multimodality image registration by maximization of mutual information a new approach to the problem of multimodality medical image registration is proposed using a basic concept from information theory mutual information mi or relative entropy as a new matching criterion the method presented in this paper applies mi to measure the statistical dependence or information redundancy between the image intensities of corresponding voxels in both images which is assumed to be maximal if the images are geometrically aligned maximization of mi is a very general and powerful criterion because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on the image content of the modalities involved the accuracy of the mi criterion is validated for rigid body registration of computed tomography ct magnetic resonance mr and photon emission tomography pet images by comparison with the stereotactic registration solution while robustness is evaluated with respect to implementation issues such as interpolation and optimization and image content including partial overlap and image degradation our results demonstrate that subvoxel accuracy with respect to the stereotactic reference solution can be achieved completely automatically and without any prior segmentation feature extraction or other preprocessing steps which makes this method very well suited for clinical applications

segmented concave least squares a nonparametric piecewise linear regression abstract in this paper segmented concave least squares scls is introduced scls is a nonparametric piecewise linear regression problem in which the estimated function is monotonic concave and the number of linear segments k is pre specified ordinary least squares k 1 and concave least squares k n the number of observations are two extreme cases of this problem an application of scls is to estimate a hedonic function using this method observations are categorized into k groups and a piecewise linear hedonic function is estimated such that there is one linear segment for every group the estimated hedonic function holds the principle of diminishing marginal utility in this paper scls is used to categorize hotels in finland into three groups a trade off between the number of groups and the goodness of fit measure is used to determine the number of groups based on the similarities of the pricing methods hotels in the sample are endogenously classified and the shadow prices for each group are calculated the results reveal that the hotels do not value hotel attributes similarly and there are significant differences among groups hedonic pricing model via scls provides a novel categorization of hotels that cannot be obtained by using ordinary least squares

nonparametric least squares estimation of a multivariate convex regression function this paper deals with the consistency of the least squares estimator of a convex regression function when the predictor is multidimensional we characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided we also consider a regression function which is known to be convex and componentwise nonincreasing and discuss the characterization computation and consistency of its least squares estimator

bayesian isotonic regression and trend analysis in many applications the mean of a response variable can be assumed to be a nondecreasing function of a continuous predictor controlling for covariates in such cases interest often focuses on estimating the regression function while also assessing evidence of an association this article proposes a new framework for bayesian isotonic regression and order restricted inference approximating the regression function with a high dimensional piecewise linear model the nondecreasing constraint is incorporated through a prior distribution for the slopes consisting of a product mixture of point masses accounting for flat regions and truncated normal densities to borrow information across the intervals and smooth the curve the prior is formulated as a latent autoregressive normal process this structure facilitates efficient posterior computation since the full conditional distributions of the parameters have simple conjugate forms point and interval estimates of the regression function and posterior probabilities of an association for different regions of the predictor can be estimated from a single mcmc run generalizations to categorical outcomes and multiple predictors are described and the approach is applied to an epidemiology application

f murtagh 2005 correspondence analysis and data coding with java and r 230 pp us 76 00 isbn 1584885289

semiconvex regression for metamodeling based optimization stochastic search involves finding a set of controllable parameters that minimizes an unknown objective function using a set of noisy observations we consider the case when the unknown function is convex and a metamodel is used as a surrogate objective function often the data are non i i d and include an observable state variable such as applicant information in a loan rate decision problem state information is difficult to incorporate into convex models we propose a new semiconvex regression method that is used to produce a convex metamodel in the presence of a state variable we show consistency for this method we demonstrate its effectiveness for metamodeling on a set of synthetic inventory management problems and a large real life auto loan dataset

principal inertia components and applications we explore properties and applications of the principal inertia components pics between two discrete random variables x and y the pics lie in the intersection of information and estimation theory and provide a fine grained decomposition of the dependence between x and y moreover the pics describe which functions of x can or cannot be reliably inferred in terms of mmse given an observation of y we demonstrate that the pics play an important role in information theory and they can be used to characterize information theoretic limits of certain estimation problems in privacy settings we prove that the pics are related to the fundamental limits of perfect privacy

labeling of ambiguous subvoxel fibre bundle configurations in high angular resolution diffusion mri whereas high angular resolution reconstruction methods for diffusion mri can estimate multiple dominant fibre orientations within a single imaging voxel they are fundamentally limited in certain cases of complex subvoxel fibre structures resulting in ambiguous local orientation distribution functions in this article we address the important problem of disambiguating such complex subvoxel fibre tract configurations with the purpose of improving the performance of fibre tractography we do so by extending a curve inference method to distinguish between the cases of curving and fanning fibre bundles using differential geometric estimates in a local neighbourhood the key benefit of this method is the inference of curves instead of only fibre orientations to model the underlying fibre bundles this in turn allows distinct fibre geometries that contain nearly identical sets of fibre orientations at a voxel to be distinguished from one another experimental results demonstrate the ability of the method to successfully label voxels into one of the above categories and improve the performance of a fibre tracking algorithm

optimal stopping of markov processes hilbert space theory approximation algorithms and an application to pricing high dimensional financial derivatives the authors develop a theory characterizing optimal stopping times for discrete time ergodic markov processes with discounted rewards the theory differs from prior work by its view of per stage and terminal reward functions as elements of a certain hilbert space in addition to a streamlined analysis establishing existence and uniqueness of a solution to bellman s equation this approach provides an elegant framework for the study of approximate solutions in particular the authors propose a stochastic approximation algorithm that tunes weights of a linear combination of basis functions in order to approximate a value function they prove that this algorithm converges almost surely and that the limit of convergence has some desirable properties the utility of the approximation method is illustrated via a computational case study involving the pricing of a path dependent financial derivative security that gives rise to an optimal stopping problem with a 100 dimensional state space

direct estimation of the fiber orientation density function from diffusion weighted mri data using spherical deconvolution diffusion weighted magnetic resonance imaging can provide information related to the arrangement of white matter fibers the diffusion tensor is the model most commonly used to derive the orientation of the fibers within a voxel however this model has been shown to fail in regions containing several fiber populations with distinct orientations a number of alternative models have been suggested such as multiple tensor fitting q space and q ball imaging however each of these has inherent limitations in this study we propose a novel method for estimating the fiber orientation distribution directly from high angular resolution diffusion weighted mr data without the need for prior assumptions regarding the number of fiber populations present we assume that all white matter fiber bundles in the brain share identical diffusion characteristics thus implicitly assigning any differences in diffusion anisotropy to partial volume effects the diffusion weighted signal attenuation measured over the surface of a sphere can then be expressed as the convolution over the sphere of a response function the diffusion weighted attenuation profile for a typical fiber bundle with the fiber orientation density function odf the fiber odf the distribution of fiber orientations within the voxel can therefore be obtained using spherical deconvolution the properties of the technique are demonstrated using simulations and on data acquired from a volunteer using a standard 1 5 t clinical scanner the technique can recover the fiber odf in regions of multiple fiber crossing and holds promise for applications such as tractography

a method of optimal scaling for multivariate ordinal data and its extensions this paper develops a method of optimal scaling for multivariate ordinal data in the framework of a generalized principal component analysis this method yields a multidimensional configuration of items a unidimensional scale of category weights for each item and optionally a multidimensional configuration of subjects the computation is performed by alternately solving an eigenvalue problem and executing a quasi newton projection method the algorithm is extended for analysis of data with mixed measurement levels or for analysis with a combined weighting of items numerical examples and simulations are provided the algorithm is discussed and compared with some related methods

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

generalized additive and index models with shape constraints summary we study generalized additive models with shape restrictions e g monotonicity convexity and concavity imposed on each component of the additive prediction function we show that this framework facilitates a non parametric estimator of each additive component obtained by maximizing the likelihood the procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals more generally our methodology can be applied to generalized additive index models here again the procedure can be justified on theoretical grounds and like the original algorithm has highly competitive finite sample performance practical utility is illustrated through the use of these methods in the analysis of two real data sets our algorithms are publicly available in the r package scar short for shape constrained additive regression

the choice of constraints in correspondence analysis a discussion of alternative constraint systems has been lacking in the literature on correspondence analysis and related techniques this paper reiterates earlier results that an explicit choice of constraints has to be made which can have important effects on the resulting scores the paper also presents new results on dealing with missing data and probabilistic category assignment

a stochastic multidimensional scaling vector threshold model for the spatial representation of pick any n data this paper presents a new stochastic multidimensional scaling vector threshold model designed to analyze pick any n choice data e g consumers rendering buy no buy decisions concerning a number of actual products a maximum likelihood procedure is formulated to estimate a joint space of both individuals represented as vectors and stimuli represented as points the relevant psychometric literature concerning the spatial treatment of such binary choice data is reviewed the nonlinear probit type model is described as well as the conjugate gradient procedure used to estimate parameters results of monte carlo analyses investigating the performance of this methodology with synthetic choice data sets are presented an application concerning consumer choices for eleven competitive brands of soft drinks is discussed finally directions for future research are presented in terms of further applications and generalizing the model to accommodate three way choice data

the geometric interpretation of correspondence analysis abstract correspondence analysis is an exploratory multivariate technique that converts a data matrix into a particular type of graphical display in which the rows and columns are depicted as points the method has a long and varied history and has appeared in different forms in the psychometric and ecological literature among others in this article we review the geometry of correspondence analysis and its geometric interpretation we also discuss various extensions of correspondence analysis to multivariate categorical data multiple correspondence analysis and a variety of other data types

on quantifying different types of categorical data in quantifying categorical data constraints play an important role in characterizing the outcome in the guttman type quantification of contingency tables and multiple choice data incidence data the trivial solution due to the marginal constraints is typically removed before quantification this removal however has the effect of distorting the shape of the total space awareness of this is important for the interpretation of the quantified outcome the present study provides some relevant formulas for those cases that are affected by the trivial solution and those cases that are not the characterization of the total space used by the guttman type quantification and pertinent discussion are presented

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

nonparametric function estimation subject to monotonicity convexity and other shape constraints this paper uses free knot and fixed knot regression splines in a bayesian context to develop methods for the nonparametric estimation of functions subject to shape constraints in models with log concave likelihood functions the shape constraints we consider include monotonicity convexity and functions with a single minimum a computationally efficient mcmc sampling algorithm is developed that converges faster than previous methods for non gaussian models simulation results indicate the monotonically constrained function estimates have good small sample properties relative to i unconstrained function estimates and ii function estimates obtained from other constrained estimation methods when such methods exist also asymptotic results show the methodology provides consistent estimates for a large class of smooth functions two detailed illustrations exemplify the ideas

singular value decomposition in additive multiplicative and logistic forms singular value decomposition svd is widely used in data processing reduction and visualization applied to a positive matrix the regular additive svd by the first several dual vectors can yield irrelevant negative elements of the approximated matrix we consider a multiplicative svd modification that corresponds to minimizing the relative errors and produces always positive matrices at any approximation step another logistic svd modification can be used for decomposition of the matrices of proportions when a regular svd can yield the elements beyond the zero one range while the modified svd decomposition produces all the elements within the correct range at any step of approximation several additional modifications of matrix approximation are also considered

the geometry of biplot scaling a simple geometry allows the main properties of matrix approximations used in biplot displays to be developed it establishes orthogonal components of an analysis of variance from which different contributions to approximations may be assessed particular attention is paid to approximations that share the same singular vectors in which case the solution space is a convex cone two and three dimensional approximations are examined in detail and then the geometry is interpreted for different forms of the matrix being approximated

trace inference curvature consistency and curve detection an approach is described for curve inference that is based on curvature information the inference procedure is divided into two stages a trace inference stage which is the subject of the present work and a curve synthesis stage it is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time and that tangent and curvature information are sufficient these make it possible to specify powerful constraints between estimated tangents to a curve in terms of a neighborhood relationship called cocircularity and between curvature estimates in terms of a curvature consistency relation because all curve information is quantized special care must be taken to obtain accurate estimates of trace points tangents and curvatures this issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint the procedure is applied to two types of images artificial images designed to evaluate curvature and noise sensitivity and natural images

flow based fiber tracking with diffusion tensor and q ball data validation and comparison to principal diffusion direction techniques in this study we evaluate the performance of a flow based surface evolution fiber tracking algorithm by means of a physical anisotropic diffusion phantom with known connectivity we introduce a novel speed function for surface evolution that is derived from either diffusion tensor dt data high angular resolution diffusion hard data or a combined dt hard hybrid approach we use the model free q ball imaging qbi approach for hard reconstruction the anisotropic diffusion phantom allows us to compare and evaluate the performance of different fiber tracking approaches in the presence of real imaging artifacts noise and subvoxel partial volume averaging of fiber directions the surface evolution approach using the full diffusion tensor as opposed to the principal diffusion direction pdd only is compared to pdd based line propagation fiber tracking additionally dt reconstruction is compared to hard reconstruction for fiber tracking both using surface evolution we show the potential for surface evolution using the full diffusion tensor to map connections in regions of subvoxel partial volume averaging of fiber directions which can be difficult to map with pdd based methods we then show that the fiber tracking results can be improved by using high angular resolution reconstruction of the diffusion orientation distribution function in cases where the diffusion tensor model fits the data poorly

multivariate convex regression with adaptive partitioning we propose a new nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function convexity constraints are common in economics statistics operations research financial engineering and optimization but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand observations we introduce convex adaptive partitioning cap which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions cap is a computationally efficient consistent method for convex regression we demonstrate empirical performance by comparing the performance of cap to other shape constrained and unconstrained regression methods for predicting weekly wages and value function approximation for pricing american basket options

probabilistic diffusion tractography with multiple fibre orientations what can we gain we present a direct extension of probabilistic diffusion tractography to the case of multiple fibre orientations using automatic relevance determination we are able to perform online selection of the number of fibre orientations supported by the data at each voxel simplifying the problem of tracking in a multi orientation field we then apply the identical probabilistic algorithm to tractography in the multi and single fibre cases in a number of example systems which have previously been tracked successfully or unsuccessfully with single fibre tractography we show that multi fibre tractography offers significant advantages in sensitivity when tracking non dominant fibre populations but does not dramatically change tractography results for the dominant pathways

approximating optimization problems over convex functions many problems of theoretical and practical interest involve finding an optimum over a family of convex functions for instance finding the projection on the convex functions in hk and optimizing functionals arising from some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and functions with positive semidefinite discrete hessian need not be convex in a discrete sense previous work has concentrated on non local descriptions of convexity making the number of constraints to grow super linearly with the number of nodes even in dimension 2 and these descriptions are very difficult to extend to higher dimensions in this paper we propose a finite difference approximation using positive semidefinite programs and discrete hessians and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to problems in two and three dimensions

a bayesian approach to non parametric monotone function estimation the paper proposes two bayesian approaches to non parametric monotone function estimation the first approach uses a hierarchical bayes framework and a characterization of smooth monotone functions given by ramsay that allows unconstrained estimation the second approach uses a bayesian regression spline model of smith and kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate the small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods asymptotic results are also given that show that bayesian methods provide consistent function estimators for a large class of smooth functions an example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple regression model where two functions are constrained to be monotone

a computational framework for multivariate convex regression and its variants we study the nonparametric least squares estimator lse of a multivariate convex regression function the lse given as the solution to a quadratic program with o n2 linear constraints n being t

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

simple structure in component analysis techniques for mixtures of qualitative and quantitative variables several methods have been developed for the analysis of a mixture of qualitative and quantitative variables and one called pcamix includes ordinary principal component analysis pca and multiple correspondence analysis mca as special cases the present paper proposes several techniques for simple structure rotation of a pcamix solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for pca in addition a recently developed technique for the analysis of mixtures of qualitative and quantitative variables called indomix is shown to construct component scores without rotational freedom maximizing the quartimax criterion over all possible sets of component scores a numerical example is used to illustrate the implication that when used for qualitative variables indomix provides axes that discriminate between the observation units better than do those generated from mca

correspondence analysis with fuzzy data the fuzzy eigenvalue problem this paper constitutes a first step towards an extension of correspondence analysis with fuzzy data fca at this stage our main objective is to lay down the algebraic foundations for this fuzzy extension of the usual correspondence analysis a two step method is introduced to convert the fuzzy eigenvalue problem to an ordinary one we consider a fuzzy matrix as the set of its cuts each such cut is an interval valued matrix viewed as a line segment in the matrix space in this way line segments of cut matrices are transformed into intervals of eigenvalues therefore the two step method is essentially a reduction of the fuzzy eigenvalue problem to an ordinary one we illustrate the fca fuzzy eigenvalue problem with a simple numerical example we hope upon the completion of this project in near future to be able to supply the necessary tools for the end user of the correspondence analysis with fuzzy data

mosaic displays for multi way contingency tables abstract mosaic displays represent the counts in a contingency table by tiles whose size is proportional to the cell count this graphical display for categorical data generalizes readily to multi way tables this article discusses extensions of the mosaic display to highlight patterns of deviations from various models for categorical data first we introduce the use of color and shading to represent sign and magnitude of standardized residuals from a specified model for unordered categorical variables we show how the perception of patterns of association can be enhanced by reordering the categories second we introduce sequential mosaics of marginal subtables together with sequential models for these tables for a class of sequential models of joint independence the individual mosaics provide a graphic representation of a partition of the overall likelihood ratio g2 for complete independence in the full table into portions attributable to hypotheses about the marginal subtables

concave regression value constrained estimation and likelihood ratio based inference we propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function based on the shape restriction of concavity alternatively convexity dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis that is studying a concave least squares estimator satisfying a further equality constraint we study this null hypothesis least squares estimator nlse here and use it to study our likelihood ratio statistic the nlse is the solution to a convex program and we find a set of inequality and equality constraints that characterize the solution we also study a corresponding limiting version of the convex program based on observing a brownian motion with drift the solution to the limit problem is a stochastic process we study the optimality conditions for the solution to the limit problem and find that they match those we derived for the solution to the finite sample problem this allows us to show the limit stochastic process yields the limit distribution of the finite sample nlse we conjecture that the likelihood ratio statistic is asymptotically pivotal meaning that it has a limit distribution with no nuisance parameters to be estimated which makes it a very effective tool for this difficult inference problem we provide a partial proof of this conjecture and we also provide simulation evidence strongly supporting this conjecture

mathematical methods for diffusion mri processing in this article we review recent mathematical models and computational methods for the processing of diffusion magnetic resonance images including state of the art reconstruction of diffusion models cerebral white matter connectivity analysis and segmentation techniques we focus on diffusion tensor images dti and q ball images qbi

a robust variational approach for simultaneous smoothing and estimation of dti abstract estimating diffusion tensors is an essential step in many applications such as diffusion tensor image dti registration segmentation and fiber tractography most of the methods proposed in the literature for this task are not simultaneously statistically robust and feature preserving techniques in this paper we propose a novel and robust variational framework for simultaneous smoothing and estimation of diffusion tensors from diffusion mri our variational principle makes use of a recently introduced total kullback leibler tkl divergence for dti regularization tkl is a statistically robust dissimilarity measure for diffusion tensors and regularization by using tkl ensures the symmetric positive definiteness of tensors automatically further the regularization is weighted by a non local factor adapted from the conventional non local means filters finally for the data fidelity we use the nonlinear least squares term derived from the stejskal tanner model we present experimental results depicting the positive performance of our method in comparison to competing methods on synthetic and real data examples

a bayesian approach for stochastic white matter tractography white matter fiber bundles in the human brain can be located by tracing the local water diffusion in diffusion weighted magnetic resonance imaging mri images in this paper a novel bayesian modeling approach for white matter tractography is presented the uncertainty associated with estimated white matter fiber paths is investigated and a method for calculating the probability of a connection between two areas in the brain is introduced the main merits of the presented methodology are its simple implementation and its ability to handle noise in a theoretically justified way theory for estimating global connectivity is also presented as well as a theorem that facilitates the estimation of the parameters in a constrained tensor model of the local water diffusion profile

correspondence analysis of multivariate categorical data by weighted least squares summary a generalization of correspondence analysis to multivariate categorical data is proposed where all two way contingency tables of a set of categorical variables are simultaneously fitted by weighted least squares an alternating least squares algorithm is developed to perform the fitting this technique has a number of advantages over the usual generalization known as multiple correspondence analysis it is also an analogue of least squares factor analysis for categorical data

singular value decomposition of the third multivariate moment abstract the third moment of a random vector is a matrix which conveniently arranges all moments of order three which can be obtained from the random vector itself we investigate some properties of its singular value decomposition in particular we show that left eigenvectors corresponding to positive singular values of the third moment are vectorized symmetric matrices we derive further properties under the additional assumptions of exchangeability reversibility and independence statistical applications deal with measures of multivariate skewness

gaussian predictive process models for large spatial data sets with scientific data available at geocoded locations investigators are increasingly turning to spatial process models for carrying out statistical inference over the last decade hierarchical models implemented through markov chain monte carlo methods have become especially popular for spatial modelling given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics however fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations rendering such models infeasible for large spatial data sets this computational burden is exacerbated in multivariate settings with several spatially dependent response variables it is also aggravated when data are collected at frequent time points and spatiotemporal process models are used with regard to this challenge our contribution is to work with what we call predictive process models for spatial and spatiotemporal data every spatial or spatiotemporal process induces a predictive process model in fact arbitrarily many of them the latter models project process realizations of the former to a lower dimensional subspace thereby reducing the computational burden hence we achieve the flexibility to accommodate non stationary non gaussian possibly multivariate possibly spatiotemporal processes in the context of large data sets we discuss attractive theoretical properties of these predictive processes we also provide a computational template encompassing these diverse settings finally we illustrate the approach with simulated and real data sets

gleaning in the field of dual scaling some historical background and preliminary technical information are first presented and then a number of hidden but important methodological aspects of dual scaling are illustrated and discussed normed versus projected weights the amount of information accounted for by each solution a perfect solution to the problem of multidimensional unfolding multidimensional quantification space graphical display number of option problems option standardization versus item standardization and asymmetry of symmetric dual scaling contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions data analysis must be carried out in such a way that common sense intuition and sound logic will prevail

an analysis and synthesis of multiple correspondence analysis optimal scaling dual scaling homogeneity analysis and other methods for quantifying categorical multivariate data we discuss a variety of methods for quantifying categorical multivariate data these methods have been proposed in many different countries by many different authors under many different names in the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data in the second major section of the paper we introduce the notion of a duality diagram and use this diagram to synthesize the many superficially different methods into a single method

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

nonparametric option pricing under shape restrictions frequently economic theory places shape restrictions on functional relationships between economic variables this paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators we apply this technique to estimate the state price density spd or risk neutral density implicit in the market prices of options the option pricing function must be monotonic and convex simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day to day option pricing once appropriate theory motivated shape restrictions are imposed using s p 500 option prices we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999 unlike the constrained estimator we propose

smooth approximations to monotone concave functions in production analysis an alternative to nonparametric concave least squares abstract estimation of banking efficiency and productivity is essential for regulatory purposes and for testing various theories in the context of banking such as the quiet life hypothesis the bad management hypothesis etc in such studies it is therefore important to place as few restrictions as possible on the functional forms subject to global satisfaction of the theoretical properties relating to monotonicity and concavity in this paper we propose an alternative to nonparametric segmented concave least squares we use a differentiable approximation to an arbitrary functional form based on smoothly mixing cobb douglas anchor functions over the data space estimation is based on bayesian techniques organized around markov chain monte carlo the approximation properties of the new functional form are investigated in a monte carlo experiment where the true functional form is a symmetric generalized mcfadden the new techniques are applied to a large u s banking data set as well as a global banking data set

parametric spherical deconvolution inferring anatomical connectivity using diffusion mr imaging the human brain forms a complex neural network with a connectional architecture that is still far from being known in full detail even at the macroscopic level the advent of diffusion mr imaging has enabled the exploration of the structural properties of white matter in vivo in this article we propose a new forward model that maps the microscopic geometry of nervous tissue onto the water diffusion process and further onto the measured mr signals our spherical deconvolution approach completely parameterizes the fiber orientation density by a finite mixture of bingham distributions in addition we define the term anatomical connectivity taking the underlying image modality into account this neurophysiological metric may represent the proportion of the nerve fibers originating in the source area which intersect a given target region the specified inverse problem is solved by bayesian statistics posterior probability maps denote the probability that the connectivity value exceeds a chosen threshold conditional upon the noisy observations these maps allow us to draw inferences about the structural organization of the cerebral cortex moreover we will demonstrate the proposed approach with diffusion weighted data sets featuring high angular resolution

registration atlas estimation and variability analysis of white matter fiber bundles modeled as currents this paper proposes a generic framework for the registration the template estimation and the variability analysis of white matter fiber bundles extracted from diffusion images this framework is based on the metric on currents for the comparison of fiber bundles this metric measures anatomical differences between fiber bundles seen as global homologous structures across subjects it avoids the need to establish correspondences between points or between individual fibers of different bundles it can measure differences both in terms of the geometry of the bundles like its boundaries and in terms of the density of fibers within the bundle it is robust to fiber interruptions and reconnections in addition a recently introduced sparse approximation algorithm allows us to give an interpretable representation of the fiber bundles and their variations in the framework of currents first we used this metric to drive the registration between two sets of homologous fiber bundles of two different subjects a dense deformation of the underlying white matter is estimated which is constrained by the bundles seen as global anatomical landmarks by contrast the alignment obtained from image registration is driven only by the local gradient of the image second we propose a generative statistical model for the analysis of a collection of homologous bundles this model consistently estimates prototype fiber bundles called template which capture the anatomical invariants in the population a set of deformations which align the geometry of the template to that of each subject and a set of residual perturbations the statistical analysis of both the deformations and the residuals describe the anatomical variability in terms of geometry stretching torque etc and texture fiber density etc third this statistical modeling allows us to simulate new synthetic bundles according to the estimated variability this gives a way to interpret the anatomical features that the model detects consistently across the subjects this may be used to better understand the bias introduced by the fiber extraction methods and eventually to give anatomical characterization of the normal or pathological variability of fiber bundles

on convex functions and the finite element method many problems of theoretical and practical interest involve finding a convex or concave function for instance optimization problems such as finding the projection on the convex functions in h k omega or some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and an adequate discrete version of the hessian must be given in this paper we propose a finite element description of the hessian and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to optimization problems

resolution of complex tissue microarchitecture using the diffusion orientation transform dot this article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion weighted magnetic resonance mr data this novel approach utilizes the fourier transform relationship between the water displacement probabilities and diffusion attenuated mr signal expressed in spherical coordinates the radial part of the fourier integral is evaluated analytically under the assumption that mr signal attenuates exponentially the values of the resulting functions are evaluated at a fixed distance away from the origin the spherical harmonic transform of these functions yields the laplace series coefficients of the probabilities on a sphere of fixed radius alternatively probability values can be computed nonparametrically using legendre polynomials orientation maps calculated from excised rat nervous tissue data demonstrate this technique s ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm this proposed methodology has a trivial extension to multiexponential diffusion weighted signal decay the developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system

deterministic and probabilistic tractography based on complex fibre orientation distributions we propose an integral concept for tractography to describe crossing and splitting fibre bundles based on the fibre orientation distribution function odf estimated from high angular resolution diffusion imaging hardi we show that in order to perform accurate probabilistic tractography one needs to use a fibre odf estimation and not the diffusion odf we use a new fibre odf estimation obtained from a sharpening deconvolution transform sdt of the diffusion odf reconstructed from q ball imaging qbi this sdt provides new insight into the relationship between the hardi signal the diffusion odf and the fibre odf we demonstrate that the sdt agrees with classical spherical deconvolution and improves the angular resolution of qbi another important contribution of this paper is the development of new deterministic and new probabilistic tractography algorithms using the full multidirectional information obtained through use of the fibre odf an extensive comparison study is performed on human brain datasets comparing our new deterministic and probabilistic tracking algorithms in complex fibre crossing regions finally as an application of our new probabilistic tracking we quantify the reconstruction of transcallosal fibres intersecting with the corona radiata and the superior longitudinal fasciculus in a group of eight subjects most current diffusion tensor imaging dti based methods neglect these fibres which might lead to incorrect interpretations of brain functions

on the complexity of detecting convexity over a box it has recently been shown that the problem of testing global convexity of polynomials of degree four is strongly np hard answering an open question of n z shor this result is minimal in the degree of the polynomial when global convexity is of concern in a number of applications however one is interested in testing convexity only over a compact region most commonly a box i e a hyper rectangle in this paper we show that this problem is also strongly np hard in fact for polynomials of degree as low as three this result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure as a byproduct our proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly np hard this problem which was previously shown to be weakly np hard by nemirovski is of independent interest in the theory of robust control

nonparametric combinatorial regression for shape constrained modeling we introduce a unified approach for calculating nonparametric shape constrained regression enforcement of the shape constraint often accounts for the impact of a physical phenomenon or a specific property it also improves the model s predicability and facilitates subsequent optimizations the regression models are built by transforming the problem into the combinatorial domain where the shape constraints are imposed by bounding the combinatorial search space we start by addressing isotonicity shape constraint using a dynamic programming algorithm and demonstrate how the problem can be mapped to the graph combinatorics domain next we show how a number of other important shape constraints including unimodality convexity limited level set and limited slope can be addressed using the same framework the flexibility of proposed framework enables solving the shape constrained regression problem with an arbitrary user defined error metric this flexibility is exploited to add robustness against outliers to the model the algorithms are described in detail and their computational complexity is established the performance and effectiveness of the shape constrained regression is evaluated on traces of temperature and humidity measurements from a deployed sensor network where a high degree of accuracy and robustness is demonstrated

consistency of multidimensional convex regression convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is possibly multidimensional such regression problems arise in operations research economics and other disciplines in which imposing a convexity constraint on the regression function is natural this paper studies a least squares estimator that is computable as the solution of a quadratic program and establishes that it converges almost surely to the true function as n under modest technical assumptions in addition to this multidimensional consistency result we identify the behavior of the estimator when the model is misspecified so that the true function is nonconvex and we extend the consistency result to settings in which the function must be both convex and nondecreasing as is needed for consumer preference utility functions

multivariate analysis with linearizable regressions we study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables this class seems more realistic than the multivariate normal or the elliptical distributions and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis in particular a two stage procedure which first scales the variables optimally and then fits a simultaneous equations model is studied in detail and is shown to have some desirable properties

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

estimation of a convex function characterizations and asymptotic theory we study nonparametric estimation of convexregression and density functions by methods of least squares in the regression and density cases and maximum likelihood in the density estimation case we provide characterizations of these estimators prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated the asymptotic distribution theory relies on the existence of an invelope function for integrated two sided brownian motion t 4 which is established in a companion paper by groeneboom jongbloed and wellner

data envelopment analysis as nonparametric least squares regression data envelopment analysis dea is known as a nonparametric mathematical programming approach to productive efficiency analysis in this paper we show that dea can be alternatively interpreted as nonparametric least squares regression subject to shape constraints on the frontier and sign constraints on residuals this reinterpretation reveals the classic parametric programming model by aigner and chu aigner d s chu 1968 on estimating the industry production function amer econom rev 58 826 839 as a constrained special case of dea applying these insights we develop a nonparametric variant of the corrected ordinary least squares cols method we show that this new method referred to as corrected concave nonparametric least squares c2nls is consistent and asymptotically unbiased the linkages established in this paper contribute to further integration of the econometric and axiomatic approaches to efficiency analysis

homogeneity analysis with k sets of variables an alternating least squares method with optimal scaling features homogeneity analysis or multiple correspondence analysis is usually applied tok separate variables in this paper we apply it to sets of variables by using sums within sets the resulting technique is called overals it uses the notion of optimal scaling with transformations that can be multiple or single the single transformations consist of three types nominal ordinal and numerical the corresponding overals computer program minimizes a least squares loss function by using an alternating least squares algorithm many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of overals an application to data from an epidemiological survey is presented

an efficient algorithm for joint correspondence analysis joint correspondence analysis is a technique for constructing reduced dimensional representations of pairwise relationships among categorical variables the technique was proposed by greenacre as an alternative to multiple correspondence analysis joint correspondence analysis differs from multiple correspondence analysis in that it focuses solely on between variable relationships greenacre described one alternating least squares algorithm for conducting joint correspondence analysis another alternating least squares algorithm is described in this article the algorithm is guaranteed to converge and does so in fewer iterations than does the algorithm proposed by greenacre a modification of the algorithm for handling heywood cases is described the algorithm is illustrated on two data sets

improved tractography using asymmetric fibre orientation distributions diffusion mri allows us to make inferences on the structural organisation of the brain by mapping water diffusion to white matter microstructure however such a mapping is generally ill defined for instance diffusion measurements are antipodally symmetric diffusion along x and x are equal whereas the distribution of fibre orientations within a voxel is generally not symmetric therefore different sub voxel patterns such as crossing fanning or sharp bending cannot be distinguished by fitting a voxel wise model to the signal however asymmetric fibre patterns can potentially be distinguished once spatial information from neighbouring voxels is taken into account we propose a neighbourhood constrained spherical deconvolution approach that is capable of inferring asymmetric fibre orientation distributions a fods importantly we further design and implement a tractography algorithm that utilises the estimated a fods since the commonly used streamline tractography paradigm cannot directly take advantage of the new information we assess performance using ultra high resolution histology data where we can compare true orientation distributions against sub voxel fibre patterns estimated from down sampled data finally we explore the benefits of a fods based tractography using in vivo data by evaluating agreement of tractography predictions with connectivity estimates made using different in vivo modalities the proposed approach can reliably estimate complex fibre patterns such as sharp bending and fanning which voxel wise approaches cannot estimate moreover histology based and in vivo results show that the new framework allows more accurate tractography and reconstruction of maps quantifying symmetric and asymmetric fibre complexity

an algorithm for restricted least squares regression abstract a commonly occurring problem in statistics is that of minimizing a least squares expression subject to side constraints here a simple iterative algorithm is presented and shown to converge to the desired solution several examples are presented including finding the closest concave convex function to a set of points and other general quadratic programming problems the dual problem to the basic problem is also discussed and a solution for it is given in terms of the algorithm finally extensions to expressions other than least squares are given

basis tensor decomposition for restoring intra voxel structure and stochastic walks for inferring brain connectivity in dt mri we present a regularized method for solving an inverse problem in diffusion tensor magnetic resonance imaging dt mri data in the case of brain images dt mr imagery technique produces a tensor field that indicates the local orientation of nerve bundles now days the spatial resolution of this technique is limited by the partial volume effect produced in voxels that contain fiber crossings or bifurcations in this paper we proposed a method for recovering the intra voxel information and inferring the brain connectivity we assume that the observed tensor is a linear combination of a given tensor basis therefore the aim of our approach is the computation of the unknown coefficients of this linear combination by regularizing the problem we introduce the needed prior information about the piecewise smoothness of nerve bundles orientation as a result we recover a multi tensor field moreover for estimating the nerve bundles trajectory we propose a method based on stochastic walks of particles through the computed multi tensor field the performance of the method is demonstrated by experiments in both synthetic and real data

a penalized method for multivariate concave least squares with application to productivity analysis abstract we propose a penalized method for the least squares estimator of a multivariate concave regression function this estimator is formulated as a quadratic programing qp problem with o n2 constraints where n is the number of observations computing such an estimator is a very time consuming task and the computational burden rises dramatically as the number of observations increases by introducing a quadratic penalty function we reformulate the concave least squares estimator as a qp with only non negativity constraints this reformulation can be adapted for estimating variants of shape restricted least squares i e the monotonic concave convex least squares the experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem the matlab and r codes for implementing the penalized problems are provided in the paper

a noniterative method of joint correspondence analysis correspondence analysis leads to a graphical representation of the associations between categories of the row and column variables of a contingency table greenacre s 1988 formulation of joint correspondence analysis is a multivariate extension which finds the optimal joint display of contingency tables between all pairs of variables in a set greenacre presented a discrepancy function and an alternating least squares algorithm for its minimization boik 1996 presented an alternative algorithm also of the alternating least squares type for minimizing the same discrepancy function in this paper a noniterative procedure not based on the minimization of any discrepancy function is described

correspondence analysis of incomplete contingency tables correspondence analysis can be described as a technique which decomposes the departure from independence in a two way contingency table in this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi independence model this form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended for example in case of missing data or structural zeros it is shown that nora s reconstitution of order zero a procedure well known in the french literature is formally identical to our correspondence analysis of incomplete tables therefore reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi independence model furthermore correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis

histogram regression estimation using data dependent partitions we establish general sufficient conditions for the l 2 conoistency of multivariate histogram regression estimates based on data dependent partitions these same conditions insure the consistency of partitioning regression estimates based on local polynomial fits and with an additional regularity assumption the consistency of histogram estimates for conditional medians our conditions require shrinking cells subexponential growth of a combinatorial complexity measure and sublinear growth of restricted cell counts it is not assumed that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points response variables are assumed to be bounded throughout our results may be applied to a variety of partitioning schemes we established the consistency of histograms regression estimates based on cubic partitions with data dependent offsets k thresholding in one dimension and empirically optimal nearest neighbor clustering schemes in addition it is shown that empirically optimal regression trees are consistent when the size of the trees grows with the number of samples at an appropriate rate

nonparametric bayesian testing for monotonicity this paper adopts a nonparametric bayesian approach to testing whether a function is monotone two new families of tests are constructed the first uses constrained smoothing splines with a hierarchical stochastic process prior that explicitly controls the prior probability of monotonicity the second uses regression splines together with two proposals for the prior over the regression coefficients via simulation the finite sample performance of the tests is shown to improve upon existing frequentist and bayesian methods the asymptotic properties of the bayes factor for comparing monotone versus nonmonotone regression functions in a gaussian model are also studied our results significantly extend those currently available which chiefly focus on determining the dimension of a parametric linear model

an objective method for regularization of fiber orientation distributions derived from diffusion weighted mri spherical deconvolution is an elegant method by which the orientation of crossing fibers in the brain can be estimated from a diffusion weighted mri measurement however higher resolution of fiber directions comes at the cost of higher susceptibility to noise in this study we describe the use of linear regularization of the fiber orientation distribution function by damped singular value decomposition furthermore the degree of regularization is optimized on a voxel by voxel basis with no user interaction using generalized cross validation we find by simulations that regularization can improve the reliability of fiber orientation determination when the signal to noise ratio is low simulations and in vivo measurements indicate that spurious peaks of the fiber orientation distribution function in regions with low anisotropy largely disappear when regularization is introduced the methods examined are fast enough to be used on a routine basis with diffusion mri data sets and may improve estimation of water diffusion properties in heterogeneous white matter and boost reliability of fiber tracking through regions of brain with complex fiber geometry

local white matter geometry from diffusion tensor gradients we introduce a mathematical framework for computing geometrical properties of white matter fibers directly from diffusion tensor fields the key idea is to isolate the portion of the gradient of the tensor field corresponding to local variation in tensor orientation and to project it onto a coordinate frame of tensor eigenvectors the resulting eigenframe centered representation then makes it possible to define scalar indices or measures that describe the local white matter geometry directly from the diffusion tensor field and its gradient without requiring prior tractography we derive new scalar indices of 1 fiber dispersion and 2 fiber curving and we demonstrate them on synthetic and in vivo data finally we illustrate their applicability to a group study on schizophrenia