relative equilibria of point vortices on the hyperbolic sphere relative equilibria of point vortices on a hyperbolic sphere with constant negative curvature is formulated and considered known symmetries for the vortex motion on the hyperbolic sphere are expounded by setting up a geometrical formulation of dynamics in particular the configuration of rings of vortices is investigated in detail

point vortices on the hyperbolic plane we investigate the dynamical system of point vortices on the hyperboloid this system has non compact symmetry sl 2 r and a coadjoint equivariant momentum map the relative equilibrium conditions are found and the trajectories of relative equilibria with non zero momentum value are described we also provide the classification of relative equilibria and the stability criteria for a number of cases focusing on 2 and 3 vortices unlike the system on the sphere this system has relative equilibria with non compact momentum isotropy subgroup and these are used to illustrate the different stability types of relative equilibria

smoothed cross validation for bandwidth selection of a kernel density estimator a generalization of the widely studied least squares cross validation method is considered the essential idea is to do a particular type of presmoothing of the data this is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance the rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions the method is more appealing than other selectors with this property because its motivation is not heavily dependent on precise asymptotic analysis and because its form is simple and intuitive theory is also given for choice of the amount of presmoothing and this is used to derive a data based method for this choice

smoothing spline density estimation a dimensionless automatic algorithm abstract as a sequel to an earlier article by gu and qiu this article describes and illustrates a dimensionless automatic algorithm for nonparametric probability density estimation using smoothing splines the algorithm is designed to calculate an adaptive finite dimensional solution to the penalized likelihood problem which was shown by gu and qiu to share the same asymptotic convergence rates as the nonadaptive infinite dimensional solution the smoothing parameter is updated jointly with the estimate in a performance oriented iteration via a cross validation performance estimate where the performance is measured by proxies of the symmetrized kullback leibler distance between the true density and the estimate simulations of limited scale are conducted to examine the relative effectiveness of the automatic smoothing parameter selection procedure and to assess the practical statistical performance of the methodology in general the method is also applied to some real data sets the algorithm is implem

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

hidden grassmann structure in the xxz model ii creation operators in this article we unveil a new structure in the space of operators of the xxz chain for each we consider the space mathcal w alpha of all quasi local operators which are products of the disorder field q alpha sum j infty 0 sigma 3 j with arbitrary local operators in analogy with cft the disorder operator itself is considered as primary field in our previous paper we have introduced the annhilation operators b c which mutually anti commute and kill the primary field here we construct the creation counterpart b c and prove the canonical anti commutation relations with the annihilation operators we conjecture that the creation operators mutually anti commute thereby upgrading the grassmann structure to the fermionic structure the bosonic operator t is the generating function of the adjoint action by local integrals of motion and commutes entirely with the fermionic creation and annihilation operators operators b c t create quasi local operators starting from the primary field we show that the ground state averages of quasi local operators created in this way are given by determinants

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

the maximal smoothing principle in density estimation abstract we propose a widely applicable method for choosing the smoothing parameters for nonparametric density estimators it has come to be realized in recent years e g see hall and marron 1987 scott and terrell 1987 that cross validation methods for finding reasonable smoothing parameters from raw data are of very limited practical value their sampling variability is simply too large the alternative discussed here the maximal smoothing principle suggests that we consider using the most smoothing that is consistent with the estimated scale of our data this greatly generalizes and exploits a phenomenon noted in terrell and scott 1985 that measures of scale tend to place upper bounds on the smoothing parameters that minimize asymptotic mean integrated squared error of density estimates such as histograms and frequency polygons the method avoids the extreme sampling variability of cross validation by using ordinary scale estimators such as the standard deviation and interquartile range which

oversmoothed nonparametric density estimates abstract the optimal histogram for a sample of size n from a density defined on the entire real line requires at least 2n 1 3 bins under mild smoothness conditions similar bounds exist for the frequency polygon and kernel estimators values near these bounds give nearly optimal results for a variety of smooth densities providing good quick density estimates

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

comparison of smoothing parameterizations in bivariate kernel density estimation abstract the basic kernel density estimator in one dimension has a single smoothing parameter usually referred to as the bandwidth for higher dimensions however there are several options for smoothing parameterization of the kernel estimator for the bivariate case there can be between one and three independent smoothing parameters in the estimator which leads to a flexibility versus complexity trade off when using this estimator in practice in this article the performances of the different possible smoothing parameterizations are compared using both the asymptotic and exact mean integrated squared error our results show that it is important to have independent smoothing parameters for each of the coordinate directions although this is enough for many situations for densities with high amounts of curvature in directions different to those of the coordinate axes substantial gains can be made by allowing the kernel mass to have arbitrary orientations the sphering approaches to choosing this o

baxter s q operator for the w algebra wn the q oscillator representation for the borel subalgebra of the affine symmetry is presented by means of this q oscillator representation we give the free field realizations of baxter s q operator for the w algebra wn we give functional relations of the t q operators including the higher rank generalization of baxter s t q relation

efficient confidence bounds for roc curves the specificity and sensitivity of a quantitative diagnostic marker depends on the selected cut off point the roc curve is generated by plotting sensitivity against specificity as the cut off point runs through the whole range of possible marker values confidence bounds for roc curves may be constructed by combining two separate confidence intervals for sensitivity and specificity in the present paper a more efficient method which is based on a statistical test introduced by greenhouse and mantel is presented the resulting 95 per cent confidence intervals are up to 40 per cent smaller a formula is given to calculate the sample size necessary to obtain a confidence interval of a stipulated length at a specified point of the roc curve as an example the performance of a tumour marker in the diagnosis of bone marrow metastases is investigated

a note on kernel polygons summary jones 1989 has pointed out that piecewise linear interpolated kernel density estimators on a sufficiently fine grid can be visually indistinguishable from the true density a simple device the kernel polygon is proposed for eliminating the evaluation of the normalisation constant of the estimator while retaining its property of being a density function as well as providing practical advantages the class of uniform and linear kernels of the kernel polygons is given finally we present a simulation study and a real data example in which we compare bandwidth selectors for the kernel polygons

data based optimal smoothing of orthogonal series density estimates let f be a density possessing some smoothness properties and let x 1 cdots x n be independent observations from f some desirable properties of orthogonal series density estimates f n m lambda of f of the form f n m lambda t sum n nu 1 frac hat f nu 1 lambda nu 2m phi nu t where phi nu is an orthonormal sequence and hat f nu 1 n sum n j 1 phi nu x j is an estimate of f nu int phi nu t f t dt are discussed the parameter lambda plays the role of a bandwidth or smoothing parameter and m controls a shape factor the major novel result of this note is a simple method for estimating lambda and m from the data in an objective manner to minimize integrated mean square error the results extend to multivariate estimates

finite temperature correlation functions from discrete functional equations we present a new approach to the static finite temperature correlation functions of the heisenberg chain based on functional equations an inhomogeneous generalization of the n site density operator is considered the lattice path integral formulation with a finite but arbitrary trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters we show that these equations yield a unique characterization of the density operator our functional equations are a discrete version of the reduced q knizhnik zamolodchikov equations which played a central role in the study of the zero temperature case as a natural result and independent of the arguments given by jimbo et al 2009 j phys a math theor 42 304018 we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature they are a sum of products of nearest neighbour correlators

uniform consistency of automatic and location adaptive delta sequence estimators the class of delta sequence estimators for a probability density includes the kernel histogram and orthogonal series types because each can be characterized as a collection of averages of some function that is indexed by a smoothing parameter there are two important extensions of this class the first allows a random smoothing parameter for example that specified by a cross validation method the second allows the smoothing parameter to be a function of location for example an estimator based on nearest neighbor distance in this paper a general method is presented which establishes uniform consistency for all of these estimators

hyperdimensional data analysis using parallel coordinates abstract this article presents the basic results of using the parallel coordinate representation as a high dimensional data analysis tool several alternatives are reviewed the basic algorithm for parallel coordinates is laid out and a discussion of its properties as a projective transformation is given several duality results are discussed along with their interpretations as data analysis tools permutations of the parallel coordinate axes are discussed and some examples are given some extensions of the parallel coordinate idea are given the article closes with a discussion of implementation and some of my experiences

kernel based object tracking a new approach toward target representation and localization the central component in visual tracking of nonrigid objects is proposed the feature histogram based target representations are regularized by spatial masking with an isotropic kernel the masking induces spatially smooth similarity functions suitable for gradient based optimization hence the target localization problem can be formulated using the basin of attraction of the local maxima we employ a metric derived from the bhattacharyya coefficient as similarity measure and use the mean shift procedure to perform the optimization in the presented tracking examples the new method successfully coped with camera motion partial occlusions clutter and target scale variations integration with motion filters and data association techniques is also discussed we describe only a few of the potential applications exploitation of background information kalman tracking using motion models and face tracking

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

computing kernel smoothed conditional quantiles from many observations abstract quantiles of a variable y conditional on another variable x when plotted against x can be a useful descriptive tool these plots give a quick impression of the functional form of the relation between x and the location spread and shape of the conditional distribution of y if several y are observed for each x then sample quantiles could be calculated for each x the resulting quantile plot may be quite noisy however and smoothing across x may be desired this article presents an algorithm that calculates kernel smoothed conditional quantiles with a cross validation choice of bandwidth for x it is computationally feasible for large data sets when x assumes a small number of values since it requires only one pass through the full data set the cross validation does not require a pass through the data because of simplifications arising from the l 1 loss function being based on absolute values the technique is illustrated by plotting the conditional quantiles of the net wealth of a sample of

reduced qkz equation and correlation functions of the xxz model correlation functions of the xxz model in the massive and massless regimes are known to satisfy a system of linear equations the main relations among them are the difference equations obtained from the qkz equation by specializing the variables 1 2n as 1 n n 1 1 1 we call it the reduced qkz equation in this article we construct a special family of solutions to this system they can be written as linear combinations of products of two transcendental functions open image in new window with coefficients being rational functions we show that correlation functions of the xxz model in the massive regime are given by these formulas with an appropriate choice of open image in new window we also present a conjectural formula in the massless regime

sar amplitude probability density function estimation based on a generalized gaussian model in the context of remotely sensed data analysis an important problem is the development of accurate models for the statistics of the pixel intensities focusing on synthetic aperture radar sar data this modeling process turns out to be a crucial task for instance for classification or for denoising purposes in this paper an innovative parametric estimation methodology for sar amplitude data is proposed that adopts a generalized gaussian gg model for the complex sar backscattered signal a closed form expression for the corresponding amplitude probability density function pdf is derived and a specific parameter estimation algorithm is developed in order to deal with the proposed model specifically the recently proposed method of log cumulants molc is applied which stems from the adoption of the mellin transform instead of the usual fourier transform in the computation of characteristic functions and from the corresponding generalization of the concepts of moment and cumulant for the developed gg based amplitude model the resulting molc estimates turn out to be numerically feasible and are also analytically proved to be consistent the proposed parametric approach was validated by using several real ers 1 xsar e sar and nasa jpl airborne sar images and the experimental results prove that the method models the amplitude pdf better than several previously proposed parametric models for backscattering phenomena

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

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

a histogram transform for probabilitydensity function estimation the estimation of multivariate probability density functions has traditionally been carried out by mixtures of parametric densities or by kernel density estimators here we present a new nonparametric approach to this problem which is based on the integration of several multivariate histograms computed over affine transformations of the training data our proposal belongs to the class of averaged histogram density estimators the inherent discontinuities of the histograms are smoothed while their low computational complexity is retained we provide a formal proof of the convergence to the real probability density function as the number of training samples grows and we demonstrate the performance of our approach when compared with a set of standard probability density estimators

on the spatial distribution of fingerprint singularities fingerprint singularities play an important role in several fingerprint recognition and classification systems although some general relationships and constraints about the location of singularities in the different fingerprint classes are well known to the best of our knowledge no statistical models have been developed until now this paper studies the spatial distributions of singularity locations in nature and derives from a representative dataset of labelled samples the probability density functions of the four main fingerprint classes the results obtained can be directly exploited to improve the accuracy of many techniques relying on the position of singularities as confirmed by the results of two experiments on fingerprint classification and synthesis

monodromy operators for higher rank we find the explicit form of the basic monodromy operators for the case of the quantum group expressions for the quantum casimir elements of the quantum groups and are obtained as a by product

regional confidence bands for roc curves the performance of a diagnostic test is characterized by its specificity and sensitivity for a quantitative diagnostic test these criteria depend on the selected cut off point the receiver operating characteristic roc curve of a quantitative diagnostic test is generated by plotting sensitivity against specificity as the cut off point runs through the whole range of possible test values in practice the roc curve is estimated from clinical data one important goal is to select an optimal cut off point for this purpose the sample variability has to be taken into account recently campbell has introduced non parametric asymptotic simultaneous confidence bands that are valid for the whole roc curve in this paper a non parametric asymptotic approach for the construction of regional confidence bands for roc curves is proposed it can be applied for any specificity interval of interest our approach is based on the asymptotic theory of empirical and quantile processes to investigate the small sample properties of the different approaches a monte carlo study was conducted using normal and log normal data a method for sample size calculation is presented finally the approaches are applied to a tumour marker in the diagnosis of bone marrow metastases

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

monte carlo study of three data based nonparametric probability density estimators abstract although the theoretical properties of modern nonparametric probability density estimators have been studied for 25 years there remains the practical problem of how to specify the amount of bias or smoothing in a density estimate based on a random sample in this paper we review and evaluate three recently developed data based algorithms that completely specify a density estimate from a random sample using monte carlo techniques we compare the statistical accuracy of these algorithms as measured by the integrated mean squared error in addition we examine the sensitivity of these algorithms to outliers and estimate computer time requirements one conclusion we draw is that the statistical accuracy of these data based algorithms seems comparable to levels predicted by theoretical models

discretized and interpolated kernel density estimates abstract the kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator nevertheless for computational and especially graphical reasons exact kernel density estimates are often presented in piecewise constant discretized or piecewise linear interpolated form using the exact estimate only at some grid of points the asymptotic integrated mean squared error properties of these modifications are studied it is seen that discretization may adversely affect the order of magnitude of this risk criterion so care needs to be taken in practice to limit the degree of discretization employed this roughness effect essentially disappears when interpolation is used instead then it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate s performance data discretization prior to applying the kernel density estimation prescription is also addressed such prebinning does not affect the smoot

hidden grassmann structure in the xxz model for the critical xxz model we consider the space mathcal w alpha of operators which are products of local operators with a disorder operator we introduce two anti commutative families of operators bf b zeta bf c zeta which act on mathcal w alpha these operators are constructed as traces over representations of the q oscillator algebra in close analogy with baxter s q operators we show that the vacuum expectation values of operators in mathcal w alpha can be expressed in terms of an exponential of a quadratic form of bf b zeta bf c zeta

integrable structure of conformal field theory ii q operator and ddv equation this paper is a direct continuation of 1 where we began the study of the integrable structures in conformal field theory we show here how to construct the operators bf q pm lambda which act in the highest weight virasoro module and commute for different values of the parameter these operators appear to be the cft analogs of the q matrix of baxter 2 in particular they satisfy baxter s famous t q equation we also show that under natural assumptions about analytic properties of the operators as the functions of the baxter s relation allows one to derive the nonlinear integral equations of destri de vega ddv 3 for the eigenvalues of the q operators we then use the ddv equation to obtain the asymptotic expansions of the q operators at large it is remarkable that unlike the expansions of the t operators of 1 the asymptotic series for q contains the dual nonlocal integrals of motion along with the local ones we also discuss an intriguing relation between the vacuum eigenvalues of the q operators and the stationary transport properties in the boundary sine gordon model on this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum hall system

multivariate density estimation by bayesian sequential partitioning consider a class of densities that are piecewise constant functions over partitions of the sample space defined by sequential coordinate partitioning we introduce a prior distribution for a density in this function class and derive in closed form the marginal posterior distribution of the corresponding partition a computationally efficient method based on sequential importance sampling is presented for the inference of the partition from this posterior distribution compared to traditional approaches such as the kernel method or the histogram the bayesian sequential partitioning bsp method proposed here is capable of providing much more accurate estimates when the sample space is of moderate to high dimension we illustrate this by simulated as well as real data examples the examples also demonstrate how bsp can be used to design new classification methods competitive with the state of the art

an automatic bandwidth selector for kernel density estimation summary to select a proper bandwidth is a critical step in kernel density estimation it is well known that the bandwidth selected by cross validation has a large variability this difficulty limits the applicability of cross validation to reduce the variability we suggested modifying the sample characteristic function beyond some cut off frequency in estimating the bias term of the mean integrated squared error it is proposed to select the cut off frequency by a generalization of cross validation for smooth density functions the asymptotic distribution of the bandwidth estimator based on the estimated cut off frequency is obtained the proposed bandwidth estimator has a relative convergence rate n which is much faster than the rate n 1110 for the bandwidth estimate selected by cross validation a modification which reduces the chance of selecting a large cut off frequency is also suggested in simulation studies the advantages of the proposed procedures are clearly demonstrated the procedures are also applied to some data sets

the influence of rounding errors on some nonparametric estimators of a density and its derivatives all continuous data is rounded to some extent and this rounding introduces errors into the estimation of the parent density and its derivatives in this paper we examine the influence of rounding errors on the two most popular types of nonparametric density estimator those based on kernels and those based on generalized fourier series our results lead directly to several important conclusions which are very relevant to the practical problems of density estimation

convergence rates for uniform b spline density estimators part i one dimension a b spline nonparametric density estimator with uniform knots convenient for solving problems in computer graphics was discussed by gehringer and redner in 1992 comm stat simulation methods 21 1992 pp 849 878 these ideas were later extended to density function estimation on metric spaces using partitions of unity in redner and gehringer in 1994 comm stat theory methods 23 1994 pp 2059 2076 in the current paper we return to the uniform b spline density estimator in one dimension we show under very natural assumptions that the b spline density estimator and all of its nontrivial derivatives converge in mean integrated squared error asymptotic rates of convergence are determined for the density estimate and its derivatives

frequency polygons theory and application abstract in this article i investigate the theoretical properties and applications of the frequency polygon which is constructed by connecting with straight lines the mid bin values of a histogram for estimating an unknown probability density function using a random sample the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error achieving the same rate of convergence to zero of the integrated mean squared error as non negative kernel estimators data based algorithms for constructing frequency polygons are discussed and illustrated one is based on a histogram with bin width equal to 2 15sn 1 5 where s is an estimate of the standard deviation from a sample of size n another approach is based on the method of generalized crossvalidation the bivariate frequency polygon is also investigated comparisons are made between the frequency polygon and other density estimators examples with data in one and two dimensions are presented

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

nonparametric testing of the existence of modes given a set of data drawn from an unknown density it is frequently desirable to estimate the number and location of modes of the density a test is proposed for the weight of evidence of individual observed modes the test statistic used is a measure of the size of the mode the absolute integrated difference between the estimated density and the same density with the mode in question excised at the level of the higher of its two surrounding antimodes samples are simulated from a conservative member of the composite null hypothesis to estimate p values within a monte carlo setting such a test can be used with the graphical mode tree of minnotte and scott to examine in a locally adaptive fashion not only the reality of individual modes but also roughly the overall number of modes of the density a proof of consistency of the test statistic is offered and simulation results are presented

nonlinear filtering using a new proposal distribution and the improved fast gauss transform with tighter performance bounds we propose a new approach to approximating the chapman kolmogorov equation cke for particle based nonlinear filtering algorithms using a new proposal distribution and the improved fast gauss transform ifgt with tighter accuracy bounds the new proposal distribution used to obtain a monte carlo mc approximation of the cke is based on the proposal distribution found in the auxiliary marginal particle filter ampf by using mc integration to approximate the integrals of the ampf proposal distribution as well as the cke we demonstrate significant improvement in terms of both error and computation time we consider the additive state noise case where the evaluation of the cke is equivalent to performing kernel density estimation kde thus fast methods such as the ifgt can be used in practice the ifgt demonstrates performance far better than that which is predicted by current error analysis therefore the existing bounds are not useful for determining the ifgt parameters which in practice have to be chosen experimentally in order to obtain satisfactory compromise between accuracy and speed of the filtering algorithm we provide in this paper much improved performance bounds for the ifgt and which unlike the previous bound are consistent with the expectation that the error decreases as the truncation order of the ifgt increases the new bounds lead to a new definition of the ifgt parameters and give important insight into the effect that these parameters have on the error therefore facilitating the choice of parameters in practice the experimental results show that we can obtain similar error to the sequential importance sampling sis particle filter while using fewer particles furthermore the choice of the new ifgt parameters remains roughly the same for all the examples that we give

a method to detect and characterize ellipses using the hough transform we describe a new technique for detecting and characterizing ellipsoidal shapes automatically from any type of image this technique is a single pass algorithm which can extract any group of ellipse parameters or characteristics which can be computed from those parameters without having to detect all five parameters for each ellipsoidal shape moreover the method can explicitly incorporate any a priori knowledge the user may have concerning ellipse parameters the method is based on techniques from projective geometry and on the hough transform this technique can significantly reduce interpretation and computation time by automatically extracting only those features or geometric parameters of interest from images and making exact use of a priori information

achieving higher order convergence rates for density estimation with binned data abstract a method is proposed for achieving fast o n 8 9 and faster asymptotic mean integrated squared error amise convergence rates in density estimation using data condensed to standard histogram bin counts and edges the method involves the two elements of b splines and the correct mass proportions property in which the probability mass for the histogram bins equals exactly the fraction of the data found in those bins it is shown that the asymptotic bias of such an estimate using a spline of order p and bin width h will be o hp resulting in optimal amise of o n 2p 2p 1 form theoretical and numeric comparisons to kernel based and other estimators show that the histospline estimator is usually comparable and often superior an extension to the multivariate density estimation setting is described

probability density function estimation with the frequency polygon transform a probability density function estimator is proposed which is based on frequency polygons its convergence to the true density is formally proved a mode finding algorithm is also proposed as an alternative to mean shift our approach outperforms histogram and kernel based estimators in synthetic and real datasets our proposal is shown to be suitable to object tracking in video sequences most current nonparametric approaches to probability density function estimation are based on the kernel density estimator also known as the parzen window estimator a usual alternative is the multivariate histogram which features a low computational complexity multivariate frequency polygons have often been neglected even though they share many of the advantages of the histograms while they are continuous unlike the histograms here we build on our previous work on histograms in order to propose a new probability density estimator which is based on averaging multivariate frequency polygons the convergence of the estimator is formally proved experiments are carried out with synthetic and real machine learning datasets finally image denoising and object tracking applications are also considered

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

preaveraged localized orthogonal polynomial estimators for surface smoothing and partial differentiation abstract we propose a multivariate smoothing method based on products of localized orthogonal polynomial series estimators for a smooth regression surface in the fixed design regression model the estimation of partial derivatives is included the proposed method provides for automatic and efficient boundary modifications near the edges of the surface assuming that the boundary of the support of the regression function satisfies some regularity conditions by allowing for a preaveraging step the corresponding algorithms are speeded up considerably and are easy to implement computation of special boundary kernels as required by the kernel method to avoid edge effects is not necessary it is shown that under sufficient smoothness assumptions the global average mean squared error has the same optimal rate of convergence as the mean squared error at an interior point that is the boundary correction is asymptotically effective the method depends on two smoothing parameters one determining the amount

exercises with the universal r matrix using the formula for the universal r matrix proposed by khoroshkin and tolstoy we give a detailed derivation of l operators for the quantum groups associated with the generalized cartan matrices a 1 1 and a 1 2

estimating information entropy for hydrological data one dimensional case there has been a recent resurgence of interest in the application of information theory to problems of system identification in the earth and environmental sciences while the concept of entropy has found increased application little attention has yet been given to the practical problems of estimating entropy when dealing with the unique characteristics of two commonly used kinds of hydrologic data rainfall and runoff in this paper we discuss four important issues of practical relevance that can bias the computation of entropy if not properly handled the first zero effect arises when precipitation and ephemeral streamflow data must be viewed as arising from a discrete continuous hybrid distribution due to the occurrence of many zero values e g days with no rain no runoff second in the widely used bin counting method for estimation of pdf s significant error can be introduced if the bin width is not carefully selected the third measurement effect arises due to the fact that continuously varying hydrologic variables can typically only be observed discretely to some degree of precision the fourth skewness effect arises when the distribution of a variable is significantly skewed here we present an approach that can deal with all four of these issues and test them with artificially generated and real hydrological data the results indicate that the method is accurate and robust

the edge frequency polygon a piecewise linear competitor to the frequency polygon the edge frequency polygon is proposed instead of joining histogram ordinates at midpoints of bins by straight lines one connects values at bin edges by straight lines those values being averages of contiguous histogram ordinates the new proposal is as simple and interpretable as the ordinary frequency polygon while holding theoretical and practical advantages

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho