persistence properties of a system of coagulating and annihilating random walkers we study a d dimensional system of diffusing particles that on contact either annihilate with probability 1 q ensuremath 1 or coagulate with probability q ensuremath 2 q ensuremath 1 in one dimension the system models the zero temperature glauber dynamics of domain walls in the q state potts model we calculate overline p m t the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly m ensuremath 1 coagulations using perturbative renormalization group analysis for dl2 we show that if the number of coagulations m is much less than the typical number m t then overline p m t ensuremath sim m ensuremath zeta d t ensuremath ensuremath theta with ensuremath theta dq q q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 ensuremath zeta 2q ensuremath 1 ensuremath epsilon 2q ensuremath 1 q ensuremath 1 1 2 aq ensuremath epsilon 2 o ensuremath epsilon 3 where q q ensuremath 1 q ensuremath epsilon 2 ensuremath d and a ensuremath 0 006 dots m t is shown to scale as m t ensuremath sim t d 2 ensuremath ensuremath delta where ensuremath delta d 1 ensuremath q q ensuremath 1 q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 in two dimensions we show that overline p m t ensuremath sim mathrm ln t q 3 ensuremath 2q mathrm ln m 2q ensuremath 1 2 t ensuremath 2q for m ensuremath ll t 2q ensuremath 1 we also derive an exact nonperturbative relation between the exponents namely ensuremath delta q ensuremath theta 1 ensuremath q the one dimensional results corresponding to ensuremath epsilon 1 are compared with results from monte carlo simulations

adaptive hausdorff estimation of density level sets consider the problem of estimating the level set g x f x of an unknown d dimensional density function f based on n independent observations x 1 x n from the density this problem has been addressed under global error criteria related to the symmetric set difference however in certain applications a spatially uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere the hausdorff error criterion provides this degree of uniformity and hence is more appropriate in such situations it is known that the minimax optimal rate of error convergence for the hausdorff metric is n logn 1 d 2 for level sets with boundaries that have a lipschitz functional form where the parameter characterizes the regularity of the density around the level of interest however the estimators proposed in previous work are nonadaptive to the density regularity and require knowledge of the parameter furthermore previously developed estimators achieve the minimax optimal rate for rather restricted classes of sets e g the boundary fragment and star shaped sets that effectively reduce the set estimation problem to a function estimation problem this characterization precludes level sets with multiple connected components which are fundamental to many applications this paper presents a fully data driven procedure that is adaptive to unknown regularity conditions and achieves near minimax optimal hausdorff error control for a class of density level sets with very general shapes and multiple connected components

riccati techniques and discrete oscillations abstract oscillation and nonoscillation criteria are obtained for the second order difference equation r n x n p n x n 1 0 where x n x n 1 x n is the forward difference operator and r n p n are sequences of real numbers the main techniques involved are riccati equations of the first and second levels

density level sets asymptotics inference and visualization abstractwe study the plug in estimator for density level sets under hausdorff loss we derive asymptotic theory for this estimator and based on this theory we develop two bootstrap confidence regions for level sets we introduce a new technique for visualizing density level sets even in multidimensions which is easy to interpret and efficient to compute supplementary materials for this article are available online

generalized density clustering we study generalized density based clustering in which sharply defined clusters such as clusters on lower dimensional manifolds are allowed we show that accurate clustering is possible even in high dimensions we propose two data based methods for choosing the bandwidth and we study the stability properties of density clusters we show that a simple graph based algorithm successfully approximates the high density clusters

scale space view of curve estimation scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation the family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface the smoothing parameter here plays the same role as that played by the scale of resolution in a visual system in this paper we study in detail various features of that surface from a statistical viewpoint weak convergence of the empirical scale space surface to its theoretical counterpart and some related asymptotic results have been established under appropriate regularity conditions our theoretical analysis provides new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data in particular we have used the scale space approach for the development of an effective exploratory data analytic tool called sizer

exact results for vicious walker models of domain walls non intersecting or vicious random walker models in one dimension can be interpreted as models of domain walls in two dimensions three problems pertaining to vicious walker models are solved the first is the exact evaluation of the partition function for the random turns model of vicious walkers on a lattice in this model at each tick of the clock a randomly chosen walker must move one step to the left or one step to the right the second problem is the calculation of the mean spacing between walls in terms of the chemical potential for a brownian motion model of continuous domain walls in a strip while the final problem solved is the calculation of the correlation between defects which occur when two domain walls meet and end without crossing the whole system

factorization of disconjugate higher order sturm liouville difference operators abstract using a recently proved equivalence between disconjugacy of the 2nth order difference equation l y k n v 0 n 1 v v k v v y k n v 0 and solvability of the corresponding riccati matrix difference equation it is shown that the equation l y 0 is disconjugate on a given interval if and only if the operator l admits the factorization of the form l y k n m c k m y k k n where m and its adjoint m are certain nth order difference operators and ck is a sequence of positive numbers

stability of density based clustering high density clusters can be characterized by the connected components of a level set l x p x of the underlying probability density function p generating the data at some appropriate level 0 the complete hierarchical clustering can be characterized by a cluster tree t l in this paper we study the behavior of a density level set estimate l and cluster tree estimate t based on a kernel density estimator with kernel bandwidth h we define two notions of instability to measure the variability of l and t as a function of h and investigate the theoretical properties of these instability measures

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

smooth discrimination analysis ties f and g can be based on the estimation of the set g x fx gx we consider applications where it is appropriate to assume that the region g has a smooth boundary or belongs to another nonparametric class of sets in particular this assumption makes sense if discrimination is used as a data analytic tool decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered their rates of convergence are obtained we show that these rules achieve optimal rates for estimation of g and optimal rates of convergence for bayes risks an interesting conclusion is that the optimal rates for bayes risks can be very fast in particular faster than the parametric root n rate these fast rates cannot be guaranteed for plug in rules

bagging for path based clustering a resampling scheme for clustering with similarity to bootstrap aggregation bagging is presented bagging is used to improve the quality of path based clustering a data clustering method that can extract elongated structures from data in a noise robust way the results of an agglomerative optimization method are influenced by small fluctuations of the input data to increase the reliability of clustering solutions a stochastic resampling method is developed to infer consensus clusters a related reliability measure allows us to estimate the number of clusters based on the stability of an optimized cluster solution under resampling the quality of path based clustering with resampling is evaluated on a large image data set of human segmentations

disconjugacy criteria for matrix difference equations abstract discrete inequalities are given and applied to the matrix jacobi equation 2yt 1 ptyt 0 to develop criteria for disconjugacy that take into account the cancellation of positive and negative parts of pt

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

alternating kinetics of annihilating random walks near a free interface the kinetics of annihilating random walks in one dimension with the half line x 0 initially filled is investigated the survival probability of the nth particle from the interface exhibits power law decay with for n 1 and all odd values of n for all n even a faster decay with is observed from consideration of the eventual survival probability in a finite cluster of particles the rigorous bound is derived while a heuristic argument gives numerically this latter value appears to be a lower bound for the average position of the first particle moves to the right approximately as with a relatively sharp and asymmetric probability distribution

equivalence of discrete euler equations and discrete hamiltonian systems abstract erbe and yan recently presented a discrete linear hamiltonian system their system is a special case of the discrete hamiltonian system y n l h z n y n z n l z n l h y n y n z n l where y n 1 y n y n 1 under certain implicit solvability hypotheses these systems are equivalent to the discrete euler equation y n yn yn l r n yn yn l a reid roundabout theorem for linear recurrence relations knyn 1 bnyn ktn 1yn 1 0 is shown to imply the corresponding result obtained by erbe and yan for discrete linear hamiltonian systems furthermore discrete linear hamiltonian systems are shown to have a symplectic transition matrix

strong consistency of k means clustering a random sample is divided into the k clusters that minimise the within cluster sum of squares conditions are found that ensure the almost sure convergence as the sample size increases of the set of means of the k clusters the result is proved for a more general clustering criterion

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

the reunions of three dissimilar vicious walkers we study the behavior of three vicious random walkers which diffuse freely in one dimension witharbitrary diffusivitiesb 1 2 b 2 2 b 3 2 except that their paths may not cross the full distribution function is calculated exactly in the continuum limit the exponent 3 governing the decay of the probability r n 3 sim 1 n psi 3 of a simultaneousreunion of all three walkers aftern steps is found to varycontinuously according to psi 3 1 pi cos 1 left b 2 2 left b 1 2 b 2 2 b 2 2 b 3 2 right 1 2 right this variation has consequences for various interfacial wetting transitions in 1 1 dimensions it may also be related heuristically to the marginality of direct interface wall interactions decaying asw0 l2 in the intermediate fluctuation regime of 1 1 dimensional wetting where exponents varying continuously withw0 have recently been found

unbinding transition in a many string system a 1 1 dimensional model of n nonintersecting strings with short range attractive interactions on a lattice is solved exactly for arbitrary finite n 2 there is a second order binding unbinding transition with the same critical exponents as for n 2 in the limit n to infinity the transition becomes first order

synchronous vs asynchronous dynamics of diffusion controlled reactions abstract an analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice at each time step either both walkers diffuse simultaneously with probability p synchronous event or one of them diffuses while the other remains immobile with complementary probability asynchronous event reaction takes place through same site occupation or position exchange we study the influence of the degree of synchronicity p of the walkers and the lattice size n on the global reaction s efficiency for odd n the purely synchronous case p 1 is always the most effective one while for even n the encounter time is minimized by a combination of synchronous and asynchronous events this new parity effect is fully confirmed by monte carlo simulations on 1d lattices as well as for 2d and 3d lattices in contrast the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of p the relevance of the model for several research areas is briefly discussed

two dimensional polymers and conformal invariance abstract the critical properties of self avoiding walks percolation clusters and critical exponents associated with the planar brownian motion are reviewed in two dimensions 2d the conformal spectrum of these geometrical critical systems is made of an infinite descrete series of scaling dimensions the latter are calculated in the framework of coulomb gas methods and conformal invariance the statistical mechanics of the same critical models on a random 2d lattice i e on a random surface namely in presence of a critically fluctuating metric in the so called 2d quantum gravity is also addressed and the above critical geometrical systems are exactly solvable in this case the new gravitational conformal spectrum is related to the standard conformal spectrum in the plane as predicted recently

on nonparametric estimation of density level sets let x 1 x n be independent identically distributed observations from an unknown probability density f consider the problem of estimating the level set g g f x r 2 f x from the sample x 1 x n under the assumption that the boundary of g has a certain smoothness we propose piecewise polynomial estimators of g based on the maximization of local empirical excess masses we show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities we find also the optimal convergence rates for estimation of convex level sets a generalization to the n dimensional case where n 2 is given

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

optimal construction of k nearest neighbor graphs for identifying noisy clusters we study clustering algorithms based on neighborhood graphs on a random sample of data points the question we ask is how such a graph should be constructed in order to obtain optimal clustering results which type of neighborhood graph should one choose mutual k nearest neighbor or symmetric k nearest neighbor what is the optimal parameter k in our setting clusters are defined as connected components of the t level set of the underlying probability distribution clusters are said to be identified in the neighborhood graph if connected components in the graph correspond to the true underlying clusters using techniques from random geometric graph theory we prove bounds on the probability that clusters are identified successfully both in a noise free and in a noisy setting those bounds lead to several conclusions first k has to be chosen surprisingly high rather of the order n than of the order logn to maximize the probability of cluster identification secondly the major difference between the mutual and the symmetric k nearest neighbor graph occurs when one attempts to detect the most significant cluster only

ordering of random walks the leader and the laggard we investigate two complementary problems related to maintaining the relative positions of n random walks on the line i the leader problem that is the probability n t that the leftmost particle remains the leftmost as a function of time and ii the laggard problem the probability n t that the rightmost particle never becomes the leftmost we map these ordering problems onto an equivalent n 1 dimensional electrostatic problem from this construction we obtain a very accurate estimate for n t for n 4 the first case that is not exactly solvable 4 t t 4 with 4 0 91342 8 the probability of being the laggard also decays algebraically n t t n we derive 2 1 2 3 3 8 and argue that n n 1 ln n as n

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

consistent procedures for cluster tree estimation and pruning for a density f on mathbb r d a high density cluster is any connected component of x f x geq lambda for some lambda 0 the set of all high density clusters forms a hierarchy called the cluster tree of f we present two procedures for estimating the cluster tree given samples from f the first is a robust variant of the single linkage algorithm for hierarchical clustering the second is based on the k nearest neighbor graph of the samples we give finite sample convergence rates for these algorithms which also imply consistency and we derive lower bounds on the sample complexity of cluster tree estimation finally we study a tree pruning procedure that guarantees under milder conditions than usual to remove clusters that are spurious while recovering those that are salient

measuring mass concentrations and estimating density contour clusters an excess mass approach by using empirical process theory the so called excess mass approach is studied it can be applied to various statistical problems especially in higher dimensions such as testing for multimodality estimating density contour clusters estimating nonlinear functionals of a density density estimation regression problems and spectral analysis we mainly consider the problems of testing for multimodality and estimating density contour clusters but the other problems also are discussed the excess mass over mathbb c is defined as a supremum of a certain functional defined on mathbb c where mathbb c is a class of subsets of the d dimensional euclidean space comparing excess masses over different classes mathbb c yields information about the modality of the underlying probability measure f this can be used to construct tests for multimodality if f has a density f the maximizing sets of the excess mass are level sets or density contour clusters of f provided they lie in mathbb c the excess mass and the density contour clusters can be estimated from the data asymptotic properties of these estimators and of the test statistics are studied for general classes mathbb c including the classes of balls ellipsoids and convex sets

fully adaptive density based clustering the clusters of a distribution are often defined by the connected components of a density level set however this definition depends on the user specified level we address this issue by proposing a simple generic algorithm which uses an almost arbitrary level set estimator to estimate the smallest level at which there are more than one connected components in the case where this algorithm is fed with histogram based level set estimates we provide a finite sample analysis which is then used to show that the algorithm consistently estimates both the smallest level and the corresponding connected components we further establish rates of convergence for the two estimation problems and last but not least we present a simple yet adaptive strategy for determining the width parameter of the involved density estimator in a data depending way

fractals in two dimensions and conformal invariance abstract the exact fractal properties of critical geometrical systems in two dimensions are briefly reviewed they are generically described by an infinite discrete spectrum of scaling dimensions associated with underlying conformal operators applications involve polymers percolation brownian motion and ising clusters a seemingly essential distinction between critical phenomena spectra and multifractal ones is given

disconjugacy for linear hamiltonian difference systems abstract in this paper we study the linear hamiltonian difference system y t b t y t 1 c t z t z t a t y t 1 b t z t where a t c t are hermitian matrices with i b t and c t invertible i is identity disconjugacy criteria analogous to those for linear hamiltonian differential systems are obtained by a discrete riccati method

consistency of single linkage for high density clusters abstract high density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form x f x c single linkage clustering is evaluated for consistency in detecting such high density clusters other standard hierarchical techniques such as average and complete linkage are hopelessly inconsistent for these clusters the asymptotic consistency of single linkage closely depends on the percolation problem of broadbent and hammersley if small spheres are removed at random from a solid at which density of spheres will water begin to flow through the solid if there is a single critical density such that no flow takes place below a certain density and flow occurs through a single connected set above that density then single linkage is consistent in separating high density clusters by disjoint single linkage clusters that include a positive fraction of sample points in the respective clusters and pass arbitrarily close to all points in the respective clusters

a discrete interpretation of reid s roundabout theorem for generalized differential systems abstract in a series of papers starting with a 1959 paper in j math mechanics 1 w t reid presented sturmian theory and asymptotic behavior for generalized differential systems these systems were equivalent to a type of linear vector riemann stieltjes integral equation reid s primary result was his roundabout theorem for this generalized setting as he pointed out if the measure is piecewise constant then results for difference equations ensue the objectives of this study are 1 i to interpret reid s results for both jacobi and riccati difference equations and 2 ii to compare those results with subsequent studies of difference equations based on discrete variational theory

survival probability of a brownian motion in a planar wedge of arbitrary angle we study the survival probability and the first passage time distribution for a brownian motion in a planar wedge with infinite absorbing edges we generalize existing results obtained for wedge angles of the form ensuremath pi n with n a positive integer to arbitrary angles which in particular cover the case of obtuse angles we give explicit and simple expressions of the survival probability and the first passage time distribution in which the difference between an arbitrary angle and a submultiple of ensuremath pi is contained in three additional terms as an application we obtain the short time development of the survival probability in a wedge of arbitrary angle

field theory of branching and annihilating random walks we develop a systematic analytic approach to the problem of branching and annihilating random walks equivalent to the diffusion limited reaction processes 2a and a m 1 a where m 1 starting from the master equation a field theoretic representation of the problem is derived and fluctuation effects are taken into account via diagrammatic and renormalization group methods for d 2 the mean field rate equation which predicts an active phase as soon as the branching process is switched on applies qualitatively for both even and odd m but the behavior in lower dimensions is shown to be quite different for these two cases for even m and d near 2 the active phase still appears immediately but with nontrivial crossover exponents which we compute in an expansion in 2 d and with logarithmic corrections in d 2 however there exists a second critical dimension d c 4 3 below which a nontrivial inactive phase emerges with asymptotic behavior characteristic of the pure annihilation process this is confirmed by an exact calculation in d 1 the subsequent transition to the active phase which represents a new nontrivial dynamic universality class is then investigated within a truncated loop expansion which appears to give a correct qualitative picture the model with m 2 is also generalized to n species of particles which provides yet another universality class and which is exactly solvable in the limit n for odd m we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d 2 in this case the transition to the active phase is in the directed percolation universality class finally we study the modification when the annihilation reaction is 3a when m 0 mod 3 the system is always in its active phase but with logarithmic crossover corrections for d 1 while the other cases should exhibit a directed percolation transition out of a fluctuation driven inactive phase