on some convergence properties of the subspace constrained mean shift abstract subspace constrained mean shift scms is a non parametric iterative algorithm that has recently been proposed to find principal curves and surfaces based on a new definition involving the gradient and hessian of a kernel probability density estimate although simulation results using synthetic and real data have demonstrated the usefulness of the scms algorithm a rigorous study of its convergence is still missing this paper aims to take initial steps in this direction by showing that the scms algorithm inherits some important convergence properties of the mean shift ms algorithm in particular the monotonicity and convergence of the density estimate values along the sequence of output values of the algorithm is shown also it is shown that the distance between consecutive points of the output sequence converges to zero as does the projection of the gradient vector onto the subspace spanned by the d d eigenvectors corresponding to the d d largest eigenvalues of the local inverse covariance matrix these last two properties provide theoretical guarantees for stopping criteria by modifying the projection step three variations of the scms algorithm are proposed and the running times and performance of the resulting algorithms are compared

online learning of single and multivalued functions with an infinite mixture of linear experts we present a supervised learning algorithm for estimation of generic input output relations in a real time online fashion the proposed method is based on a generalized expectation maximization approach to fit an infinite mixture of linear experts imle to an online stream of data samples this probabilistic model while not fully bayesian can efficiently choose the number of experts that are allocated to the mixture this way effectively controlling the complexity of the resulting model the result is an incremental online and localized learning algorithm that performs nonlinear multivariate regression on multivariate outputs by approximating the target function by a linear relation within each expert input domain and that can allocate new experts as needed a distinctive feature of the proposed method is the ability to learn multivalued functions one to many mappings that naturally arise in some robotic and computer vision learning domains using an approach based on a bayesian generative model for the predictions provided by each of the mixture experts as a consequence it is able to directly provide forward and inverse relations from the same learned mixture model we conduct an extensive set of experiments to evaluate the proposed algorithm performance and the results show that it can outperform state of the art online function approximation algorithms in single valued regression while demonstrating good estimation capabilities in a multivalued function approximation context

dimensionality reduction for data of unknown cluster structure dimensionality reduction that preserves certain characteristics of data is needed for numerous reasons in this work we focus on data coming from a mixture of gaussian distributions and we propose a method that preserves the distinctness of the clustering structure although this structure is assumed to be yet unknown the rationale behind the method is the following i had one known the clusters classes within the data one could facilitate further analysis and reduce space dimensionality by projecting the data to the fisher s linear subspace which by definition best preserves the structure of the given classes ii under some reasonable assumptions this can be done albeit approximately without prior knowledge of the clusters classes in this paper we show how this approach works we present a method of preliminary data transformation that brings the directions of largest overall variability close to the directions of the best between class separation hence for the transformed data simple pca provides an approximation to the fisher s subspace we show that the transformation preserves the distinctness of the unknown structure in the data to a great extent

zealots tame oscillations in the spatial rock paper scissors game the rock paper scissors game is a paradigmatic model for biodiversity with applications ranging from microbial populations to human societies research has shown however that mobility jeopardizes biodiversity by promoting the formation of spiral waves especially if there is no conservation law in place for the total number of competing players first we show that even if such a conservation law applies mobility still jeopardizes biodiversity in the spatial rock paper scissors game if only a small fraction of links of the square lattice is randomly rewired secondly we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and or interaction randomness and this regardless of whether the later is quenched or annealed while even a tiny fraction of zealots brings significant benefits at 5 occupancy zealots practically destroy all oscillations regardless of the intensity of mobility and regardless of the type and strength of randomness in the interaction structure interestingly by annealed randomness the impact of zealots is qualitatively the same as by mobility which highlights that fast diffusion does not necessarily destroy the coexistence of species and that zealotry thus helps to recover the stable mean field solution our results strengthen the important role of zealots in models of cyclic dominance and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior

asymptotic theory for density ridges the large sample theory of estimators for density modes is wellunderstood in this paper we consider density ridges which are a higher dimensional extension of modes modes correspond to zerodimensional local high density regions in point clouds density ridges correspond to s dimensional local high density regions in point clouds we establish three main results first we show that under appropriate regularity conditions the local variation of the estimated ridge can be approximated by an empirical process second we show that the distribution of the estimated ridge converges to a gaussian process third we establish that the bootstrap leads to valid confidence sets for density ridges

infection promotes species coexistence rock paper scissors game with epidemic on graphs abstract we combine the rock paper scissors rps game with sis epidemic model we study the effect of infection on population dynamics of rps games with epidemic by using the metapopulation dynamic model on graphs via infection an individual r changes to i with infection rate an infected individual i returns to r with recovery rate all agents move by random walk between patches subpopulations and the rps game with epidemic is performed in each patch the dynamics are represented by the reaction diffusion equations in diffusively coupled reactors we obtain the solutions of the reaction diffusion equations for the mean field dynamics in a single patch analytically the numerical solutions are obtained for metapopulation dynamics on graphs with three nodes the analytical solutions are also obtained for metapopulation dynamics on complete graph when the recovery rate is lower than the critical value infected individuals survive the dynamics exhibit stable focuses and all species coexist infection promotes the coexistence of species

euler integration over definable functions we extend the theory of euler integration from the class of constructible functions to that of tame r valued functions definable with respect to an o minimal structure the corresponding integral operator has some unusual defects it is not a linear operator however it has a compelling morse theoretic interpretation in addition it is an advantageous setting in which to integrate in applications to diffused and noisy data in sensor networks

invasion controlled pattern formation in a generalized multispecies predator prey system rock scissors paper game as the simplest model of intransitive relation between competing agents is a frequently quoted model to explain the stable diversity of competitors in the race of surviving when increasing the number of competitors we may face a novel situation because beside the mentioned unidirectional predator prey like dominance a balanced or peer relation can emerge between some competitors by utilizing this possibility in the present work we generalize a four state predator prey type model where we establish two groups of species labeled by even and odd numbers in particular we introduce different invasion probabilities between and within these groups which results in a tunable intensity of bidirectional invasion among peer species our study reveals an exceptional richness of pattern formations where five quantitatively different phases are observed by varying solely the strength of the mentioned inner invasion the related transition points can be identified with the help of appropriate order parameters based on the spatial autocorrelation decay on the fraction of empty sites and on the variance of the species density furthermore the application of diverse alliance specific inner invasion rates for different groups may result in the extinction of the pair of species where this inner invasion is moderate these observations highlight that beyond the well known and intensively studied cyclic dominance there is an additional source of complexity of pattern formation that has not been explored earlier

finding cluster centers and sizes via multinomial parameterization the clustering problem consists in dividing a data set into groups of observations that are similar within but different across this paper presents a method for assessment the clusters centers and sizes in a non linear least squares optimization with multinomial parameterization the method is especially useful for large data sets as it operates on the summary statistics only this approach also works for the problem of finding clusters centers and sizes by the covariance matrix when the original data is not available estimation of the clusters centers and sizes can be followed by actual clustering example of application to marketing research problem is discussed

predominance of the weakest species in lotka volterra and may leonard formulations of the rock paper scissors model we revisit the problem of the predominance of the weakest species in the context of lotka volterra and may leonard formulations of a spatial stochastic rock paper scissors model in which one of the species has its predation probability reduced by 0l mathcal p w l1 we show that despite the different population dynamics and spatial patterns these two formulations lead to qualitatively similar results for the late time values of the relative abundances of the three species as a function of mathcal p w as long as the simulation lattices are sufficiently large for coexistence to prevail the weakest species generally having an advantage over the others specially over its predator however for smaller simulation lattices we find that the relatively large oscillations at the initial stages of simulations with random initial conditions may result in a significant dependence of the probability of species survival on the lattice size

clusterability assessment for gaussian mixture models there are numerous measures designed to evaluate quality of a given data grouping in terms of its distinctness and between cluster separation however there seems to be no efficient method to assess distinctness of the intrinsic structure within data clusterability before actual clustering is determined based on recent findings we propose such measure in terms of covariance matrix decomposition for appropriately transformed data the data is assumed to come from a gaussian mixture model the transformation reshapes the data so that unsupervised technique of principal component analysis is able to uncover information directly indicative of the data clusterability characteristics in this work we propose the measure and explain the motivation as well as the relation to supervised structure distinctness coefficients we also show how the measure can be applied for number of clusters and feature selection tasks

the topography of multivariate normal mixtures multivariate normal mixtures provide a flexible method of fitting high dimensional data it is shown that their topography in the sense of their key features as a density can be analyzed rigorously in lower dimensions by use of a ridgeline manifold that contains all critical points as well as the ridges of the density a plot of the elevations on the ridgeline shows the key features of the mixed density in addition by use of the ridgeline we uncover a function that determines the number of modes of the mixed density when there are two components being mixed a followup analysis then gives a curvature function that can be used to prove a set of modality theorems

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

on the relation between the linear factor model and the latent profile model the relationship between linear factor models and latent profile models is addressed within the context of maximum likelihood estimation based on the joint distribution of the manifest variables although the two models are well known to imply equivalent covariance decompositions in general they do not yield equivalent estimates of the unconditional covariances in particular a 2 class latent profile model with gaussian components underestimates the observed covariances but not the variances when the data are consistent with a unidimensional gaussian factor model in explanation of this phenomenon we provide some results relating the unconditional covariances to the goodness of fit of the latent profile model and to its excess multivariate kurtosis the analysis also leads to some useful parameter restrictions related to symmetry

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

diffusively coupled lotka volterra system stabilized by heterogeneous graphs abstract we study the effect of the network structure on the dynamic stability in the diffusively coupled lotka volterra system here we present a metapopulation model for the lotka volterra system on various graphs the total population is assumed to consist of several subpopulations nodes each individual migrates by random walk the destination of migration is randomly determined from reaction diffusion equations we obtain the population dynamics the numerical analyses are performed only for a few and characteristic values of the parameters representing typical behaviors it is found that the dynamics highly depends on network structures when a network is homogeneous the dynamics are neutrally stable each node has a periodic solution with neutral stability and the oscillations synchronize in all nodes however when a network is heterogeneous the dynamics approach stable focus and all nodes reach equilibriums with different densities hence the heterogeneity of the network induces dynamic stabilization

on choosing mixture components via non local priors choosing the number of mixture components remains an elusive challenge model selection criteria can be either overly liberal or conservative and return poorly separated components of limited practical use we formalize non local priors nlps for mixtures and show how they lead to well separated components with non negligible weight interpretable as distinct subpopulations we also propose an estimator for posterior model probabilities under local priors and nlps showing that bayes factors are ratios of posterior to prior empty cluster probabilities the estimator is widely applicable and helps to set thresholds to drop unoccupied components in overfitted mixtures we suggest default prior parameters based on multimodality for normal t mixtures and minimal informativeness for categorical outcomes we characterize theoretically the nlp induced sparsity derive tractable expressions and algorithms we fully develop normal binomial and product binomial mixtures but the theory computation and principles hold more generally we observed a serious lack of sensitivity of the bayesian information criterion insufficient parsimony of the akaike information criterion and a local prior and a mixed behaviour of the singular bayesian information criterion we also considered overfitted mixtures their performance was competitive but depended on tuning parameters under our default prior elicitation nlps offered a good compromise between sparsity and power to detect meaningfully separated components

kernels degrees of freedom and power properties of quadratic distance goodness of fit tests in this article we study the power properties of quadratic distance based goodness of fit tests first we introduce the concept of a root kernel and discuss the considerations that enter the selection of this kernel we derive an easy to use normal approximation to the power of quadratic distance goodness of fit tests and base the construction of a noncentrality index an analogue of the traditional noncentrality parameter on it this leads to a method akin to the neyman pearson lemma for constructing optimal kernels for specific alternatives we then introduce a midpower analysis as a device for choosing optimal degrees of freedom for a family of alternatives of interest finally we introduce a new diffusion kernel called the pearson normal kernel and study the extent to which the normal approximation to the power of tests based on this kernel is valid supplementary materials for this article are available online

a nonparametric statistical approach to clustering via mode identification a new clustering approach based on mode identification is developed by applying new optimization techniques to a nonparametric density estimator a cluster is formed by those sample points that ascend to the same local maximum mode of the density function the path from a point to its associated mode is efficiently solved by an em style algorithm namely the modal em mem this method is then extended for hierarchical clustering by recursively locating modes of kernel density estimators with increasing bandwidths without model fitting the mode based clustering yields a density description for every cluster a major advantage of mixture model based clustering moreover it ensures that every cluster corresponds to a bump of the density the issue of diagnosing clustering results is also investigated specifically a pairwise separability measure for clusters is defined using the ridgeline between the density bumps of two clusters the ridgeline is solved for by the ridgeline em rem algorithm an extension of mem based upon this new measure a cluster merging procedure is created to enforce strong separation experiments on simulated and real data demonstrate that the mode based clustering approach tends to combine the strengths of linkage and mixture model based clustering in addition the approach is robust in high dimensions and when clusters deviate substantially from gaussian distributions both of these cases pose difficulty for parametric mixture modeling a c package on the new algorithms is developed for public access at http www stat psu edu jiali hmac

migration difference in diffusively coupled prey predator system on heterogeneous graphs abstract when the migration rate of predators is definitely different from that of prey the metapopulation dynamics on heterogeneous graphs change greatly with migration rate we study the effect of the migration difference on the metapopulation dynamics in the diffusively coupled prey predator system on homogeneous and heterogeneous graphs we present a metapopulation model on various graphs for the prey predator system with different migration rates the total population is assumed to consist of several subpopulations patches or nodes each individual migrates by random walk the destination of migration is randomly determined the migration rates of prey and predators are represented by different diffusion constants from reaction diffusion equations with two different diffusion constants we obtain the population dynamics the numerical analyzes are performed only for a few and characteristic values of the parameters representing typical behaviors when a network is homogeneous the dynamics are neutrally stable and the equilibrium point does not depend on diffusion constants however when a network is heterogeneous the dynamics approach stable focus and the equilibrium values of prey s and predator s numbers vary with two diffusion constants it is found that the metapopulation dynamics on heterogeneous graphs depend highly on two diffusion constants of prey and predators