a wavelet shrinkage approach to tomographic image reconstruction abstract a method is proposed for reconstructing images from tomographic data with respect to a two dimensional wavelet basis the wavelet vaguelette decomposition wvd is used as a framework within which expressions for the necessary wavelet coefficients may be derived these coefficients are calculated using a version of the filtered back projection algorithm as a computational tool in a multiresolution fashion the necessary filters are defined in terms of the underlying wavelets denoising is accomplished through an adaptation of the wavelet shrinkage ws approach of donoho et al and amounts to a form of regularization combining these two steps yields the proposed wvd ws reconstruction algorithm which is compared to the traditional filtered backprojection method in a small study

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

mri artifact cancellation due to rigid motion in the imaging plane a post processing technique has been developed to suppress the magnetic resonance imaging mri artifact arising from object planar rigid motion in two dimensional fourier transform 2 dft mri rotational and translational motions of the target during magnetic resonance magnetic resonance mr scan respectively impose nonuniform sampling and a phase error an the collected mri signal the artifact correction method introduced considers the following three conditions 1 for planar rigid motion with known parameters a reconstruction algorithm based on bilinear interpolation and the super position method is employed to remove the mri artifact 2 for planar rigid motion with known rotation angle and unknown translational motion including an unknown rotation center first a super position bilinear interpolation algorithm is used to eliminate artifact due to rotation about the center of the imaging plane following which a phase correction algorithm is applied to reduce the remaining phase error of the mri signal and 3 to estimate unknown parameters of a rigid motion a minimum energy method is proposed which utilizes the fact that planar rigid motion increases the measured energy of an ideal mr image outside the boundary of the imaging object by using this property all unknown parameters of a typical rigid motion are accurately estimated in the presence of noise to confirm the feasibility of employing the proposed method in a clinical setting the technique was used to reduce unknown rigid motion artifact arising from the head movements of two volunteers

wavelet deconvolution in a periodic setting in this paper we present an inverse estimation procedure which combines fourier analysis with wavelet expansion in the periodic setting our method can recover a blurred function observed in white noise the blurring process is achieved through a convolution operator which can either be smooth polynomial decay of the fourier transform or irregular such as the convolution with a box car the proposal is non linear and does not require any prior knowledge of the smoothness class it enjoys fast computation and is spatially adaptive this contrasts with more traditional ltering methods which demand a certain amount of regularisation and often fail to recover non homogeneous functions a ne tuning of our method is derived via asymptotic minimax theory which reveals some key dierences with the direct case of donoho et al 1995 a band limited wavelet families have nice theoretical and computing features b the high frequency cut o depends on the spectral characteristics of the convolution kernel c thresholds are level dependent in a geometric fashion we tested our method using simulated lidar data for underwater remote sensing both visual and numerical results show an improvement over existing methods finally the theory behind our estimation paradigm gives a complete characterisation of the maxiset of the method i e the set of functions where the method attains a near optimal rate of convergence for a variety of l p loss functions

a logistic normal multinomial regression model for microbiome compositional data analysis changes in human microbiome are associated with many human diseases next generation sequencing technologies make it possible to quantify the microbial composition without the need for laboratory cultivation one important problem of microbiome data analysis is to identify the environmental biological covariates that are associated with different bacterial taxa taxa count data in microbiome studies are often over dispersed and include many zeros to account for such an over dispersion we propose to use an additive logistic normal multinomial regression model to associate the covariates to bacterial composition the model can naturally account for sampling variabilities and zero observations and also allow for a flexible covariance structure among the bacterial taxa in order to select the relevant covariates and to estimate the corresponding regression coefficients we propose a group l1 penalized likelihood estimation method for variable selection and estimation we develop a monte carlo expectation maximization algorithm to implement the penalized likelihood estimation our simulation results show that the proposed method outperforms the group l1 penalized multinomial logistic regression and the dirichlet multinomial regression models in variable selection we demonstrate the methods using a data set that links human gut microbiome to micro nutrients in order to identify the nutrients that are associated with the human gut microbiome enterotype

estimating fetal age by ultrasound measurements an example of multivariate calibration a problem in the estimation of gestational age by ultrasound measurement of fetal bone lengths the femur length f and the biparietal diameter bpd is described by means of multivariate calibration techniques f and bpd are combined to give an index of gestational age which is significantly more accurate than existing indices based on f or bpd alone a detailed description of the statistical analysis is given ranging from the initial steps of data analysis and model choice to the final step of determining the index and its accuracy a number of interesting aspects of multivariate calibration theory are illustrated

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

data driven and optimal denoising of a signal and recovery of its derivative using multiwavelets multiwavelets are relative newcomers into the world of wavelets thus it has not been a surprise that the used methods of denoising are modified universal thresholding procedures developed for uniwavelets on the other hand the specific of a multiwavelet discrete transform is that typical errors are not identically distributed and correlated whereas the theory of the universal thresholding is based on the assumption of identically distributed and independent normal errors thus we suggest an alternative denoising procedure based on the efromovich pinsker algorithm we show that this procedure is optimal over a wide class of noise distributions moreover together with a new cristina class of biorthogonal multiwavelets which is introduced in this paper the procedure implies an optimal method for recovering the derivative of a noisy signal a monte carlo study supports these conclusions

spatial classification using fuzzy membership models in the usual statistical approach to spatial classification it is assumed that each pixel belongs to precisely one of a small number of known groups this framework is extended to include mixed pixel data then only a proportion of each pixel belongs to each group two models based on multivariate gaussian random fields are proposed to model this fuzzy membership process the problems of predicting the group membership and estimating the parameters are discussed some simulations are presented to study the properties of this approach and an example is given using landsat remote sensing data

a general class of distributions on the simplex summary the dirichlet class of distributions on the simplex is overstructured for practical work because of its many strong independence properties a persistent problem of distribution theory has thus been to extend this class to include members free of these independence properties an alternative approach has been the recent introduction of the logistic normal class which contains members with and without the independence properties unfortunately this class is separate from the dirichlet class and so cannot sustain discussion of some strong forms of independence these difficulties are overcome in this paper by the introduction of a more general class which includes as special cases the dirichlet and logistic normal classes the role of this new class in relation to the analysis of compositional data in particular the investigation of independence hypotheses is discussed and illustrated

wavelet deconvolution with noisy eigenvalues over the last decade there has been a lot of interest in wavelet vaguelette methods for the recovery of noisy signals or images in motion blur nonlinear wavelet estimators are known to have good adaptive properties and to outperform linear approximations over a wide range of signals and images see e g the recent waved method of johnstone kerkyacharian picard and raimondo 2004 or the forward method of neelamani choi and baraniuk 2004 and also fan and koo 2002 in the density setting in the deblurring setting wavelet vaguelette methods rely on the complete knowledge of a convolution operator s eigenvalues this is an unlikely situation in practice however a more realistic scenario such as would arise when passing the fourier basis as an input signal through a linear time invariant system is to imagine that one also observes a set of noisy eigenvalues in this paper we define a version of the waved estimator which is near optimal when used with noisy eigenvalues a key feature of our method includes a data driven method for choosing the fine resolution level in waved estimation asymptotic theory is illustrated with a wide range of finite sample examples

group factor analysis factor analysis fa provides linear factors that describe the relationships between individual variables of a data set we extend this classical formulation into linear factors that describe the relationships between groups of variables where each group represents either a set of related variables or a data set the model also naturally extends canonical correlation analysis to more than two sets in a way that is more flexible than previous extensions our solution is formulated as a variational inference of a latent variable model with structural sparsity and it consists of two hierarchical levels 1 the higher level models the relationships between the groups and 2 the lower models the observed variables given the higher level we show that the resulting solution solves the group factor analysis gfa problem accurately outperforming alternative fa based solutions as well as more straightforward implementations of gfa the method is demonstrated on two life science data sets one on brain activation and the other on systems biology illustrating its applicability to the analysis of different types of high dimensional data sources

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

principal component analysis of compositional data summary compositional data consisting of vectors of proportions have proved difficult to handle statistically because of the awkward constraint that the components of each vector must sum to unity moreover such data sets frequently display marked curvature so that linear techniques such as standard principal component analysis are likely to prove inadequate from a critical reexamination of previous approaches we evolve through adaptation of recently introduced transformation techniques for compositional data analysis a log linear contrast form of principal component analysis and illustrate its advantages in applications

a method for discriminating between models describing compositional data summary until recently only the dirichlet class of distributions was available to describe compositional data but now there is an alternative more flexible class of logistic normal distributions to choose between these two classes requires methods for testing separate families of distributions but application of the methods described in the literature involves extremely tedious calculation in this paper a conceptually straightforward and computationally simpler method is introduced and illustrated

large covariance estimation for compositional data via composition adjusted thresholding high dimensional compositional data arise naturally in many applications such as metagenomic data analysis the observed data lie in a high dimensional simplex and conventional statistical methods

a model of text for experimentation in the social sciences abstractstatistical models of text have become increasingly popular in statistics and computer science as a method of exploring large document collections social scientists often want to move beyond exploration to measurement and experimentation and make inference about social and political processes that drive discourse and content in this article we develop a model of text data that supports this type of substantive research our approach is to posit a hierarchical mixed membership model for analyzing topical content of documents in which mixing weights are parameterized by observed covariates in this model topical prevalence and topical content are specified as a simple generalized linear model on an arbitrary number of document level covariates such as news source and time of release enabling researchers to introduce elements of the experimental design that informed document collection into the model within a generally applicable framework we demonstrate the proposed methodology by analyzi

bayesian analysis of case control studies with categorical covariates in a case control study the appropriate likelihood is the retrospective likelihood i e the likelihood of exposure given disease for the classical frequentist analysis the prospective likelihood i e the likelihood of disease given exposure and the retrospective likelihood produce the same odds ratio estimators for exposure and so logistic regression may be used for both the bayesian analysis is not so simple but the bayesian framework for case control studies offers flexible possibilities for the hierarchical modelling that is needed in many contexts we review the bayesian approaches to the analysis of case control studies developed so far show how to extend these approaches to the situation of a study with any number of categorical or discretised continuous exposure variables and identifying suitable priors we then show how the resulting models may be fitted using markov chain monte carlo methods and provide an illustration based on genotype data

an autoregressive process for beta random variables two stationary first order autoregressive processes with beta marginal distributions are presented they are both linear additive processes but the coefficients are beta random variables their autocorrelation functions are investigated one is positive and the other alternates in sign the usefulness of the models in simulation is discussed the bivariate beta distributions of two consecutive observations are considered in some detail several examples are given including a bivariate uniform process which is also examined in detail the relationship of these bivariate beta distributions to the dirichlet distribution is discussed

log contrast models for experiments with mixtures summary a new form of expected response function involving log contrasts of the proportions is introduced for experiments with mixtures the advantages and disadvantages of log contrast models are discussed and illustrated in applications in particular the parameters of the model are related to certain useful hypotheses about mixtures and some relevant contrasts are identified

generalized linear models with linear constraints for microbiome compositional data motivated by regression analysis for microbiome compositional data this article considers generalized linear regression analysis with compositional covariates where a group of linear constraints on regression coefficients are imposed to account for the compositional nature of the data and to achieve subcompositional coherence a penalized likelihood estimation procedure using a generalized accelerated proximal gradient method is developed to efficiently estimate the regression coefficients a de biased procedure is developed to obtain asymptotically unbiased and normally distributed estimates which leads to valid confidence intervals of the regression coefficients simulations results show the correctness of the coverage probability of the confidence intervals and smaller variances of the estimates when the appropriate linear constraints are imposed the methods are illustrated by a microbiome study in order to identify bacterial species that are associated with inflammatory bowel disease ibd and to predict ibd using fecal microbiome

variational inference in nonconjugate models mean field variational methods are widely used for approximate posterior inference in many probabilistic models in a typical application mean field methods approximately compute the posterior with a coordinate ascent optimization algorithm when the model is conditionally conjugate the coordinate updates are easily derived and in closed form however many models of interest like the correlated topic model and bayesian logistic regression are nonconjugate in these models mean field methods cannot be directly applied and practitioners have had to develop variational algorithms on a case by case basis in this paper we develop two generic methods for nonconjugate models laplace variational inference and delta method variational inference our methods have several advantages they allow for easily derived variational algorithms with a wide class of nonconjugate models they extend and unify some of the existing algorithms that have been derived for specific models and they work well on real world data sets we studied our methods on the correlated topic model bayesian logistic regression and hierarchical bayesian logistic regression

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

two sample tests of high dimensional means for compositional data summary compositional data are ubiquitous in many scientific endeavours motivated by microbiome and metagenomic research we consider a two sample testing problem for high dimensional compositional data and formulate a testable hypothesis of compositional equivalence for the means of two latent log basis vectors we propose a test through the centred log ratio transformation of the compositions the asymptotic null distribution of the test statistic is derived and its power against sparse alternatives is investigated a modified test for paired samples is also considered simulations show that the proposed tests can be significantly more powerful than tests that are applied to the raw and log transformed compositions the usefulness of our tests is illustrated by applications to gut microbiome composition in obesity and crohn s disease

a dirichlet tree multinomial regression model for associating dietary nutrients with gut microorganisms summary understanding the factors that alter the composition of the human microbiota may help personalized healthcare strategies and therapeutic drug targets in many sequencing studies microbial communities are characterized by a list of taxa their counts and their evolutionary relationships represented by a phylogenetic tree in this article we consider an extension of the dirichlet multinomial distribution called the dirichlet tree multinomial distribution for multivariate over dispersed and tree structured count data to address the relationships between these counts and a set of covariates we propose the dirichlet tree multinomial regression model for which we develop a penalized likelihood method for estimating parameters and selecting covariates for efficient optimization we adopt the accelerated proximal gradient approach simulation studies are presented to demonstrate the good performance of the proposed procedure an analysis of a data set relating dietary nutrients with bacterial counts is used to show that the incorporation of the tree structure into the model helps increase the prediction power

centile estimation for a proportion response variable this paper introduces two general models for computing centiles when the response variable y can take values between 0 and 1 inclusive of 0 or 1 the models developed are more flexible alternatives to the beta inflated distribution the first proposed model employs a flexible four parameter logit skew student t logitsst distribution to model the response variable y on the unit interval 0 1 excluding 0 and 1 this model is then extended to the inflated logitsst distribution for y on the unit interval including 1 the second model developed in this paper is a generalised tobit model for y on the unit interval including 1 applying these two models to 1 y rather than y enables modelling of y on the unit interval including 0 rather than 1 an application of the new models to real data shows that they can provide superior fits

mobius deconvolution on the hyperbolic plane with application to impedance density estimation in this paper we consider a novel statistical inverse problem on the poincare or lobachevsky upper complex half plane here the riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via mobius transformations our approach is based on a deconvolution technique which relies on the helgason fourier calculus adapted to this hyperbolic space this gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random mobius transform a motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the poincare plane exactly describes the physical system that is of statistical interest

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

extending the scope of wavelet regression methods by coefficient dependent thresholding abstract various aspects of the wavelet approach to nonparametric regression are considered with the overall aim of extending the scope of wavelet techniques to irregularly spaced data to regularly spaced datasets of arbitrary size to heteroscedastic and correlated data and to data that contain outliers the core of the methodology is an algorithm for finding ail of the variances and within level covariances in the wavelet table of a sequence with given covariance structure if the original covariance matrix is band limited then the algorithm is linear in the length of the sequence the variance calculation algorithm allows data on any set of independent variable values to be treated by first interpolating to a fine regular grid of suitable length and then constructing a wavelet expansion of the gridded data various thresholding methods are discussed and investigated exact risk formulas for the mean square error of the methodology for given design are derived good performance is obtained by nois

wavelet decomposition approaches to statistical inverse problems summary a wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise primary interest is in some function f t but data are accessible only about some linear transform corrupted by noise the usual linear methods for such inverse problems do not perform satisfactorily when f t is spatially inhomogeneous one existing nonlinear alternative is the wavelet vaguelette decomposition method based on the expansion of the unknown f t in wavelet series in the vaguelette wavelet decomposition method proposed here the observed data are expanded directly in wavelet series the performances of various methods are compared through exact risk calculations in the context of the estimation of the derivative of a function observed subject to noise a result is proved demonstrating that with a suitable universal threshold somewhat larger than that used for standard denoising problems both the wavelet based approaches have an ideal spatial adaptivity property

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed