cluster analysis for cognitive diagnosis theory and applications latent class models for cognitive diagnosis often begin with specification of a matrix that indicates which attributes or skills are needed for each item then by imposing restrictions that take this into account along with a theory governing how subjects interact with items parametric formulations of item response functions are derived and fitted cluster analysis provides an alternative approach that does not require specifying an item response model but does require an item by attribute matrix after summarizing the data with a particular vector of sum scores k means cluster analysis or hierarchical agglomerative cluster analysis can be applied with the purpose of clustering subjects who possess the same skills asymptotic classification accuracy results are given along with simulations comparing effects of test length and method of clustering an application to a language examination is provided to illustrate how the methods can be implemented in practice

influence of observational noise on the recurrence quantification analysis in this paper we estimate the errors due to observational noise on the recurrence quantification analysis rqa based on this estimation we present ways to minimize these errors we give a criterion to choose the threshold e needed for the optimal computation of the recurrence plot rp one important point is to show the limits of interpretability of the results of the rqa if it is applied to measured time series we show that even though the rqa is very susceptible to observational noise it can yield reliable results for an optimal choice of e if the noise level is not too high we apply the results to typical models such as white noise the logistic map and the lorenz system and to experimental laser data

consistency of spectral clustering consistency is a key property of all statistical procedures analyzing randomly sampled data surprisingly despite decades of work little is known about consistency of most clustering algorithms in this paper we investigate consistency of the popular family of spectral clustering algorithms which clusters the data with the help of eigenvectors of graph laplacian matrices we develop new methods to establish that for increasing sample size those eigenvectors converge to the eigenvectors of certain limit operators as a result we can prove that one of the two major classes of spectral clustering normalized clustering converges under very general conditions while the other unnormalized clustering is only consistent under strong additional assumptions which are not always satisfied in real data we conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering

piecewise linear skeletonization using principal curves proposes an algorithm to find piecewise linear skeletons of handwritten characters by using principal curves the development of the method was inspired by the apparent similarity between the definition of principal curves smooth curves which pass through the middle of a cloud of points and medial axes smooth curves that run equidistantly from the contours of a character image the central fitting and smoothing step of the algorithm is an extension of the polygonal line algorithm which approximates principal curves of data sets by piecewise linear curves the polygonal line algorithm is extended to find principal graphs and complemented with two steps specific to the task of skeletonization an initialization method to capture the approximate topology of the character and a collection of restructuring operations to improve the structural quality of the skeleton produced by the initialization method an advantage of our approach over existing methods is that we optimize the skeleton graph by minimizing an intuitive and explicit objective function that captures the two competing criteria of smoothing the skeleton and fitting it closely to the pixels of the character image we tested the algorithm on isolated handwritten digits and images of continuous handwriting the results indicated that the proposed algorithm can find a smooth medial axis in the great majority of a wide variety of character templates and that it substantially improves the pixel wise skeleton obtained by traditional thinning methods

classification of non parametric regression functions in longitudinal data models summary we investigate a longitudinal data model with non parametric regression functions that may vary across the observed individuals in a variety of applications it is natural to impose a group structure on the regression curves specifically we may suppose that the observed individuals can be grouped into a number of classes whose members all share the same regression function we develop a statistical procedure to estimate the unknown group structure from the data moreover we derive the asymptotic properties of the procedure and investigate its finite sample performance by means of a simulation study and a real data example

nonlinear dynamics delay times and embedding windows in order to construct an embedding of a nonlinear time series one must choose an appropriate delay time d often d is estimated using the autocorrelation function however this does not treat the nonlinearity appropriately and it may yield an incorrect value for d on the other hand the correct value of d can be found from the mutual information but this process is rather cumbersome computationally here we suggest a simpler method for estimating d using the correlation integral we call this the c c method and we test it on several nonlinear time series obtaining estimates of d in agreement with those obtained using the mutual information furthermore some researchers have suggested that one should not choose a fixed delay time d independent of the embedding dimension m but rather one should choose an appropriate value for the delay time window w d m 1 which is the total time spanned by the components of each embedded point unfortunately w cannot be estimated using the autocorrelation function or the mutual information and no standard procedure for estimating w has emerged however we show that the c c method can also be used to estimate w basically w is the optimal time for independence of the data while d is the first locally optimal time as tests we apply the c c method to the lorenz system a three dimensional irrational torus the rossler system and the rabinovich fabrikant system we also demonstrate the robustness of this method to the presence of noise c 1999 elsevier science b v all rights reserved

k means inverse regression li suggested the method of sliced inverse regression for dimension reduction in regressions with a univariate response in this article we extend that method to multivariate regressions by introducing a new way of performing the slicing this method applies for any number of response variables and may be particularly useful at the outset of an analysis before positing a multivariate model the emphasis is on application no new asymptotic theory is presented

the diastatic exponential of a symmetric space let m g be a real analytic kahler manifold we say that a smooth map expp w m from a neighbourhood w of the origin of tpm into m is a diastatic exponential at p if it satisfies begin array lll left d rm exp p right 0 rm id t pm d p left rm exp p left v right right g p left v v right forall v in w end array where dp is calabi s diastasis function at p the usual exponential expp obviously satisfied these equations when dp is replaced by the square of the geodesics distance from p in this paper we prove that for every point p of an hermitian symmetric space of noncompact type m there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks an analogous result holds true in an open dense neighbourhood of every point of m the compact dual of m we also provide a geometric interpretation of the symplectic duality map recently introduced in di scala and loi adv math 217 2336 2352 2008 in terms of diastatic exponentials as a byproduct of our analysis we show that the symplectic duality map pulls back the reproducing kernel of m to the reproducing kernel of m

robustness properties of k means and trimmed k means abstract the generalized k means method is based on the minimization of the discrepancy between a random variable or a sample of this random variable and a set with k points measured through a penalty function as in the m estimators setting k 1 a penalty function with unbounded derivative naturally leads to nonrobust generalized k means however surprisingly the lack of robustness extends also to the case of bounded that is generalized k means do not inherit the robustness properties of the m estimator from which they came attempting to robustify the generalized k means method the generalized trimmed k means method arises from combining k means idea with a so called impartial trimming procedure in this article study generalized k means and generalized trimmed k means performance from the viewpoint of hampel s robustness criteria that is we investigate the influence function breakdown point and qualitative robustness confirming the superiority provided by the trimming we inc

a central limit theorem for multivariate generalized trimmed k means a central limit theorem for generalized trimmed k means is obtained in a very general framework that covers the multivariate setting general penalty functions and general k 1 several applications including the location estimator case k 1 for elliptical distributions and the construction of multivariate not necessarily connected tolerance zones are also given

a versatile technique for the optimal approximation of random processes by functional quantization this paper presents the mathematical foundation of a novel and versatile technique for the approximation of random processes using the functional quantization concept in this approach random processes are represented by optimal quantizers that is finite collections of deterministic functions and corresponding probability masses that are carefully constructed to attain certain optimal properties although the computational cost of obtaining such optimal quantizers is not negligible their use in applications is simple and relatively inexpensive the paper has two objectives first we present an accessible overview of functional quantization theory second we introduce a novel methodology to compute optimal quantizers which is based on the classical lloyd s method and monte carlo simulation for validation purposes the proposed methodology is tested using two fundamental gaussian random processes brownian motion and fractional brownian motion for which optimal quantizers are known the results obtained compare very well with previous results reported in the literature then to demonstrate the unique versatility of the methodology the application to a non gaussian process is presented

adaptive noisy clustering the problem of adaptive noisy clustering is investigated given a set of noisy observations z i x i epsilon i i 1 ldots n the goal is to design clusters associated with the law of x i s with unknown density f with respect to the lebesgue measure since we observe a corrupted sample a direct approach as the popular k means is not suitable in this case in this paper we propose a noisy k means minimization which is based on the k means loss function and a deconvolution estimator of the density f in particular this approach suffers from the dependence on a bandwidth involved in the deconvolution kernel fast rates of convergence for the excess risk are proposed for a particular choice of the bandwidth which depends on the smoothness of the density f then we turn out into the main issue of this paper the data driven choice of the bandwidth we state an adaptive upper bound using a modified version of lespki s method called empirical risk comparison where empirical risks associated with different bandwidths are compared eventually we illustrate that the selection rule can be used in many statistical problems of m estimation where the empirical risk depends on a nuisance parameter

continuum limit of total variation on point clouds we consider point clouds obtained as random samples of a measure on a euclidean domain a graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect our goal is to develop mathematical tools needed to study the consistency as the number of available data points increases of graph based machine learning algorithms for tasks such as clustering in particular we study when the cut capacity and more generally total variation on these graphs is a good approximation of the perimeter total variation in the continuum setting we address this question in the setting of convergence we obtain almost optimal conditions on the scaling as the number of points increases of the size of the neighborhood over which the points are connected by an edge for the convergence to hold taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds

approximate confidence intervals for the number of clusters abstract we consider clustering for the purpose of data reduction similar objects are grouped together in clusters so that one can then work with the few cluster descriptors instead of the many data points the quality of any given clustering is measured by a loss function that takes into account both the parsimony of the clustering and the loss of information due to clustering an optimal clustering can be obtained by minimizing the theoretical loss function it is shown that a sample version of the loss function and optimal clustering converge strongly to their theoretical counterparts as the sample size tends to infinity we then develop a bootstrap based procedure for obtaining approximate confidence bounds on the number of clusters in the best clustering the effectiveness of this procedure is evaluated in a simulation study an application is presented

constraint local principal curve existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice since these algorithms are mainly based on global optimization in this paper we present the concept of constraint local principal curve clpc which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity in addition we propose three clpc algorithms by local optimization and adaptive radius to expand the range of applications and increase the solution quality the first algorithm i e clpcg is based on greedy thinking the second algorithm i e clpcs uses one dimensional search and the last algorithm clpcc combines the greedy thinking and one dimensional search then we define six performance indices to evaluate the performance of the clpc algorithms finally we present some numerical experiments with three simulation data sets and two gps measured data sets in both highway and railway the results indicate that all of the three clpc algorithms can obtain high accuracy data from multiple low accuracy data efficiently the clpc algorithms can improve the accuracy and computational speed compared with the existing k segment principal curve kpc algorithm in addition clpcc outperforms clpcg and clpcs according to the comprehensive experiments while clpcg runs much faster than other ones we present the concept of constraint local principal curve clpc to reduce the computational complexity we propose three clpc algorithms that combine local optimization and adaptive radius to expand the range of applications and increase the solution quality we present some numerical experiments with three simulation data sets and two measured gps data sets in highway and railway the clpc algorithms can improve the accuracy and computational speed compared with the existing kpc algorithms the features of the each clpc algorithm are analyzed according to the comprehensive experiments

sharp asymptotics of the functional quantization problem for gaussian processes the sharp asymptotics for the l 2 quantization errors of gaussian measures on a hilbert space and in particular for gaussian processes is derived the condition imposed is regular variation of the eigenvalues

holomorphic isometries of mathbb b m into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals we study general properties of images of holomorphic isometric embeddings of complex unit balls mathbb b m into irreducible bounded symmetric domains varomega of rank at least 2 in particular we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections varlambda of the compact dual x c of varomega the question naturally arises as to which linear sections z varlambda cap varomega are actually images of holomorphic isometries of complex unit balls we study the latter question in the case of bounded symmetric domains varomega of type iv alias lie balls i e bounded symmetric domains dual to hyperquadrics we completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains varomega especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by mok proc am math soc 144 4515 4525 2016 from varieties of minimal rational tangents and that moreover any holomorphic isometric embedding f mathbb b m rightarrow varomega extends to a holomorphic isometric embedding f mathbb b n 1 rightarrow varomega dim varomega n the case of lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type iv domain are expected to be more rigid

on the path density of a gradient field we consider the problem of reliably finding filaments in point clouds realistic data sets often have numerous filaments of various sizes and shapes statistical techniques exist for finding one or a few filaments but these methods do not handle noisy data sets with many filaments other methods can be found in the astronomy literature but they do not have rigorous statistical guarantees we propose the following method starting at each data point we construct the steepest ascent path along a kernel density estimator we locate filaments by finding regions where these paths are highly concentrated formally we define the density of these paths and we construct a consistent estimator of this path density

similarity preserving principal curve an optimal 1 d feature extractor for data representation this paper discusses the problem of what kind of learning model is suitable for the tasks of feature extraction for data representation and suggests two evaluation criteria for nonlinear feature extractors reconstruction error minimization and similarity preservation based on the suggested evaluation criteria a new type of principal curve similarity preserving principal curve sppc is proposed sppcs minimize the reconstruction error under the condition that the similarity between similar samples are preserved in the extracted features thus giving researchers effective and reliable cognition of the inner structure of data sets the existence and properties of sppcs are analyzed a practical learning algorithm is proposed and high dimensional extensions of sppcs are also discussed experimental results show the virtues of sppcs in preserving inner structures of data sets and discovering manifolds with high nonlinearity

clustering functional data into groups by using projections we show that in the functional data context by appropriately exploiting the functional nature of the data it is possible to cluster the observations asymptotically perfectly we demonstrate that this level of performance can sometimes be achieved by the k means algorithm as long as the data are projected on a carefully chosen finite dimensional space in general the notion of an ideal cluster is not clearly defined we derive our results in the setting where the data come from two populations whose distributions differ at least in terms of means and where an ideal cluster corresponds to one of these two populations we propose an iterative algorithm to choose the projection functions in a way that optimizes clustering performance where to avoid peculiar solutions we use a weighted least squares criterion we apply our iterative clustering procedure on simulated and real data where we show that it works well

termination and continuity of greedy growing for tree structured vector quantizers tree structured vector quantizers tsvq provide a computationally efficient variable rate method of compressing vector valued data in applications the problem of designing a tsvq from empirical training data is critical greedy growing algorithms are a common and effective approach to the design problem they are recursive procedures that produce a tsvq one node at a time by optimizing a simple splitting criterion at each step while unsupervised greedy growing algorithms are well understood from an experimental point of view there has been little theory to support their use or to examine their behavior on large training sets the authors present a rigorous analysis of a greedy growing algorithm proposed by riskin 1990 riskin and gray 1991 and balakrishnan 1991 the first part of the paper is a description of the algorithm and an examination of its asymptotic behavior as it applies to a fixed absolutely continuous distribution the second part of the paper establishes the structural consistency of the algorithm with respect to a convergent sequence of distributions as an application the authors obtain results concerning the large sample empirical behavior of the algorithm when it is applied to stationary ergodic training vectors

a space quantization method for numerical integration abstract we propose a new method sqm for numerical integration of c functions 0 2 defined on a convex subset c of rd with respect to a continuous distribution it relies on a space quantization of c by a n tuplex x1 xn cn fd is approximated by a weighted sum of the f xi s the integration error bound depends on the distortion en x of the voronoi tessellation of x this notion comes from information theoretists its main properties existence of a minimizing n tuple in cn asymptotics of mincn en as n are presented for a wide class of measures a simple stochastic optimization procedure is proposed to compute in any dimension d x and the characteristics of its voronoi tessellation some new results on the competitive learning vector quantization algorithm when 2 are obtained as a by product some tests simulations and provisional remarks are proposed as a conclusion

quantization based on a novel sample adaptive product quantizer sapq in this paper we propose a novel feedforward adaptive quantization scheme called the sample adaptive product quantizer sapq this is a structurally constrained vector quantizer that uses unions of product codebooks sapq is based on a concept of adaptive quantization to the varying samples of the source and is very different from traditional adaptation techniques for nonstationary sources sapq quantizes each source sample using a sequence of quantizers even when using scalar quantization in sapq we can achieve performance comparable to vector quantization with the complexity still close to that of scalar quantization we also show that important lattice based vector quantizers can be constructed using scalar quantization in sapq we mathematically analyze sapq and propose a algorithm to implement it we numerically study sapq for independent and identically distributed gaussian and laplacian sources through our numerical study we find that sapq using scalar quantizers achieves typical gains of 13 db in distortion over the lloyd max quantizer we also show that sapq can he used in conjunction with vector quantizers to further improve the gains

learning and design of principal curves principal curves have been defined as self consistent smooth curves which pass through the middle of a d dimensional probability distribution or data cloud they give a summary of the data and also serve as an efficient feature extraction tool we take a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution the new definition makes it possible to theoretically analyze principal curve learning from training data and it also leads to a new practical construction our theoretical learning scheme chooses a curve from a class of polygonal lines with k segments and with a given total length to minimize the average squared distance over n training points drawn independently convergence properties of this learning scheme are analyzed and a practical version of this theoretical algorithm is implemented in each iteration of the algorithm a new vertex is added to the polygonal line and the positions of the vertices are updated so that they minimize a penalized squared distance criterion simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity and is more robust to varying data models

the strong law of large numbers for sequential decisions under uncertainty combines optimization and ergodic theory to characterize the optimum long run average performance that can be asymptotically attained by nonanticipating sequential decisions let x sub t be a stationary ergodic process and suppose an action b sub t must be selected in a space spl bscr with knowledge of the t past x sub 0 spl middot spl middot spl middot x sub t 1 at the beginning of every period t spl ges 0 action b sub t will incur a loss l b sub t x sub t at the end of period t when the random variable x sub t is revealed the author proves under mild integrability conditions that the optimum strategy is to select actions that minimize the conditional expected loss given the currently available information at each step the minimum long run average loss per decision can be approached arbitrarily closely by strategies that are finite order markov and under certain continuity conditions it is equal to the minimum expected loss given the infinite past if the loss l b x is bounded and continuous and if the space spl bscr is compact then the minimum can be asymptotically attained even if the distribution of the process x sub t is unknown a priori and must be learned from experience

on the bernstein von mises theorem with infinite dimensional parameters if there are many independent identically distributed observations governed by a smooth finite dimensional statistical model the bayes estimate and the maximum likelihood estimate will be close furthermore the posterior distribution of the parameter vector around the posterior mean will be close to the distribution of the maximum likelihood estimate around truth thus bayesian confidence sets have good frequentist coverage properties and conversely however even for the simplest infinite dimensional models such results do not hold the object here is to give some examples

reconstruction of dynamical and geometrical properties of chaotic attractors from threshold crossing interspike intervals we reconstruct the largest lyapunov exponent and fractal dimension of a chaotic attractor using thresholdcrossing interspike intervals alone we show that in certain cases one may reconstruct from this data a set looking very similar to the initial attractor we also give an explanation of this possibility based on the concept of instantaneous frequency s1063 651x 98 51207 x pacs number s 05 45 1b the problem of extracting information about a dynamical system from experimental data is of great importance for all those who study the systems of natural origin that may be thought of as black boxes the traditional way of obtaining experimental data is to fix the sampling stepdt and store the values of some state variable s t corresponding to the time moments idt the fundamental works 1 state that the delay method applied to such time series often allows one to reconstruct a set being topologically equivalent to the initial system attractor on the base of these works a series of methods have been developed allowing one to evaluate attractor dimension 2 and the largest lyapunov characteristic exponent lce 3 to detect unstable equilibrium points 4 and unstable periodic orbits 5 and finally to create in some cases global models 6 or different predictors 7 another way of getting experimental data which is often preferred by biologists is to generate spike train data 8 one imposes a certain condition on the value of state variable s t and records the interspike intervals isi ti between the time moments for which the condition is fulfilled there are two general methods for recording spike train data the first one is to generate a spike when s t crosses some threshold level u in one direction and measure isi as an alternative one may measure the time intervals between the local maxima of s t fig 1 a the values of ti can be treated as the times of trajectory return to the surface defined by the condition imposed to the initial s t the condition s t 5u means the definition of a plane in the phase space while the condition s t 50 s t being a realization x j t o f a dynamical system dx w dt5f

self organization as an iterative kernel smoothing process kohonen s self organizing map when described in a batch processing mode can be interpreted as a statistical kernel smoothing problem the batch som algorithm consists of two steps first the training data are partitioned according to the voronoi regions of the map unit locations second the units are updated by taking weighted centroids of the data falling into the voronoi regions with the weighing function given by the neighborhood then the neighborhood width is decreased and steps 1 2 are repeated the second step can be interpreted as a statistical kernel smoothing problem where the neighborhood function corresponds to the kernel and neighborhood width corresponds to kernel span to determine the new unit locations kernel smoothing is applied to the centroids of the voronoi regions in the topological space this interpretation leads to some new insights concerning the role of the neighborhood and dimensionality reduction it also strengthens the algorithm s connection with the principal curve algorithm a generalized self organizing algorithm is proposed where the kernel smoothing step is replaced with an arbitrary nonparametric regression method

nearest neighbor clustering a baseline method for consistent clustering with arbitrary objective functions clustering is often formulated as a discrete optimization problem the objective is to find among all partitions of the data set the best one according to some quality measure however in the statistical setting where we assume that the finite data set has been sampled from some underlying space the goal is not to find the best partition of the given sample but to approximate the true partition of the underlying space we argue that the discrete optimization approach usually does not achieve this goal and instead can lead to inconsistency we construct examples which provably have this behavior as in the case of supervised learning the cure is to restrict the size of the function classes under consideration for appropriate small function classes we can prove very general consistency theorems for clustering optimization schemes as one particular algorithm for clustering with a restricted function space we introduce nearest neighbor clustering similar to the k nearest neighbor classifier in supervised learning this algorithm can be seen as a general baseline algorithm to minimize arbitrary clustering objective functions we prove that it is statistically consistent for all commonly used clustering objective functions

functional quantization of gaussian processes quantization consists in studying the lr error induced by the approximation of a random vector x by a vector quantized version taking a finite number n of values for rm valued random vectors the theory and practice is quite well established and in particular the asymptotics as n of the resulting minimal quantization error for nonsingular distributions is well known it behaves like c x r m n 1 m this paper is a transposition of this problem to random vectors in an infinite dimensional hilbert space and in particular to stochastic processes xt t 0 1 viewed as l2 0 1 dt valued random vectors for gaussian vectors and the l2 error we present detailed results for stationary and optimal quantizers we further establish a precise link between the rate problem and shannon kolmogorov s entropy of x this allows us to compute the exact rate of convergence to zero of the minimal l2 quantization error under rather general conditions on the eigenvalues of the covariance operator typical rates are o logn a a 0 they are obtained for instance for the fractional brownian motion and the fractional ornstein uhlenbeck process the exponent a is closely related with the l2 regularity of the process

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

individual convergence rates in empirical vector quantizer design we consider the rate of convergence of the expected distortion redundancy of empirically optimal vector quantizers earlier results show that the mean squared distortion of an empirically optimal quantizer designed from n independent and identically distributed i i d source samples converges uniformly to the optimum at a rate of o 1 spl radic n and that this rate is sharp in the minimax sense we prove that for any fixed distribution supported on a given finite set the convergence rate is o 1 n faster than the minimax lower bound where the corresponding constant depends on the source distribution for more general source distributions we provide conditions implying a little bit worse o logn n rate of convergence although these conditions in general are hard to verify we show that sources with continuous densities satisfying certain regularity properties similar to the ones of pollard that were used to prove a central limit theorem for the code points of the empirically optimal quantizers are included in the scope of this result in particular scalar distributions with strictly log concave densities with bounded support such as the truncated gaussian distribution satisfy these conditions

spherical principal curves this paper presents a new approach for dimension reduction of data observed on spherical surfaces several dimension reduction techniques have been developed in recent years for non euclidean data analysis as a pioneer work hauberg 2016 attempted to implement principal curves on riemannian manifolds however this approach uses approximations to process data on riemannian manifolds resulting in distorted results this study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces our approach lies in the same line of hastie and stuetzle 1989 that proposed principal curves for data on euclidean space we further investigate the stationarity of the proposed principal curves that satisfy the self consistency on spherical surfaces the results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach

a new algorithm and theory for penalized regression based clustering clustering is unsupervised and exploratory in nature yet it can be performed through penalized regression with grouping pursuit as demonstrated in pan et al 2013 in this paper we develop a more efficient algorithm for scalable computation and a new theory of clustering consistency for the method this algorithm called dc admm combines difference of convex dc programming with the alternating direction method of multipliers admm this algorithm is shown to be more computationally efficient than the quadratic penalty based algorithm of pan et al 2013 because of the former s closed form updating formulas numerically we compare the dc admm algorithm with the quadratic penalty algorithm to demonstrate its utility and scalability theoretically we establish a finite sample mis clustering error bound for penalized regression based clustering with the l0 constrained regularization in a general setting on this ground we provide conditions for clustering consistency of the penalized clustering method as an end product we put r package prclust implementing prclust with various loss and grouping penalty functions available on github and cran

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

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

self consistency and principal component analysis abstract i examine the self consistency of a principal component axis that is when a distribution is centered about a principal component axis a principal component axis of a random vector x is self consistent if each point on the axis corresponds to the mean of x given that x projects orthogonally onto that point a large class of symmetric multivariate distributions are examined in terms of self consistency of principal component subspaces elliptical distributions are characterized by the preservation of self consistency of principal component axes after arbitrary linear transformations a lack of fit test is proposed that tests for self consistency of a principal axis the test is applied to two real datasets

bootstrapping for significance of compact clusters in multidimensional datasets this article proposes a bootstrap approach for assessing significance in the clustering of multidimensional datasets the procedure compares two models and declares the more complicated model a better candidate if there is significant evidence in its favor the performance of the procedure is illustrated on two well known classification datasets and comprehensively evaluated in terms of its ability to estimate the number of components via extensive simulation studies with excellent results the methodology is also applied to the problem of k means color quantization of several standard images in the literature and is demonstrated to be a viable approach for determining the minimal and optimal numbers of colors needed to display an image without significant loss in resolution additional illustrations and performance evaluations are provided in the online supplementary material

symplectic duality of symmetric spaces let be a complex n dimensional hermitian symmetric space endowed with the hyperbolic form hyp denote by m fs the compact dual of m hyp where fs is the fubini study form on m our first result is theorem 1 1 where with the aid of the theory of jordan triple systems we construct an explicit symplectic duality namely a diffeomorphism satisfying and for the pull back of m where 0 is the restriction to m of the flat kahler form of the hermitian positive jordan triple system associated to m amongst other properties of the map m we also show that it takes complete complex and totally geodesic submanifolds of m through the origin to complex linear subspaces of as a byproduct of the proof of theorem 1 1 we get an interesting characterization theorem 5 3 of the bergman form of a hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin

clustering based on pairwise distances when the data is of mixed dimensions in the context of clustering we consider a generative model in a euclidean ambient space with clusters of different shapes dimensions sizes and densities in an asymptotic setting where the number of points becomes large we obtain theoretical guaranties for some emblematic methods based on pairwise distances a simple algorithm based on the extraction of connected components in a neighborhood graph hierarchical clustering with single linkage and the spectral clustering method of ng jordan and weiss the methods are shown to enjoy some near optimal properties in terms of separation between clusters and robustness to outliers the local scaling method of zelnik manor and perona is shown to lead to a near optimal choice for the scale in the first and third methods we also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method

robust estimation of mixture complexity in many applications it is important to find the mixture with fewest number of components known as the mixture complexity that provides a satisfactory fit to the data this article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification we treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a hellinger information criterion this estimator is shown to be consistent for any parametric family of mixtures when the model is correctly specified monte carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity the basic construction being firmly rooted in the minimum hellinger distance approach

a central limit theorem for k means clustering a set of n points in euclidean space is partitioned into the k groups that minimize the within groups sum of squares under the assumption that the n points come from independent sampling on a fixed distribution conditions are found to assure asymptotic normality of the vector of means of the k groups the method of proof makes novel application of a functional central limit theorem for empirical processes a generalization of donsker s theorem due to dudley

principal points and self consistent points of elliptical distributions the k principal points of a p variate random vector x are those points 1 k r p which approximate the distribution of x by minimizing the expected squared distance of x from the nearest of the j any set of k points y 1 y k partitions r p into domains of attraction d 1 d k according to minimal distance following hastie and stuetzle we call y 1 y k self consistent if e xx d j y j for j 1 k principal points are a special case of self consistent points in this paper we study principal points and self consistent points of p variate elliptical distributions the main results are the following 1 if k self consistent points of x span a subspace of dimension q p then this subspace is also spanned by q principal components that is self consistent points of elliptical distributions exist only in principal component subspaces 2 the subspace spanned by k principal points of x is identical with the subspace spanned by the principal components associated with the largest roots this proves a conjecture of flury we also discuss implications of our results for the computation and estimation of principal points

improved minimax bounds on the test and training distortion of empirically designed vector quantizers it has been shown by earlier results that the minimax expected test distortion redundancy of empirical vector quantizers with three or more levels designed from n independent and identically distributed i i d data points is at least omega 1 radicn for the class of distributions on a bounded set in this correspondence a much simpler construction and proof for this are given with much better constants there are similar bounds for the training distortion of the empirically optimal vector quantizer with three or more levels these rates however do not hold for a one level quantizer here the two level quantizer case is clarified showing that it already shares the behavior of the general case given that the minimax bounds are proved using a construction that involves discrete distributions one suspects that for the class of distributions with uniformly bounded continuous densities the expected distortion redundancy might decrease as o 1 radicn uniformly it is shown as well that this is not so proving that the lower bound for the expected test distortion remains true for these subclasses

optimal approach for fast object template matching this paper proposes a novel algorithm for an optimal reduction of object description for object matching purposes our aim is to decrease the computation needs by considering simplified objects thus reducing the number of pixels involved in the matching process we develop the appropriate theoretical background based on centroidal voronoi tessellations its use within the chamfer matching framework is also discussed we present experimental results regarding the performance of this approach for 2 d contour and region like object matching as a special case we investigate how the snake based representation of target objects can be employed in chamfer matching the experimental results concern the use of object part matching for recognizing humans and show how the proposed simplification leads to valid replacements of the original templates

empirical risk minimization for heavy tailed losses the purpose of this paper is to discuss empirical risk minimization when the losses are not necessarily bounded and may have a distribution with heavy tails in such situations usual empirical averages may fail to provide reliable estimates and empirical risk minimization may provide large excess risk however some robust mean estimators proposed in the literature may be used to replace empirical means in this paper we investigate empirical risk minimization based on a robust estimate proposed by catoni we develop performance bounds based on chaining arguments tailored to catoni s mean estimator

trimmed k means an attempt to robustify quantizers a class of procedures based on impartial trimming self determined by the data is introduced with the aim of robustifying k means hence the associated clustering analysis we include a detailed study of optimal regions showing that only nonpathological regions can arise from impartial trimming procedures the asymptotic results provided in the paper focus on strong consistency of the suggested methods under widely general conditions a section is devoted to exploring the performance of the procedure to detect anomalous data in simulated data sets

numerical studies of macqueen s k means algorithm for computing the centroidal voronoi tessellations abstract we study a probabilistic algorithm for the computation of the centroidal voronoi tessellation which is a voronoi tessellation of a given set such that the associated generating points are centroids centers of mass of the corresponding voronoi regions we discuss various issues related to the implementation of the algorithm and provide numerical results some measures to improve the performance are also presented

sample weighted clustering methods although there have been many researches on cluster analysis considering feature or variable weights little effort has been made regarding sample weights in clustering in practice not every sample in a data set has the same importance in cluster analysis therefore it is interesting to obtain the proper sample weights for clustering a data set in this paper we consider a probability distribution over a data set to represent its sample weights we then apply the maximum entropy principle to automatically compute these sample weights for clustering such method can generate the sample weighted versions of most clustering algorithms such as k means fuzzy c means fcm and expectation maximization em etc the proposed sample weighted clustering algorithms will be robust for data sets with noise and outliers furthermore we also analyze the convergence properties of the proposed algorithms this study also uses some numerical data and real data sets for demonstration and comparison experimental results and comparisons actually demonstrate that the proposed sample weighted clustering algorithms are effective and robust clustering methods

incremental nonlinear dimensionality reduction by manifold learning understanding the structure of multidimensional patterns especially in unsupervised cases is of fundamental importance in data mining pattern recognition and machine learning several algorithms have been proposed to analyze the structure of high dimensional data based on the notion of manifold learning these algorithms have been used to extract the intrinsic characteristics of different types of high dimensional data by performing nonlinear dimensionality reduction most of these algorithms operate in a batch mode and cannot be efficiently applied when data are collected sequentially in this paper we describe an incremental version of isomap one of the key manifold learning algorithms our experiments on synthetic data as well as real world images demonstrate that our modified algorithm can maintain an accurate low dimensional representation of the data in an efficient manner

clustering massive datasets with application in software metrics and tomography clustering datasets is not an easy problem in general and the difficulty is compounded for massive datasets this article develops under gaussian assumptions a multistage algorithm that clusters an initial sample filters out observations that can be reasonably classified by these clusters and iterates the preceding procedure on the remainder a final step uses the estimated class probabilities and dispersions to classify each observation in the dataset results on test experiments indicate good performance application to datasets from software metrics and positron emission tomography required no more than five stages each suggesting that the procedure is practical to implement

quantization and the method of k means asymptotic results from the statistical theory of k means clustering are applied to problems of vector quantization the behavior of quantizers constructed from long training sequences of data is analyzed by relating it to the consistency problem for k means

chi square diagnostic tests for econometric models introduction and applications abstract this paper extends the pearson chi square testing method to non dynamic parametric econometric models in particular models with covariates the paper introduces the test and discusses a variety of applications these applications include testing for goodness of fit of a parametric model as well as testing particular aspects of a parametric model the cells used to construct the test statistic may be random and can be specified in a general form

lagrangian empirical design of variable rate vector quantizers consistency and convergence rates the lagrangian formulation of variable rate vector quantization is known to yield useful necessary conditions for quantizer optimality and generalized lloyd algorithms for quantizer design the lagrangian formulation is demonstrated to provide a convenient framework for analyzing the empirical design of variable rate vector quantizers in particular the consistency of empirical design based on minimizing the lagrangian performance over a stationary and ergodic training sequence is shown for sources with finite second moment the finite sample performance is also studied for independent training data and sources with bounded support

on testing trained vector quantizer codebooks this paper discusses a criterion for testing a vector quantizer vq codebook that is obtained by training when a vq codebook is designed by a clustering algorithm using a training set time average distortion which is called the training set distortion tsd is usually calculated in each iteration of the algorithm since the input probability function is unknown in general and cumbersome to deal with the algorithm stops when the tsd ceases to significantly decrease in order to test the resultant codebook validating set distortion vsd is calculated on a separate validating set vs codebooks that yield small difference between the tsd and the vsd are regarded as good ones however the difference vsd tsd is not necessarily a desirable criterion for testing a trained codebook unless certain conditions are satisfied a condition that is previously assumed to be important is that the vs has to be quite large to well approximate the source distribution this condition implies greater computational burden of testing a codebook in this paper we first discuss the condition under which the difference vsd tsd is a meaningful codebook testing criterion then convergence properties of the vsd a time average quantity are investigated finally we show that for large codebooks a vs size as small as the size of the codebook is sufficient to evaluate the vsd this paper consequently presents a simple method to test trained codebooks for vq s experimental results on synthetic data and real images supporting the analysis are also provided and discussed

consistency of cheeger and ratio graph cuts this paper establishes the consistency of a family of graph cut based algorithms for clustering of data clouds we consider point clouds obtained as samples of a ground truth measure we investigate approaches to clustering based on minimizing objective functionals defined on proximity graphs of the given sample our focus is on functionals based on graph cuts like the cheeger and ratio cuts we show that minimizers of these cuts converge as the sample size increases to a minimizer of a corresponding continuum cut which partitions the ground truth measure moreover we obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the consistency to hold we provide results for two way and for multiway cuts furthermore we provide numerical experiments that illustrate the results and explore the optimality of scaling in dimension two

mixed rates asymptotics a general method is presented for deriving the limiting behavior of estimators that are defined as the values of parameters optimizing an empirical criterion function the asymptotic behavior of such estimators is typically deduced from uniform limit theorems for rescaled and reparametrized criterion functions the new method can handle cases where the standard approach does not yield the complete limiting behavior of the estimator the asymptotic analysis depends on a decomposition of criterion functions into sums of components with different rescalings the method is explained by examples from lasso type estimation k means clustering shorth estimation and partial linear models

on the performance of clustering in hilbert spaces based on randomly drawn vectors in a separable hilbert space one may construct a k means clustering scheme by minimizing an empirical squared error we investigate the risk of such a clustering scheme defined as the expected squared distance of a random vector x from the set of cluster centers our main result states that for an almost surely bounded the expected excess clustering risk is o 1 n since clustering in high or even infinite dimensional spaces may lead to severe computational problems we examine the properties of a dimension reduction strategy for clustering based on johnson lindenstrauss type random projections our results reflect a tradeoff between accuracy and computational complexity when one uses k means clustering after random projection of the data to a low dimensional space we argue that random projections work better than other simplistic dimension reduction schemes

quantifying the closeness of fractal measures numerically fractal measures are represented by finite sets of discrete points e g those generated by trajectories we introduce a cross correlation integral to evaluate the distinctness and overlap of two such sets and present a criterion to determine up to what precision the two sets are realizations of the same measure

a unified model for probabilistic principal surfaces principal curves and surfaces are nonlinear generalizations of principal components and subspaces respectively they can provide insightful summary of high dimensional data not typically attainable by classical linear methods solutions to several problems such as proof of existence and convergence faced by the original principal curve formulation have been proposed in the past few years nevertheless these solutions are not generally extensible to principal surfaces the mere computation of which presents a formidable obstacle consequently relatively few studies of principal surfaces are available we previously 2000 proposed the probabilistic principal surface pps to address a number of issues associated with current principal surface algorithms pps uses a manifold oriented covariance noise model based on the generative topographical mapping gtm which can be viewed as a parametric formulation of kohonen s self organizing map building on the pps we introduce a unified covariance model that implements pps 0 1 by varying the clamping parameter spl alpha then we comprehensively evaluate the empirical performance of pps gtm and the manifold aligned gtm on three popular benchmark data sets it is shown in two different comparisons that the pps outperforms the gtm under identical parameter settings convergence of the pps is found to be identical to that of the gtm and the computational overhead incurred by the pps decreases to 40 percent or less for more complex manifolds these results show that the generalized pps provides a flexible and effective way of obtaining principal surfaces

cats clustering after transformation and smoothing cats clustering after transformation and smoothing is a technique for nonparametrically estimating and clustering a large number of curves our motivating example is a genetic microarray experiment but the method is very general the method includes transformation and smoothing multiple curves multiple nonparametric testing for screening out flat curves clustering curves with similar shape and nonparametrically inferring the clustering estimation error rate

small ball probabilities for gaussian random fields and tensor products of compact operators we find the logarithmic l 2 small ball asymptotics for a large class of zero mean gaussian fields with covariances having the structure of tensor product the main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of gaussian random functions including the fractional brownian sheet ornstein uhlenbeck sheet etc so we get the far reaching generalizations of well known results by csaki 1982 and by li 1992 another class of gaussian fields considered is the class of additive fields studied under the supremum norm by chen and li 2003 our theorems are based on new results on spectral asymptotics for the tensor products of compact self adjoint operators in hilbert space which are of independent interest

optimal partitioning for linear mixed effects models applications to identifying placebo responders a longstanding problem in clinical research is distinguishing drug treated subjects that respond due to specific effects of the drug from those that respond to nonspecific or placebo effects of the treatment linear mixed effect models are commonly used to model longitudinal clinical trial data in this paper we present a solution to the problem of identifying placebo responders using an optimal partitioning methodology for linear mixed effects models since individual outcomes in a longitudinal study correspond to curves the optimal partitioning methodology produces a set of prototypical outcome profiles the optimal partitioning methodology can accommodate both continuous and discrete covariates the proposed partitioning strategy is compared and contrasted with the growth mixture modeling approach the methodology is applied to a two phase depression clinical trial where subjects in a first phase were treated openly for 12 weeks with fluoxetine followed by a double blind discontinuation phase where

minimum hellinger distance estimation for finite mixture models abstract minimum hellinger distance estimates are considered for finite mixture models when the exact forms of the component densities are unknown in detail but are thought to be close to members of some parametric family minimum hellinger distance estimates are asymptotically efficient if the data come from a member of the parametric family and are robust to certain departures from the parametric family a new algorithm is introduced that is similar to the em algorithm a specialized adaptive density estimate is also introduced standard measures of robustness are discussed some difficulties are noted the robustness and asymptotic efficiency of the estimators are illustrated using simulations

the mean shift algorithm and its relation to kernel regression we investigate the connection between the asymptotic bias of the well known nadaraya watson kernel regression and the mean shift ms vector with the gaussian kernel we first show that the asymptotic bias for the univariate nadaraya watson kernel regression can be estimated using the ms scalar then we investigate the general d dimensional case d 1 and derive a formula for the asymptotic bias as a function of the ms vector we show that when the regression function is a linear function of its entries then the asymptotic bias can be represented as a linear function of the ms vector the ms algorithm for the univariate and the linear cases can be used to find points where the nadaraya watson kernel regression gives an unbiased estimate furthermore this connection suggests that exploiting the theoretical properties of the asymptotic bias of the nadaraya watson kernel regression may be helpful to show the convergence of the ms algorithm through the simulations we show that how the given theoretical results can be used to estimate the bias of the estimated clean signal by just observing the noisy signal having access to the bias of the estimator enables us to evaluate the accuracy of the estimated clean data which later will be used to design a quantizer

asymmetric clustering using the alpha beta divergence we propose the use of an asymmetric dissimilarity measure in centroid based clustering the dissimilarity employed is the alpha beta divergence ab divergence which can be asymmetrized using its parameters we compute the degree of asymmetry of the ab divergence on the basis of the within cluster variances in this way the proposed approach is able to flexibly model even clusters with significantly different variances consequently this method overcomes one of the major drawbacks of the standard symmetric centroid based clustering

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

the strong law of large numbers for k means and best possible nets of banach valued random variables let b be a uniformly convex banach space x a b valued random variable and k a given positive integer number a random sample of x is substituted by the set of k elements which minimizes a criterion we found conditions to assure that this set converges a s as the sample size increases to the set of k elements which minimizes the same criterion for x

sharp asymptotics of the kolmogorov entropy for gaussian measures abstract let u denote the unit ball of the cameron martin space of a gaussian measure on a hilbert space the sharp asymptotics for the kolmogorov metric entropy numbers of u is derived the condition imposed is regular variation of the eigenvalues of the covariance operator a consequence is a precise link including constants to the functional quantization problem

functional clustering and identifying substructures of longitudinal data summary a functional clustering fc method k centres fc for longitudinal data is proposed the k centres fc approach accounts for both the means and the modes of variation differentials between clusters by predicting cluster membership with a reclassification step the cluster membership predictions are based on a non parametric random effect model of the truncated karhunen loeve expansion coupled with a non parametric iterative mean and covariance updating scheme we show that under the identifiability conditions derived the k centres fc method proposed can greatly improve cluster quality as compared with conventional clustering algorithms moreover by exploring the mean and covariance functions of each cluster thek centres fc method provides an additional insight into cluster structures which facilitates functional cluster analysis practical performance of the k centres fc method is demonstrated through simulation studies and data applications including growth curve and gene expression profile data

comparison of three nonlinear seizure prediction methods by means of the seizure prediction characteristic epilepsy is characterized by the spontaneous and unforeseeable occurrence of seizures during which the perception or behavior of patients is disturbed the predictability of these seizures would render novel therapeutic approaches possible several prediction methods have claimed to be able to predict seizures based on eeg recordings minutes in advance however the term seizure prediction is not unequivocally defined different criteria to assess prediction methods exist and only little attention has been paid to issues of sensitivity and false prediction rate we introduce an assessment criterion called the seizure prediction characteristicthat incorporates the assessment of sensitivity and false prediction rate within this framework three nonlinear seizure prediction methods were evaluated on a large eeg data pool of 21 patients altogether 582 h intracranial eeg data and 88 seizures were examined with a rate of 1 3 6 false predictions per day the dynamical similarity index achieves a sensitivity between 21 and 42 which was the best result of the three methods sensitivity was between 18 and 31 for the extended prospective version of the accumulated energy and between 13 and 30 for the effective correlation dimension these results still are not sufficient for clinical applications

nonlinear similarity analysis for epileptic seizures prediction abstract the prediction of epileptic seizures can promise a new diagnostic application and a novel approach for seizure control this paper proposes an improved dynamical similarity measure to predict epileptic seizures in electroencephalographic eeg first mutual information and cao s method are employed to reconstruct a phase space of preprocessed eeg recordings by using the positive zero crossing method second a gaussian function replaces the heavyside function within correlation integral at calculating a similarity index the crisp boundary of the heavyside function is eliminated because of the gaussian function s smooth boundary third an adaptive detection method based on the similarity index is proposed to predict the epileptic seizures in light of test results of eeg recordings of rats it is found that the new dynamical similarity index is insensitive to the selection of the radius value of gaussian function and the size of segmented eeg recordings comparing with the dynamical similarity index proposed by le van quyen et al anticipating epileptic seizures in real time by a non linear analysis of similarity between eeg recordings neuroreport 10 1999 2149 2155 the tests of twelve rats show the new dynamical similarity index is better to predict the epileptic seizures

recursive partitioning to reduce distortion adaptive partitioning of a multidimensional feature space plays a fundamental role in the design of data compression schemes most partition based design methods operate in an iterative fashion seeking to reduce distortion at each stage of their operation by implementing a linear split of a selected cell the operation and eventual outcome of such methods is easily described in terms of binary tree structured vector quantizers this paper considers a class of simple growing procedures for tree structured vector quantizers of primary interest is the asymptotic distortion of quantizers produced by the unsupervised implementation of the procedures it is shown that application of the procedures to a convergent sequence of distributions with a suitable limit yields quantizers whose distortion tends to zero analogous results are established for tree structured vector quantizers produced from stationary ergodic training data the analysis is applicable to procedures employing both axis parallel and oblique splitting and a variety of distortion measures the results of the paper apply directly to unsupervised procedures that may be efficiently implemented on a digital computer

strong consistency of the prototype based clustering in probabilistic space in this paper we formulate in general terms an approach to prove strong consistency of the empirical risk minimisation inductive principle applied to the prototype or distance based clustering this approach was motivated by the divisive information theoretic feature clustering model in probabilistic space with kullback leibler divergence which may be regarded as a special case within the clustering minimisation framework

regularized principal manifolds many settings of unsupervised learning can be viewed as quantization problems the minimization of the expected quantization error subject to some restrictions this allows the use of tools such as regularization from the theory of supervised risk minimization for unsupervised learning this setting turns out to be closely related to principal curves the generative topographic map and robust coding we explore this connection in two ways 1 we propose an algorithm for finding principal manifolds that can be regularized in a variety of ways and 2 we derive uniform convergence bounds and hence bounds on the learning rates of the algorithm in particular we give bounds on the covering numbers which allows us to obtain nearly optimal learning rates for certain types of regularization operators experimental results demonstrate the feasibility of the approach