nonparametric multivariate density estimation a comparative study the paper algorithmically and empirically studies two major types of nonparametric multivariate density estimation techniques where no assumption is made about the data being drawn from any of known parametric families of distribution the first type is the popular kernel method and several of its variants which uses locally tuned radial basis e g gaussian functions to interpolate the multidimensional density the second type is based on an exploratory projection pursuit technique which interprets the multidimensional density through the construction of several 1d densities along highly interesting projections of multidimensional data performance evaluations using training data from mixture gaussian and mixture cauchy densities are presented the results show that the curse of dimensionality and the sensitivity of control parameters have a much more adverse impact on the kernel density estimators than on the projection pursuit density estimators

the motion of a vortex on a closed surface of constant negative curvature the purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one the algorithm is based on a relation between the laplace beltrami green function and the heat kernel the algorithm is used to compute the motion of a vortex on the bolza surface this is the first determination of the orbits of a vortex on a closed surface of genus greater than one the numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the bolza surface some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc the following theorem is proved a weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

rate of convergence in density estimation using neural networks given n i i d observations xi ni 1 taking values in a compact subset of rd such that p denotes their common probability density function we estimate p from an exponential family of densities based on single hidden layer sigmoidal networks using a certain minimum complexity density estimation scheme assuming that p possesses a certain exponential representation we establish a rate of convergence independent of the dimension d for the expected hellinger distance between the proposed minimum complexity density estimator and the true underlying density p

perturbative analysis of the method of particular solutions for improved inclusion of high lying dirichlet eigenvalues the dirichlet eigenvalue or drum problem in a domain omega subset mathbb r 2 becomes numerically challenging at high eigenvalue frequency e in this regime the method of particular solutions mps gives spectral accuracy for many domain shapes it requires a number of degrees of freedom scaling as sqrt e the number of wavelengths on the boundary in contrast to direct discretization for which this scaling is e our main result is an inclusion bound on eigenvalues that is a factor o sqrt e tighter than the classical bound of moler payne and that is optimal in that it reflects the true slopes of curves appearing in the mps we also present an mps variant that cures a normalization problem in the original method while evaluating basis functions only on the boundary this method is efficient at high frequencies where we show that in practice our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort

feasibility of multivariate density estimates summary the curse of dimensionality has been interpreted as suggesting that kernel methods have limited applicability in more than several dimensions in this note qualitative and quantitative performance measures for multivariate density estimates are examined optimal pointwise and global window widths for mean absolute and mean squared errors are compared for multivariate data one result is that the optimal pointwise absolute and squared error window widths are nearly equal for all dimensions we also show that sample size requirements predicted by absolute rather than squared error criterion are substantially less further reductions are realized by using a coefficient of variation criterion finally an example of a 10 dimensional kernel density estimate is given it is suggested that the true nature of the curse of dimensionality is as much the lack of full rank as sparseness of the data

approximations on the peano river network application of the horton strahler hierarchy to the case of low connections a network analysis is used to investigate the low connections of natural river channels at the basin scale the river networks are analyzed according to the horton strahler hierarchy we propose a quantitative criterion for the average junction degree as a function of a fixed hierarchical order of the network and independent of the usual scaling laws the numerical results of this analysis are compared with exact results of the peano river network showing differences of the order of 10 ensuremath 3 this aspect is especially relevant for the characterization of transport and diffusion processes at the basin scale

on largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces we apply topological methods to study the smallest non zero number lambda 1 in the spectrum of the laplacian on finite area hyperbolic surfaces for closed hyperbolic surfaces of genus two we show that the set s in mathcal m 2 lambda 1 s frac 1 4 is unbounded and disconnects the moduli space mathcal m 2 using this for genus g ge 3 we show the existence of eigenbranches that start as lambda 1 and eventually becomes frac 1 4

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of genus two riemann surfaces we study extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of compact genus two riemann surfaces by a combination of analytical and numerical methods we identify four non degenerate critical points of this function and compute the signature of the hessian at these points the curve with the maximal number of automorphisms the burnside curve turns out to be the point of the absolute maximum our results agree with the mass formula for virtual euler characteristics of the moduli space a similar analysis is performed for bolza s strata of symmetric riemann surfaces of genus two

definition of new equivalent indices of horton strahler ratios for the derivation of the geomorphological instantaneous unit hydrograph 1 the horton strahler concept has been important in river basin geomorphology it describes scaling properties and identifies giuh but has also attracted criticisms because it depends on the threshold area s which is used to extract the channel network from digital elevation models dem on the position of the outlet and on the few number of ratios used to identify the scaling laws to overcome these limitations this paper proposes new indices independently of s and which have similar properties than the horton strahler ratios applications were conducted on seven french basins the methodology consists on analyzing the morphometric properties such as the drained area the number of sources and the length of the channel network as a function of s the results show that horton strahler ratios vary considerably with s particularly while the order length l is a staircase function of s results show that the channel network is articulated around two main nodes independent of s which are descriptors of l then we compared the properties of horton strahler laws to those obtained from self similarity analysis which enables to define new indices independent of s these indices are calculated automatically from dem have similar properties as the horton strahler ratios and are used to calculate the giuh independently of s the four new descriptors rbe rle rae and le which have similar properties as rb rl ra and l respectively are equivalent to those of horton strahler s laws and can be used as indicators of hydrological similarity for catchment comparison and regionalization

regularized mixture density estimation with an analytical setting of shrinkage intensities in this paper we propose a method for p variate probability density estimation assuming a gaussian mixture model gmm our method exploits a regularization technique for improving the estimation accuracy of the gmm component covariance matrices we derive an expectation maximization algorithm for fitting our regularized gmm rgmm which exploits an analytical ledoit wolf type shrinkage estimation of the covariance matrices our method is compared with recent model based and variational bayes approximation methods using synthetic and real data sets the obtained results show that the proposed rgmm method achieves a significant improvement in the performance of multivariate probability density estimation with respect to other methods on both the synthetic and the real data sets

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

ambient constructions for sasakian η einstein manifolds abstract the theory of ambient spaces is useful to define cr invariant objects such as cr invariant powers of the sub laplacian the p prime operator and q prime curvature however in general it is difficult to write down these objects in terms of the tanaka webster connection in this paper we give those explicit formulas for cr manifolds satisfying an einstein condition called sasakian einstein manifolds as an application we study properties of the first and the second variation of the total q prime curvature at sasakian einstein manifolds

strong approximation of density estimators from weakly dependent observations by density estimators from independent observations we derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors we construct on a sufficiently rich probability space such a pairing of the random variables of both experiments that the set of observations x 1 x n from the time series model is nearly the same as the set of observations y 1 y n from the i i d model with a high probability all sets of the form x 1 x n delta y 1 y n boolean and a 1 b 1 x x a d b d contain no more than o n 1 2 pi b i a i 1 log n elements respectively although this does not imply very much for parametric problems it has important implications in nonparametric statistics it yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i i d model moreover it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap using these results we derive simultaneous confidence bands as well as supremum type nonparametric tests based on reasoning for the i i d model

projection pursuit mixture density estimation in this paper we seek a gaussian mixture model gmm of an n variate probability density function usually the parameters of gmms are determined in the original n dimensional space by optimizing a maximum likelihood ml criterion a practical deficiency of this method of fitting gmms is its poor performance when dealing with high dimensional data since a large sample size is needed to match the accuracy that is possible in low dimensions we propose a method for fitting the gmm based on the projection pursuit strategy this gmm is highly constrained and hence its ability to model structure in subspaces is enhanced compared to a direct ml fitting of a gmm in high dimensions our method is closely related to recently developed independent factor analysis ifa mixture models the comparisons with ml fitting of gmm in n dimensions and ifa mixtures show that the proposed method is an attractive choice for fitting gmms using small sizes of training sets

projection pursuit density estimation abstract the projection pursuit methodology is applied to the multivariate density estimation problem the resulting nonparametric procedure is often less biased than the kernel and near neighbor methods in addition graphical information is produced that can be used to help gain geometric insight into the multivariate data distribution

boundary quasi orthogonality and sharp inclusion bounds for large dirichlet eigenvalues we study eigenfunctions phi j and eigenvalues e j of the dirichlet laplacian on a bounded domain omega subset mathbb r n with piecewise smooth boundary we bound the distance between an arbitrary parameter e 0 and the spectrum e j in terms of the boundary l 2 norm of a normalized trial solution u of the helmholtz equation delta e u 0 we also bound the l 2 norm of the error of this trial solution from an eigenfunction both of these results are sharp up to constants hold for all e greater than a small constant and improve upon the best known bounds of moler payne by a factor of the wavenumber sqrt e one application is to the solution of eigenvalue problems at high frequency via for example the method of particular solutions in the case of planar strictly star shaped domains we give an inclusion bound where the constant is also sharp we give explicit constants in the theorems and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy the proof makes use of a new quasi orthogonality property of the boundary normal derivatives of the eigenmodes theorem 1 3 of interest in its own right namely the operator norm of the sum of rank 1 operators partial n phi j langle partial n phi j cdot rangle over all e j in a spectral window of width sqrt e a sum with about e n 1 2 terms is at most a constant factor independent of e larger than the operator norm of any one individual term

statistical dynamics of early river networks abstract based on local erosion rule and fluctuations in rainfall geology and parameters of a river channel a generalized langevin equation is proposed to describe the random prolongation of a river channel this equation is transformed into the fokker plank equation to follow the early evolution of a river network and the variation of probability distribution of channel lengths the general solution of the equation is in the product form of two terms one term is in power form and the other is in exponent form this distribution shows a complete history of a river network evolving from its infancy to adulthood the infancy is characterized by the gaussian distribution of the channel lengths while the adulthood is marked by a power law distribution of the channel lengths the variation of the distribution from the gaussian to the power law displays a gradual developing progress of the river network the distribution of basin areas is obtained by means of hack s law these provide us with new understandings towards river networks

characterization of spatial networklike patterns from junction geometry we propose a method for quantitative characterization of spatial networklike patterns with loops such as surface fracture patterns leaf vein networks and patterns of urban streets such patterns are not well characterized by purely topological estimators also patterns that both look different and result from different morphogenetic processes can have similar topology a local geometric cue the angles formed by the different branches at junctions can complement topological information and allow the quantification of the large scale spatial coherence of the pattern for patterns that grow over time such as fracture lines on the surface of ceramics the rank assigned by our method to each individual segment of the pattern approximates the order of appearance of that segment we apply the method to various networklike patterns and find a continuous but sharp dichotomy between two classes of spatial networks hierarchical and homogeneous the former class results from a sequential growth process and presents large scale organization and the latter presents local but not global organization

the minimum risk principle that underlies the criteria of bounded component analysis this paper studies the problem of the blind extraction of a subset of bounded component signals from the observations of a linear mixture in the first part of this paper we analyze the geometric assumptions of the observations that characterize the problem and their implications on the mixing matrix and latent sources in the second part we solve the problem by adopting the principle of minimizing the risk which refers to the encoding complexity of the observations in the worst admissible situation this principle provides an underlying justification of several bounded component analysis bca criteria including the minimum normalized volume criterion of the estimated sources or the maximum negentropy likelihood criterion with a uniform reference model for the estimated sources this unifying framework can explain the differences between the criteria in accordance with their considered hypotheses for the model of the observations this paper is first presented for the case of the extraction of a complex and multidimensional source and later is particularized for the case of the extraction of subsets of 1 d complex sources the results also hold true in the case of real signals where the obtained criteria for the extraction of a set of 1 d sources usually coincide with the existing bca criteria

a reformulation of horton s laws for large river networks in terms of statistical self similarity the well known horton s laws are empirical observations on how the means of measurements for river networks and basins vary with horton strahler order it is now known that these laws are a consequence of an average sense self similarity in the bifurcation structure of river networks in this paper we present a reformulation of horton s laws which generalizes the familiar scaling of first moments or means to scaling of entire distributions we also present extensive data analysis which supports this reformulation and show that this feature is also exhibited by shreve s well known random topology model

a variant of the acceptance rejection method for computer generation of random variables abstract a variant of the common acceptance rejection method for generating random variables guarantees that steps are not repeated this approach affords considerable flexibility in tailoring the method efficiently to the specified density an algorithm for generating cauchy random variables illustrates the use of this approach

periodic orbit sum rules for the hadamard gutzwiller model abstract it is shown how a variety of periodic orbit sum rules can be used to extract information about a quantum mechanical system whose classical counterpart is completely chaotic from knowledge only of the classical system and vice versa the basis is the selberg trace formula an exact analogue for the hadamard gutzwiller model of the semiclassical periodic orbit theory of gutzwiller which relates the quantal energies to the lengths of the periodic orbits of the classical system statistical properties of the quantal energies in the low energy region are studied where we restrict ourselves to the level spacing and spectral rigidity

geometry of river networks i scaling fluctuations and deviations this paper is the first in a series of three papers investigating the detailed geometry of river networks branching networks are a universal structure employed in the distribution and collection of material large scale river networks mark an important class of two dimensional branching networks being not only of intrinsic interest but also a pervasive natural phenomenon in the description of river network structure scaling laws are uniformly observed reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists to improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure here we report a theoretical and empirical study of fluctuations about and deviations from scaling we examine data for continent scale river networks such as the mississippi and the amazon and draw inspiration from a simple model of directed random networks we center our investigations on the scaling of the length of a subbasin s dominant stream with its area a characterization of basin shape known as hack s law we generalize this relationship to a joint probability density and provide observations and explanations of deviations from scaling we show that fluctuations about scaling are substantial and grow with system size we find strong deviations from scaling at small scales which can be explained by the existence of a linear network structure at intermediate scales we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate at large scales we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape the extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution

hyperbolic octagons and teichmuller space in genus 2 in this paper we study different models of the teichmuller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems they are on the one hand complex enough to show the relevant features and on the other hand they possess a simple mathematical structure allowing practical implementations we give a new description of teichmuller space and the mapping class group in terms of geometric data of the octagons this provides modellings based on the vertices and also on the generators of the associated isometry group in addition we explicitly connect our approach with other models currently used in the literature helling matrices fenchel nielsen parameters the result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others

nonparametric density estimation with a parametric start the traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes the present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities for example the normal while not losing much in precision when the true density is far from the parametric class the idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor this works well in cases where the correction factor function is less rough than the original density itself extensive comparisons with the kernel estimator are carried out including exact analysis for the class of all normal mixtures the new method with a normal start wins quite often even in many cases where the true density is far from normal procedures for choosing the smoothing parameter of the estimator are also discussed the new estimator should be particularly useful in higher dimensions where the usual nonparametric methods have problems the idea is also spelled out for nonparametric regression

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

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

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

from classical periodic orbits to the quantization of chaos we present a quantitative analysis of selberg s trace formula viewed as an exact version of gutzwiller s semiclassical periodic orbit theory for the quantization of classically chaotic systems two main applications of the trace formula are discussed in detail i the periodic orbit sum rules giving a smoothing of the quantal energy level density ii the selberg zeta function as a prototype of a dynamical zeta function defined as an euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a dirichlet series it is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an orbit selection function numerical results are presented for the free motion of a point particle on compact riemann surfaces hadamard gutzwiller model which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos our results demonstrate clearly the crucial role played by the long periodic orbits an exact rule for quantizing chaos is derived for such systems where the dirichlet series representing the selberg zeta function converges on the critical line explicit formulae are given for the computation of the abscissae of absolute and conditional convergence respectively of these dynamical dirichlet series for the two riemann surfaces considered it turns out that one can cross the entropy barrier but that the critical line cannot be reached by a convergent dirichlet series it would seem that this is the main reason why the riemann siegel lookalike formula recently conjectured by m v berry and j p keating fails in generating the lower lying quantal energies for these strongly chaotic systems

cross validation and the estimation of conditional probability densities many practical problems especially some connected with forecasting require nonparametric estimation of conditional densities from mixed data for example given an explanatory data vector x for a prospective customer with components that could include the customer s salary occupation age sex marital status and address a company might wish to estimate the density of the expenditure y that could be made by that person basing the inference on observations of x y for previous clients choosing appropriate smoothing parameters for this problem can be tricky not in the least because plug in rules take a particularly complex form in the case of mixed data an obvious difficulty is that there exists no general formula for the optimal smoothing parameters more insidiously and more seriously it can be difficult to determine which components of x are relevant to the problem of conditional inference for example if the jth component of x is independent of y then that component is irrelevant to es

multimodality exploration by an unsupervised projection pursuit neural network graphical inspection of multimodality is demonstrated using unsupervised lateral inhibition neural networks three projection pursuit indexes are compared on low dimensional simulated and real world data principal components legendre polynomial and projection pursuit network

ridge polynomial networks this paper presents a polynomial connectionist network called ridge polynomial network rpn that can uniformly approximate any continuous function on a compact set in multidimensional input space r sup d with arbitrary degree of accuracy this network provides a more efficient and regular architecture compared to ordinary higher order feedforward networks while maintaining their fast learning property the ridge polynomial network is a generalization of the pi sigma network and uses a special form of ridge polynomials it is shown that any multivariate polynomial can be represented in this form and realized by an rpn approximation capability of the rpn s is shown by this representation theorem and the weierstrass polynomial approximation theorem the rpn provides a natural mechanism for incremental network growth simulation results on a surface fitting problem the classification of high dimensional data and the realization of a multivariate polynomial function are given to highlight the capability of the network in particular a constructive learning algorithm developed for the network is shown to yield smooth generalization and steady learning

multidimensional density estimation by tomography the paper explores the application of the filtered backprojection algorithm of computerized tomography to statistical smoothing of multidimensional densities the method makes use of automated one dimensional density smoothers to obtain an automated smoothed multidimensional density no structural assumptions such as additivity or log additivity are used an important computational aspect of this method is its inherently parallel structure we demonstrate linear speed up on a parallel computer architecture various practical details are discussed together with some illustrative two dimensional examples

new results for self similar trees with applications to river networks in a little known series of papers beginning in 1966 tokunaga introduced an infinite class of tree graphs based on the strahler ordering scheme as recognized by tokunaga 1984 these trees are characterized by a self similarity property so we will refer to them as self similar trees or ssts ssts are defined in terms of a generator matrix which acts as a blueprint for constructing different trees many familiar tree constructions are absorbed as special cases however in tokunaga s work an additional assumption is imposed which restricts from ssts to a much smaller class we will refer to this subclass as tokunaga s trees this paper presents several new and unifying results for ssts in particular the conditions under which ssts have well defined horton strahler stream ratios are given as well as a general method for computing these ratios it is also shown that the diameters of ssts grow like m where m is the number of leaves in contrast to many other tree constructions here need not equal 1 2 thus ssts offer an explanation for hack s law finally it is demonstrated that large discrepancies exist between the predictions of shreve s well known model and detailed measurements for large river networks while other ssts fit the data quite well other potential applications of the sst framework include diffusion limited aggregation dla lightning bronchial passages neural networks and botanical trees

on the forward and backward algorithms of projection pursuit this article provides a historic review of the forward and backward projection pursuit algorithms previously thought to be equivalent and points out an important difference between the two in doing so a small error in the original exploratory projection pursuit article by friedman j amer statist assoc 82 1987 249 266 is corrected the implication of the difference is briefly discussed in the context of an application in which projection pursuit density estimation is used as a building block for nonparametric discriminant analysis

local likelihood density estimation local likelihood was introduced by tibshirani and hastie as a method of smoothing by local polynomials in non gaussian regression models in this paper an extension of these methods to density estimation is discussed and comparison with other methods of density estimation presented the local likelihood method has particularly strong advantages over kernel methods when estimating tails of densities and in multivariate settings suppose constraints are incorporated in a simple manner asymptotic properties of the estimate are discussed a method for computing the estimate is outlined c code to implement the estimation procedure described in this paper together with s interfaces for graphical display of results are available at http cm bell labs com stat project locfit index html

the discrimination power of projection pursuit with different density estimators summary we explore the properties of projection pursuit discriminant analysis this discriminant method is very powerful but relies heavily on a univariate density estimate we show that the procedure based on wavelets maintains the same rate of convergence as with univariate wavelet density estimation we also show the bayes risk strong consistency of both the kernel and wavelet based methods simulated data and real data concerning character recognition show that the method is effective and robust against the curse of dimensionality the wavelet alternative seems more likely than the kernel counterpart to find an interesting projection wavelets are often criticised for giving too wiggly an estimate and for being too localised to give good global properties in the above context these potential drawbacks do not weaken the method but the use of wavelets seems to enhance it a multiple projection generalisation is also considered

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

fourier tauberian theorems and applications abstract the main aim of the paper is to present a general version of the fourier tauberian theorem for monotone functions this result together with berezin s inequality allows us to obtain a refined version of the li yau estimate for the counting function of the dirichlet laplacian which implies the li yau estimate itself and at the same time the asymptotic results obtained by the variational method

estimating the direction in which a data set is most interesting estimation of orientation is a key operation at each step in projection pursuit since projection pursuit is a nonparametric algorithm and since even low dimensional approximations to the target function must converge to their limits at rates considerably slower than n 1 2 where n is sample size then it might be thought that the same is true of orientation estimates it is shown in the present paper that this is not the case and that estimation of orientation is a parametric operation in the sense that under mild nonparametric assumptions correctly chosen kernel type orientation estimates converge to their limits at rate n 1 2 this property is not enjoyed by standard projection pursuit orientation estimates which converge at a slower rate than n 1 2 most attention in the present paper is focussed on the case of projection pursuit density approximation but it is pointed out that our arguments hold generally an important practical conclusion is that data should be smoothed less when estimating orientation than when constructing the final projection pursuit approximation

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

power law distribution of discharge in ideal networks data for several river networks show algebraically decaying distribution functions for the mean annual discharge and the dissipation rate rodriguez iturbe et al 1992 we derive a relation between the exponent a in the integrated mean annual discharge distribution p q q q and the topological dimension dt of the network 1 1 dt using the experimentally determined value dt 1 8 tarboton et al 1988 we find 0 45 in good agreement with the data of rodriguez iturbe et al 1992 for the random model shreve 1967 1969 smart and werner 1974 we find an exponent of 1 2

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos

semiparametric single index versus fixed link function modelling discrete choice models are frequently used in statistical and econometric practice standard models such as logit models are based on exact knowledge of the form of the link and linear index function semiparametric models avoid possible misspecification but often introduce a computational burden especially when optimization over nonparametric and parametric components are to be done iteratively it is therefore interesting to decide between approaches here we propose a test of semiparametric versus parametric single index modelling our procedure allows the linear index of the semiparametric alternative to be different from that of the parametric hypothesis the test is proved to be rate optimal in the sense that it provides rate minimal distance between hypothesis and alternative for a given power function

pattern classification using projection pursuit abstract this article discusses the adaptation of recently developed regression techniques to classifier design apart from finite sample effects projection pursuit pp regression may be used to model a desired response class as a sum of ridge functions according to a minimum expected squared error criterion this approach can be shown to furnish an optimal discriminant function which can satisfy the neyman pearson criterion over all possible thresholds basis function expansions are used instead of smoothed histograms to reduce computation since good approximation of a discriminant by a linear combination of moderate number of ridge functions may not be easy we introduce an improved method utilizing a nonlinear weighting function

on projection pursuit measures of multivariate location and dispersion 0 introduction projection pursuit pp techniques for the analysis of high dimensional data were introduced by kruskal 1972 and switzer 1970 switzer and wright 1971 and friedman and tukey 1974 and later applied by friedman and stuetzle to a variety of problems including regression 1980a classification 1980b and density estimation 1981 the idea underlying pp is to select interesting low dimensional projections by iteratively maximizing an appropriate projection index usually with the aid of a computer huber 1981b and li and chen 1981 have proposed the use of pp to construct and estimate measures of location and dispersion for multivariate distributions when is such a measure equivariant under affine tranformations in essence we show that a pp location dispersion measure is equivariant if and only if the univariate measure from which it is built is linear quadratic in an appropriate sense lemmas 1 4 this equivariance holds for all distributions only when the univariate measure is expectation standard deviation theorems 2 5 for a fixed multivariate distribution all pp location dispersion measures will be equivariant only when the distribution is symmetric through its center elliptically contoured theorems 3 6 we emphasize that these theorems pertain to functionals on probability distributions corresponding estimators are constructed in the usual way by evaluating these functionals at the empirical distribution thus pp measures of location and dispersion possess limited equivariance properties nevertheless for data generated from elliptically contoured distributions the pp dispersion estimators of li and chen 1981 perform promisingly our results are developed in section 1 location and 2 dispersion and proved in sections 4 and 5 respectively section 3 contains further comments and discussion of some implications for data analysis

a semiparametric density estimation approach to pattern classification a new multivariate density estimator suitable for pattern classifier design is proposed the data are first transformed so that the pattern vector components with the most non gaussian structure are separated from the gaussian components nonparametric density estimation is then used to capture the non gaussian structure of the data while parametric gaussian conditional density estimation is applied to the rest of the components both simulated and real data sets are used to demonstrate the potential usefulness of the proposed approach

convergence of the iterative proportional fitting procedure the iterative proportional fitting procedure ipfp was introduced in 1940 by deming and stephan to estimate cell probabilities in contingency tables subject to certain marginal constraints its convergence and statistical properties have been investigated since then by several authors and by several different methods a natural extension of the ipfp to the case of bivariate densities has been introduced by ireland and kullback it has been conjectured that also in the general case the ipfp converges to the minimum discrimination projection on the class of distributions with given marginals we verify this conjecture under some regularity conditions

geometry of river networks ii distributions of component size and number the structure of a river network may be seen as a discrete set of nested subnetworks built out of individual stream segments these network components are assigned an integral stream order via a hierarchical and discrete ordering method exponential relationships known as horton s laws between stream order and ensemble averaged quantities pertaining to network components are observed we extend these observations to incorporate fluctuations and all higher moments by developing functional relationships between distributions the relationships determined are drawn from a combination of theoretical analysis analysis of real river networks including the mississippi amazon and nile and numerical simulations on a model of directed random networks underlying distributions of stream segment lengths are identified as exponential combinations of these distributions form single humped distributions with exponential tails the sums of which are in turn shown to give power law distributions of stream lengths distributions of basin area and stream segment frequency are also addressed the calculations identify a single length scale as a measure of size fluctuations in network components this article is the second in a series of three addressing the geometry of river networks