conformal invariance and self avoiding walks in restricted geometries the prediction of conformal invariance for the statistics of self avoiding random walks restricted to both semi infinite and wedge shaped geometries are tested by extrapolating exact enumerations close agreement is found both for the angular distribution of the end to end vector and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge

the box cox metric for nearest neighbour classification improvement abstract the nearest neighbour rule is a distribution free classification method in the literature it has been shown however that the performance of the method improves if complex often locally defined metrics are chosen in this paper we demonstrate that in many real world discrimination problems comparable or even better classification results can be obtained with a simple global metric based on the box cox transformation in three case studies this is demonstrated synthetic data the iris data and real radar data in the latter experiment a large reduction in classification error of more than a factor four is achieved copyright 1997 pattern recognition society

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

an optimal global nearest neighbor metric a quadratic metric dao x y x y t ao x y is proposed which minimizes the mean squared error between the nearest neighbor asymptotic risk and the finite sample risk under linearity assumptions a heuristic argument is given which indicates that this metric produces lower mean squared error than the euclidean metric a nonparametric estimate of ao is developed if samples appear to come from a gaussian mixture an alternative parametrically directed distance measure is suggested for nearness decisions within a limited region of space examples of some two class gaussian mixture distributions are included

adaptive local linear regression with application to printer color management local learning methods such as local linear regression and nearest neighbor classifiers base estimates on nearby training samples neighbors usually the number of neighbors used in estimation is fixed to be a global ldquooptimalrdquo value chosen by cross validation this paper proposes adapting the number of neighbors used for estimation to the local geometry of the data without need for cross validation the term enclosing neighborhood is introduced to describe a set of neighbors whose convex hull contains the test point when possible it is proven that enclosing neighborhoods yield bounded estimation variance under some assumptions three such enclosing neighborhood definitions are presented natural neighbors natural neighbors inclusive and enclosing k nn the effectiveness of these neighborhood definitions with local linear regression is tested for estimating lookup tables for color management significant improvements in error metrics are shown indicating that enclosing neighborhoods may be a promising adaptive neighborhood definition for other local learning tasks as well depending on the density of training samples

locally adaptive metric nearest neighbor classification nearest neighbor classification assumes locally constant class conditional probabilities this assumption becomes invalid in high dimensions with finite samples due to the curse of dimensionality severe bias can be introduced under these conditions when using the nearest neighbor rule we propose a locally adaptive nearest neighbor classification method to try to minimize bias we use a chi squared distance analysis to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations neighborhoods are elongated along less relevant feature dimensions and constricted along most influential ones as a result the class conditional probabilities are smoother in the modified neighborhoods whereby better classification performance can be achieved the efficacy of our method is validated and compared against other techniques using both simulated and real world data

the optimal distance measure for nearest neighbor classification a local distance measure is shown to optimize the performance of the nearest neighbor two class classifier for a finite number of samples the difference between the finite sample error and the asymptotic error is used as the criterion of improvement this new distance measure is compared to the well known euclidean distance an algorithm for practical implementation is introduced this algorithm is shown to be computationally competitive with the present nearest neighbor procedures and is illustrated experimentally a closed form for the corresponding second order moment of this criterion is found finally the above results are extended to

conformal off diagonal boundary density profiles on a semi infinite strip the off diagonal profile odb v associated with a local operator v order parameter or energy density close to the boundary of a semi infinite strip with width l is obtained at criticality using conformal methods it involves the surface exponent xs and displays a simple universal behaviour which crosses over from surface finite size scaling when v l is held constant to corner finite size scaling when v l 0

lda svm driven nearest neighbor classification nearest neighbor nn classification relies on the assumption that class conditional probabilities are locally constant this assumption becomes false in high dimensions with finite samples due to the curse of dimensionality the nn rule introduces severe bias under these conditions we propose a locally adaptive neighborhood morphing classification method to try to minimize bias we use local support vector machine learning to estimate an effective metric for producing neighborhoods that are elongated along less discriminant feature dimensions and constricted along most discriminant ones as a result the class conditional probabilities can be expected to be approximately constant in the modified neighborhoods whereby better classification performance can be achieved the efficacy of our method is validated and compared against other competing techniques using a number of datasets

adaptive quasiconformal kernel nearest neighbor classification nearest neighbor classification assumes locally constant class conditional probabilities this assumption becomes invalid in high dimensions due to the curse of dimensionality severe bias can be introduced under these conditions when using the nearest neighbor rule we propose an adaptive nearest neighbor classification method to try to minimize bias we use quasiconformal transformed kernels to compute neighborhoods over which the class probabilities tend to be more homogeneous as a result better classification performance can be expected the efficacy of our method is validated and compared against other competing techniques using a variety of data sets

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

the numerical solution of the semi explicit idaes by discontinuous piecewise polynomial approximation abstract in this paper we consider a semi explicit form of volterra integro differential algebraic equations idaes and investigate the existence and uniqueness of solution of these systems by using differentiability index numerical method based on discontinuous piecewise polynomial approximation is proposed for the solution of the semi explicit idaes and global convergence results are established the performance of the numerical scheme is illustrated by means of some test problems

universal order parameter profiles in confined critical systems with boundary fields the authors consider systems at the bulk critical temperature with an ordering field applied at the boundary for various boundary geometries universal expressions for the spatial dependence of the order parameter are obtained by conformal mapping of the known results for the half space analytical results for strip wedge and other two dimensional geometries and for systems with spherical boundaries in arbitrary dimension are given

a fuzzy version of the k nn method abstract a version for increasing of the classification accuracy of one of the basic statistical nonparametrical methods the k nn method is proposed here the idea of including fuzzy information in the k nn method is applied in a new way the distances are modified by means of the pattern degrees of membership to the classes to which the pattern belongs thus for each of the labeled samples its typicalness is taken into consideration it is proved that independently of the metrics used the classification error in case of modified distances is less than or equal to the error in case of the nonmodified distances furthermore fuzzy information is used to form fuzzy decision rules that is done to increase the classification accuracy and to establish reasonable limits for fuzzy approach applications the proposed version is tested with clinical data of neonates which suffer from respiratory distress syndrome rds on the basis of theoretical and experimental results a computer consulting system for rds diagnostics is designed

the numerical solution of integral algebraic equations of index 1 by polynomial spline collocation methods in this paper we study polynomial spline collocation methods applied to a particular class of integral algebraic equations of volterra type we analyse mixed systems of second and first kind integral equations global convergence and local superconvergence results are established

conformal invariance and the ising model on a spheroid we formulate conformal mappings between an infinite plane and a spheroid and one between a semi infinite plane and a half spheroid special cases of the spheroid include the surface of an infinitely long cylinder of a sphere and of a flat disc these mappings are applied to the critical ising model for the case of the sphere and the flat disc we derive analytical expressions for the second and the fourth moments of the magnetization density and thus for the binder cumulant next we investigate ising models on spheroids and half spheroids by means of a continuous cluster monte carlo method for simulations in curved geometries fixed and free boundary conditions are imposed for half spheroids the monte carlo data are analyzed by finite size scaling critical values of the binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above at criticality we also sample two and one point correlation functions on spheroids on half spheroids respectively the magnetic and temperature scaling dimensions as determined from the monte carlo data and the theory of conformal invariance are in good agreement with exact results

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

the multi class metric problem in nearest neighbour discrimination rules abstract nearest neighbour rules classify a new data point on the basis of the class of the closest design set point a prerequisite for this is a choice of metric by which distance may be calculated metrics which have been suggested for the two class case fall into two classes global and local in this paper local metrics are described and are generalized to the multi class case properties of various possible generalizations are compared and the results of a simulation study are described

random forests and adaptive nearest neighbors in this article we study random forests through their connection with a new framework of adaptive nearest neighbor methods we introduce a concept of potential nearest neighbors k pnns and show that random forests can be viewed as adaptively weighted k pnn methods various aspects of random forests can be studied from this perspective we study the effect of terminal node sizes on the prediction accuracy of random forests we further show that random forests with adaptive splitting schemes assign weights to k pnns in a desirable way for the estimation at a given target point these random forests assign voting weights to the k pnns of the target point according to the local importance of different input variables we propose a new simple splitting scheme that achieves desirable adaptivity in a straightforward fashion this simple scheme can be combined with existing algorithms the resulting algorithm is computationally faster and gives comparable results other possible aspects of random forests such

differential algebraic equations indices and integral algebraic equations in a recent report hairer lubich and roche report ch 1211 dept de mathematiques universite de geneve geneve switzerland 1988 define the index of differential algebraic equations daes by considering the effect of perturbations of the equations on the solutions this index which will be called the perturbation index p i is one more than the number of derivatives of the perturbation that must appear in any estimate of the bound of the change in the solution an earlier form of index used by a number of authors including gear and petzold siam j numer anal 21 1984 pp 716 728 is determined by the number of differentiations of the daes that are required to generate an ordinary differential equation ode satisfied by the solution this will be called the differential index d i hairer lubich and roche give an example whose differential index is one and perturbation index is two and other examples where they are identical it will be shown that d i leqq p i leqq d i 1 and th

optimal convergence results of piecewise polynomial collocation solutions for integral algebraic equations of index 3 in this work we will consider piecewise polynomial collocation methods for a new class of integral algebraic equations of index 3 an analysis of the global convergence properties of the collocation methods is given and the optimal order of convergence results is established the performance of the proposed scheme is illustrated by means of two numerical examples and confirm that we can achieve the expected theoretical orders of convergence

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

integral algebraic equations theory of collocation methods i our analysis of collocation solutions for general systems of linear integral algebraic equations iaes is based on the notions of the tractability index and the nu smoothing property of a volterra integral operator these are used to decouple the given iae system into the inherent system of regular second kind volterra integral equations vies and a system of first kind vies this decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions numerical examples illustrate the theoretical results

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems