscale models of cortical activity

wavelet generalized least squares a new blu estimator of linear regression models with 1 f errors long memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors classical autoregressive moving average arma methods can adequately address the problem of linear time invariant short memory errors but may be inefficient and or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function here we introduce a novel method called wavelet generalized least squares wls which is to a good approximation the best linear unbiased blu estimator of regression model parameters in the context of long memory errors the method also provides maximum likelihood ml estimates of the hurst exponent which can be readily translated to the fractal dimension or spectral exponent characterizing the correlational structure of the errors and the error variance the algorithm exploits the whitening or kar

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

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

analysis of fmri time series revisited again friston et al 1995 neuroimage 2 45 53 presented a method for detecting activations in fmr1 time series based on the general linear model and a heuristic analysis of the effective degrees of freedom in this communication we present corrected results that replace those of the previous paper and solve the same problem without recourse to heuristic arguments specifically we introduce a proper and unbiased estimator for the error terms and provide a more generally correct expression for the effective degrees of freedom the previous estimates of error variance were biased and in some instances could have led to a 10 20 overestimate of z values although the previous results are almost correct for the random regressors chosen for validation the present theoretical results are exact for any covariate or waveform we comment on some aspects of experimental design and data analysis in the light of the theoretical framework discussed here

fractional gaussian noise functional mri and alzheimer s disease fractional gaussian noise fgn provides a parsimonious model for stationary increments of a self similar process parameterised by the hurst exponent h and variance sigma2 fractional gaussian noise with h 0 5 demonstrates 1 f long memory or persistent behaviour and the special case of fgn with h 0 5 corresponds to classical gaussian white noise we comparatively evaluate four possible estimators of fgn parameters one method implemented in the time domain and three in the wavelet domain we show that a wavelet based maximum likelihood ml estimator yields the most efficient estimates of h and sigma2 in simulated fgn with 0 0 5 tend to be concentrated in cortex whereas voxels with h 0 5 are more frequently located in ventricles and sulcal csf the wavelet ml estimator can be generalised to estimate the parameter vector beta for general linear modelling glm of a physiological response to experimental stimulation and we demonstrate nominal type i error control in multiple testing of beta divided by its standard error in simulated and biological data under the null hypothesis beta 0 we illustrate these methods principally by showing that there are significant differences between patients with early alzheimer s disease ad and age matched comparison subjects in the persistence of fgn in the medial and lateral temporal lobes insula dorsal cingulate medial premotor cortex and left pre and postcentral gyrus patients with ad had greater persistence of resting fmri noise larger h in these regions comparable abnormalities in the ad patients were also identified by a permutation test of local differences in the first order autoregression ar 1 coefficient which was significantly more positive in patients however we found that the hurst exponent provided a more sensitive metric than the ar 1 coefficient to detect these differences perhaps because neurophysiological changes in early ad are naturally better described in terms of abnormal salience of long memory dynamics than a change in the strength of association between immediately consecutive time points we conclude that parsimonious mapping of fmri noise properties in terms of fgn parameters efficiently estimated in the wavelet domain is feasible and can enhance insight into the pathophysiology of alzheimer s disease

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

denoising functional mr images a comparison of wavelet denoising and gaussian smoothing we present a general wavelet based denoising scheme for functional magnetic resonance imaging fmri data and compare it to gaussian smoothing the traditional denoising method used in fmri analysis one dimensional wavelab thresholding routines were adapted to two dimensional 2 d images and applied to 2 d wavelet coefficients to test the effect of these methods on the signal to noise ratio snr we compared the snr of 2 d fmri images before and after denoising using both gaussian smoothing and wavelet based methods we simulated a fmri series with a time signal in an active spot and tested the methods on noisy copies of it the denoising methods were evaluated in two ways by the average temporal snr inside the original activated spot and by the shape of the spot detected by thresholding the temporal snr maps denoising methods that introduce much smoothness are better suited for low snrs but for images of reasonable quality they are not preferable because they introduce heavy deformations wavelet based denoising methods that introduce less smoothing preserve the sharpness of the images and retain the original shapes of active regions we also performed statistical parametric mapping on the denoised simulated time series as well as on a real fmri data set false discovery rate control was used to correct for multiple comparisons the results show that the methods that produce smooth images introduce more false positives the less smoothing wavelet based methods although generating more false negatives produce a smaller total number of errors than gaussian smoothing or wavelet based methods with a large smoothing effect

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

variable precision registration via wavelets optimal spatial scales for inter subject registration of functional mri abstract the detection of significantly activated brain regions in multi subject functional magnetic resonance imaging fmri studies almost invariably entails the coregistration of individual subjects data in a standard space here we investigate how sensitivity to detect loci of generic activation in such studies may be conditioned by the precision of anatomical registration we describe a novel algorithm implemented in the wavelet domain for inhomogeneous deformation of individual images to match a template the algorithm matches anatomical features in a coarse to fine fashion first minimising a cost function in terms of relatively coarse spatial features and then proceeding iteratively to match the images in terms of progressively more detailed anatomical features applying the method to data acquired from two groups of 12 healthy volunteers with mean age 27 and 70 years respectively during performance of a paired associate learning task we show that geometrical overlap between template and individual images is monotonically improved compared to an affine transform by additional inhomogeneous deformations informed by more detailed features likewise sensitivity to detect activated voxels can be substantially improved by a factor of 4 or more if wavelet mediated deformations informed by medium sized anatomical features are applied in addition to a preliminary affine transform however sensitivity to detect activated voxels was reduced by over registering data or matching anatomical features at the finest scales of the wavelet transform the benefits of variable precision registration are particularly salient for data acquired in older subjects which showed evidence of greater inter subject anatomic variability and generally required more extensive local deformation to achieve a satisfactory match to the template image we conclude that major benefits in sensitivity to detect functional activation in multi subject fmri studies can be attained with an inhomogeneous deformation applied over appropriate spatial scales

wavelet based estimation for seasonal long memory processes we introduce the multiscale analysis of seasonal persistent processes that is time series models with a singularity in their spectral density function at one or more frequencies in 0 1 2 the discrete wavelet packet transform dwpt and a nondecimated version of it known as the maximal overlap dwpt modwpt are introduced as alternative methods to fourier based techniques for analyzing time series that exhibit seasonal long memory the approximate log linear relationship between the wavelet packet variance and frequency is used to produce a least squares estimator of the fractional difference parameter approximate maximum likelihood estimation is performed by replacing the variance covariance matrix with a diagonalized matrix based on the dwpt simulations are performed to compare the wavelet based techniques with the spectral estimate based techniques for both least squares and maximum likelihood procedures an application of this methodology to atmospheric and economic time series is used for demon

sparse code shrinkage denoising of nongaussian data by maximum likelihood estimation sparse coding is a method for finding a representation of data in which each of the components of the representation is only rarely significantly active such a representation is closely related to redundancy reduction and independent component analysis and has some neurophysiological plausibility in this article we show how sparse coding can be used for denoising using maximum likelihood estimation of nongaussian variables corrupted by gaussian noise we show how to apply a soft thresholding shrinkage operator on the components of sparse coding so as to reduce noise our method is closely related to the method of wavelet shrinkage but it has the important benefit over wavelet methods that the representation is determined solely by the statistical properties of the data the wavelet representation on the other hand relies heavily on certain mathematical properties like self similarity that may be only weakly related to the properties of natural data

a method to produce evolving functional connectivity maps during the course of an fmri experiment using wavelet based time varying granger causality functional magnetic resonance imaging fmri is widely used to identify neural correlates of cognitive tasks however the analysis of functional connectivity is crucial to understanding neural dynamics although many studies of cerebral circuitry have revealed adaptative behavior which can change during the course of the experiment most of contemporary connectivity studies are based on correlational analysis or structural equations analysis assuming a time invariant connectivity structure in this paper a novel method of continuous time varying connectivity analysis is proposed based on the wavelet expansion of functions and vector autoregressive model wavelet dynamic vector autoregressive dvar the model also allows identification of the direction of information flow between brain areas extending the granger causality concept to locally stationary processes simulation results show a good performance of this approach even using short time intervals the application of this new approach is illustrated with fmri data from a simple ab motor task experiment

on stationarizability for nonstationary 2 d random fields using discrete wavelet transforms the emphasis in this article is on the study of nonstationary two dimensional 2 d random fields with wide sense stationary increments wide sense stationary jumps and 2 d fractional brownian motion fbm fields the effort made in this work is to develop a realizable method of stationarization provided for nonstationary 2 d random fields we also present the correlation functions of the discrete wavelet transform relating to 2 d fbm fields that will decay hyperbolically fast

multidimensional scaling of integrated neurocognitive function and schizophrenia as a disconnexion disorder multidimensional scaling mds is a multivariate statistical technique that can be used to define sub systems of functionally connected brain regions based on the analysis of functional magnetic resonance im aging fmri data here we introduce three way mul tidimensional scaling as a method for the analysis of a group of fmri data which yields both a generic inter regional configuration in low dimensional space and a measure of each individual s deviation from the ge neric configuration the distance between two generic interregional configurations obtained by mds of two groups of data can be minimized by generalized pro crustes analysis and the probability under the null hypothesis that the two groups are sampled from the same population of any residual group difference in interregional configurations can be assessed by a per mutation test these methods are developed and ap plied to activated fmri time series acquired from 19 patients with schizophrenia and 20 normal compari son subjects during the performance of a semantic categorization and subvocal rehearsal task the first three scaling dimensions are interpretable in terms of the major anatomical or functional subsystems of the activated system left right input processing other and subvocal output other we found no sig nificant global or local differences between groups in interregional configurations in this 3d space how ever there was significantly greater variability of interregional configurations within the group of patients with schizophrenia the implications for schizophrenia as a disconnexion disorder are

bivariate shrinkage functions for wavelet based denoising exploiting interscale dependency most simple nonlinear thresholding rules for wavelet based denoising assume that the wavelet coefficients are independent however wavelet coefficients of natural images have significant dependencies we only consider the dependencies between the coefficients and their parents in detail for this purpose new non gaussian bivariate distributions are proposed and corresponding nonlinear threshold functions shrinkage functions are derived from the models using bayesian estimation theory the new shrinkage functions do not assume the independence of wavelet coefficients we show three image denoising examples in order to show the performance of these new bivariate shrinkage rules in the second example a simple subband dependent data driven image denoising system is described and compared with effective data driven techniques in the literature namely visushrink sureshrink bayesshrink and hidden markov models in the third example the same idea is applied to the dual tree complex wavelet coefficients

validating cluster size inference random field and permutation methods cluster size tests used in analyses of brain images can have more sensitivity compared to intensity based tests the random field rf theory has been widely used in implementation of such tests however the behavior of such tests is not well understood especially when the rf assumptions are in doubt in this paper we carried out a simulation study of cluster size tests under varying smoothness thresholds and degrees of freedom comparing rf performance to that of the permutation test which is known to be exact for gaussian images we find that the rf methods are generally conservative especially for low smoothness and low threshold for t images the rf tests are found to be conservative at lower thresholds and do not perform well unless the threshold is high and images are sufficiently smooth the permutation test performs well for any settings though the discreteness in cluster size must be accounted for we make specific recommendations on when permutation tests are to be preferred to rf tests

conditions for optimality over h the fundamental optimization problem in worst case frequency domain design where stability is the key constraint is left text opt a n right qquad begin array 20 c text given gamma text a text map text from mathbb t times mathbb c n text to mathbb r text find gamma 0 text and f in a n text such text that gamma mathop inf limits f in a n mathop sup limits theta gamma left e i theta f left e i theta right right mathop sup limits theta gamma left e i theta f left e i theta right right end array here a n denotes the mathbb c n valued functions on the unit circle mathbb t with analytic continuation to the unit disk mathbb d that are continuous on the closed unit disk the special case where the sublevel sets of gamma in z are disks in mathbb c n is the main mathematical problem in the area called h

a direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

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

spatiotemporal nonlinearity in resting state fmri of the human brain in this work the spatiotemporal nonlinearity in resting state fmri data of the human brain was detected by nonlinear dynamics methods nine human subjects during resting state were imaged using single shot gradient echo planar imaging on a 1 5t scanner eigenvalue spectra for the covariance matrix correlation dimensions and spatiotemporal lyapunov exponents were calculated to detect the spatiotemporal nonlinearity in resting state fmri data by simulating adjusting and comparing the eigenvalue spectra of pure correlated noise with the corresponding real fmri data the intrinsic dimensionality was estimated the intrinsic dimensionality was used to extract the first few principal components from the real fmri data using principal component analysis which will preserve the correct phase dynamics while reducing both computational load and noise level of the data then the phase space was reconstructed using the time delay embedding method for their principal components and the correlation dimension was estimated by the grassberger procaccia algorithm of multiple variable series the spatiotemporal lyapunov exponents were calculated by using the method based on coupled map lattices through nonlinearity testing there are significant differences of correlation dimensions and spatiotemporal lyapunov exponents between fmri data and their surrogate data the fractal dimension and the positive spatiotemporal lyapunov exponents characterize the spatiotemporal nonlinear dynamics property of resting state fmri data therefore the results suggest that fluctuations presented in resting state may be an inherent model of basal neural activation of human brain cannot be fully attributed to noise

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

to smooth or not to smooth bias and efficiency in fmri time series analysis this paper concerns temporal filtering in fmri time series analysis whitening serially correlated data is the most efficient approach to parameter estimation however if there is a discrepancy between the assumed and the actual correlations whitening can render the analysis exquisitely sensitive to bias when estimating the standard error of the ensuing parameter estimates this bias although not expressed in terms of the estimated responses has profound effects on any statistic used for inference the special constraints of fmri analysis ensure that there will always be a misspecification of the assumed serial correlations one resolution of this problem is to filter the data to minimize bias while maintaining a reasonable degree of efficiency in this paper we present expressions for efficiency of parameter estimation and bias in estimating standard error in terms of assumed and actual correlation structures in the context of the general linear model we show that i whitening strategies can result in profound bias and are therefore probably precluded in parametric fmri data analyses ii band pass filtering and implicitly smoothing has an important role in protecting against inferential bias

a constrained approximation problem arising in parameter identification abstract we pose and solve an extremal problem in the hardy class h 2 of the disc involving the best approximation of a function on a subarc of the circle by a h 2 function subject to a constraint on its imaginary part on the complementary arc a constructive algorithm is presented for the computation of such a best approximant and the method is illustrated by a numerical example the whole problem is motivated by boundary parameter identification problems arising in non destructive control

signal processing with fractals a wavelet based approach wavelet transmission statistically self similar signals detection and estimation with 1 processes deterministically self similar signals fractal modulation linear self similar signals

a multi domain spectral computation of three dimensional laminar horseshoe vortex flow using incompressible navier stokes equations abstract the laminar horseshoe vortex flow at a strut wall intersection is numerically simulated by direct solution of the 3 dimensional incompressible time dependent navier stokes equations using multi domain spectral methods flow dependence in the direction normal to the wall is represented by direct expansion in chebyshev polynomials thus yielding good resolution of the viscous layer at the endwall however spatial discretization of flow dependence on the plane parallel to the endwall is done via spectral element method such implementation offers the advantages of high order accuracy minimal dispersion errors a degree of geometrical flexibility and parallelism the horseshoe vortical flow is characterized by 3 dimensionality unsteadiness and a reynolds number that is large but having viscous effects which are critical to the flow evolution time evolution is implemented using a fractional time stepping technique results at low and moderate reynolds numbers will be presented to demonstrate the application of the method in these computations a simple extrapolation outflow boundary condition is used difficulty associated with the imposition of the outflow boundary conditions especially in the endwall flow region near the outflow boundary is discussed

the role of exchangeability in inference abstract this paper is concerned with basic problems of statistical inference the thesis is in three parts 1 that inference is a procedure whereby one passes from a population or sample to a new individual 2 that this connection can be established using de finetti s idea of exchangeability or fisher s concept of a subpopulation and 3 in making the connection use must be made of the appropriate probability these three principles are used in a variety of situations and the topics discussed include analysis of variance and covariance contingency tables and calibration some comments on randomization are also included author

wavelet based multifractal analysis of fmri time series functional magnetic resonance imaging fmri time series are investigated with a multifractal method based on the wavelet modulus maxima wtmm method to extract local singularity fractal exponents the spectrum of singularity exponents of each fmri time series is quantified by spectral characteristics including its maximum and the corresponding dimension we found that the range of holder exponents in voxels with activation is close to 1 whereas exponents are close to 0 5 in white matter voxels without activation the maximum dimension decreases going from white matter to gray matter and is lower still for activated time series the full width at half maximum of the spectra is higher in activated areas the proposed method becomes particularly effective when combining these spectral characteristics into a single parameter using these multifractal parameters it is possible to identify activated areas in the human brain in both hybrid and in vivo fmri data sets without knowledge of the stimulation paradigm applied

hardy spaces of the conjugate beltrami equation we study hardy spaces of solutions to the conjugate beltrami equation with lipschitz coefficient on dini smooth simply connected planar domains in the range of exponents 1 infty we analyse their boundary behaviour and certain density properties of their traces we derive on the way an analog of the fatou theorem for the dirichlet and neumann problems associated with the equation mbox div sigma nabla u 0 with l p boundary data

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

a wavelet method for the characterization of spatiotemporal patterns this paper introduces a wavelet based method for the characterization of spatiotemporal patterns based on the wavelet multiresolution analysis two wavelet indices multiscale accumulative density mad and multiscale accumulative change mac are proposed for the characterization of the dynamics of the spatiotemporal patterns both indices are constructed by using orthogonal wavelet projection operators the mad is a measure of the spatial complexity of a pattern at a given time whereas the mac characterizes the spatial complexity of instantaneous change of the spatiotemporal patterns at a given time the ratio of the mad indices between the lowest and the highest scales reflects the order of coherence in a pattern the time series of both mad and mac provide the dynamical information of morphological pattern evolutions numerical experiments based on the cahn hilliard equation indicate that the proposed method is efficient for quantitatively characterizing the dynamics of the spatiotemporal patterns 2002 published by elsevier science b v

on the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes in this correspondence we develop an asymptotic theory for the correlations of wavelet coefficients of the discrete wavelet transform dwt for fractionally differenced processes it provides a theoretical justification for the approximate decorrelation property of the dwt for fractionally differenced processes in addition it provides insights on how the length of the wavelet filter affects the within scale correlations and the between scale correlations differently for within scale correlations increasing the length of the wavelet filter increases the rate of decay as the two wavelet coefficients get further apart while for between scale correlations using a wavelet filter that is long enough can reduce the between scale correlations even for wavelet coefficients that are close together

multiple hypothesis mapping of functional mri data in orthogonal and complex wavelet domains we are interested in methods for multiple hypothesis testing that optimize power to refute the null hypothesis while controlling the false discovery rate fdr the wavelet transform of a spatial map of brain activation statistics can be tested in two stages to achieve this objective first a set of possible wavelet coefficients to test is reduced and second each hypothesis in the remaining subset is formally tested we show that a bayesian bivariate shrinkage operator baybishrink for the first step provides a powerful and expedient alternative to a subband adaptive chi squared test or an enhanced fdr algorithm based on the generalized degrees of freedom we also investigate the dual tree complex wavelet transform cwt as an alternative basis to the orthogonal discrete wavelet transform dwt we design and validate a test for activation based on the magnitude of the complex wavelet coefficients and show that this confers improved specificity for mapping spatial signals the methods are applied to simulated and experimental data including a pharmacological magnetic resonance imaging mri study we conclude that using baybishrink to define a reduced set of complex wavelet coefficients and testing the magnitude of each complex pair to control the fdr represents a competitive solution for multiple hypothesis mapping in fmri

on bias in the estimation of autocorrelations for fmri voxel time series analysis for fmri time series analysis to be statistically valid it is important to deal correctly with temporal autocorrelation in the noise most of the approaches in the literature adopt a two stage approach in which the autocorrelation structure is estimated using the residuals of an initial model fit this estimate is then used to prewhiten the data and the model before the model is refit to obtain final activation parameter estimates an assumption implicit in this scheme is that the residuals from the initial model fit represent a realization of the true noise process in general this assumption will not be correct as certain components of the noise will be removed by the model fit in this paper we examine i the form of the bias induced by the initial model fit ii methods of correcting for the bias and iii the impact of bias correction on the model parameter estimates we find that while bias correction does result in more accurate estimates of the correlation structure this does not translate into improved estimates of the model parameters in fact estimates of the model parameters and their standard errors are seen to be so accurate that we conclude that bias correction is unnecessary

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

determining significant connectivity by 4d spatiotemporal wavelet packet resampling of functional neuroimaging data an active area of neuroimaging research involves examining functional relationships between spatially remote brain regions when determining whether two brain regions exhibit significant correlation due to true functional connectivity one must account for the background spatial correlation inherent in neuroimaging data we define background correlation as spatiotemporal correlation in the data caused by factors other than neurophysiologically based functional associations such as scanner induced correlations and image preprocessing we develop a 4d spatiotemporal wavelet packet resampling method which generates surrogate data that preserves only the average background spatial correlation within an axial slice across axial slices and through each voxel time series while excluding the specific correlations due to true functional relationships we also extend an amplitude adjustment algorithm which adjusts our surrogate data to closely match the amplitude distribution of the original data our method improves upon existing wavelet based methods and extends them to 4d we apply our resampling technique to determine significant functional connectivity from resting state and motor task fmri datasets

1 f noise in diffuse optical imaging and wavelet based response estimation in diffuse optical imaging doi data analysis the functional response is contaminated with physiological noise as in functional magnetic resonance imaging fmri in this work we extend a previously proposed method for fmri to estimate the parameters of a linear model of doi time series the regression is performed in the wavelet domain to infer drift coefficients at different scales and to estimate the strength of the hemodynamic response function hrf this multiresolution approach benefits from the whitening property of the discrete wavelet transform dwt which approximately decorrelates long memory noise processes we also show that a more accurate estimation is obtained by removing some regressors correlating with the protocol moreover we observe that this improvement is related to a quantitative measure of 1 f noise the performances of the method are first evaluated against a standard spline cosine drift approach with simulated hrf and real background physiology lastly the technique is applied to experimental event related data acquired by near infrared spectroscopy nirs

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

global voxel and cluster tests by theory and permutation for a difference between two groups of structural mr images of the brain the authors describe almost entirely automated procedures for estimation of global voxel and cluster level statistics to test the null hypothesis of zero neuroanatomical difference between two groups of structural magnetic resonance imaging mri data theoretical distributions under the null hypothesis are available for 1 global tissue class volumes 2 standardized linear model analysis of variance anova and ancova coefficients estimated at each voxel and 3 an area of spatially connected clusters generated by applying an arbitrary threshold to a two dimensional 2 d map of normal statistics at voxel level the authors describe novel methods for economically ascertaining probability distributions under the null hypothesis with fewer assumptions by permutation of the observed data nominal type i error control by permutation testing is generally excellent whereas theoretical distributions may be over conservative permutation has the additional advantage that it can be used to test any statistic of interest such as the sum of suprathreshold voxel statistics in a cluster or cluster mass regardless of its theoretical tractability under the null hypothesis these issues are illustrated by application to mri data acquired from 18 adolescents with hyperkinetic disorder and 16 control subjects matched for age and gender

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

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

a comparative evaluation of wavelet based methods for hypothesis testing of brain activation maps wavelet based methods for hypothesis testing are described and their potential for activation mapping of human functional magnetic resonance imaging fmri data is investigated in this approach we emphasise convergence between methods of wavelet thresholding or shrinkage and the problem of hypothesis testing in both classical and bayesian contexts specifically our interest will be focused on the trade off between type i probability error control and power dissipation estimated by the area under the roc curve we describe a technique for controlling the false discovery rate at an arbitrary level of error in testing multiple wavelet coefficients generated by a 2d discrete wavelet transform dwt of spatial maps of fmri time series statistics we also describe and apply change point detection with recursive hypothesis testing methods that can be used to define a threshold unique to each level and orientation of the 2d dwt and bayesian methods incorporating a formal model for the anticipated sparseness of wavelet coefficients representing the signal or true image the sensitivity and type i error control of these algorithms are comparatively evaluated by analysis of null images acquired with the subject at rest and an experimental data set acquired from five normal volunteers during an event related finger movement task we show that all three wavelet based algorithms have good type i error control the fdr method being most conservative and generate plausible brain activation maps the bayesian method being most powerful we also generalise the formal connection between wavelet based methods for simultaneous multiresolution denoising hypothesis testing and methods based on monoresolution gaussian smoothing followed by statistical testing of brain activation maps

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

a wavelet based estimator of the degrees of freedom in denoised fmri time series for probabilistic testing of functional connectivity and brain graphs connectome mapping using techniques such as functional magnetic resonance imaging fmri has become a focus of systems neuroscience there remain many statistical challenges in analysis of functional connectivity and network architecture from bold fmri multivariate time series one key statistic for any time series is its effective degrees of freedom df which will generally be less than the number of time points or nominal degrees of freedom n if we know the df then probabilistic inference on other fmri statistics such as the correlation between two voxel or regional time series is feasible however we currently lack good estimators of df in fmri time series especially after the degrees of freedom of the raw data have been modified substantially by denoising algorithms for head movement here we used a wavelet based method both to denoise fmri data and to estimate the effective df of the denoised process we show that seed voxel correlations corrected for locally variable df could be tested for false positive connectivity with better control over type i error and greater specificity of anatomical mapping than probabilistic connectivity maps using the nominal degrees of freedom we also show that wavelet despiked statistics can be used to estimate all pairwise correlations between a set of regional nodes assign a p value to each edge and then iteratively add edges to the graph in order of increasing p these probabilistically thresholded graphs are likely more robust to regional variation in head movement effects than comparable graphs constructed by thresholding correlations finally we show that time windowed estimates of df can be used for probabilistic connectivity testing or dynamic network analysis so that apparent changes in the functional connectome are appropriately corrected for the effects of transient noise bursts wavelet despiking is both an algorithm for fmri time series denoising and an estimator of the effective df of denoised fmri time series accurate estimation of df offers many potential advantages for probabilistically thresholding functional connectivity and network statistics tested in the context of spatially variant and non stationary noise code for wavelet despiking seed correlational testing and probabilistic graph construction is freely available to download as part of the brainwavelet toolbox at www brainwavelet org

a general statistical analysis for fmri data we propose a method for the statistical analysis of fmri data that seeks a compromise between efficiency generality validity simplicity and execution speed the main differences between this analysis and previous ones are a simple bias reduction and regularization for voxel wise autoregressive model parameters the combination of effects and their estimated standard deviations across different runs sessions subjects via a hierarchical random effects analysis using the em algorithm overcoming the problem of a small number of runs session subjects using a regularized variance ratio to increase the degrees of freedom

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

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

non parametric model selection for subject specific topological organization of resting state functional connectivity abstract in recent years graph theory has been successfully applied to study functional and anatomical connectivity networks in the human brain most of these networks have shown small world topological characteristics high efficiency in long distance communication between nodes combined with highly interconnected local clusters of nodes moreover functional studies performed at high resolutions have presented convincing evidence that resting state functional connectivity networks exhibits exponentially truncated scale free behavior such evidence however was mostly presented qualitatively in terms of linear regressions of the degree distributions on log log plots even when quantitative measures were given these were usually limited to the r 2 correlation coefficient however the r 2 statistic is not an optimal estimator of explained variance when dealing with truncated power law models recent developments in statistics have introduced new non parametric approaches based on the kolmogorov smirnov test for the problem of model selection in this work we have built on this idea to statistically tackle the issue of model selection for the degree distribution of functional connectivity at rest the analysis performed at voxel level and in a subject specific fashion confirmed the superiority of a truncated power law model showing high consistency across subjects moreover the most highly connected voxels were found to be consistently part of the default mode network our results provide statistically sound support to the evidence previously presented in literature for a truncated power law model of resting state functional connectivity

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

spatiotemporal wavelet analysis for functional mri characterizing the spatiotemporal behavior of the bold signal in functional magnetic resonance imaging fmri is a central issue in understanding brain function while the nature of functional activation clusters is fundamentally heterogeneous many current analysis approaches use spatially invariant models that can degrade anatomic boundaries and distort the underlying spatiotemporal signal furthermore few analysis approaches use true spatiotemporal continuity in their statistical formulations to address these issues we present a novel spatiotemporal wavelet procedure that uses a stimulus convolved hemodynamic signal plus correlated noise model the wavelet fits computed by spatially constrained maximum likelihood estimation provide efficient multiscale representations of heterogeneous brain structures and give well identified parsimonious spatial activation estimates that are modulated by the temporal fmri dynamics in a study of both simulated data and actual fmri memory task experiments our new method gave lower mean squared error and seemed to result in more localized fmri activation maps compared to models using standard wavelet or smoothing techniques our spatiotemporal wavelet framework suggests a useful tool for the analysis of fmri studies

empirical analyses of bold fmri statistics i spatially unsmoothed data collected under null hypothesis conditions temporal autocorrelation spatial coherency and their effects on voxel wise parametric statistics were examined in bold fmri null hypothesis or noise datasets seventeen normal young subjects were scanned using bold fmri while not performing any time locked experimental behavior temporal autocorrelation in these datasets was described well by a 1 frequency relationship voxel wise statistical analysis of these noise datasets which assumed independence i e ignored temporal autocorrelation rejected the null hypothesis at a higher rate than specified by the nominal alpha temporal smoothing in conjunction with the use of a modified general linear model worsley and friston 1995 neuroimage 2 173 182 brought the false positive rate closer to the nominal alpha it was also found that the noise fmri datasets contain spatially coherent time signals this observed spatial coherence could not be fully explained by a continuously differentiable spatial autocovariance function and was much greater for lower temporal frequencies its presence made voxel wise test statistics in a given noise dataset dependent and thus shifted their distributions to the right or left of 0 inclusion of a global signal covariate in the general linear model reduced this dependence and consequently stabilized i e reduced the variance of dataset false positive rates

constrained approximation and interpolation in hilbert function spaces are ofimportance in systems and control and certain identification problems arenaturally solved by optimization 1 17 25 then again the study ofwxinverse 2d dirichlet neumann problems leads naturally to questions ofharmonic approximation 4 a third example is to be found in signalwxprocessing 10 17 where the paleywx wiener spaces are of crucial impor tance and again certain sampling problems can be expressed in terms ofhilbert space optimization we consider here a family of bounded extremal problems and ofinterpolation problems expressed using two distinct orthogonal decomposi tions of some hilbert space