estimating the gap of finite metric spaces of strict p negative type abstract let x d be a finite metric space this paper first discusses the spectrum of the p distance matrix of a finite metric space of p negative type and then gives upper and lower bounds for the so called gap x p of a finite metric space of strict p negative type furthermore estimations for x p under a certain gluing construction for finite metric spaces are given and finally be applied to finite ultrametric spaces

an investigation of statistical power for continuous arterial spin labeling imaging at 1 5 t variance estimates can be used in conjunction with scientifically meaningful effect sizes to design experiments with type ii error control here we present estimates of intra and inter subject variances for region of interest roi from resting cerebral blood flow cbf maps obtained using whole brain spin echo echoplanar se epi continuous arterial spin labeling casl imaging on 52 elderly subjects age 70 5 7 9 years 29 males there was substantial intrasubject systematic variability in cbf of gray matter rois corresponding to a range of standard deviations 39 168 ml 100g min this variability was mainly due to two factors 1 an expected inverse relationship between roi volume and intrasubject variance and 2 an increased effective post labeling delay for more superior slices acquired later in the sequence for example intrasubject variance in brodmann area 4 ba 4 was 8 times larger than in hippocampus despite their similar gray matter volumes estimated roi wise power was computed for various numbers of acquired cbf images numbers of subjects and cbf effect sizes for two experimental designs independent samples t test and paired t test the theoretical effects of pulse sequence and field strength on general applicability of these results are discussed

on the generalized roundness of finite metric spaces in this paper we show that the generalized roundness of a finite metric space can be bounded from below by a strictly positive constant that only depends upon the number of points in the space

multilevel linear modelling for fmri group analysis using bayesian inference functional magnetic resonance imaging studies often involve the acquisition of data from multiple sessions and or multiple subjects a hierarchical approach can be taken to modelling such data with a general linear model glm at each level of the hierarchy introducing different random effects variance components inferring on these models is nontrivial with frequentist solutions being unavailable a solution is to use a bayesian framework one important ingredient in this is the choice of prior on the variance components and top level regression parameters due to the typically small numbers of sessions or subjects in neuroimaging the choice of prior is critical to alleviate this problem we introduce to neuroimage modelling the approach of reference priors which drives the choice of prior such that it is noninformative in an information theoretic sense we propose two inference techniques at the top level for multilevel hierarchies a fast approach and a slower more accurate approach we also demonstrate that we can infer on the top level of multilevel hierarchies by inferring on the levels of the hierarchy separately and passing summary statistics of a noncentral multivariate t distribution between them

trees and ultrametric spaces a categorical equivalence abstract there is a well known correspondence between infinite trees and ultrametric spaces that comes from considering the end space of the tree the correspondence is interpreted here as an equivalence between two categories one of which encodes the geometry of trees at infinity and the other encodes the micro geometry of complete ultrametric spaces

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

mixed effects and fmri studies abstract this note concerns mixed effect mfx analyses in multisession functional magnetic resonance imaging fmri studies it clarifies the relationship between mixed effect analyses and the two stage summary statistics procedure holmes a p friston k j 1998 generalisability random effects and population inference neuroimage 7 s754 that has been adopted widely for analyses of fmri data at the group level we describe a simple procedure based on restricted maximum likelihood reml estimates of covariance components that enables full mixed effects analyses in the context of statistical parametric mapping using this procedure we compare the results of a full mixed effects analysis with those obtained from the simpler two stage procedure and comment on the situations when the two approaches may give different results

newton raphson and em algorithms for linear mixed effects models for repeated measures data abstract we develop an efficient and effective implementation of the newton raphson nr algorithm for estimating the parameters in mixed effects models for repeated measures data we formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by jennrich and schluchter 1986 to speed convergence and ensure a positive definite covariance matrix for the random effects at each iteration we use matrix decompositions to develop efficient and computationally stable implementations of both the nr algorithm and an em algorithm laird and ware 1982 for this model we compare the two methods em vs nr in terms of computational order and performance on two sample data sets and conclude that in most situations a well implemented nr algorithm is preferable to the em algorithm or em algorithm with aitken s acceleration the term repeated measures refers to experimental designs where there are several individuals and several

asymptotic negative type properties of finite ultrametric spaces abstract negative type inequalities arise in the study of embedding properties of metric spaces but they often reduce to intractable combinatorial problems in this paper we study more quantitative versions of these inequalities involving the so called p negative type gap in particular we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining let x d be a given finite ultrametric space with minimum non zero distance then the p negative type gap x p of x d is positive for all p 0 in this paper we compute the value of the limit x lim p x p p it turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula this formula allows us to characterize when the ratio x p p is a constant independent of p the determination of x also leads to new asymptotically sharp families of enhanced p negative type inequalities for x d indeed suppose that g 0 x then for all sufficiently large p the inequality g p 2 k 1 n k 2 j i 1 n d z j z i p j i 0 holds for each finite subset z 1 z n x and each scalar n tuple 1 n r n that satisfies 1 n 0 notably these results do not extend to general finite metric spaces

enhanced negative type for finite metric trees abstract a finite metric tree is a finite connected graph that has no cycles endowed with an edge weighted path metric finite metric trees are known to have strict 1 negative type in this paper we introduce a new family of inequalities 1 that encode the best possible quantification of the strictness of the non trivial 1 negative type inequalities for finite metric trees these inequalities are sufficiently strong to imply that any given finite metric tree t d must have strict p negative type for all p in an open interval 1 1 where 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on t in particular if the edges of the tree are not weighted then it follows that depends only upon the number of vertices in the tree we also give an example of an infinite metric tree that has strict 1 negative type but does not have p negative type for any p 1 this shows that the maximal p negative type of a metric space can be strict

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

asymptotic a priori estimates for the error in the nonlinear filtering problem corresp the minimal mean square error for the scalar nonlinear filtering problem is considered asymptotic lower and upper bounds on the error are derived for the case where the intensity of the observation noise tends to zero

estimating and testing variance components in a multi level glm most analysis of multi subject fmri data is concerned with determining whether there exists a significant population wide activation in a comparison between two or more conditions this is typically assessed by testing the average value of a contrast of parameter estimates cope against zero in a general linear model glm analysis however important information can also be obtained by testing whether there exist significant individual differences in effect magnitude between subjects i e whether the variance of a cope is significantly different from zero intuitively such a test amounts to testing whether inter individual differences are larger than would be expected given the within subject error variance we compare several methods for estimating variance components including a a naive estimate using ordinary least squares ols b linear mixed effects in r lmer c a novel matlab implementation of iterative generalized least squares igls and its restricted maximum likelihood variant rigls all methods produced reasonable estimates of within and between subject variance components with igls providing an attractive balance between sensitivity and appropriate control of false positives finally we use the igls method to estimate inter subject variance in a perfusion fmri study n 18 of social evaluative threat and show evidence for significant inter individual differences in ventromedial prefrontal cortex vmpfc amygdala hippocampus and medial temporal lobes insula and brainstem with predicted inverse coupling between vmpfc and the midbrain periaqueductal gray only when high inter individual variance was used to define the seed for functional connectivity analyses in sum tests of variance provides a way of selecting regions that show significant inter individual variability for subsequent analyses that attempt to explain those individual differences

on the gap of finite metric spaces of p negative type let x d be a metric space of p negative type recently i doust and a weston introduced a quantification of the p negative type property the so called gap of x this paper gives some formulas for the gap of a finite metric space of strict p negative type and applies them to evaluate for some concrete finite metric spaces

polygonal equalities and virtual degeneracy in lp spaces abstract suppose 0 p 2 and that is a measure space for which l p is at least two dimensional the central results of this paper provide a complete description of the subsets of l p that have strict p negative type in order to do this we study non trivial p polygonal equalities in l p these are equalities that can after appropriate rearrangement and simplification be expressed in the form j i 1 n j i z j z i p p 0 where z 1 z n is a subset of l p and 1 n are non zero real numbers that sum to zero we provide a complete classification of the non trivial p polygonal equalities in l p the cases p 2 and p 2 are substantially different and are treated separately the case p 1 generalizes an elegant result of elsner han koltracht neumann and zippin another reason for studying non trivial p polygonal equalities in l p is due to the fact that they preclude the existence of certain types of isometry for example our techniques show that if x d is a metric space that has strict q negative type for some q p then 1 x d is not isometric to any linear subspace w of l p that contains a pair of disjointly supported non zero vectors and 2 x d is not isometric to any subset of l p that has non empty interior furthermore in the case p 2 it also follows that x d is not isometric to any affinely dependent subset of l 2 more generally we show that if y is a metric space whose generalized roundness is finite and if x d is a metric space that has strict q negative type for some q then x d is not isometric to any metric subspace of y that admits a non trivial p 1 polygonal equality for some p 1 q it is notable in all of these statements that the metric space x d can for instance be any ultrametric space as a result we obtain new insights into sophisticated embedding theorems of lemin and shkarin we conclude the paper by constructing some pathological infinite dimensional linear subspaces of l p that do not have strict p negative type

finite metric spaces of strictly negative type abstract we prove that if a finite metric space is of strictly negative type then its transfinite diameter is uniquely realized by the infinite extender load vector finite metric spaces that have this property include all spaces on two three or four points all trees and all finite subspaces of euclidean spaces we prove that if the distance matrix is both hypermetric and regular then it is of strictly negative type we show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points in connection with an open problem raised by kelly we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular and hence of strictly negative type

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v