empirical analyses of bold fmri statistics ii spatially smoothed data collected under null hypothesis and experimental conditions in the companion to this paper e zarahn g k aguirre and m d esposito 1997 neuroimage 179 197 we describe an implementation of a general linear model for autocorrelated observations in which the voxel wise false positive rates in fmri noise datasets were stabilized and brought close to theoretical values here implementations of the model are tested for use with statistical parametric mapping analysis of spatially smoothed fmri data analyses using varying models of intrinsic temporal autocorrelation and either including or excluding a global signal covariate were conducted upon human subject data collected under null hypothesis as well as under experimental conditions we found that smoothing with an empirically derived impulse response function irf combined with a model of the intrinsic temporal autocorrelation in spatially smoothed fmri data resulted in a map wise false positive rate which did not exceed a 5 level when a nominal alpha 0 05 tabular threshold was applied use of other models of intrinsic temporal autocorrelation resulted in map wise false positive rates that significantly exceeded this level fmri data collected while subjects performed a behavioral task were used to examine a task dependent global signal changes and b the dependence of sensitivity on the temporal smoothing kernel and inclusion exclusion of a global signal covariate the global signal changes within an fmri dataset were shown to be influenced by the performance of a behavioral task however the inclusion of this measure as a covariate did not have an adverse affect upon our measure of sensitivity finally use of an empirically derived estimate of the irf of the system was shown to result in greater map wise sensitivity for signal changes than the use of a broader in time poisson parameter 8 s kernel

false discovery control in large scale spatial multiple testing summary the paper develops a unified theoretical and computational framework for false discovery control in multiple testing of spatial signals we consider both pointwise and clusterwise spatial analyses and derive oracle procedures which optimally control the false discovery rate false discovery exceedance and false cluster rate a data driven finite approximation strategy is developed to mimic the oracle procedures on a continuous spatial domain our multiple testing procedures are asymptotically valid and can be effectively implemented using bayesian computational algorithms for analysis of large spatial data sets numerical results show that the procedures proposed lead to more accurate error control and better power performance than conventional methods we demonstrate our methods for analysing the time trends in tropospheric ozone in eastern usa

a distributed spatio temporal eeg meg inverse solver we propose a novel l1l2 norm inverse solver for estimating the sources of eeg meg signals based on the standard l1 norm inverse solvers this sparse distributed inverse solver integrates the l1 norm spatial model with a temporal model of the source signals in order to avoid unstable activation patterns and spiky reconstructed signals often produced by the currently used sparse solvers the joint spatio temporal model leads to a cost function with an l1l2 norm regularizer whose minimization can be reduced to a convex second order cone programming socp problem and efficiently solved using the interior point method the efficient computation of the socp problem allows us to implement permutation tests for estimating statistical significance of the inverse solution validation with simulated and human meg data shows that the proposed solver yields source time course estimates qualitatively similar to those obtained through dipole fitting but without the need to specify the number of dipole sources in advance furthermore the l1l2 norm solver achieves fewer false positives and a better representation of the source locations than the conventional l2 minimum norm estimates

interpolation on a triangulated 3d surface abstract an interpolation method for scalar functions on a rectangular grid on a planar surface is extended to the interpolation function on a closed three dimensional triangulated surface of arbitrary shape two variants are considered the first one constrains the laplacian of the function to be zero at points where the function values are unknown the second one minimizes the laplacian at all points of the surface considered some illustrative examples of both variants are given in applications to the display of potential distributions on the boundary surface of an electrical volume conductor

controlling familywise error rate for matched subspace detection in dynamic fdg pet detection of small lesions in fluorodeoxyglucose fdg positron emission tomography pet is limited by image resolution and low signal to noise ratio we have previously described a matched subspace detection method that uses the time activity curve to distinguish tumors from background in dynamic fdg pet applying this algorithm on a voxel by voxel basis throughout the dynamic image produces a test statistic image or ldquomaprdquo which on thresholding indicates the potential locations of secondary or metastatic tumors in this paper we describe a thresholding method that controls familywise error rate fwer for the matched subspace detection statistical map the method involves three steps first the pet image is segmented into several homogeneous regions then the statistical map is normalized to a zero mean unit variance gaussian random field finally the images are thresholded at a fixed fwer by estimating their spatial smoothness and applying a random field theory maximum statistic approach we evaluate this thresholding method using digital phantoms generated from clinical dynamic images we also present an application of the proposed approach to clinical pet data from a breast cancer patient with metastatic disease

recursive parameter estimation using incomplete data abstract stochastic approximation procedures are considered for the estimation of parameters using incomplete data one procedure is stated and illustrated which often leads to asymptotically efficient estimators others are developed which although possibly not optimal in the above sense will be very much easier to apply this will be particularly advantageous when quick recursive estimates are required examples are given and a link is made between one of the sub optimal methods and the em algorithm author

multiple testing via fdrl for large scale imaging data the multiple testing procedure plays an important role in detecting the presence of spatial signals for large scale imaging data typically the spatial signals are sparse but clustered this paper provides empirical evidence that for a range of commonly used control levels the conventional fdr procedure can lack the ability to detect statistical significance even if the p values under the true null hypotheses are independent and uniformly distributed more generally ignoring the neighboring information of spatially structured data will tend to diminish the detection effectiveness of the fdr procedure this paper first introduces a scalar quantity to characterize the extent to which the lack of identification phenomenon lip of the fdr procedure occurs second we propose a new multiple comparison procedure called fdrl to accommodate the spatial information of neighboring p values via a local aggregation of p values theoretical properties of the fdrl procedure are investigated under weak dependence of p values it is shown that the fdrl procedure alleviates the lip of the fdr procedure thus substantially facilitating the selection of more stringent control levels simulation evaluations indicate that the fdrl procedure improves the detection sensitivity of the fdr procedure with little loss in detection specificity the computational simplicity and detection effectiveness of the fdrl procedure are illustrated through a real brain fmri dataset

on the benjamini hochberg method we investigate the properties of the benjamini hochberg method for multiple testing and of a variant of storey s generalization of it extending and complementing the asymptotic and exact results available in the literature results are obtained under two different sets of assumptions and include asymptotic and exact expressions and bounds for the proportion of rejections the proportion of incorrect rejections out of all rejections and two other proportions used to quantify the efficacy of the method

large deviations for the empirical measure of a markov chain with an application to the multivariate empirical measure on demontre les proprietes de grandes deviations uniformes pour la mesure empirique et la mesure empirique multivariable d une chaine de markov qui prend ses valeurs dans un espace metrique separable complet

sulcal set optimization for cortical surface registration flat mapping based cortical surface registration constrained by manually traced sulcal curves has been widely used for inter subject comparisons of neuroanatomical data even for an experienced neuroanatomist manual sulcal tracing can be quite time consuming with the cost increasing with the number of sulcal curves used for registration we present a method for estimation of an optimal subset of size nc from n possible candidate sulcal curves that minimizes a mean squared error metric over all combinations of nc curves the resulting procedure allows us to estimate a subset with a reduced number of curves to be traced as part of the registration procedure leading to optimal use of manual labeling effort for registration to minimize the error metric we analyze the correlation structure of the errors in the sulcal curves by modeling them as a multivariate gaussian distribution for a given subset of sulci used as constraints in surface registration the proposed model estimates registration error based on the correlation structure of the sulcal errors the optimal subset of constraint curves consists of the nc sulci that jointly minimize the estimated error variance for the subset of unconstrained curves conditioned on the nc constraint curves the optimal subsets of sulci are presented and the estimated and actual registration errors for these subsets are computed

determining the optimal level of smoothing in cortical thickness analysis a hierarchical approach based on sequential statistical thresholding article i nfo the extent of smoothing applied to cortical thickness maps critically influences sensitivity anatomical precision and resolution of statistical change detection theoretically it could be optimized by increasing the trade off between vertex wise sensitivity and specificity across several levels of smoothing but to date neither parametric nor nonparametric methods are able to control the error at the vertex level if the null hypothesis is rejected after smoothing of cortical thickness maps to overcome these drawbacks we applied sequential statistical thresholding based on a simple hierarchical model this methodology aims at controlling erroneous detections firstly at the level of clusters over smoothed statistical maps and secondly at the vertex level over unsmoothed statistical maps by applying an adaptive false discovery rate fdr procedure to clusters previously detected the superior performance of the proposed methodology over other conventional procedures was demonstrated in simulation studies as expected only the hierarchical method yielded a predictable false discovery proportion near the predefined fdr q value for any smoothing level at the same time as being as sensitive as the others at the optimal setting it was therefore the only method able to approximate the optimal size of spatial smoothing when the true change was assumed unknown the hierarchical method was further validated in a cross sectional study comparing moderate alzheimer s disease ad patients with healthy elderly subjects results suggest that the extent of cortical thinning reported in previous ad studies might be artificially inflated by the choice of inadequate smoothing in these cases interpretation should be based on the location of local maxima of suprathreshold regions rather than on the spatial extent of the detected signal in the statistical parametric map

weighted false discovery rate control in large scale multiple testing abstractthe use of weights provides an effective strategy to incorporate prior domain knowledge in large scale inference this article studies weighted multiple testing in a decision theoretical framework we develop oracle and data driven procedures that aim to maximize the expected number of true positives subject to a constraint on the weighted false discovery rate the asymptotic validity and optimality of the proposed methods are established the results demonstrate that incorporating informative domain knowledge enhances the interpretability of results and precision of inference simulation studies show that the proposed method controls the error rate at the nominal level and the gain in power over existing methods is substantial in many settings an application to a genome wide association study is discussed supplementary materials for this article are available online

false discovery control for random fields this article extends false discovery rates to random fields for which there are uncountably many hypothesis tests we develop a method for finding regions in the field s domain where there is a significant signal while controlling either the proportion of area or the proportion of clusters in which false rejections occur the method produces confidence envelopes for the proportion of false discoveries as a function of the rejection threshold from the confidence envelopes we derive threshold procedures to control either the mean or the specified tail probabilities of the false discovery proportion an essential ingredient of this construnction is a new algorithm to compute a confidence superset for the set of all true null locations we demonstrate our method with applications to scan statistics and functional neuroimaging

comparing connectomes across subjects and populations at different scales brain connectivity can be represented by a network that enables the comparison of the different patterns of structural and functional connectivity among individuals in the literature two levels of statistical analysis have been considered in comparing brain connectivity across groups and subjects 1 the global comparison where a single measure that summarizes the information of each brain is used in a statistical test 2 the local analysis where a single test is performed either for each node connection which implies a multiplicity correction or for each group of nodes connections where each subset is summarized by one single test in order to reduce the number of tests to avoid a penalizing multiplicity correction we comment on the different levels of analysis and present some methods that have been proposed at each scale we highlight as well the possible factors that could influence the statistical results and the questions that have to be addressed in such an analysis

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

a bottom up approach to testing hypotheses that have a branching tree dependence structure with error rate control modern statistical analyses often involve testing large numbers of hypotheses in many situations these hypotheses may have an underlying tree structure that both helps determine the order that te

a note on adaptive bonferroni and holm procedures under dependence summary hochberg benjamini 1990 first presented adaptive procedures for controlling familywise error rate however until now it has not been proved that these procedures control the familywise error rate we introduce a simplified version of hochberg benjamini s adaptive bonferroni and holm procedures assuming a conditional dependence model we prove that the former procedure controls the familywise error rate in finite samples while the latter controls it approximately

strong control conservative point estimation and simultaneous conservative consistency of false discovery rates a unified approach summary the false discovery rate fdr is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses benjamini and hochberg introduced this quantity and proved that a particular step up p value method controls the fdr storey introduced a point estimate of the fdr for fixed significance regions the former approach conservatively controls the fdr at a fixed predetermined level and the latter provides a conservatively biased estimate of the fdr for a fixed predetermined significance region in this work we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent in particular the fdr point estimates can be used to define valid fdr controlling procedures in the asymptotic setting we also show that the point estimates can be used to estimate the fdr conservatively over all significance regions simultaneously which is equivalent to controlling the fdr at all levels simultaneously the main tool that we use is to translate existing fdr methods into procedures involving empirical processes this simplifies finite sample proofs provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence

a large deviation principle for join the shortest queue we consider a join the shortest queue model which is as follows there are k single fifo servers and m arrival processes the customers from a given arrival process can be served only by the servers from a certain subset of all servers the actual destination is the server with the smallest weighted queue length the arrival processes are assumed to obey a large deviation principle while service is exponential a large deviation principle is established for the queue length process the action functional is expressed in terms of solutions to mathematical programming problems the large deviation limit point is identified as a weak solution to a system of idempotent equations uniqueness of the weak solution follows by trajectorial uniqueness

susan a new approach to low level image processing this paper describes a new approach to low level image processing in particular edge and corner detection and structure preserving noise reduction non linear filtering is used to define which parts of the image are closely related to each individual pixel each pixel has associated with it a local image region which is of similar brightness to that pixel the new feature detectors are based on the minimization of this local image region and the noise reduction method uses this region as the smoothing neighbourhood the resulting methods are accurate noise resistant and fast details of the new feature detectors and of the new noise reduction method are described along with test results

on line em algorithm for the normalized gaussian network a normalized gaussian network ngnet moody darken 1989 is a network of local linear regression units the model softly partitions the input space by normalized gaussian functions and each local unit linearly approximates the output within the partition in this article we propose a new on line em algorithm for the ngnet which is derived from the batch em algorithm xu jordan hinton 1995 by introducing a discount factor we show that the on line em algorithm is equivalent to the batch em algorithm if a specific scheduling of the discount factor is employed in addition we show that the on line em algorithm can be considered as a stochastic approximation method to find the maximum likelihood estimator a new regularization method is proposed in order to deal with a singular input distribution in order to manage dynamic environments where the input output distribution of data changes over time unit manipulation mechanisms such as unit production unit deletion and unit division are also introduced based on probabilistic interpretation experimental results show that our approach is suitable for function approximation problems in dynamic environments we also apply our on line em algorithm to robot dynamics problems and compare our algorithm with the mixtures of experts family

correlated z values and the accuracy of large scale statistical estimates we consider large scale studies in which there are hundreds or thousands of correlated cases to investigate each represented by its own normal variate typically a z value a familiar example is provided by a microarray experiment comparing healthy with sick subjects expression levels for thousands of genes this paper concerns the accuracy of summary statistics for the collection of normal variates such as their empirical cdf or a false discovery rate statistic it seems like we must estimate an n by n correlation matrix n the number of cases but our main result shows that this is not necessary good accuracy approximations can be based on the root mean square correlation over all n n 1 2 pairs a quantity often easily estimated a second result shows that z values closely follow normal distributions even under nonnull conditions supporting application of the main theorem practical application of the theory is illustrated for a large leukemia microarray study

multiple testing of composite null hypotheses in heteroscedastic models in large scale studies the true effect sizes often range continuously from zero to small to large and are observed with heteroscedastic errors in practical situations where the failure to reject small deviations from the null is inconsequential specifying an indifference region or forming composite null hypotheses can greatly reduce the number of unimportant discoveries in multiple testing the heteroscedasticity issue poses new challenges for multiple testing with composite nulls in particular the conventional framework in multiple testing which involves rescaling or standardization is likely to distort the scientific question we propose the concept of a composite null distribution for heteroscedastic models and develop an optimal testing procedure that minimizes the false nondiscovery rate subject to a constraint on the false discovery rate the proposed approach is different from conventional methods in that the effect size statistical significance and multiplicity issues are addressed in

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

operating characteristics and extensions of the false discovery rate procedure summary we investigate the operating characteristics of the benjamini hochberg false discovery rate procedure for multiple testing this is a distribution free method that controls the expected fraction of falsely rejected null hypotheses among those rejected the paper provides a framework for understanding more about this procedure we first study the asymptotic properties of the deciding point d that determines the critical p value from this we obtain explicit asymptotic expressions for a particular risk function we introduce the dual notion of false non rejections and we consider a risk function that combines the false discovery rate and false non rejections we also consider the optimal procedure with respect to a measure of conditional risk

hierarchical recognition of sparse patterns in large scale simultaneous inference summary we study how to accurately separate signals from noisy data and determine the patterns of the selected signals controlling the inflation of false positive errors is an important issue in largescale simultaneous inference but has not been addressed in the pattern recognition literature we develop a decision theoretic framework and formulate the sparse pattern recognition problem as a simultaneous inference problem with multiple decision trees oracle and adaptive classifiers are proposed for maximizing the expected number of true positives subject to a constraint on the overall false positive rate existing results on multiple testing are extended by allowing more than two states of nature hierarchical decision making and new error rate concepts

optimal screening and discovery of sparse signals with applications to multistage high throughput studies summary a common feature in large scale scientific studies is that signals are sparse and it is desirable to narrow down significantly the focus to a much smaller subset in a sequential manner we consider two related data screening problems one is to find the smallest subset such that it virtually contains all signals and another is to find the largest subset such that it essentially contains only signals these screening problems are closely connected to but distinct from the more conventional signal detection or multiple testing problems we develop phase transition diagrams to characterize the fundamental limits in simultaneous inference and derive data driven screening procedures which control the error rates with near optimality properties applications in the context of multistage high throughput studies are discussed

recursive estimation procedures for missing data problems abstract titterington in a previous paper proposed recursive methods for dealing with incomplete data this present paper concentrates on versions of these for multiparameter problems involving missing data theorems are outlined from which asymptotic properties of the recursive procedures can be established and versions of the recursions are written down for problems in which the missing data are missing at random after illustration with exponential family models the case of multivariate normal data is considered in detail numerical comparisons of the various methods are obtained using bivariate normal data whereas the previous paper discussed incomplete data in general the present one restricts attention to the problem of missing values typically each experimental unit should have records of the values of several characteristics associated with it statistical analysis is made difficult if one or more of those values are missing on some units to combat the heavy analysis required for a proper analysis of the data comparatively simple recursive procedures are outlined in which the data are incorporated sequentially into the estimation scheme some comments are made about theoretical properties and special emphasis is laid on the case of data from multivariate normal distributions

hierarchical control of false discovery rate for phase locking measures of eeg synchrony computing phase locking values plvs between eeg signals is becoming a popular measure for quantifying functional connectivity because it affords a more detailed picture of the synchrony relationships between channels at different times and frequencies however the accompanying increase in data dimensionality incurs a serious multiple testing problem for determining plv significance standard methods for controlling type i error which treat all hypotheses as belonging to a single family can fail to detect any significant discoveries instead we propose a novel application of a hierarchical fdr method which subsumes multiple families for detecting significant plv effects for simulations and experimental data we show that the proposed hierarchical fdr method is most powerful this method revealed significant synchrony effects in the expected regions at an acceptable error rate of 5 where other methods including standard fdr correction failed to reveal any significant effects

recursive estimation in mixture models with markov regime a recursive algorithm is proposed for estimation of parameters in mixture models where the observations are governed by a hidden markov chain the often badly conditioned information matrix is estimated and its inverse is incorporated into the algorithm the performance of the algorithm is studied by simulations of a symmetric normal mixture the algorithm seems to be stable and produce approximately normally distributed estimates provided the adaptive matrix is kept well conditioned some numerical examples are included

morphology based hypothesis testing in discrete random fields a non parametric method to address the multiple comparison problem in neuroimaging a new method for detecting activations in random fields which may be useful for addressing the issue of multiple comparisons in neuroimaging is presented this method is based on some constructs of mathematical morphology specifically morphological erosions and dilations that enable the detection of active regions in random fields possessing moderate activation levels and relatively large spatial extension which may not be detected by the standard methods that control the family wise error rate the method presented here permits an appropriate control of the false positive errors without having to adjust any threshold values other than the significance level the method is easily adapted to permutation based procedures with the usual restrictions and therefore does not require strong assumptions about the distribution and spatio temporal correlation structure of the data some examples of applications to synthetic data including realistic fmri simulations as well as to real fmri and electroencephalographic data are presented illustrating the power of the presented technique comparisons with other methods that combine voxel intensity and cluster size as well as some extensions of the method presented here based on their basic ideas are presented as well

action functional for diffusions in discontinuous media the action functional i e the rate function governing the large deviations is obtained for a family of stochastic processes with discontinuous drift and small diffusion a well known method of continuous mapping is developed which proves to be efficient in a so called stable case

random field union intersection tests for eeg meg imaging electrophysiological eeg meg imaging challenges statistics by providing two views of the same spatiotemporal data topographic and tomographic until now statistical tests for these two situations have developed separately this work introduces statistical tests for assessing simultaneously the significance of spatiotemporal event related potential event related field erp erf components and that of their sources the test for detecting a component at a given time instant is provided by a hotelling s t2 statistic this statistic is constructed in such a manner to be invariant to any choice of reference and is based upon a generalized version of the average reference transform of the data as a consequence the proposed test is a generalization of the well known global field power statistic consideration of tests at all time instants leads to a multiple comparison problem addressed by the use of random field theory rft the union intersection ui principle is the basis for testing hypotheses about the topographic and tomographic distributions of such erp erf components the performance of the method is illustrated with actual eeg recordings obtained from a visual experiment of pattern reversal stimuli

threshold free cluster enhancement addressing problems of smoothing threshold dependence and localisation in cluster inference many image enhancement and thresholding techniques make use of spatial neighbourhood information to boost belief in extended areas of signal the most common such approach in neuroimaging is cluster based thresholding which is often more sensitive than voxel wise thresholding however a limitation is the need to define the initial cluster forming threshold this threshold is arbitrary and yet its exact choice can have a large impact on the results particularly at the lower e g t z 4 cluster forming thresholds frequently used furthermore the amount of spatial pre smoothing is also arbitrary given that the expected signal extent is very rarely known in advance of the analysis in the light of such problems we propose a new method which attempts to keep the sensitivity benefits of cluster based thresholding and indeed the general concept of clusters of signal while avoiding or at least minimising these problems the method takes a raw statistic image and produces an output image in which the voxel wise values represent the amount of cluster like local spatial support the method is thus referred to as threshold free cluster enhancement tfce we present the tfce approach and discuss in detail roc based optimisation and comparisons with cluster based and voxel based thresholding we find that tfce gives generally better sensitivity than other methods over a wide range of test signal shapes and snr values we also show an example on a real imaging dataset suggesting that tfce does indeed provide not just improved sensitivity but richer and more interpretable output than cluster based thresholding

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

some nonasymptotic results on resampling in high dimension i confidence regions we study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure the random vector is supposed to be either gaussian or to have a symmetric and bounded distribution the dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non asymptotic control of the confidence level following ideas inspired by recent results in learning theory we consider two approaches the first based on a concentration principle valid for a large class of resampling weights and the second on a direct resampled quantile specifically using rademacher weights several intermediate results established in the approach based on concentration principles are of self interest we also discuss the question of accuracy when using monte carlo approximations of the resampled quantities we present an application of these results to the one sided and two sided multiple testing problem in which we derive several resampling based step down procedures providing a non asymptotic fwer control we compare our different procedures in a simulation study and we show that they can outperform bonferroni s or holm s procedures as soon as the observed vector has sufficiently correlated coordinates

a comparison of random field theory and permutation methods for the statistical analysis of meg data we describe the use of random field and permutation methods to detect activation in cortically constrained maps of current density computed from meg data the methods are applicable to any inverse imaging method that maps event related meg to a coregistered cortical surface these approaches also extend directly to images computed from event related eeg data we determine statistical thresholds that control the familywise error rate fwer across space or across both space and time both random field and permutation methods use the distribution of the maximum statistic under the null hypothesis to find fwer thresholds the former methods make assumptions on the distribution and smoothness of the data and use approximate analytical solutions the latter resample the data and rely on empirical distributions both methods account for spatial and temporal correlation in the cortical maps unlike previous nonparametric work in neuroimaging we address the problem of nonuniform specificity that can arise without a gaussianity assumption we compare and evaluate the methods on simulated data and experimental data from a somatosensory evoked response study we find that the random field methods are conservative with or without smoothing though with a 5 vertex fwhm smoothness they are close to exact our permutation methods demonstrated exact specificity in simulation studies in real data the permutation method was not as sensitive as the rf method although this could be due to violations of the random field theory assumptions

a general interactive framework for false discovery rate control under structural constraints we propose a general framework based on selectively traversed accumulation rules star for interactive multiple testing with generic structural constraints on the rejection set it combines accumulation tests from ordered multiple testing with data carving ideas from post selection inference allowing for highly flexible adaptation to generic structural information our procedure defines an interactive protocol for gradually pruning a candidate rejection set beginning with the set of all hypotheses and shrinking with each step by restricting the information at each step via a technique we call masking our protocol enables interaction while controlling the false discovery rate fdr in finite samples for any data adaptive update rule that the analyst may choose we suggest update rules for a variety of applications with complex structural constraints show that star performs well for problems ranging from convex region detection to fdr control on directed acyclic graphs and show how to extend it to regression problems where knockoff statistics are available in lieu of p values

selective inference on multiple families of hypotheses summary in many complex multiple testing problems the hypotheses are divided into families given the data families with evidence for true discoveries are selected and hypotheses within them are tested neither controlling the error rate in each family separately nor controlling the error rate over all hypotheses together can assure some level of confidence about the filtration of errors within the selected families we formulate this concern about selective inference in its generality for a very wide class of error rates and for any selection criterion and present an adjustment of the testing level inside the selected families that retains control of the expected average error over the selected families

large deviation lower bounds for additive functionals of markov processes let x 1 x 2 ldots be a markov process with state space e a polish space let l n omega a n 1 sum n 1 j 0 1 a x j omega denote the normalized occupation time measure if mu is a probability measure on e g is a weak neighborhood of mu and if v subset e then we obtain asymptotic lower bounds for probabilities p x lbrack l n omega cdot in g x j omega in v 0 leq j leq n 1 rbrack in terms of i mu the rate function of donsker and varadhan our assumptions are weaker than those imposed by donsker and varadhan and the proof works without any essential change in the continuous time case as well in fact the same proofs apply to certain bounded additive functionals let r geq 0 and let f omega rightarrow mathbf b be bounded mathscr f 0 r measurable where omega is the sample space with the product topology skorohod topology in the continuous time case and mathbf b is a separable banach space let theta k omega rightarrow omega be the shift operator i e theta k omega j omega k j then we get lower bounds for probabilities involving n 1 f omega f theta 1 omega cdots f theta n 1 omega in place of l n omega cdot in this latter situation the rate function has to be the entropy function h q of donsker and varadhan

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

large scale multiple testing under dependence summary the paper considers the problem of multiple testing under dependence in a compound decision theoretic framework the observed data are assumed to be generated from an underlying two state hidden markov model we propose oracle and asymptotically optimal datadriven procedures that aim to minimize the false non discovery rate fnr subject to a constraint on the false discovery rate fdr it is shown that the performance of a multiple testing procedure can be substantially improved by adaptively exploiting the dependence structure among hypotheses and hence conventional fdr procedures that ignore this structural information are inefficient both theoretical properties and numerical performances of the procedures proposed are investigated it is shown that the procedures proposed control fdr at the desired level enjoy certain optimality properties and are especially powerful in identifying clustered non null cases the new procedure is applied to an influenza like illness surveillance study for detecting the timing of epidemic periods

large deviations for vector valued functionals of a markov chain lower bounds on obtient des bornes inferieures que l on exprime en termes des parametres de convergence de certains noyaux on donne une application aux mesures empiriques de chaines de markov

combining spatial extent and peak intensity to test for activations in functional imaging within the framework of statistical mapping there are up to now only two tests used to assess the regional significance in functional images one is based on the magnitude of the foci and tends to detect high intensity signals while the second is based on the spatial extent of regions defined by a simple thresholding of the statistical map a test that is more sensitive to extended signals the aim of this paper is to combine the two tests into a single test that is more sensitive to a wider range of signals this combined test is based on an analytical approximation of the distribution of these two parameters size and height and is applied in the context of statistical maps the risk of error in noise only 2d or 3d volumes is assessed under a wide range of experimental conditions obtained by varying both the resolution of the map and the threshold at which clusters are defined in addition we have investigated this new test on simulated signals and applied it to an experimental pet dataset the experimental risk of error is close to the predicted one and the overall sensitivity increases when analyzing a volume containing different types of signals

adaptive linear step up procedures that control the false discovery rate we provide a new two stage procedure in which the linear step up procedure is used in stage one to estimate mo providing a new level q which is used in the linear step up procedure in the second stage we prove that a general form of the two stage procedure controls the false discovery rate at the desired level q this framework enables us to study analytically the properties of other procedures that exist in the literature a simulation study is presented that shows that two stage adaptive procedures improve in power over the original procedure mainly because they provide tighter control of the false discovery rate we further study the performance of the current suggestions some variations of the procedures and previous suggestions in the case where the test statistics are positively dependent a case for which the original procedure controls the false discovery rate in the setting studied here the newly proposed two stage procedure is the only one that controls the false discovery rate the procedures are illustrated with two examples of biological importance

discovering the false discovery rate summary i describe the background for the paper controlling the false discovery rate a new and powerful approach to multiple comparisons by benjamini and hochberg that was published in the journal of the royal statistical society series b in 1995 i review the progress since made on the false discovery rate as well as the major conceptual developments that followed

multiple testing of local maxima for detection of peaks in 1d a topological multiple testing scheme for one dimensional domains is proposed where rather than testing every spatial or temporal location for the presence of a signal tests are performed only at the local maxima of the smoothed observed sequence assuming unimodal true peaks with finite support and gaussian stationary ergodic noise it is shown that the algorithm with bonferroni or benjamini hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate and is power consistent as the search space and the signal strength get large where the search space may grow exponentially faster than the signal strength simulations show that error levels are maintained for nonasymptotic conditions and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks akin to the matched filter theorem in signal processing the methods are illustrated in an analysis of electrical recordings of neuronal cell activity

statistical tests for fmri based on experimental randomization statistical parametric mapping spm analysis of fmri data requires specifying correctly the temporal and spatial noise covariance structure this is a difficult if not impossible task when these assumptions are not satisfied statistical inference can be invalid or inefficient permutation tests are free of strong assumptions on the distribution of signal noise we propose permutation tests of fmri data based on experimental randomization of the stimulus sequences smooth hemodynamic response curves are estimated using quadratic b splines we study fmri data obtained from event related potential erp oddball paradigm tests of two tone and three tone stimulus sequences are conducted to illustrate the application of the proposed method comparisons of the spm and proposed method show that a permutation test is more specific and less susceptible to artifacts

false discovery rates for spatial signals the problem of multiple testing for the presence of signal in spatial data can involve numerous locations traditionally each location is tested separately for signal presence but then the findings are reported in terms of clusters of nearby locations this is an indication that the units of interest for testing are clusters rather than individual locations the investigator may know a priori these more natural units or an approximation to them we suggest testing these cluster units rather than individual locations thus increasing the signal to noise ratio within the unit tested as well as reducing the number of hypothesis tests conducted because the signal may be absent from part of each cluster we define a cluster as containing a signal if the signal is present somewhere within the cluster we suggest controlling the false discovery rate fdr on clusters i e the expected proportion of clusters rejected erroneously out of all clusters rejected or its extension to general weights wfdr we int

a stochastic process approach to false discovery control this paper extends the theory of false discovery rates fdr pioneered by benjamini and hochberg j roy statist soc ser b 57 1995 289 300 we develop a framework in which the false discovery proportion fdp the number of false rejections divided by the number of rejections is treated as a stochastic process after obtaining the limiting distribution of the process we demonstrate the validity of a class of procedures for controlling the false discovery rate the expected fdp we construct a confidence envelope for the whole fdp process from these envelopes we derive confidence thresholds for controlling the quantiles of the distribution of the fdp as well as controlling the number of false discoveries we also investigate methods for estimating the p value distribution

multiple testing of local maxima for detection of peaks in random fields a topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic gaussian noise where tests are performed at local maxima of the smoothed observed signals the procedure generalizes the one dimensional scheme of 31 to euclidean domains of arbitrary dimension two methods are developed according to two different ways of computing p values i using the exact distribution of the height of local maxima available explicitly when the noise field is isotropic 9 10 ii using an approximation to the overshoot distribution of local maxima above a pre threshold applicable when the exact distribution is unknown such as when the stationary noise field is non isotropic 9 the algorithms combined with the benjamini hochberg procedure for thresholding p values provide asymptotic strong control of the false discovery rate fdr and power consistency with specific rates as the search space and signal strength get large the optimal smoothing bandwidth and optimal pre threshold are obtained to achieve maximum power simulations show that fdr levels are maintained in non asymptotic conditions the methods are illustrated in the analysis of functional magnetic resonance images of the brain

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

on line expectation maximization algorithm for latent data models in this contribution we propose a generic online also sometimes called adaptive or recursive version of the expectation maximisation em algorithm applicable to latent variable models of independent observations compared to the algorithm of titterington 1984 this approach is more directly connected to the usual em algorithm and does not rely on integration with respect to the complete data distribution the resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the kullback leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate i e that of the maximum likelihood estimator in addition the proposed approach is also suitable for conditional or regression models as illustrated in the case of the mixture of linear regressions model

adaptive false discovery rate control under independence and dependence in the context of multiple hypothesis testing the proportion 0 of true null hypotheses in the pool of hypotheses to test often plays a crucial role although it is generally unknown a priori a testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive in this paper we focus on the issue of false discovery rate fdr control and we present new adaptive multiple testing procedures with control of the fdr in a first part assuming independence of the p values we present two new procedures and give a unified review of other existing adaptive procedures that have provably controlled fdr we report extensive simulation results comparing these procedures and testing their robustness when the independence assumption is violated the new proposed procedures appear competitive with existing ones the overall best though is reported to be storey s estimator albeit for a specific parameter setting that does not appear to have been considered before in a second part we propose adaptive versions of step up procedures that have provably controlled fdr under positive dependence and unspecified dependence of the p values respectively in the latter case while simulations only show an improvement over non adaptive procedures in limited situations these are to our knowledge among the first theoretically founded adaptive multiple testing procedures that control the fdr when the p values are not independent

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

some nonasymptotic results on resampling in high dimension ii multiple tests in the context of correlated multiple tests we aim to nonasymptotically control the family wise error rate fwer using resampling type procedures we observe repeated realizations of a gaussian random vector in possibly high dimension and with an unknown covariance matrix and consider the one and two sided multiple testing problem for the mean values of its coordinates we address this problem by using the confidence regions developed in the companion paper ann statist 2009 to appear which lead directly to single step procedures these can then be improved using step down algorithms following an established general methodology laid down by romano and wolf j amer statist assoc 100 2005 94 108 this gives rise to several different procedures whose performances are compared using simulated data

martin boundary of a reflected random walk on a half space the complete representation of the martin compactification for reflected random walks on a half space mathbb z d times mathbb n is obtained it is shown that the full martin compactification is in general not homeomorphic to the radial compactification obtained by ney and spitzer for the homogeneous random walks in mathbb z d convergence of a sequence of points z n in mathbb z d 1 times mathbb n to a point of on the martin boundary does not imply convergence of the sequence zn zn on the unit sphere sd our approach relies on the large deviation properties of the scaled processes and uses pascal s method combined with the ratio limit theorem the existence of non radial limits is related to non linear optimal large deviation trajectories

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

topological fdr for neuroimaging in this technical note we describe and validate a topological false discovery rate fdr procedure for statistical parametric mapping this procedure is designed to deal with signal that is continuous and has in principle unbounded spatial support we therefore infer on topological features of the signal such as the existence of local maxima or peaks above some threshold using results from random field theory we assign a p value to each maximum in an spm and identify an adaptive threshold that controls false discovery rate using the benjamini and hochberg bh procedure 1995 this provides a natural complement to conventional family wise error fwe control on local maxima we use simulations to contrast these procedures both in terms of their relative number of discoveries and their spatial accuracy via the distribution of the euclidian distance between true and discovered activations we also assessed two other procedures cluster wise and voxel wise fdr procedures our results suggest that a fdr control of maxima or peaks is more sensitive than fwe control of peaks with minimal cost in terms of false positives b voxel wise fdr is substantially less accurate than topological fwe or fdr control finally we present an illustrative application using an fmri study of visual attention

comparing spatial predictions under a general loss function we develop a hypothesis test to determine whether a significant difference in the spatial predictions produced by two competing models exists on average across the entire spatial domain of interest the null hypothesis is that of no difference and a spatial loss differential is created based on the observed data the two sets of predictions and the loss function chosen by the researcher the test assumes only isotropy and short range spatial dependence of the loss differential but does allow it to be non gaussian non zero mean and spatially correlated constant and nonconstant spatial trends in the loss differential are treated in two separate cases monte carlo simulations illustrate the size and power properties of this test and an example based on daily average wind speeds in oklahoma is used for illustration supplemental results are available online

mapping human brain function with meg and eeg methods and validation we survey the field of magnetoencephalography meg and electroencephalography eeg source estimation these modalities offer the potential for functional brain mapping with temporal resolution in the millisecond range however the limited number of spatial measurements and the ill posedness of the inverse problem present significant limits to our ability to produce accurate spatial maps from these data without imposing major restrictions on the form of the inverse solution here we describe approaches to solving the forward problem of computing the mapping from putative inverse solutions into the data space we then describe the inverse problem in terms of low dimensional solutions based on the equivalent current dipole ecd and high dimensional solutions in which images of neural activation are constrained to the cerebral cortex we also address the issue of objective assessment of the relative performance of inverse procedures by the free response receiver operating characteristic froc curve we conclude with a discussion of methods for assessing statistical significance of experimental results through use of the bootstrap for determining confidence regions in dipole fitting methods and random field rf and permutation methods for detecting significant activation in cortically constrained imaging studies

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

controlling false positive rates in mass multivariate tests for electromagnetic responses we address the problem of controlling false positive rates in mass multivariate tests for electromagnetic responses in compact regions of source space we show that mass univariate thresholds based on sensor level multivariate thresholds approximated using roy s union intersection principle are unduly conservative we then consider a bonferroni correction for source level tests based on the number of unique lead field extrema for a given source space the sensor indices corresponding to the maxima and minima for each dipolar lead field are listed and the number of unique extrema is given by the number of unique pairs in this list using a multivariate beamformer formulation we validate this heuristic against empirical permutation thresholds for mass univariate and mass multivariate tests of induced and evoked responses for a variety of source spaces using simulated and real data we also show that the same approximations hold when dealing with a cortical manifold rather than a volume and for mass multivariate minimum norm solutions we demonstrate that the mass multivariate framework is not restricted to tests on a single contrast of effects cf roy s maximum root but also accommodates multivariate effects cf wilk s lambda

enhanced signal detection in neuroimaging by means of regional control of the global false discovery rate in the context of neuroimaging experiments it is essential to account for the multiple comparisons problem when thresholding statistical mappings various methods are in use to deal with this issue but they differ in their signal detection power for small and large scale effects in this paper we comprehensively describe a new method that is based on control of the false discovery rate fdr our method increases sensitivity by exploiting the spatially clustered nature of neuroimaging effects this is achieved by using a sliding window technique in which fdr control is first applied at a regional level thus a new statistical map that is related to the regionally achieved fdr is derived from the available voxelwise p values on the basis of receiver operating characteristic roc curves thresholding based on this map is demonstrated to have better discriminatory power than conventional thresholding based on p values secondly it is shown that the resulting maps can be thresholded at a level that results in control of the global fdr by means of statistical arguments and numerical simulations under widely varying conditions our method is validated characterized and compared to some other common voxel based methods uncorrected thresholding bonferroni correction and conventional fdr control it is found that our method shows considerably higher sensitivity as compared to conventional fdr control while still controlling the achieved fdr at the same level or better finally our method is applied to two diverse neuroimaging experiments to assess its practical merits resulting in substantial improvements as compared to the other methods

distributed maximum likelihood for simultaneous self localization and tracking in sensor networks we show that the sensor self localization problem can be cast as a static parameter estimation problem for hidden markov models and we implement fully decentralized versions of the recursive maximum likelihood and on line expectation maximization algorithms to localize the sensor network simultaneously with target tracking for linear gaussian models our algorithms can be implemented exactly using a distributed version of the kalman filter and a novel message passing algorithm the latter allows each node to compute the local derivatives of the likelihood or the sufficient statistics needed for expectation maximization in the non linear case a solution based on local linearization in the spirit of the extended kalman filter is proposed in numerical examples we demonstrate that the developed algorithms are able to learn the localization parameters

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

large deviations for markov processes with discontinuous statistics ii random walks let 1 and 2 be borel probability measures on d with finite moment generating functions the main theorem in this paper proves the large deviation principle for a random walk whose transition mechanism is governed by 1 when the walk is in the left halfspace 1 x d x 1 0 and whose transition mechanism is governed by 2 when the walk is in the right halfspace 2 x d x 1 0 when the measures 1 and 2 are equal the main theorem reduces to cramer s theorem