incorporating spatial dependence into bayesian multiple testing of statistical parametric maps in functional neuroimaging abstract the analysis of functional neuroimaging data often involves the simultaneous testing for activation at thousands of voxels leading to a massive multiple testing problem this is true whether the data analyzed are time courses observed at each voxel or a collection of summary statistics such as statistical parametric maps spms it is known that classical multiplicity corrections become strongly conservative in the presence of a massive number of tests some more popular approaches for thresholding imaging data such as the benjamini hochberg step up procedure for false discovery rate control tend to lose precision or power when the assumption of independence of the data does not hold bayesian approaches to large scale simultaneous inference also often rely on the assumption of independence we introduce a spatial dependence structure into a bayesian testing model for the analysis of spms by using spms rather than the voxel time courses much of the computational burden of bayesian analysis is mitigated increased power is demonstrated by using the dependence model to draw inference on a real dataset collected in a fmri study of cognitive control the model also is shown to lead to improved identification of neural activation patterns known to be associated with eye movement tasks

limits of nonlinear discrete time control systems with fast subsystems we investigate the asymptotic properties of nonlinear discrete time control systems with fast subsystems in particular we use the averaging method in order to construct a limiting system for the slow subsystem we show continuous dependence of the slow trajectories on the perturbation parameter measuring the ratio of the different step sizes as a byproduct we get an exponential formula for the reachable set of nonlinear continuous time control systems with finite control range

efficient representation of induced currents on large scatterers using the generalized pencil of function method it is shown that the currents induced over large and smooth scatterers can often be represented by a series of complex exponentials with only a few terms the generalized pencil of function method is employed to extract these exponentials from the numerical solution of the scattering problem derived by using the method of moments illustrative examples include a partially coated two dimensional scatterer modeling a helicopter blade and the three dimensional problem of scattering by a thin plate the ultimate objective of this effort is to solve a class of large body scattering problems for arbitrary angles of incidence including the grazing angle by utilizing the numerically derived complex exponential type of basis functions extrapolated to higher frequencies it is well known that the physical optics approximation for the induced current becomes inaccurate when the angle of incidence is close to grazing and the asymptotic methods e g the gtd can become unwieldy for scatterers with edge treatments and those with complex geometrical shapes the approach based upon the use of numerically derived entire domain basis functions on a portion of the body is proposed as a means to circumventing this difficulty

classical and bayesian inference in neuroimaging theory this paper reviews hierarchical observation models used in functional neuroimaging in a bayesian light it emphasizes the common ground shared by classical and bayesian methods to show that conventional analyses of neuroimaging data can be usefully extended within an empirical bayesian framework in particular we formulate the procedures used in conventional data analysis in terms of hierarchical linear models and establish a connection between classical inference and parametric empirical bayes peb through covariance component estimation this estimation is based on an expectation maximization or em algorithm the key point is that hierarchical models not only provide for appropriate inference at the highest level but that one can revisit lower levels suitably equipped to make bayesian inferences bayesian inferences eschew many of the difficulties encountered with classical inference and characterize brain responses in a way that is more directly predicated on what one is interested in the motivation for bayesian approaches is reviewed and the theoretical background is presented in a way that relates to conventional methods in particular restricted maximum likelihood reml this paper is a technical and theoretical prelude to subsequent papers that deal with applications of the theory to a range of important issues in neuroimaging these issues include i estimating nonsphericity or variance components in fmri time series that can arise from serial correlations within subject or are induced by multisubject i e hierarchical studies ii spatiotemporal bayesian models for imaging data in which voxels specific effects are constrained by responses in other voxels iii bayesian estimation of nonlinear models of hemodynamic responses and iv principled ways of mixing structural and functional priors in eeg source reconstruction although diverse all these estimation problems are accommodated by the peb framework described in this paper

electromagnetic scattering from extended wires and two and three dimensional surfaces efficient numerical solutions for the electromagnetic scattering for classes of electrically large one two and three dimensional perfectly conducting scatterers are presented the formulation is based on solution of the electric field integral equation efie using the method of moments mm an entire domain galerkin representation is used for wires and two dimensional surfaces and a combination of entire and subdomain representations is applied to surfaces in three dimensions the analysis is extendable to corrugated surfaces formed from sections of surfaces of translation or rotation numerical results are presented for wires infinite strips and finite strips or plates the behavior of the solutions with the number of terms in the entire domain expansion is examined the reconstruction of the traveling wave contribution to the scattering cross section using various approximations is discussed and representative examples are given

a c density theorem for differential inclusions with lipschitz continuous right hand sides abstract by definition solutions of differential inclusions are absolutely continuous functions with l 1 derivatives we prove that at least for a class of lipschitz continuous right hand sides the c solutions are dense in the set of all solutions with respect to the supremum norm

graph partitioned spatial priors for functional magnetic resonance images abstract spatial models of functional magnetic resonance imaging fmri data allow one to estimate the spatial smoothness of general linear model glm parameters and eschew pre process smoothing of data entailed by conventional mass univariate analyses recently diffusion based spatial priors harrison l m penny w daunizeau j and friston k j 2008 diffusion based spatial priors for functional magnetic resonance images neuroimage were proposed which provide a way to formulate an adaptive spatial basis where the diffusion kernel of a weighted graph laplacian wgl is used as the prior covariance matrix over glm parameters an advantage of these is that they can be used to relax the assumption of isotropy and stationarity implicit in smoothing data with a fixed gaussian kernel the limitation of diffusion based models is purely computational due to the large number of voxels in a brain volume one solution is to partition a brain volume into slices using a spatial model for each slice this reduces computational burden by approximating the full wgl with a block diagonal form where each block can be analysed separately while fmri data are collected in slices the functional structures exhibiting spatial coherence and continuity are generally three dimensional calling for a more informed partition we address this using the graph laplacian to divide a brain volume into sub graphs whose shape can be arbitrary their shape depends crucially on edge weights of the graph which can be based on the euclidean distance between voxels isotropic or on glm parameters anisotropic encoding functional responses the result is an approximation the full wgl that retains its 3d form and also has potential for parallelism we applied the method to high resolution 1 mm3 fmri data and compared models where a volume was divided into either slices or graph partitions models were optimized using expectation maximization and the approximate log evidence computed to compare these different ways to partition a spatial prior the high resolution fmri data presented here had greatest evidence for the graph partitioned anisotropic model which was best able to preserve fine functional detail

hybrid dirichlet mixture models for functional data summary in functional data analysis curves or surfaces are observed up to measurement error at a finite set of locations for say a sample of n individuals often the curves are homogeneous except perhaps for individual specific regions that provide heterogeneous behaviour e g damaged areas of irregular shape on an otherwise smooth surface motivated by applications with functional data of this nature we propose a bayesian mixture model with the aim of dimension reduction by representing the sample of n curves through a smaller set of canonical curves we propose a novel prior on the space of probability measures for a random curve which extends the popular dirichlet priors by allowing local clustering non homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations hybrids of a few canonical curves more precisely the prior proposed envisions a conceptual hidden factor with k levels that acts locally on each curve we discuss several models incorporating this prior and illustrate its performance with simulated and real data sets we examine theoretical properties of the proposed finite hybrid dirichlet mixtures specifically their behaviour as the number of the mixture components goes to 1 and their connection with dirichlet process mixtures

a bayesian discovery procedure we discuss a bayesian discovery procedure for multiple comparison problems we show that under a coherent decision theoretic framework a loss function combining true positive and false positive counts leads to a decision rule based on a threshold of the posterior probability of the alternative under a semi parametric model for the data we show that the bayes rule can be approximated by the optimal discovery procedure odp recently introduced by storey 2007a improving the approximation leads us to a bayesian discovery procedure bdp which exploits the multiple shrinkage in clusters implied by the assumed nonparametric model we compare the bdp and the odp estimates in a simple simulation study and in an assessment of differential gene expression based on microarray data from tumor samples we extend the setting of the odp by discussing modifications of the loss function that lead to different single thresholding statistics finally we provide an application of the previous arguments to dependent spatial data

second order discrete approximations to strongly convex differential inclusions we present a runge kutta type scheme for discretization of strongly convex differential inclusions the accuracy of the approximation to the set of trajectories of the differential inclusion on a given interval is proven to be of second order with respect to the step length

accurate solution of square scatterer as benchmark for validation of electromagnetic modeling of plate structures an infinitesimally thin square scatterer of size spl lambda spl times spl lambda excited normally by an incident plane wave which is polarized along a scatterer edge is analyzed the accurate solution of its current distribution is found in the form of a double series of basis functions which automatically satisfy the continuity equation at the plate edges and include the edge effect the coefficients that multiply basis functions are determined starting from the electric field integral equation by using the galerkin method the solution obtained for the order of approximation n 8 is adopted as a benchmark the corresponding coefficients are tabulated and graphs of such obtained current distribution are given the solution adopted as a benchmark is applied for comparison of rooftop basis functions and polynomial entire domain basis functions the relative error of the mean absolute value of current deviation is used as an error metric

posterior probability maps and spms this technical note describes the construction of posterior probability maps that enable conditional or bayesian inferences about regionally specific effects in neuroimaging posterior probability maps are images of the probability or confidence that an activation exceeds some specified threshold given the data posterior probability maps ppms represent a complementary alternative to statistical parametric maps spms that are used to make classical inferences however a key problem in bayesian inference is the specification of appropriate priors this problem can be finessed using empirical bayes in which prior variances are estimated from the data under some simple assumptions about their form empirical bayes requires a hierarchical observation model in which higher levels can be regarded as providing prior constraints on lower levels in neuroimaging observations of the same effect over voxels provide a natural two level hierarchy that enables an empirical bayesian approach in this note we present a brief motivation and the operational details of a simple empirical bayesian method for computing posterior probability maps we then compare bayesian and classical inference through the equivalent ppms and spms testing for the same effect in the same data

efficient simulation of rectangular dielectric resonators using volume mpie mom formulation with combined entire domain and subdomain basis functions an efficient method of moments mom formulation using combined entire domain and subdomain basis functions is proposed for simulating the resonant modes of a rectangular dielectric resonator in free space an approximate analytical solution for the resonant mode of the dr is used as the entire domain basis function while a set of tetrahedral basis functions are used as subdomain basis functions since the main profile of the resonant mode can be well represented by the entire domain basis function only small number of subdomain basis functions are needed for further refinement of the representation compared with the mom formulations using only subdomain or entire domain basis functions this method is much faster for the same accuracy numerical results are given and compared with other methods

determination of a natural basis function set on thick structured slabs using prony s method the calculation of currents on a thick structured slab such as a thick slab of honeycomb is discussed unfortunately for certain applications the slab can be of the order of several wavelengths so that the straightforward application of the method of moments using subdomain basis functions is too expensive the authors discuss how to apply prony s method to the currents calculated for a thin structured slab to obtain a natural set of basis functions to represent the currents in the interior of a thick slab prony s method represents the currents as a series of complex exponential functions the thick slab problem is then solved by the method of moments using subdomain basis functions near the slab interfaces and one or two of the complex exponentials as basis functions within the slab

sampling method using prefiltered band limited green s functions for the solution of electromagnetic integral equations in the context of far field scattering the spatial spectrum of fields or currents on a scatterer may be approximated as band limited on this premise one can greatly simplify the computation of radiation integrals by eliminating in advance those spectral components of the green s function not expected to be present in the fields this approach enables one to dispense with the notion of basis functions and instead represent fields and currents as well as the impedance matrix representing the green s function entirely in terms of a sparse set of samples thereby obviating the need for integration moreover by taking full advantage of the band limited character of the solution this sampling theoretical approach yields an efficient representation with a minimal number of degrees of freedom the method was explored for scattering from two dimensional perfect electric conductors including ellipsoidal cylinders and flat strips in both polarizations using uniform as well as nonuniform sampling and yielded surprisingly good results

bayesian spatiotemporal model of fmri data this research describes a new bayesian spatiotemporal model to analyse block design bold fmri studies in the temporal dimension we parameterise the hemodynamic response function s hrf shape with a potential increase of signal and a subsequent exponential decay in the spatial dimension we use gaussian markov random fields gmrf priors on activation characteristics parameters location and magnitude that embody our prior knowledge that evoked responses are spatially contiguous and locally homogeneous the result is a spatiotemporal model with a small number of parameters all of them interpretable simulations from the model are performed in order to ascertain the performance of the sampling scheme and the ability of the posterior to estimate model parameters as well as to check the model sensitivity to signal to noise ratio results are shown on synthetic data and on real data from a block design fmri experiment

a technique for extrapolating numerically rigorous solutions of electromagnetic scattering problems to higher frequencies and their scaling properties the possibility of extrapolating the current distribution on two dimensional scatterers to high frequencies from knowledge of the solution at two or more lower frequencies is investigated in this paper a simple extrapolation algorithm is developed in which the current distribution is first calculated at two lower frequencies and then split into propagating or decaying traveling wave components in the lit and shadow regions these components are scaled to higher frequencies by using simple operations such as stretching of the magnitude and linear extrapolation of the phase this technique enables one to solve a class of large body scattering problems well beyond the range of rigorous numerical techniques furthermore the extrapolated solution is rapidly constructed over a very wide range of frequencies typically by utilizing the rigorous solution at only two lower frequencies the application of the extrapolation algorithm is demonstrated for several examples viz an ellipse with a high aspect ratio and wing shaped geometries with rounded and sharp edges the robustness of the technique is illustrated by considering grazing angles of incidence where the asymptotic techniques typically break down

convergence of the forward euler method for nonconvex differential inclusions the convergence of reachable sets for nonconvex differential inclusions is considered when the right hand side in the differential inclusion is a compact valued lipschitz continuous set valued function it is shown that the convergence in hausdorff distance of reachable sets for a forward euler discretization is linear in the time step

the exponential formula for the reachable set of a lipschitz differential inclusion the main goal of this paper is to prove a formula for the reachable set of a lipschitz differential inclusion with convex values the formula involves a kuratowski limit of sets that resembles a standard approach of defining the exponential of a matrix this explains the title the proof of the main theorem partially relies on a c 1 approximation result due to filippov for which a new proof is given a new approach of characterizing the value function associated with a mayer optimal control problem is given as an application

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

mixed domain galerkin expansions in scattering problems entire domain galerkin expansions provide a computationally efficient representation in method of moments mm solutions for electrically extended surfaces forming part of a separable coordinate geometry for surfaces not conforming to such geometries such as those with discontinuities subdomain representations have been used for many classes of scattering problems involving electrically large two dimensional scatterers with localized discontinuities a mixed domain galerkin expansion is shown to lead to a computationally efficient formulation electromagnetic scattering solutions are developed for such configurations convergence of these solutions is illustrated with choices of mixed domain expansions as a function of surface discontinuity the computed results are given for both transverse magnetic tm and transverse electric te polarization and are compared with published results

combining an extrapolation technique with the method of moments for solving large scattering problems involving bodies of revolution the purpose of this work is to combine an extrapolation technique with the method of moments mom to solve scattering problems involving large bodies it has been shown in a previous work that the current induced on the smooth parts of large scatterers may be represented as a series of complex exponential functions with a few terms based on this concept a hybrid set of basis functions is constructed using entire domain functions of complex exponential type on the smooth portion of the scatterer complemented by subdomain basis functions near edges and discontinuities an extrapolation procedure is developed in which the scattering problem is first solved for a portion of the scatterer using the conventional mom next a set of entire domain basis functions whose behaviour could be extrapolated with an increase in the size of the scatterer is extracted from this original solution the procedure outlined has the very desirable feature that the total number of basis functions remains unchanged even as the scatterer size is increased allowing for large scatterers to be handled with a relatively small number of unknowns the extrapolation technique is applied to scattering problems from bodies of revolution bors and numerical results for an open cylinder and a barrel shaped bor are presented

population level inferences for distributed meg source localization under multiple constraints application to face evoked fields we address some key issues entailed by population inference about responses evoked in distributed brain systems using magnetoencephalography meg in particular we look at model selection issues at the within subject level and feature selection issues at the between subject level using responses evoked by intact and scrambled faces around 170 ms m170 we compared the face validity of subject specific forward models and their summary statistics in terms of how estimated responses reproduced over subjects at the within subject level we focused on the use of multiple constraints or priors for inverting distributed source models we used restricted maximum likelihood reml estimates of prior covariance components in both sensor and source space and show that their relative importance is conserved over subjects at the between subject level we used standard anatomical normalization methods to create posterior probability maps that furnish inference about regionally specific population responses we used these to compare different summary statistics namely i whether to test for differences between condition specific source estimates or whether to test the source estimate of differences between conditions and ii whether to accommodate differences in source orientation by using signed or unsigned absolute estimates of source activity

note on interpolational basis functions in the method of moments em scattering efficient basis sets for the method of moments may be obtained using quasi localized bandlimited interpolational functions that broadly speaking are defined relative to a mean sampling rate that is adjusted to curvature and proximity to edges thus reflecting the local spatial frequency bandwidth computed scattering data in a number of structures including perfectly conducting circular and elliptical two dimensional cylinders as well as a flat infinite strip indicate that reasonable accuracy can be obtained with an average rate of between 2 5 and 3 basis functions per wavelength average sampling rates need not correspond strictly to the bandwidth of the basis functions and there is considerable latitude with respect to random variation of sampling intervals although each basis function typically extends over several sample points required integrals can be obtained speedily by the use of standard sampling theoretical methods

diffusion based spatial priors for imaging we describe a bayesian scheme to analyze images which uses spatial priors encoded by a diffusion kernel based on a weighted graph laplacian this provides a general framework to formulate a spatial model whose parameters can be optimized the application we have in mind is a spatiotemporal model for imaging data we illustrate the method on a random effects analysis of fmri contrast images from multiple subjects this simplifies exposition of the model and enables a clear description of its salient features typically imaging data are smoothed using a fixed gaussian kernel as a pre processing step before applying a mass univariate statistical model e g a general linear model to provide images of parameter estimates an alternative is to include smoothness in a multivariate statistical model penny w d trujillo barreto n j friston k j 2005 bayesian fmri time series analysis with spatial priors neuroimage 24 350 362 the advantage of the latter is that each parameter field is smoothed automatically according to a measure of uncertainty given the data in this work we investigate the use of diffusion kernels to encode spatial correlations among parameter estimates nonlinear diffusion has a long history in image processing in particular flows that depend on local image geometry romeny b m t 1994 geometry driven diffusion in computer vision kluwer academic publishers can be used as adaptive filters this can furnish a non stationary smoothing process that preserves features which would otherwise be lost with a fixed gaussian kernel we describe a bayesian framework that incorporates non stationary adaptive smoothing into a generative model to extract spatial features in parameter estimates critically this means adaptive smoothing becomes an integral part of estimation and inference we illustrate the method using synthetic and real fmri data

improving reliability of subject level resting state fmri parcellation with shrinkage estimators abstract a recent interest in resting state functional magnetic resonance imaging rsfmri lies in subdividing the human brain into anatomically and functionally distinct regions of interest for example brain parcellation is often a necessary step for defining the network nodes used in connectivity studies while inference has traditionally been performed on group level data there is a growing interest in parcellating single subject data however this is difficult due to the inherent low signal to noise ratio of rsfmri data combined with typically short scan lengths a large number of brain parcellation approaches employ clustering which begins with a measure of similarity or distance between voxels the goal of this work is to improve the reproducibility of single subject parcellation using shrinkage based estimators of such measures allowing the noisy subject specific estimator to borrow strength in a principled manner from a larger population of subjects we present several empirical bayes shrinkage estimators and outline methods for shrinkage when multiple scans are not available for each subject we perform shrinkage on raw inter voxel correlation estimates and use both raw and shrinkage estimates to produce parcellations by performing clustering on the voxels while we employ a standard spectral clustering approach our proposed method is agnostic to the choice of clustering method and can be used as a pre processing step for any clustering algorithm using two datasets a simulated dataset where the true parcellation is known and is subject specific and a test retest dataset consisting of two 7 minute resting state fmri scans from 20 subjects we show that parcellations produced from shrinkage correlation estimates have higher reliability and validity than those produced from raw correlation estimates application to test retest data shows that using shrinkage estimators increases the reproducibility of subject specific parcellations of the motor cortex by up to 30