minimal attractors in digraph system models of neuronal networks abstract we study a class of discrete dynamical systems models of neuronal networks in these models each neuron is represented by a finite number of states and there are rules for how a neuron transitions from one state to another in particular the rules determine when a neuron fires and how this affects the state of other neurons in an earlier paper d terman s ahn x wang w just reducing neuronal networks to discrete dynamics physica d 237 2008 324 338 we demonstrate that a general class of excitatory inhibitory networks can in fact be rigorously reduced to the discrete model in the present paper we analyze how the connectivity of the network influences the dynamics of the discrete model for randomly connected networks we find two major phase transitions if the connection probability is above the second but below the first phase transition then starting in a generic initial state most but not all cells will fire at all times along the trajectory as soon as they reach the end of their refractory period above the first phase transition this will be true for all cells in a typical initial state thus most states will belong to a minimal attractor of oscillatory behavior in a sense that is defined precisely in the paper the exact positions of the phase transitions depend on intrinsic properties of the cells including the lengths of the cells refractory periods and the thresholds for firing existence of these phase transitions is both rigorously proved for sufficiently large networks and corroborated by numerical experiments on networks of moderate size

high breakdown linear discriminant analysis abstract the classification rules of linear discriminant analysis are defined by the true mean vectors and the common covariance matrix of the populations from which the data come because these true parameters are generally unknown they are commonly estimated by the sample mean vector and covariance matrix of the data in a training sample randomly drawn from each population however these sample statistics are notoriously susceptible to contamination by outliers a problem compounded by the fact that the outliers may be invisible to conventional diagnostics high breakdown estimation is a procedure designed to remove this cause for concern by producing estimates that are immune to serious distortion by a minority of outliers regardless of their severity in this article we motivate and develop a high breakdown criterion for linear discriminant analysis and give an algorithm for its implementation the procedure is intended to supplement rather than replace the usual sample moment methodology of discri

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

stability of boolean networks the joint effects of topology and update rules we study the stability of orbits in large boolean networks we treat the case in which the network has a given complex topology and we do not assume a specific form for the update rules which may be correlated with local topological properties of the network while recent past work has addressed the separate effects of complex network topology and certain classes of update rules on stability only crude results exist about how these effects interact we present a widely applicable solution to this problem numerical simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems

spatiotemporal coding in an electrochemical oscillatory network a network consisting of n relaxation electrochemical oscillators mutually coupled by all to all inhibitory connections can have n ensuremath 1 coexisting out of phase states each state being a permutation of a periodic spiking sequence the modification of the out of phase states by shots of laser pulse perturbations is shown in such networks the phase relation of the oscillators is stored as a coded pattern the ability of the network to function as a rewritable memory of n ensuremath 1 different spatiotemporal patterns is demonstrated experimentally for n 4

the effect of network topology on the stability of discrete state models of genetic control boolean networks have been proposed as potentially useful models for genetic control an important aspect of these networks is the stability of their dynamics in response to small perturbations previous approaches to stability have assumed uncorrelated random network structure real gene networks typically have nontrivial topology significantly different from the random network paradigm to address such situations we present a general method for determining the stability of large boolean networks of any specified network topology and predicting their steady state behavior in response to small perturbations additionally we generalize to the case where individual genes have a distribution of expression biases and we consider a nonsynchronous update as well as extension of our method to non boolean models in which there are more than two possible gene states we find that stability is governed by the maximum eigenvalue of a modified adjacency matrix and we test this result by comparison with numerical simulations we also discuss the possible application of our work to experimentally inferred gene networks

on connectedness a solution based on oscillatory correlation a long standing problem in neural computation has been the problem of connectedness first identified by minsky and papert 1969 this problem served as the cornerstone for them to establish analytically that perceptrons are fundamentally limited in computing geometrical topological properties a solution to this problem is offered by a different class of neural networks oscillator networks to solve the problem the representation of oscillatory correlation is employed whereby one pattern is represented as a synchronized block of oscillators and different patterns are represented by distinct blocks that desynchronize from each other oscillatory correlation emerges from legion locally excitatory globally inhibitory oscillator network whose architecture consists of local excitation and global inhibition among neural oscillators it is further shown that these oscillator networks exhibit sensitivity to topological structure which may lay a neurocomputational foundation for explaining the psychophysical phenomenon of topological perception

the random effects model in discriminant analysis abstract in this article the characteristics of classical discriminant analysis under the random effects model are investigated assuming that the elements within any randomly selected population are normally distributed with mean vector and common covariance matrix and that over different populations has a normal distribution with mean vector and covariance matrix t the distribution of the population based and sample based mahalanobis distances between two different populations as well as those between an observation and a randomly selected population are derived from these expressions and bounds are derived for the expected probabilities of misclassification under classical discriminant analysis applied to two and multiple population problems respectively when either the population based or the sample based linear discriminant functions are used

coordination complexity of parallel price directive decomposition the general block angular convex resource sharing problem in k blocks and m nonnegative block separable coupling constraints is considered applications of this model are in combinatorial optimization network flows scheduling communication networks engineering design and finance this paper studies the coordination complexity of approximate price directive decomposition pdd for this problem i e the number of iterations required to solve the problem to a fixed relative accuracy as a function of k and m first a simple pdd method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in m in the class of all pdd methods that work with the original unrestricted blocks it is then shown that logarithmic and exponential potentials generate a polylogarithmically optimal algorithm for a wider class of pdd methods which can restrict the blocks by the coupling constraints as an application the fastest currently known deterministic approximation algorithm for minimum cost multicommodity flows is obtained

approximate max min resource sharing for structured concave optimization we present a lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an varepsilon approximate solution of the general max min resource sharing problem with m nonnegative concave constraints on a convex set b we show that this algorithm runs in o m varepsilon 2 rm ln m iterations a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy varepsilon in 0 1 in the block angular case b is the product of k convex sets blocks and each constraint function is block separable for such models an iteration of our method requires a theta varepsilon approximate solution of k independent block maximization problems which can be computed in parallel

distribution of the kurtosis statistic b2 for normal samples summary d agostino pearson 1973 gave percentage points of the distribution of b2 for independent observations from a common univariate normal distribution their results can be adequately approximated when the first three moments of the distribution of b2 have been determined by fitting a linear function of the reciprocal of a x2 variable and then using the wilson hilferty transformation evidence is presented suggesting that the same method of approximation is satisfactory for the b2 statistic calculated from any set of linear least squares residuals on the hypothesis of normal homoscedastic errors

a class of invariant procedures for assessing multivariate normality summary distribution theory pertaining to a class of invariant procedures for assessing multivariate normality is described a cramer von mises type statistic belonging to this class is investigated in greater detail its asymptotic distribution is tabulated and compared with a monte carlo simulation of its finite sample distribution and a set of quadrivariate data is examined with it

a near optimal solution to a two dimensional cutting stock problem we present an asymptotic fully polynomial approximation scheme for strip packing or packing rectangles into a rectangle of fixed width and minimum height a classicalnp hard cutting stock problem the algorithm based on a new linear programming relaxation finds a packing ofn rectangles whose total height is within a factor of 1 e of optimal up to an additive term and has running time polynomial both in n and in 1 e

a test to determine the multivariate normality of a data set a test is described for multivariate normality that is useful in pattern recognition the test is based on the friedman rafsky 1979 multivariate extension of the wald wolfowitz runs test the test data are combined with a multivariate swarm of points following the normal distribution generated with mean vector and covariance matrix estimated from the test data the minimal spanning tree of this resultant ensemble of points is computed and the count of the interpopulation edges in the minimal spanning tree is used as a test statistic the simulation studied both the null case of the test and one simple deviation from normality two conclusions are made from this study first the test can be conservatively applied by using the asymptotic normality of the test statistic even for small sample sizes second the power of the test appears reasonable especially in high dimensions monte carlo experiments were performed to determine if the test is reliable in high dimensions with moderate sample size the method is compared to other such tests available in the literature

range image segmentation using a relaxation oscillator network a locally excitatory globally inhibitory oscillator network legion is constructed and applied to range image segmentation where each oscillator has excitatory lateral connections to the oscillators in its local neighborhood as well as a connection with a global inhibitor a feature vector consisting of depth surface normal and mean and gaussian curvatures is associated with each oscillator and is estimated from local windows at its corresponding pixel location a context sensitive method is applied in order to obtain more reliable and accurate estimations the lateral connection between two oscillators is established based on a similarity measure of their feature vectors the emergent behavior of the legion network gives rise to segmentation due to the flexible representation through phases our method needs no assumption about the underlying structures in image data and no prior knowledge regarding the number of regions more importantly the network is guaranteed to converge rapidly under general conditions these unique properties may lead to a real time approach for range image segmentation in machine perception

synchronization and desynchronization in a network of locally coupled wilson cowan oscillators a network of wilson cowan wc oscillators is constructed and its emergent properties of synchronization and desynchronization are investigated by both computer simulation and formal analysis the network is a 2d matrix where each oscillator is coupled only to its neighbors we show analytically that a chain of locally coupled oscillators the piecewise linear approximation to the wc oscillator synchronizes and we present a technique to rapidly entrain finite numbers of oscillators the coupling strengths change on a fast time scale based on a hebbian rule a global separator is introduced which receives input from and sends feedback to each oscillator in the matrix the global separator is used to desynchronize different oscillator groups unlike many other models the properties of this network emerge from local connections that preserve spatial relationships among components and are critical for encoding gestalt principles of feature grouping the ability to synchronize and desynchronize oscillator groups within this network offers a promising approach for pattern segmentation and figure ground segregation based on oscillatory correlation

a test of the gaussian ness of a data set using clustering the properties of the magnifying glass method of clustering are discussed these properties which include unbiased and consistent estimation of the mean for gaussian distributions and biased and inconsistent estimation of the mean for non gaussian distributions lead to the development of a technique for testing data to determine whether or not it is gaussian an example of a non gaussian distribution is given to show the sensitivity of the proposed gaussian test

real time computation at the edge of chaos in recurrent neural networks depending on the connectivity recurrent networks of simple computational units can show very different types of dynamics ranging from totally ordered to chaotic we analyze how the type of dynamics ordered or chaotic exhibited by randomly connected networks of threshold gates driven by a time varying input signal depends on the parameters describing the distribution of the connectivity matrix in particular we calculate the critical boundary in parameter space where the transition from ordered to chaotic dynamics takes place employing a recently developed framework for analyzing real time computations we show that only near the critical boundary can such networks perform complex computations on time series hence this result strongly supports conjectures that dynamical systems that are capable of doing complex computational tasks should operate near the edge of chaos that is the transition from ordered to chaotic dynamics

pixel clustering by adaptive pixel moving and chaotic synchronization in this paper a network of coupled chaotic maps for pixel clustering is proposed time evolutions of chaotic maps in the network corresponding to a pixel cluster are synchronized with each other those synchronized trajectories are desynchronized with respect to the time evolutions of chaotic maps corresponding to other pixel clusters in the same image a pixel motion mechanism is also introduced which makes each group of pixels more compact and consequently makes the model robust enough to classify ambiguous pixels another feature of the proposed model is that the number of pixel clusters does not need to be previously known

estimation of allocation rates in a cluster analysis context abstract a sample of multivariate observations is assumed to be drawn from a mixture of a given number of underlying populations the mixture likelihood approach to clustering is used to allocate each individual in the sample to its population of origin on the basis of the estimated posterior probabilities of population membership estimation of the correct allocation rate is considered for individual populations as well as for the overall mixture by averaging functions of the maximum of these posterior probabilities the estimates of the correct allocation rates provide a means of assessing the performance of the mixture approach to clustering the bootstrap technique is investigated for its effectiveness in reducing the bias of the estimates so obtained results are reported for three real data sets and a simulation study it is demonstrated that the proposed estimates generally provide useful information on the unobservable allocation rates of the mixture approach encouraging results are obtained for

a new test for multivariate normality and homoscedasticity given data x ij j 1 to n i i 1 to g one may wish to test whether the x ij are normal with common covariance matrix letting x i denote the mean of the ith group and s the pooled covariance matrix the quantities v ij x ij x i s 1 x ij x i are used in the test the exact null distribution of v ij is noted and the anderson darling statistic is proposed for testing whether the data fit this null distribution the new test has acceptable power is flexible and is easily implemented

dynamics of two mutually coupled slow inhibitory neurons abstract inhibition in oscillatory networks of neurons can have apparently paradoxical effects sometimes creating dispersion of phases sometimes fostering synchrony in the network we analyze a pair of biophysically modeled neurons and show how the rates of onset and decay of inhibition interact with the timescales of the intrinsic oscillators to determine when stable synchrony is possible we show that there are two different regimes in parameter space in which different combinations of the time constants and other parameters regulate whether the synchronous state is stable we also discuss the construction and stability of nonsynchronous solutions and the implications of the analysis for larger networks the analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow flows and fast jumps

dimension selection for feature selection and dimension reduction with principal and independent component analysis this letter is concerned with the problem of selecting the best or most informative dimension for dimension reduction and feature extraction in high dimensional data the dimension of the data is reduced by principal component analysis subsequent application of independent component analysis to the principal component scores determines the most nongaussian directions in the lower dimensional space a criterion for choosing the optimal dimension based on bias adjusted skewness and kurtosis is proposed this new dimension selector is applied to real data sets and compared to existing methods simulation studies for a range of densities show that the proposed method performs well and is more appropriate for nongaussian data than existing methods

nonlinear dynamics of pattern formation and pattern recognition in the rabbit olfactory bulb abstract a mathematical model of the process of pattern recognition in the first olfactory sensory cortex of the rabbit is presented it explains the formation and alteration of spatial patterns in neural activity observed experimentally during classical pavlovian conditioning on each inspiration of the animal a surge of receptor input enters the olfactory bulb eeg activity recorded at the surface of the bulb undergoes a transition from a low amplitude background state of temporal disorder to coherent oscillation there is a distinctive spatial pattern of rms amplitude in this oscillation which changes reliably to a second pattern during each successful recognition by the animal of a conditioned stimulus odor when a new odor is paired as conditioned stimulus these patterns are replaced by new patterns that stabilize as the animal adapts to the new environment i will argue that a unification of the theories of pattern formation and associative memory is required to account for these observations this is achieved in a model of the bulb as a discrete excitable medium with spatially inhomogeneous coupling expressed by a connection matrix the theory of multiple hopf bifurcations is employed to find coupled equations for the amplitudes of competing unstable oscillatory modes these may be created in the system by proper coupling and selectively evoked by specific classes of inputs this allows a view of limit cycle attractors as stored fixed points of a gradient vector field and thereby recovers the more familiar dynamical systems picture of associative memory

fast approximation schemes for convex programs with many blocks and coupling constraints this paper presents block coordinate descent algorithms for the approximate solution of large structured convex programming problems the constraints of such problems consist of k disjoint convex compact sets b k called blocks and m nonnegative valued convex block separable inequalities called coupling or resource constraints the algorithms are based on an exponential potential function reduction technique it is shown that feasibility as well as min mix resource sharing problems for such constraints can be solved to a relative accuracy varepsilon in o k ln m varepsilon 2 ln k iterations each of which solves k block problems to a comparable accuracy either sequentially or in parallel the same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration an extension to objective and constraint functions of arbitrary sign is also presented the above results yield fast approximatio

image segmentation based on oscillatory correlation we study image segmentation on the basis of locally excitatory globally inhibitory oscillator networks legion whereby the phases of oscillators encode the binding of pixels we introduce a lateral potential for each oscillator so that only oscillators with strong connections from their neighborhood can develop high potentials based on the concept of the lateral potential a solution to remove noisy regions in an image is proposed for legion so that it suppresses the oscillators corresponding to noisy regions but without affecting those corresponding to major regions we show that the resulting oscillator network separates an image into several major regions plus a background consisting of all noisy regions and we illustrate network properties by computer simulation the network exhibits a natural capacity in segmenting images the oscillatory dynamics leads to a computer algorithm which is applied successfully to segmenting real gray level images a number of issues regarding biological plausibility and perceptual organization are discussed we argue that legion provides a novel and effective framework for image segmentation and figure ground segregation

complexity measures of supervised classification problems we studied a number of measures that characterize the difficulty of a classification problem focusing on the geometrical complexity of the class boundary we compared a set of real world problems to random labelings of points and found that real problems contain structures in this measurement space that are significantly different from the random sets distributions of problems in this space show that there exist at least two independent factors affecting a problem s difficulty we suggest using this space to describe a classifier s domain of competence this can guide static and dynamic selection of classifiers for specific problems as well as subproblems formed by confinement projection and transformations of the feature vectors

synchronization rate of synchronized coupled systems abstract synchronization phenomena an emergent property in networks of interacting dynamical elements are widely observed in nature and have become the subject of intense research here we will investigate the synchronization rate problem in coupled limit cycle and chaotic oscillators based on the mode decomposition method and gershgorin s discs theorem some sufficient conditions for synchronization of coupled systems are obtained and a synchronization rate is then derived such a synchronization rate indicates that the error functions between state variables of underlying individual systems tend to zero in the exponential form as time tends to the infinity several numerical examples are also given to validate the theoretical results

testing for uniformity in multidimensional data testing for uniformity in multidimensional data is important in exploratory pattern analysis statistical pattern recognition and image processing the goal of this paper is to determine whether the data follow the uniform distribution over some compact convex set in k dimensional space called the sampling window we first provide a simple computationally efficient method for generating a uniformly distributed sample over a set which approximates the convex hul of the data we then test for uniformity by comparing this generated sample to the data by using friedman rafsky s minimal spanning tree mst based test experiments with both simulated and real data indicate that this mst based test is useful in deciding if data are uniform

two statistics for testing for multivariate normality malkovich afifi 1973 proposed measures of multivariate skewness and kurtosis in the present paper distributional properties of the two statistics are examined a method is given for calculating approximate percentage points of their distributions for all sample sizes greater than 8 for dimensions 2 3 and 4 and in principle for all higher dimensions the percentage points are derived from appropriate transformations of the asymptotic distributions a simulation study shows the approximations to be good for sample sizes of 25 and above

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

an approximation algorithm for the general max min resource sharing problem we propose an approximation algorithm for the general max min resource sharing problem with m nonnegative concave constraints on a convex set b the algorithm is based on a lagrangian decomposition method and it uses a c approximation algorithm called approximate block solver for a simpler maximization problem over the convex set b we show that our algorithm achieves within iterations or calls to the approximate block solver a solution for the general max min resource sharing problem with approximation ratio the algorithm is faster and simpler than the previous known approximation algorithms for the problem 12 13

weakly connected quasi periodic iscillators fm interactions and multiplexing in the brain we prove that weakly connected networks of quasi periodic multifrequency oscillators can be transformed into a phase model by a continuous change of variables the phase model has the same form as the one for periodic oscillators with the exception that each phase variable is a vector when the oscillators have mutually nonresonant frequency rotation vectors the phase model uncouples this implies that such oscillators do not interact even though there might be physical connections between them when the frequency vectors have mutual low order resonances the oscillators interact via phase deviations this mechanism resembles that of the fm radio with a shared feature multiplexing of signals possible applications to neuroscience are discussed

misclassification among methods used for multiple group discrimination the effects of distributional properties methods of multiple group discriminant analysis have not been fully studied with respect to classification into more than two populations when the covariate distributions are normal or non normal the present study examines the classification performance of several multiple discrimination methods under a variety of simulated continuous normal and non normal covariate distributions the methods include polychotomous logistic regression multiple group linear discriminant analysis kernel density estimation and rank transformations of the data as input into the linear function the parameters of interest were distance among populations configuration of population mean vectors collinear or forming the vertices of a regular simplex skewness kurtosis and bimodality simulation of the last three parameters was by log normal sinh 1 normal and a two component mixture of normal distributions respectively results with three trivariate populations show that for all distributions logistic discrimination classifies close to the optimal under neyman pearson allocation these results suggest that logistic discrimination is preferable to other widely used methods for multiple group classification with non normal data and is comparable to classification by multiple linear discrimination with normal data

synchrony and desynchrony in integrate and fire oscillators due to many experimental reports of synchronous neural activity in the brain there is much interest in understanding synchronization in networks of neural oscillators and its potential for computing perceptual organization contrary to hopfield and herz 1995 we find that networks of locally coupled integrate and fire oscillators can quickly synchronize furthermore we examine the time needed to synchronize such networks we observe that these networks synchronize at times proportional to the logarithm of their size and we give the parameters used to control the rate of synchronization inspired by locally excitatory globally inhibitory oscillator network legion dynamics with relaxation oscillators terman wang 1995 we find that global inhibition can play a similar role of desynchronization in a network of integrate and fire oscillators we illustrate that a legion architecture with integrate and fire oscillators can be similarly used to address image analysis

tail probabilities of the maxima of multilinear forms and their applications let z be a k way array consisting of independent standard normal variables for column vectors h1 hk define a multilinear form of degree k by h 1 otimes cdots otimes h k vec z we derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the hi s under the condition that the hi s are unit vectors and of its standardized statistic obtained by dividing by the norm of z we also give formulas for the maximum of a symmetric multilinear form h 1 otimes cdots otimes h k sym z where sym z denotes the symmetrization of z with respect to indices these classes of statistics are used for testing hypotheses in the analysis of variance of multiway layout data and for testing multivariate normality in order to derive the tail probabilities we employ a geometric approach developed by hotelling weyl and sun upper and lower bounds for the tail probabilities are given by reexamining sun s results some numerical examples are given to illustrate the practical usefulness of the obtained formulas including the upper and lower bounds

classification error for a very large number of classes classification error is analyzed for a situation where the number of possible classes may be on the order of a hundred or more the error associated with classifying to a single class is shown to depend mainly on average nearest neighbor distance between class means noise level and effective dimensionality of the class mean distribution and not much on other aspects of the distribution noise correlation or number of classes since single class error is large separation of classes into groups is also explored group classification error has the same properties as single class error but the size of the error is moderated by the bayes overlap between groups standard curves are provided to predict single class and group error also discussed are the effect of pattern blurring on classification error and the nearest neighbor distance statistics throughout a distribution

faster and simpler approximation algorithms for mixed packing and covering problems we propose an algorithm for approximately solving the mixed packing and covering problem given a convex compact set 0 b r n either compute x b such that f x 1 e a and g x 1 e b or decide that x b f x a g x b 0 here f g b r m are vectors whose components are m non negative convex and concave functions respectively and a b r m are constant positive vectors our algorithm requires an efficient feasibility oracle or block solver which given vectors c d r m and a r computes x b such that c tf x d tg x a or correctly decides that no such x b exists our algorithm which is based on the lagrangian or price directive decomposition method generalizes the result from k jansen approximation algorithm for the mixed fractional packing and covering problem in proceedings of 3rd ifip conference on theoretical computer science ifip tcs 2004 kluwer 2004 pp 223 236 siam journal on optimization 17 2006 331 352 and needs only o m lnm e 2ln e 1 iterations or calls to the feasibility oracle furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound

bayes error estimation using parzen and k nn procedures the use of k nearest neighbor k nn and parzen density estimates to obtain estimates of the bayes error is investigated under limited design set conditions by drawing analogies between the k nn and parzen procedures new procedures are suggested and experimental results are given which indicate that these procedures yield a significant improvement over the conventional k nn and parzen procedures we show that by varying the decision threshold many of the biases associated with the k nn or parzen density estimates may be compensated and successful error estimation may be performed in spite of these biases experimental results are given which demonstrate the effect of kernel size and shape parzen the size of k k nn and the number of samples in the design set

discriminant analysis through a semiparametric model summary we consider a semiparametric generalisation of normal theory discriminant analysis the semiparametric model assumes that after unspecified univariate monotone transformations the class distributions are multivariate normal we introduce an estimation procedure based on the distribution quantiles in which the parameters of the semiparametric model are estimated directly without estimating the nonparametric transformations the procedure is computationally fast and the estimation accuracy is shown to have the usual parametric rate the relationship between the method and more general nonparametric discriminant analysis is discussed the semiparametric specification of the class densities is a submodel of the nonparametric log density functional analysis of variance model in which the main effects are completely nonparametric but the interaction terms are specified semiparametrically simulations and real examples are used to illustrate the procedure

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

recurrent network with large representational capacity a recurrent network is proposed with the ability to bind image features into a unified surface representation within a single layer and without capacity limitations or border effects a group of cells belonging to the same object or surface is labeled with the same activity amplitude while cells in different groups are kept segregated due to lateral inhibition labeling is achieved by activity spreading through local excitatory connections in order to prevent uncontrolled spreading a separate network computes the intensity difference between neighboring locations and signals the presence of the surface boundary which constrains local excitation the quality of surface representation is not compromised due to the self excitation the model is also applied on gray level images in order to remove small noisy regions a feedforward network is proposed that computes the size of surfaces size estimation is based on the difference of dendritic inhibition in lateral excitatory and inhibitory pathways which allows the network to selectively integrate signals only from cells with the same activity amplitude when the output of the size estimation network is combined with the recurrent network good segmentation results are obtained both networks are based on biophysically realistic mechanisms such as dendritic inhibition and multiplicative integration among different dendritic branches

generalized radius processes for elliptically contoured distributions the use of mahalanobis distances has a long history in statistics given a sample of size n and general location and scatter estimators mn and n we can define generalized radii as if we wish to trim observations based on the estimators mn and n then it is natural to first remove the most remote ones i e those with the largest s with this in mind we define a process that maps the trimming proportion in 0 1 to the generalized radius of the observation that has just been removed by this level of trimming we analyze the asymptotic behavior of this process for elliptically contoured distributions we show that the limit law depends only on the elliptical family considered and how n serves to estimate the underlying scale factor through its determinant we carry out monte carlo simulations for finite sample sizes and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded

parameter estimation by ensemble kalman filters with transformed data approach and application to hydraulic tomography 1 ensemble kalman filters enkfs are a successful tool for estimating state variables in atmospheric and oceanic sciences recent research has prepared the enkf for parameter estimation in groundwater applications enkfs are optimal in the sense of bayesian updating only if all involved variables are multivariate gaussian subsurface flow and transport state variables however generally do not show gaussian dependence on hydraulic log conductivity and among each other even if log conductivity is multi gaussian to improve enkfs in this context we apply nonlinear monotonic transformations to the observed states rendering them gaussian gaussian anamorphosis ga similar ideas have recently been presented by beal et al 2010 in the context of state estimation our work transfers and adapts this methodology to parameter estimation additionally we address the treatment of measurement errors in the transformation and provide several multivariate analysis tools to evaluate the expected usefulness of ga beforehand for illustration we present a first time application of an enkf to parameter estimation from 3 d hydraulic tomography in multi gaussian log conductivity fields results show that 1 ga achieves an implicit pseudolinearization of drawdown data as a function of log conductivity and 2 this makes both parameter identification and prediction of flow and transport more accurate combining enkfs with ga yields a computationally efficient tool for nonlinear inversion of data with improved accuracy this is an attractive benefit given that linearization free methods such as particle filters are computationally extremely demanding

pattern segmentation in associative memory the goal of this paper is to show how to modify associative memory such that it can discriminate several stored patterns in a composite input and represent them simultaneously segmention of patterns takes place in the temporal domain components of one pattern becoming temporally correlated with each other and anticorrelated with the components of all other patterns correlations are created naturally by the usual associative connections in our simulations temporal patterns take the form of oscillatory bursts of activity model oscillators consist of pairs of local cell populations connected appropriately transition of activity from one pattern to another is induced by delayed self inhibition or simply by noise

synchronization in two neurons results for a two component dynamical model with time delayed inhibition abstract we study synchronization in two neurons using two dimensional neuronal models coupled through threshold delayed inhibition acting for a fixed duration t we explain the mechanism of synchronization between the two coupled units by means of phase plane analysis using a piecewise linear approximation we compute the firing times of two mutually coupled neurons which allows a quantitative study of synchronization the results are in good agreement both quantitatively and qualitatively with the numerical simulations of the full nonlinear coupled system describing the two neurons we first show that with simple inhibition only phase lagged synchronization can be achieved except for a particular set of t values we then show that coincidence synchronization can rapidly be achieved with an improved coupling model which takes into account the increase of membrane conductance during inhibition

collective dynamics in heterogeneous networks of neuronal cellular automata abstract we examine the collective dynamics of heterogeneous random networks of model neuronal cellular automata each automaton has b active states a single silent state and r b 1 refractory states and can show spiking or bursting behavior depending on the values of b we show that phase transitions that occur in the dynamical activity can be related to phase transitions in the structure of erdos renyi graphs as a function of edge probability different forms of heterogeneity allow distinct structural phase transitions to become relevant we also show that the dynamics on the network can be described by a semi annealed process and as a result can be related to the boolean lyapunov exponent

approximation algorithm for the mixed fractional packing and covering problem we propose an approximation algorithm based on the lagrangian or price directive decomposition method to compute an epsilon approximate solution of the mixed fractional packing and covering problem find x in b such that f x le 1 epsilon a g x ge 1 epsilon b where f x g x are vectors with m nonnegative convex and concave functions a and b are m dimensional nonnegative vectors and b is a convex set that can be queried by an optimization oracle we propose an algorithm that needs only o m epsilon 2 ln m epsilon 1 iterations or calls to the oracle the main contribution is that the algorithm solves the general mixed fractional packing and covering problem in contrast to pure fractional packing and covering problems and to the special mixed packing and covering problem with b rz n and runs in time independent of the so called width of the problem

majority vote cellular automata ising dynamics and p completeness we study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood these are equivalent for a restricted set of initial conditions to nonzero probability transitions in single spin flip dynamics of the ising model at zero temperature we show that in three or more dimensions these systems can simulate boolean circuits of and and or gates and are therefore p complete that is predicting their state t time steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer therefore unless a widely believed conjecture in computer science is false it is impossible even with parallel computation to predict majority vote cellular automata or zero temperature single spin flip ising dynamics qualitatively faster than by explicit simulation

on assessing multivariate normality summary we discuss procedures for assessing multivariate normality based on properties of radii and angles of the multivariate normal distribution and suggest that different procedures may be efficiently combined in order to achieve omnibus tests the assumption of population multivariate normality underlies many multivariate data analytic procedures unfortunately however there are relatively few formal methods available for assess ing the validity of this underlying assumption gnanadesikan 1977 section 5 4 2 and cox and small 1978 have extensively reviewed various formal and informal procedures proposed for testing the multinormality hypothesis and it is not our purpose to survey the procedures anew rather we suggest in this note that different characteristics of multivariate normality can be examined with different procedures which may thereafter be combined to obtain omnibus tests for the multinormality hypothesis

texture segmentation using gaussian markov random fields and neural oscillator networks we propose an image segmentation method based on texture analysis our method is composed of two parts the first part determines a novel set of texture features derived from a gaussian markov random fields gmrf model unlike a gmrf based approach our method does not employ model parameters as features or require the extraction of features for a fixed set of texture types a priori the second part is a 2d array of locally excitatory globally inhibitory oscillator networks legion after being filtered for noise suppression features are used to determine the local couplings in the network when legion runs the oscillators corresponding to the same texture tend to synchronize whereas different texture regions tend to correspond to distinct phases in simulations a large system of differential equations is solved for the first time using a recently proposed method for integrating relaxation oscillator networks we provide results on real texture images to demonstrate the performance of our method

bias of nearest neighbor error estimates the bias of the finite sample nearest neighbor nn error from its asymptotic value is examined expressions are obtained which relate the bias of the nn and 2 nn errors to sample size dimensionality metric and distributions these expressions isolate the effect of sample size from that of the distributions giving an explicit relation showing how the bias changes as the sample size is increased experimental results are given which suggest that the expressions accurately predict the bias it is shown that when the dimensionality of the data is high it may not be possible to estimate the asymptotic error simply by increasing the sample size a new procedure is suggested to alleviate this problem this procedure involves measuring the mean nn errors at several sample sizes and using our derived relationship between the bias and the sample size to extrapolate an estimate of the asymptotic nn error the results are extended to the multiclass problem the choice of an optimal metric to minimize the bias is also discussed

synchronization in networks of excitatory and inhibitory neurons with sparse random connectivity in model networks of e cells and i cells excitatory and inhibitory neurons respectively synchronous rhythmic spiking often comes about from the interplay between the two cell groups the e cells synchronize the i cells and vice versa under ideal conditions homogeneity in relevant network parameters and all to all connectivity for instance this mechanism can yield perfect synchronization we find that approximate imperfect synchronization is possible even with very sparse random connectivity the crucial quantity is the expected number of inputs per cell as long as it is large enough more precisely as long as the variance of the total number of synaptic inputs per cell is small enough tight synchronization is possible the desynchronizing effect of random connectivity can be reduced by strengthening the e i synapses more surprising it cannot be reduced by strengthening the i e synapses however the decay time constant of inhibition plays an important role faster decay yields tighter synchrony in particular in models in which the inhibitory synapses are assumed to be instantaneous the effects of sparse random connectivity cannot be seen

scaling behavior in probabilistic neuronal cellular automata we study a neural network model of interacting stochastic discrete two state cellular automata on a regular lattice the system is externally tuned to a critical point which varies with the degree of stochasticity or the effective temperature there are avalanches of neuronal activity namely spatially and temporally contiguous sites of activity a detailed numerical study of these activity avalanches is presented and single joint and marginal probability distributions are computed at the critical point we find that the scaling exponents for the variables are in good agreement with a mean field theory

generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena the integrate and fire if neuron model and the piecewise linear version of fitzhugh nagumo pl neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena although the two models are regarded to be different type we show a certain equivalence between them by deriving the coupled if model of an improved version with a firing duration from the recovery variable coupled system of the pl model under taking the limit of mu 0 without a loss of any coupling properties in the coupled if model with the duration time the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model synaptic strength decaying relaxation rate of the synaptic coupling and the parameter exhibiting the firing duration we find some irregularly synchronous behavior with or without constant firing order alternations we show that the duration of an impulse plays an important role in synchronization phenomena

missing data mechanisms and homogeneity of means and variances covariances unless data are missing completely at random mcar proper methodology is crucial for the analysis of incomplete data consequently methods for effectively testing the mcar mechanism become important and procedures were developed via testing the homogeneity of means and variances covariances across the observed patterns e g kim bentler in psychometrika 67 609 624 2002 little in j am stat assoc 83 1198 1202 1988 the current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data and the normal distribution based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random although the values may not equal those of the underlying population distribution the results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the mcar mechanism even when the population distribution is multivariate normal

bounds on the bayes classification error based on pairwise risk functions upper and lower bounds on the bayes risk for multiple composite hypothesis classification are obtained bounds on the bayes risk for m simple classes are derived in terms of the risk functions for m 1 classes and so on until the desired result depends only on the pairwise m 2 bayes risks a method of computing upper and lower bounds on the pairwise bayes risk for composite classes is developed algorithms for computing the upper and lower bounds for the general m class case and for composite hypothesis classes are presented numerical examples of the application of the bounding techniques to a problem involving the classification of aircraft are discussed results for the bounds and other performance measures are compared for the most interesting cases

alternating and synchronous rhythms in reciprocally inhibitory model neurons we study pacemaker rhythms generated by two nonoscillatory model cells that are coupled by inhibitory synapses a minimal ionic model that exhibits postinhibitory rebound pir is presented when the post synaptic conductance depends instantaneously on presynaptic potential the classical alternating rhythm is obtained using phase plane analysis we identify two underlying mechanisms release and escape for the out of phase oscillation when the postsynaptic conductance is not instantaneous but decays slowly the two cells can oscillate synchronously with no phase difference in each case different stable activity patterns can coexist over a substantial parameter range

the role of weight normalization in competitive learning the effect of different kinds of weight normalization on the outcome of a simple competitive learning rule is analyzed it is shown that there are important differences in the representation formed depending on whether the constraint is enforced by dividing each weight by the same amount divisive enforcement or subtracting a fixed amount from each weight subtractive enforcement for the divisive cases weight vectors spread out over the space so as to evenly represent typical inputs whereas for the subtractive cases the weight vectors tend to the axes of the space so as to represent extreme inputs the consequences of these differences are examined

computing the width of a set for a set of points p in three dimensional space the width of p w p is defined as the minimum distance between parallel planes of support of p it is shown that w p can be computed in o n log n i time and o n space where i is the number of antipodal pairs of edges of the convex hull of p and n is the number of vertices in the worst case i o n sup 2 for a convex polyhedra the time complexity becomes o n i if p is a set of points in the plane the complexity can be reduced to o nlog n for simple polygons linear time suffices

tests of homoscedasticity normality and missing completely at random for incomplete multivariate data test of homogeneity of covariances or homoscedasticity among several groups has many applications in statistical analysis in the context of incomplete data analysis tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random mcar these tests of mcar require large sample sizes n and or large group sample sizes n i and they usually fail when applied to non normal data hawkins 1981 proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when n i are small this paper proposes a modification of this test for complete data to improve its performance and extends its application to test of homoscedasticity and mcar when data are multivariate normal and incomplete moreover it is shown that the statistic used in the hawkins test in conjunction with a nonparametric k sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and non normal data it is explained how a combination of the proposed normal theory hawkins test and the nonparametric test can be employed to test for homoscedasticity mcar and multivariate normality simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of type i error rejection rates also a power study of the proposed tests indicates good power the proposed methods use appropriate missing data imputations to impute missing data methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups

synchronization in relaxation oscillator networks with conduction delays we study locally coupled networks of relaxation oscillators with excitatory connections and conduction delays and propose a mechanism for achieving zero phase lag synchrony our mechanism is based on the observation that different rates of motion along different nullclines of the system can lead to synchrony in the presence of conduction delays we analyze the system of two coupled oscillators and derive phase compression rates this analysis indicates how to choose nullclines for individual relaxation oscillators in order to induce rapid synchrony the numerical simulations demonstrate that our analytical results extend to locally coupled networks with conduction delays and that these networks can attain rapid synchrony with appropriately chosen nullclines and initial conditions the robustness of the proposed mechanism is verified with respect to different nullclines variations in parameter values and initial conditions

global competition and local cooperation in a network of neural oscillators abstract an architecture of locally excitatory globally inhibitory oscillator networks is proposed and investigated both analytically and by computer simulation the model for each oscillator corresponds to a standard relaxation oscillator with two time scales oscillators are locally coupled by a scheme that resembles excitatory synaptic coupling and each oscillator also inhibits other oscillators through a common inhibitor oscillators are driven to be oscillatory by external stimulation the network exhibits a mechanism of selective gating whereby an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern while preventing the other oscillators from jumping up we show analytically that with the selective gating mechanism the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks computer simulations demonstrate the model s promising ability for segmenting multiple input patterns in real time this model lays a physical foundation for the oscillatory correlation theory of feature binding and may provide an effective computational framework for scene segmentation and figure ground segregation

self organization of pulse coupled oscillators with application to clustering we introduce an efficient synchronization model that organizes a population of integrate and fire oscillators into stable and structured groups each oscillator fires synchronously with all the others within its group but the groups themselves fire with a constant phase difference the structure of the synchronized groups depends on the choice of the coupling function we show that by defining the interaction between oscillators according to the relative distance between them our model can be used as a general clustering algorithm unlike existing models our model incorporates techniques from relational and prototype based clustering methods and results in a clustering algorithm that is simple efficient robust unbiased by the size of the clusters and that can find an arbitrary number of clusters in addition to helping the model self organize into stable groups the synergy between clustering and synchronization reduces the computational complexity significantly the resulting clustering algorithm has several advantages over conventional clustering techniques in particular it can generate a nested sequence of partitions and it can determine the optimum number of clusters in an efficient manner moreover since our approach does not involve optimizing an objective function it is not sensitive to initialization and it can incorporate nonmetric similarity measures we illustrate the performance of our algorithms with several synthetic and real data sets

continuum of weakly coupled oscillatory mckean neurons the mckean model of a neuron possesses a one dimensional fast voltage like variable and a slow recovery variable a recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve s coombes 2001 phase locking in networks of synaptically coupled mckean relaxation oscillators physica d vol 160 173 188 here we use tools from coupled oscillator theory to study weakly coupled networks of mckean neurons using numerical techniques we show that the mckean system has traveling wave phase locked solutions consistent with that of a network of more biophysically detailed hodgkin huxley neurons

computing with arrays of coupled oscillators an application to preattentive texture discrimination recent experimental findings gray et al 1989 eckhorn et al 1988 seem to indicate that rapid oscillations and phase lockings of different populations of cortical neurons play an important role in neural computations in particular global stimulus properties could be reflected in the correlated firing of spatially distant cells here we describe how simple coupled oscillator networks can be used to model the data and to investigate whether useful tasks can be performed by oscillator architectures a specific demonstration is given for the problem of preattentive texture discrimination texture images are convolved with different sets of gabor filters feeding into several corresponding arrays of coupled oscillators after a brief transient the dynamic evolution in the arrays leads to a separation of the textures by a phase labeling mechanism the importance of noise and of long range connections is briefly discussed

population dynamics of spiking neurons fast transients asynchronous states and locking an integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate and fire type is discussed it is analytically shown that transients from a state of incoherent firing can be immediate the stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived the response of a population of noisy integrate and fire neurons to an input current of small amplitude is calculated and characterized by a linear filter l the stability of perfectly synchronized locked solutions is analyzed

synchronization rates in classes of relaxation oscillators relaxation oscillators arise frequently in physics electronics mathematics and biology their mathematical definitions possess a high degree of flexibility in the sense that through appropriate parameter choices relaxation oscillators can be made to exhibit qualitatively different kinds of oscillations we study numerically four different classes of relaxation oscillators through their synchronization rates in one dimensional chains with a heaviside step function interaction and obtain the following results relaxation oscillators in the sinusoidal and relaxation regime both exhibit an average time to synchrony spl sim n where n is the chain length relaxation oscillators in the singular limit exhibit spl sim n sup p where p is a numerically obtained value less than 0 5 relaxation oscillators in the singular limit with parameters modified so that they resemble spike oscillations exhibit spl sim log n in chains and spl sim log l in two dimensional square networks of length l finally using a sigmoid interaction results in spl sim n sup 2 for relaxation oscillators in the sinusoidal and relaxation regimes indicating that the form of the coupling is a controlling factor in the synchronization rate