fractal aggregates and the current lines of their electrostatic potentials abstract we have constructed diffusion limited aggregates off lattice that is using brownian motion instead of random walk and the equipotential and the current lines have also been computed in the continuous plane this construction describes as being off off lattice is illustrated for the composite of a circular and a cylindrical dla cluster the striking new finding we report is that the current lines form trees with nearly straight branches

likelihood based analysis of causal effects of job training programs using principal stratification government sponsored job training programs must be subject to evaluation to assess whether their effectiveness justifies their cost to the public the evaluation usually focuses on employment and total earnings although the effect on wages is also of interest because this effect reflects the increase in human capital due to the training program whereas the effect on total earnings may be simply reflecting the increased likelihood of employment without any effect on wage rates estimating the effects of training programs on wages is complicated by the fact that even in a randomized experiment wages are truncated or less accurately censored by nonemployment that is they are only observed and well defined for individuals who are employed in this article we develop a likelihood based approach to estimate the wage effect of the us federally funded job corps training program using principal stratification our estimands are formulated in terms of 1 the effect of the training program on wages

on a mixture autoregressive conditional heteroscedastic model we propose a mixture autoregressive conditional heteroscedastic mar arch model for modeling nonlinear time series the models consist of a mixture of k autoregressive components with autoregressive conditional heteroscedasticity that is the conditional mean of the process variable follows a mixture ar mar process whereas the conditional variance of the process variable follows a mixture arch process in addition to the advantage of better description of the conditional distributions from the mar model the mararch model allows a more flexible squared autocorrelation structure the stationarity conditions autocorrelation function and squared autocorrelation function are derived construction of multiple step predictive distributions is discussed the estimation can be easily done through a simple em algorithm and the model selection problem is addressed the shape changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distr

self similarity of the branching structure in very large dla clusters and other branching fractals we study the branching structure of very large dla clusters of up to 100 million particles the horton strahler ordering of the branches in these clusters shows a relaxation towards a state with the stream numbers forming a geometric series this behaviour is compared with those of several self similar trees it indicates that dla clusters converge to topologically self similar objects

mice multiple peak identification characterization and estimation summary mice multiple peak identification characterization and estimation is a procedure for estimating a lower bound of the number of frequency peaks and for estimating the frequency peak parameters the leading application is protein structure determination using nuclear magnetic resonance nmr experiments nmr frequency data are multiple peak data where each frequency peak corresponds to two connected atoms in the three dimensional protein structure we analyze the nmr frequency data through a series of steps a preliminary step for separating the signal from the background followed by identification of local maxima up to a noise level dependent threshold estimation of the frequency peak parameters using an iterative algorithm and detection of mixtures of peaks using hypothesis testing

a new sequential classifier using information criterion window abstract we propose a model based parametric unsupervised sequential classifier with an information criterion window icw based on the assumption that the number of clusters changes smoothly from a stage to the following stage to determine the number of clusters at each stage and to perform classification on the observed mixture data the proposed approach provides possible solutions for two problems of pattern recognition adaptive classification and sequential clustering validation simulation results demonstrate that the proposed algorithm provides good results for pure mixed data and successfully performs clustering for the mixed gaussian data in the large sample limit

finite mixture models for proportions summary six data sets recording fetal control mortality in mouse litters are presented the data are clearly overdispersed and a standard approach would be to describe the data by means of a beta binomial model or to use quasi likelihood methods for five of the examples we show that the beta binomial model provides a reasonable description but that the fit can be significantly improved by using a mixture of a beta binomial model with a binomial distribution this mixture provides two alternative solutions in one of which the binomial component indicates a high probability of death but is selected infrequently this accounts for outlying litters with high mortality the influence of the outliers on the beta binomial fits is also demonstrated the location and nature of the two main maxima to the likelihood are investigated through profile log likelihoods comparisons are made with the performance of finite mixtures of binomial distributions data describing fetal deaths in litters are frequently overdispersed relative to the binomial distribution due to differential litter effects in some cases sample sizes may be too small for a significant effect of overdispersion to be detected in others however data sets are certainly large enough and six illustrations of such data sets are given in tables 1 6 the data of tables 3 6 have been published before while those of tables 1 and 2 have been formed by pooling smaller sets of data that have previously been analyzed by james and smith 1982 we shall refer to the data sets in order as el e2 hs1 hs2 hs3 avss different data sets highlight different aspects of the work of this paper statisticians analyzing such data might adopt a parametric approach for example by fitting a beta binomial or a correlated binomial distribution the more robust approach provided by quasilikelihood is an alternative see liang and hanfelt 1994 the various options are described in morgan 1992 chapter 6 and model details are given in section 2 also shiown in tables 1 6 are the expected values from fitting the beta binomial distribution the fits appear to be reasonably good except for the small numbers of litters with high mortality because of small cell frequencies we use simulation to test goodness of fit taking the el data set as an example we can simulate from the fitted beta binomial model a data set that matches el with regard to numbers of litters of the various different sizes observed and that uses as parameter values the maximum likelihood estimates resulting from fitting the model to the real data the model can then be fitted to the simulated data and the maximum log likelihood recorded this process may then be repeated many times to provide a goodness of fit reference for the maximum

newton s method for solving a class of nonlinear matrix equations in this paper a class of nonlinear matrix equations x q a h i x c 1a is considered an equivalent form of this equation is derived and the properties of the solutions are studied then newton s iterative method is applied to solve this equivalent one finally two numerical examples are given to illustrate the efficiency of newton s method

testing in normal mixture models when the proportions are known summary the aim of this paper is to study the behaviour of the likelihood ratio test statistic in mixture models when the proportions of the distributions are known a priori the results depend on the value of the proportion the dimensionality of the problem and whether or not the variance of the distributions is known

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

conjugate mixture models for clustering multimodal data the problem of multimodal clustering arises whenever the data are gathered with several physically different sensors observations from different modalities are not necessarily aligned in the sense there there is no obvious way to associate or compare them in some common space a solution may consist in considering multiple clustering tasks independently for each modality the main difficulty with such an approach is to guarantee that the unimodal clusterings are mutually consistent in this letter we show that multimodal clustering can be addressed within a novel framework conjugate mixture models these models exploit the explicit transformations that are often available between an unobserved parameter space objects and each of the observation spaces sensors we formulate the problem as a likelihood maximization task and derive the associated conjugate expectation maximization algorithm the convergence properties of the proposed algorithm are thoroughly investigated several local and global optimization techniques are proposed in order to increase its convergence speed two initialization strategies are proposed and compared a consistent model selection criterion is proposed the algorithm and its variants are tested and evaluated within the task of 3d localization of several speakers using both auditory and visual data

random multifractals negative dimensions and the resulting limitations of the thermodynamic formalism the story described in this paper has started with the death or survival criterion which the author published in 1972 1974 and had obtained in 1968 while investigating kolmogorov s hypothesis that the turbulent dissipation epsilon dx in a box is log normally distributed using this criterion the present paper discusses the concrete significance of negative fractal dimensions they arise in those random multifractal measures for which the cramer function f alpha the spectrum of singularities satisfies f alpha 0 and concerns three aspects of such measures a the fine grained multifractal properties which are non random and the same for almost all realizations b the properties obtained by using the partition function formalism and c the typical coarse grained multifractal properties however the f alpha s in the range where f alpha 0 say nothing about the variability of coarse grained properties between samples a description of these fluctuations hence a fuller multifractal description of the measure is shown to be provided by the values of f alpha in the range where f alpha 0 we prefer to reserve the term thermodynamic for the fine grained and partition functional properties and to say that the coarse grained properties go beyond the thermodynamics i e are not macroscopic but mesoscopic

potential theory and analytic properties of a cantor set the authors consider the electrostatic potential due to a uniform distribution of charge on a cantor set mellin transforms are used to find expansions for the potential at a distance r from the end of the charge distribution it is found that the potential is a power law multiplied by a function periodic in ln r together with a power series in r a recursion relation for the moments of the distribution is used to reveal a similar structure in these moments finally fourier transform techniques are used to find explicit representations for the distribution and its moments

analytical representations of the spread harmonic measure density we study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface for euclidean domains in which brownian motion can be split into independent lateral and transverse displacements we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior this analysis is applicable to slab domains general cylindrical domains and a half space we investigate the spreading effect due to multiple reflections on the surface and the underlying role of finite reactivity we discuss further extensions and applications of analytical results to describe laplacian transfer phenomena such as permeation through semipermeable membranes secondary current distribution on partially blocking electrodes and surface relaxation in nuclear magnetic resonance

application of the em algorithm for the multitarget multisensor tracking problem an important problem in surveillance and reconnaissance systems is the tracking of multiple moving targets in cluttered noise environments using outputs from a number of sensors possessing wide variations in individual characteristics and accuracies a number of approaches have been proposed for this multitarget multisensor tracking problem ranging from reasonably simple though ad hoc schemes to fairly complex but theoretically optimum approaches in this paper we describe an iterative procedure for time recursive multitarget multisensor tracking based on use of the expectation maximization em algorithm more specifically we pose the multitarget multisensor tracking problem as an incomplete data problem with the observable sensor outputs representing the incomplete data whereas the target associated sensor outputs constitute the complete data target updates at each time use an em based approach that calculates the maximum a posteriori map estimate of the target states under the assumption of appropriate motion models based on the outputs of disparate sensors the approach uses a markov random field mrf model of the associations between observations and targets and allows for estimation of joint association probabilities without explicit enumeration the advantage of this em based approach is that it provides a computationally efficient means for approaching the performance offered by theoretically optimum approaches that use explicit enumeration of the joint association probabilities we provide selected results illustrating the performance complexity characteristics of this em based approach compared with competing schemes

a note on the aitkin rubin approach to hypothesis testing in mixture models on montre que les conditions de regularite habituellement utilisees pour obtenir des resultats asymptotiques standards ne conviennent pas meme en utilisant une distribution a priori sur le vecteur des proportions de melange

on hermitian positive definite solution of the matrix equation x σ m i 1 a i x r a i q a conjecture that the nonlinear matrix equation email protected i 1 ma i x ra i q email protected r 0 or 0

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains we present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains the key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations we prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non intrusively computed either at the element or the global level the resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics the proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three dimensional gas turbine blade model with uncertainty in the cooling core geometry the numerical results are compared against monte carlo simulations and it is shown that the proposed approach provides high quality approximations for the first two statistical moments at modest computational effort

self similarity of harmonic measure on dla abstract the right hand side of the curve of the harmonic measure on dla is undefined this does not necessarily imply that the harmonic measure and the dla geometry are not self similar we show for off lattice dla that the right hand tail satisfies a different rescaling rule this cauchy rescaling is compatible with self similarity the analysis is done on off off lattice dla in which both the brownian motion and the laplace equation are off lattice the cluster sizes range between 32 and 50 000 atoms the square lattice used to numerically estimate the laplacian potential introduces a lower cutoff on the spatial resolution of this potential we find a dependence of the right tail of the distribution of holders on this ultraviolet cutoff whereas the shape of the tail does depend on this ultraviolet lattice cutoff the applicability of the collapse rules do not

potential theory and analytic properties of self similar fractal and multifractal distributions by the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential moments and fourier transform of exactly self similar fractal and multifractal charge distributions three specific examples are given the binomial distribution on the middle third cantor set which is a multifractal distribution the uniform distribution on the menger sponge which illustrates the added complication of higher dimensionality and the uniform distribution on the von koch snowflake which illustrates the effect of rotations in the defining transformations

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

a maximum likelihood method for latent class regression involving a censored dependent variable the standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored this model is generalized by developing a conditional mixture maximum likelihood method for latent class censored regression the proposed method simultaneously estimates separate regression functions and subject membership in k latent classes or groups given a censored dependent variable for a cross section of subjects maximum likelihood estimates are obtained using an em algorithm the proposed method is illustrated via a consumer psychology application

harmonic measure around a linearly self similar tree the authors use the concept of random multiplicative processes to help describe and understand the distribution of the harmonic measure on growing fractal boundaries the laplacian potential around a linearly self similar square koch tree is studied in detail the multiplicative nature of this potential and the consequent multifractality of the harmonic measure are discussed on prefractal stages the density d mu of the harmonic measure and the corresponding holder alpha ln d mu are well defined along the boundary except in the folds where the tangent is undefined a regularization scheme is introduced to eliminate these local effects they then consider the probability distributions p alpha d alpha of successive stages and discuss their collapse into an f alpha curve both the left and right hand sides of this curve show good convergence other studies indicate that for dla the right hand tail does not converge a brief comparison is made between the multifractality of these two cases

modeling turbulent flow over fractal trees with renormalized numerical simulation high reynolds number flow over tree like fractals is considered with emphasis on the drag forces produced fractal objects display large scale disparity and complexity while being amenable to a simple and standardized description hence they offer an elegant idealization of the actual boundaries in practical applications where turbulence interacts with boundaries that are characterized by multiple length scales first using large eddy simulation of flow over prefractal shapes with increasing numbers of branch generations the dependence of the tree drag on the inner cutoff scale of the fractal is studied it is found that the convergence of the drag coefficient towards a value that is independent of inner cutoff scale is very slow in order to address this fundamental difficulty and avoid the need to resolve all the small scale branches of the fractal a new numerical modeling technique called renormalized numerical simulation rns is introduced rns models the drag of the unresolved branches using drag coefficients measured from both resolved branches and unresolved branches as modeled in previous iterations of the procedure the rns technique and its convergence properties are tested by means of a series of simulations using different levels of resolution then rns is used to investigate the influence of the tree fractal dimension on the drag coefficient the increase of the drag with fractal dimension is quantified for two types of tree geometry in two flow configurations results illustrate that rns enables numerical modeling of physical processes associated with fractal geometries using affordable computational resolution

on the nonlinear matrix equation x i 1mai xδiai q abstract based on fixed point theorems for monotone and mixed monotone operators in a normal cone we prove that the nonlinear matrix equation x i 1 m a i x i a i q 0 i 1 always has a unique positive definite solution a conjecture which is proposed in x g liu h gao on the positive definite solutions of the matrix equation x s a t x t a i n linear algebra appl 368 2003 83 97 is solved multi step stationary iterative method is proposed to compute the unique positive definite solution numerical examples show that this iterative method is feasible and effective

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

a latent class mixture model for incomplete longitudinal gaussian data in the analyses of incomplete longitudinal clinical trial data there has been a shift away from simple methods that are valid only if the data are missing completely at random to more principled ignorable analyses which are valid under the less restrictive missing at random assumption the availability of the necessary standard statistical software nowadays allows for such analyses in practice while the possibility of data missing not at random mnar cannot be ruled out it is argued that analyses valid under mnar are not well suited for the primary analysis in clinical trials rather than either forgetting about or blindly shifting to an mnar framework the optimal place for mnar analyses is within a sensitivity analysis context one such route for sensitivity analysis is to consider next to selection models pattern mixture models or shared parameter models the latter can also be extended to a latent class mixture model the approach taken in this article the performance of the so obtained flexible model is assessed through simulations and the model is applied to data from a depression trial

new anomalous multiplicative multifractals left sided ƒ α and the modelling of dla abstract a sharp distinction is drawn between general multiplicative multifractals as originally introduced by the author and the more familiar but more restricted class defined by frisch and parisi and by halsey et al all the general multiplicative multifractals are exactly renormalizable by design however this is important and perhaps suprising it is shown that the scaling relations that serve to define the restricted class fail to extend to certain multiplicative multifractals as a result in addition to the familiar restricted class the general multifractals are shown to allow for a wide variety of anomalous behaviors several of which are described we believe that suitable examples of these anomalies are of the most important concrete applications to fully developed turbulence and to diffusion limited aggregates dla this paper concerns several new classes of multifractals we have recently proposed as tools to model the anomalies of dla they are left sided that is characterized by max with max d 0 related multifractals characterized by max 0 and even by max and min 0 are encountered along the way

stochastic model for dielectric breakdown we discuss a model for the development of discharge patterns in dielectric breakdown based on the laplace equation associated with a probability field the model gives rise to random fractals with well defined hausdorff dimensions the relations of this model with the diffusion limited aggregation are discussed in detail the possibility of application to other stochastic phenomena like fracture propagation is proposed

a graphical technique for determining the number of components in a mixture of normals abstract when a population is assumed to be composed of a finite number of subpopulations a natural model to choose is the finite mixture model it will often be the case however that the number of component distributions is unknown and must be estimated this problem can be difficult for instance the density of two mixed normals is not bimodal unless the means are separated by at least 2 standard deviations hence modality of the data per se can be an insensitive approach to component estimation we demonstrate that a mixture of two normals divided by a normal density having the same mean and variance as the mixed density is always bimodal this analytic result and other related results form the basis for a diagnostic and a test for the number of components in a mixture of normals the density is estimated using a kernel density estimator under the null hypothesis the proposed diagnostic can be approximated by a stationary gaussian process under the alternative hypothesis components in the mixtu

influence based diagnostics for finite mixture models summary in the fitting of finite mixtures of distributions to empirical data it is often felt necessary to exclude certain observations in order to achieve a satisfactory fit this is particularly true of fisheries lengthfrequency data where the object is to model a discrete number of age classes by a mixture of normal components and the presence of outliers in the sample can have a large effect on the parameter estimates and the fit this paper develops diagnostic tools to measure the effect of individual observations on the parameter estimates and the fit in order to make the data rejection decision of the data analyst less ad hoc specifically we show how to compute the parameter influence curves and introduce a statistic similar to cook s distance used in regression diagnostics

bayesian winner take all reconstruction of intermediate views from stereoscopic images this paper presents a new algorithm for the reconstruction of intermediate views from a pair of still stereoscopic images the algorithm is designed to address the issue of blur caused by linear filtering often employed in such reconstruction the proposed algorithm is block based and to reconstruct the intermediate views employs nonlinear disparity compensated filtering by means of a winner take all strategy the reconstructed image is modeled as a tiling by fixed size blocks coming from various positions disparity compensation of either the left or right images while the tiling map itself is modeled by a binary decision field in addition to that an observation model relating the left and right images via a disparity field and a disparity field model are used all models are probabilistic and are combined into a maximum a posteriori probability criterion the intermediate intensities disparities and the binary decision field are estimated jointly using the expectation maximization algorithm the new approach is compared experimentally on complex natural images with a reference block based algorithm employing linear filtering although the improvements are localized and often subtle they demonstrate that a high quality intermediate view reconstruction for complex scenes is feasible

maximum likelihood estimation of the multivariate normal mixture model the hessian of the multivariate normal mixture model is derived and estimators of the information matrix are obtained thus enabling consistent estimation of all parameters and their precisions the usefulness of the new theory is illustrated with two examples and some simulation experiments the newly proposed estimators appear to be superior to the existing ones

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

a nonlinear matrix equation connected to interpolation theory in this paper we study the matrix equation x q a x c 1a where q is an n n positive definite matrix c is an mn mn positive semidefinite matrix a is an arbitrary mn n matrix and x is the m m block diagonal matrix with on each diagonal entry the n n matrix x we are interested in the existence and uniqueness of solutions which are contained in a certain subset of the set of the positive definite matrices under the condition that c q these solutions play a role in an optimal interpolation theory problem interpolation theory and its applications mathematics and its applications vol 428 kluwer academic dordrecht 1997 chapter 7

the analysis of repeated measures data on schizophrenic reaction times using mixture models reaction times for schizophrenic individuals in a simple visual tracking experiment can be substantially more variable than for non schizophrenic individuals current psychological theory suggests that at least some of this extra variability arises from an attentional lapse that delays some but not all of each schizophrenic s reaction times based on this theory we pursue models in which measurements from non schizophrenics arise from a normal linear model with a separate mean for each individual whereas measurements from schizophrenics arise from a mixture of i a component analogous to the distribution of response times for non schizophrenics and ii a mean shifted component we fit four mixture models within this framework where the distinctions between models arise from assumptions about the variance of the shifted observations and the exchangeability of schizophrenic individuals some of these models can be fit by maximum likelihood using the em algorithm and all can be fit using the ecm algorithm where the covariance matrices associated with the parameters are calculated by the sem and secm algorithms respectively bayesian model monitoring using posterior predictive checks is invoked to discard models that fail to reproduce certain observed features of the data and to stimulate the development of better models

calibrating the excess mass and dip tests of modality summary nonparametric tests of modality are a distribution free way of assessing evidence about inhomogeneity in a population provided that the potential subpopulations are sufficiently well separated they include the excess mass and dip tests which are equivalent in univariate settings and are alternatives to the bandwidth test only very conservative forms of the excess mass and dip tests are available at present however and for that reason they are generally not competitive with the bandwidth test in the present paper we develop a practical approach to calibrating the excess mass and dip tests to improve their level accuracy and power substantially our method exploits the fact that the limiting distribution of the excess mass statistic under the null hypothesis depends on unknowns only through a constant which may be estimated our calibrated test exploits this fact and is shown to have greater power and level accuracy than the bandwidth test has the latter tends to be quite conservative even in an asymptotic sense moreover the calibrated test avoids difficulties that the bandwidth test has with spurious modes in the tails which often must be discounted through subjective intervention of the experimenter

a note on the fixed point iteration for the matrix equations x a x 1a i the fixed point iteration is a simple method for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a i the plus minus equations the convergence of this method may be very slow if the initial matrix is not chosen carefully a strategy for choosing better initial matrices has been recently proposed by ivanov hasanov and uhlig they proved that this strategy can improve the convergence in general and observed from numerical experiments that dramatic improvement happens for the plus equation with some matrices a it turns out that the matrices a are normal for those examples in this note we prove a result that explains the dramatic improvement in convergence for normal and thus nearly normal matrices for the plus equation a similar result is also proved for the minus equation ams classification 15a24 65f30 65h10

extremal classical interpolation problems matrix case in this paper we try to find amongst the solutions w z of the corresponding interpolation problem the solution which satisfies an additional extremal condition we show that degenerate interpolation problems play an important role in the theory of extremal interpolation problems at the end of the paper we accomplish a comparison of our approach with former known approaches

the em mpm algorithm for segmentation of textured images analysis and further experimental results in this paper we present new results relative to the expectation maximization maximization of the posterior marginals em mpm algorithm for simultaneous parameter estimation and segmentation of textured images the em mpm algorithm uses a markov random field model for the pixel class labels and alternately approximates the mpm estimate of the pixel class labels and estimates parameters of the observed image model the goal of the em mpm algorithm is to minimize the expected value of the number of misclassified pixels we present new theoretical results in this paper which show that the algorithm can be expected to achieve this goal to the extent that the em estimates of the model parameters are close to the true values of the model parameters we also present new experimental results demonstrating the performance of the em mpm algorithm

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

behaviour of the harmonic measure at the bottom of fjords diffusion limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords to estimate the harmonic measure at the bottom of a fjord seems a prohibitive task but the authors find that a new mathematical equality due to beurling carleson and jones makes it easy they find that the harmonic measure at the bottom of a fjord as a function of its euclidean depth can exhibit a wide range of behaviours they introduce an infinite family of model fjords for which the equality takes a very simple form in this family the decay of the harmonic measure at their bottoms can be for example power law semi exponential stretched exponential and exponentially stretched exponential they show that self affinity or randomness can lead to faster than power law decays of the minimal growth probability on boundaries

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

estimation and hypothesis testing in finite mixture models summary finite mixture models are a useful class of models for application to data when sample sizes are not large and the number of underlying densities is in question likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter x and the parameter of the underlying densities 0 are problematical our approach places a prior distribution on x and estimates 0 by maximizing the likelihood of the data given 0 with x integrated out advantages of this approach computational issues using the em algorithm and directions for further work are discussed the technique is applied to two examples

a finite mixture of nonlinear random coefficient models for continuous repeated measures data nonlinear random coefficient models nrcms for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development or growth over time in measured variables as an extension of this model this study considers the finite mixture of nrcms that combine features of nrcms with the idea of finite mixture or latent class models the efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations thus allowing the simultaneous investigation of intra individual within person variability inter individual between person variability and subpopulation heterogeneity effectiveness of this model to work under real data analytic conditions was examined by executing a monte carlo simulation study the simulation study was carried out using an r routine specifically developed for the purpose of this study the r routine used maximum likelihood with the expectation maximization algorithm the design of the study mimicked the output obtained from running a two class mixture model on task completion data

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

maximin effects in inhomogeneous large scale data large scale data are often characterized by some degree of inhomogeneity as data are either recorded in different time regimes or taken from multiple sources we look at regression models and the effect of randomly changing coefficients where the change is either smoothly in time or some other dimension or even without any such structure fitting varying coefficient models or mixture models can be appropriate solutions but are computationally very demanding and often return more information than necessary if we just ask for a model estimator that shows good predictive properties for all regimes of the data then we are aiming for a simple linear model that is reliable for all possible subsets of the data we propose the concept of maximin effects and a suitable estimator and look at its prediction accuracy from a theoretical point of view in a mixture model with known or unknown group structure under certain circumstances the estimator can be computed orders of magnitudes faster than standard penalized regression estimators making computations on large scale data feasible empirical examples complement the novel methodology and theory

dynamic trees for unsupervised segmentation and matching of image regions we present a probabilistic framework namely multiscale generative models known as dynamic trees dt for unsupervised image segmentation and subsequent matching of segmented regions in a given set of images beyond these novel applications of dts we propose important additions for this modeling paradigm first we introduce a novel dt architecture where multilayered observable data are incorporated at all scales of the model second we derive a novel probabilistic inference algorithm for dts structured variational approximation sva which explicitly accounts for the statistical dependence of node positions and model structure in the approximate posterior distribution thereby relaxing poorly justified independence assumptions in previous work finally we propose a similarity measure for matching dynamic tree models representing segmented image regions across images our results for several data sets show that dts are capable of capturing important component subcomponent relationships among objects and their parts and that dts perform well in segmenting images into plausible pixel clusters we demonstrate the significantly improved properties of the sva algorithm both in terms of substantially faster convergence rates and larger approximate posteriors for the inferred models when compared with competing inference algorithms furthermore results on unsupervised object recognition demonstrate the viability of the proposed similarity measure for matching dynamic structure statistical models

a method for calibrating false match rates in record linkage abstract specifying a record linkage procedure requires both 1 a method for measuring closeness of agreement between records typically a scalar weight and 2 a rule for deciding when to classify records as matches or nonmatches based on the weights here we outline a general strategy for the second problem that is for accurately estimating false match rates for each possible cutoff weight the strategy uses a model where the distribution of observed weights are viewed as a mixture of weights for true matches and weights for false matches an em algorithm for fitting mixtures of transformed normal distributions is used to find posterior modes associated posterior variability is due to uncertainty about specific normalizing transformations as well as uncertainty in the parameters of the mixture model the latter being calculated using the sem algorithm this mixture model calibration method is shown to perform well in an applied setting with census data further a simulation experiment reveals that

notes on two perturbation estimates of the extreme solutions to the equations x a x 1 a q two perturbation estimates of the maximal positive definite solutions to the matrix equations x a x 1a q and x a x 1a q are considered these estimates are like to the estimates discussed by hasanov and ivanov 8 the conditions x l 1a 2 1 and x 1a 2 1 in 8 are not always satisfied we replace this conditions by px l 1ap 1 2 1 and px 1ap email protected 2 1 respective where p is positive definite matrix the theoretical results are illustrated by numerical examples

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

mixtures of conditional mean and covariance structure models models and parameters of finite mixtures of multivariate normal densities conditional on regressor variables are specified and estimated we consider mixtures of multivariate normals where the expected value for each component depends on possibly nonnormal regressor variables the expected values and covariance matrices of the mixture components are parameterized using conditional mean and covariance structures we discuss the construction of the likelihood function estimation of the mixture model with regressors using three different em algorithms estimation of the asymptotic covariance matrix of parameters and testing for the number of mixture components in addition to simulation studies data on food preferences are analyzed

on the regression analysis of tumour recurrence rates regression models with mixture random components are proposed for the statistical analysis of recurrent events when waiting times between successive events are unknown these models allow adjustment of parameter estimates for unobserved heterogeneity in the population due for example to missing covariates or overdispersion resulting from inexact distributional assumptions the models are illustrated by a study of recurrence rates of superficial bladder cancer in men

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

multifractality of the harmonic measure on fractal aggregates and extended self similarity abstract we show that dla follows a surprising new scaling rule it expresses that the screened region in which the harmonic measure is tiny increases more than proportionately as the cluster grows this scaling rule also gives indirect evidence that the harmonic measure of lattice dla follows a hyperbolic probability distribution of exponent equal to 1 this distribution predicts that sample moments behave erratically hence explains why the common restricted multifractal formalism fails to apply to dla

cluster validation for unsupervised stochastic model based image segmentation image segmentation is an important and early processing stage in many image analysis problems often this must be done in an unsupervised fashion in that training data is not available and the class conditioned feature vectors must be estimated directly from the data a major problem in such applications is the determination of the number of classes actually present in an image this problem called the cluster validation problem remains essentially unsolved we investigate the cluster validation problem associated with the use of a previously developed unsupervised segmentation algorithm based upon the expectation maximization em algorithm more specifically we consider several well known information theoretic criteria ics as candidate solutions to the validation problem when used in conjunction with this em based segmentation scheme we show that these criteria generally provide inappropriate solutions due to the domination of the penalty term by the associated log likelihood function as an alternative we propose a model fitting technique in which the complete data log likelihood functional is modeled as an exponential function in the number of classes acting the estimated number of classes are then determined in a manner similar to finding the rise time of the exponential function this new validation technique is shown to be robust and outperform the ics in our experiments experimental results for both synthetic and real world imagery are detailed

heat transport through rough channels abstract we investigate the two dimensional transport of heat through viscous flow between two parallel rough interfaces with a given fractal geometry the flow and heat transport equations are solved through direct numerical simulations and for different conduction convection conditions compared with the behavior of a channel with smooth interfaces the results for the rough channel at low and moderate values of the peclet number indicate that the effect of roughness is almost negligible on the efficiency of the heat transport system this is explained here in terms of makarov s theorem using the notion of active zone in laplacian transport at sufficiently high peclet numbers where convection becomes the dominant mechanism of heat transport the role of the interface roughness is to generally increase both the heat flux across the wall as well as the active length of heat exchange when compared with the smooth channel finally we show that this last behavior is closely related with the presence of recirculation zones in the re entrant regions of the fractal geometry

theory of laplacian fractals diffusion limited aggregation and dielectric breakdown model abstract we describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the laplace equation and a stochastic field this new theory provides a systematic method for the calculation of the fractal dimension d and of the multifractal spectrum of the growth probability a detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented our approach exploits the scale invariance of the laplace equation that implies that the structure is self similar both under growth and scale transformation this allows one to introduce a fixed scale transformation instead of coarse graining as in the renormalization group theory that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process for the calculation of the matrix elements of this transformation one has to consider an infinite but rapidly convergent number of processes that occurs outside a considered diagram

density estimation with confidence sets exemplified by superclusters and voids in the galaxies abstract a method is presented for forming both a point estimate and a confidence set of semiparametric densities the final product is a three dimensional figure that displays a selection of density estimates for a plausible range of smoothing parameters the boundaries of the smoothing parameter are determined by a nonparametric goodness of fit test that is based on the sample spacings for each value of the smoothing parameter our estimator is selected by choosing the normal mixture that maximizes a function of the sample spacings a point estimate is selected from this confidence set by using the method of cross validation an algorithm to find the mixing distribution that maximizes the spacings functional is presented these methods are illustrated with a data set from the astronomy literature the measurements are velocities at which galaxies in the corona borealis region are moving away from our galaxy if the galaxies are clustered the velocity density will be multimodal with clusters correspond

renormalisation group for dla and fixed point distribution a renormalization group method is presented to analyse the multifractal structure of the growth probability in the diffusion limited aggregation dla a renormalisation group transformation is derived for the probability distribution of the growth bond s conductance after repeated scaling an invariant distribution is reached as a fixed point distribution the growth probability assigned to each growth bond is represented by a random multiplicative process of the cell s growth probability with fixed point distribution a hierarchy of generalised dimensions d q is calculated and the alpha f spectrum is found

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

segmentation of textured images using a multiresolution gaussian autoregressive model we present a new algorithm for segmentation of textured images using a multiresolution bayesian approach the new algorithm uses a multiresolution gaussian autoregressive mgar model for the pyramid representation of the observed image and assumes a multiscale markov random field model for the class label pyramid the models used in this paper incorporate correlations between different levels of both the observed image pyramid and the class label pyramid the criterion used for segmentation is the minimization of the expected value of the number of misclassified nodes in the multiresolution lattice the estimate which satisfies this criterion is referred to as the multiresolution maximization of the posterior marginals mmpm estimate and is a natural extension of the single resolution maximization of the posterior marginals mpm estimate previous multiresolution segmentation techniques have been based on the maximum a posterior map estimation criterion which has been shown to be less appropriate for segmentation than the mpm criterion it is assumed that the number of distinct textures in the observed image is known the parameters of the mgar model the means prediction coefficients and prediction error variances of the different textures are unknown a modified version of the expectation maximization em algorithm is used to estimate these parameters the parameters of the gibbs distribution for the label pyramid are assumed to be known experimental results demonstrating the performance of the algorithm are presented

testing the number of components in a normal mixture we demonstrate that under a theorem proposed by vuong the likelihood ratio statistic based on the kullback leibler information criterion of the null hypothesis that a random sample is drawn from a k 0 component normal mixture distribution against the alternative hypothesis that the sample is drawn from a k 1 component normal mixture distribution is asymptotically distributed as a weighted sum of independent chi squared random variables with one degree of freedom under general regularity conditions we report simulation studies of two cases where we are testing a single normal versus a two component normal mixture and a two component normal mixture versus a three component normal mixture an empirical adjustment to the likelihood ratio statistic is proposed that appears to improve the rate of convergence to the limiting distribution

dissipation in laplacian fields across irregular boundaries the entropy production associated to a laplacian field distributed across irregular boundaries is studied in the context of the active zone approximation an explicit expression is given for the entropy production in terms of geometry whose relation to the variational formulation is discussed it is shown that the entropy production diminishes for successive prefractal generations of the same fractal generator so that the final fractal object is expected to dissipate less than all previous ones the relevance of this result in the abundance of fractal surfaces or interfaces observed in nature is discussed