segmentation and classification of range images the recognition of objects in three dimensional space is a desirable capability of a computer vision system range images which directly measure 3 d surface coordinates of a scene are well suited for this task in this paper we report a procedure to detect connected planar convex and concave surfaces of 3 d objects this is accomplished in three stages the first stage segments the range image into surface patches by a square error criterion clustering algorithm using surface points and associated surface normals the second stage classifies these patches as planar convex or concave based on a non parametric statistical test for trend curvature values and eigenvalue analysis in the final stage boundaries between adjacent surface patches are classified as crease or noncrease edges and this information is used to merge compatible patches to produce reasonable faces of the object s this procedure has been successfully applied to a large number of real and synthetic images four of which we present in this paper

a fuzzy extension of the silhouette width criterion for cluster analysis the present paper proposes a new cluster validity measure as an additional criterion to help the decision making process in fuzzy cluster analysis this measure named fuzzy silhouette is a generalization to the fuzzy case of the average silhouette width criterion originally conceived to assess crisp non fuzzy data partitions the fuzzy silhouette is more appealing than its crisp counterpart in the context of fuzzy cluster analysis since it makes explicit use of the fuzzy partition matrix provided by the clustering algorithm in addition it has been designed to improve performance of the original silhouette criterion in detecting regions with higher data density when the data set involves overlapping clusters the performance of the fuzzy silhouette is evaluated and compared to that of five well known cluster validity measures six data sets are used to illustrate different scenarios in which the proposed fuzzy silhouette performs similar to or better than these other criteria thus becoming eligible to join a pool of measures to be used all together in fuzzy cluster analysis

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

dynamic clustering detection through multi valued descriptors of dermoscopic images this paper introduces a dynamic clustering methodology based on multi valued descriptors of dermoscopic images the main idea is to support medical diagnosis to decide if pigmented skin lesions belonging to an uncertain set are nearer to malignant melanoma or to benign nevi melanoma is the most deadly skin cancer and early diagnosis is a current challenge for clinicians most data analysis algorithms for skin lesions discrimination focus on segmentation and extraction of features of categorical or numerical type as an alternative approach this paper introduces two new concepts first it considers multi valued data that scalar variables not only describe but also intervals or histogram variables second it introduces a dynamic clustering method based on wasserstein distance to compare multi valued data the overall strategy of analysis can be summarized into the following steps first a segmentation of dermoscopic images allows to identify a set of multi valued descriptors second we performed a discriminant analysis on a set of images where there is an a priori classification so that it is possible to detect which features discriminate the benign and malignant lesions and third we performed the proposed dynamic clustering method on the uncertain cases which need to be associated to one of the two previously mentioned groups results based on clinical data show that the grading of specific descriptors associated to dermoscopic characteristics provides a novel way to characterize uncertain lesions that can help the dermatologist s diagnosis

decoupling of multivariable control systems over unique factorization domains necessary and sufficient conditions are established for the existence of a state variable feedback decoupling of an m input m output time invariant linear control system over a unique factorization domain an explicit computation is provided for the feedback and the feedforward gain matrix also necessary and sufficient conditions for the existence of a stability preserving state feedback decoupling are given the results are illustrated by some examples

a new sequential classifier using information criterion window abstract we propose a model based parametric unsupervised sequential classifier with an information criterion window icw based on the assumption that the number of clusters changes smoothly from a stage to the following stage to determine the number of clusters at each stage and to perform classification on the observed mixture data the proposed approach provides possible solutions for two problems of pattern recognition adaptive classification and sequential clustering validation simulation results demonstrate that the proposed algorithm provides good results for pure mixed data and successfully performs clustering for the mixed gaussian data in the large sample limit

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

a transfer matrix approach to feedback invariants of linear systems application to decoupling abstract the transfer matrix approach allows one to exhibit fairly easily fundamental invariants of multivariable systems under the action of transformation groups these transformation groups include the classical state feedback state and input changes of coordinates then the solvability of numerous classical control problems e g decouplability disturbance rejection model following can be expressed directly in terms of these invariants in this expository paper we present in a unified way three lists of feedback invariants related to the infinite behavior of linear systems these invariants turn out to be crucial in studying feedback decoupling problems the main tools for this study are factorization of transfer matrices over the p i d of proper rational functions when dealing with stability we factor transfer matrices over the p i d of proper rational stable functions we illustrate the use of these invariants by studying some decoupling problems

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

on the complexity of computing the k restricted edge connectivity of a graph the k restricted edge connectivity of a graph g denoted by k g is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices this graph invariant which can be seen as a generalization of a minimum edge cut has been extensively studied from a combinatorial point of view however very little is known about the complexity of computing k g very recently in the parameterized complexity community the notion of good edge separation of a graph has been defined which happens to be essentially the same as the k restricted edge connectivity motivated by the relevance of this invariant from both combinatorial and algorithmic points of view in this article we initiate a systematic study of its computational complexity with special emphasis on its parameterized complexity for several choices of the parameters we provide a number of np hardness and w 1 hardness results as well as fpt algorithms

a modified k means algorithm for circular invariant clustering several important pattern recognition applications are based on feature vector extraction and vector clustering directional patterns are commonly represented by rotation variant vectors f sub d formed from features uniformly extracted in m directions it is often desirable that pattern recognition algorithms are invariant under pattern rotation this paper introduces a distance measure and a k means based algorithm namely circular k means ck means to cluster vectors containing directional information such as f sub d in a circular shift invariant manner a circular shift of f sub d corresponds to pattern rotation thus the algorithm is rotation invariant an efficient fourier domain representation of the proposed measure is presented to reduce computational complexity a split and merge approach smck means suited to the proposed ck means technique is proposed to reduce the possibility of converging at local minima and to estimate the correct number of clusters experiments performed for textural images illustrate the superior performance of the proposed algorithm for clustering directional vectors f sub d compared to the alternative approach that uses the original k means and rotation invariant feature vectors transformed from f sub d

designing fpt algorithms for cut problems using randomized contractions we introduce a new technique for designing fixed parameter algorithms for cut problems called randomized contractions we apply our framework to obtain the first fixed parameter algorithms fpt algorithms with exponential speed up for the steiner cut and node multiway cut uncut problems we prove that the parameterized version of the unique label cover problem which is the base of the unique games conjecture can be solved in 2 o k 2 log sigma n 4 log n deterministic time even in the stronger vertex deletion variant where k is the number of unsatisfied edges and sigma is the size of the alphabet as a consequence we show that one can in polynomial time solve instances of unique games where the number of edges allowed not to be satisfied is upper bounded by o sqrt log n to optimality which improves over the trivial o 1 upper bound we prove that the steiner cut problem can be solved in 2 o k 2 log k n 4 log n deterministic time and tilde o 2 o k 2 log k n 2 random

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

bootstrap technique in cluster analysis abstract we define a method to estimate the number of clusters in a data set e using the bootstrap technique this approach involves the generation of several fake data sets by sampling patterns with replacement in e bootstrapping for each number k of clusters a measure of stability of the k cluster partitions over the bootstrap samples is used to characterize the significance of the k cluster partition for the original data set the value of k which provides the most stable partitions is the estimate of the number of clusters in e the performance of this new technique is demonstrated on both synthetic and real data and is applied to the segmentation of range images

solution of the static state feedback decoupling problem for linear systems with two outputs we give for linear systems with two outputs a necessary and sufficient condition for the solvability of the restricted decoupling problem rdp in terms of some structural invariants a geometric characterization is also given which provides the least dynamical extension needed when rdp is not solvable

matrix pencil characterization of almost a z invariant subspaces a classification of geometric concepts the equivalence between the algebraic matrix pencil characterization of the sub spaces of the extended geometric theory and their dynamic characterization 13 established as a result a complete classification of almost a b invariant a b invariant almost controllability and controllability subspaces is derived in terms of matrix pencil invariants the frequency propagation aspects of infinite spectrum a b invariant subspaces are investigated and it is shown that they are limits of closed loop eigenspaces with arbitrarily largo eigenvalues finally the importance of the infinite frequency subspaces in the study of the asymptotic behaviour of the closed loop eigenspaces and eigenvalues under scalar gain output feedback is discussed

the minimal delay decoupling problem feedback implementation with stability in this paper we solve the minimal delay decoupling problem of linear multivariable systems we look for feedback implementable solutions which moreover guarantee closed loop stability we prove that dynamic state feedback decoupling with stability is achievable if the number of independent inputs is large enough to compensate the intrinsic nondecouplability of the system we introduce new feedback invariants characterizing the minimal number of infinite and unstable zeros of the decoupled system

curvature based method for determining the number of clusters abstract determining the number of clusters is one of the research questions attracting considerable interests in recent years majority of the existing methods require parametric assumptions and substantiated computations in this paper we propose a simple yet powerful method for determining the number of clusters based on curvature our technique is computationally efficient and straightforward to implement we compare our method with 6 other approaches on a wide range of simulated and real world datasets theoretical motivation underlying the proposed method is also presented

comparative evaluation of two superior stopping rules for hierarchical cluster analysis a split sample replication stopping rule for hierarchical cluster analysis is compared with the internal criterion previously found superior by milligan and cooper 1985 in their comparison of 30 different procedures the number and extent of overlap of the latent population distributions was systematically varied in the present evaluation of stopping rule validity equal and unequal population base rates were also considered both stopping rules correctly identified the actual number of populations when there was essentially no overlap and clusters occupied visually distinct regions of the measurement space the replication criterion which is evaluated by clustering of cluster means from preliminary analyses that are accomplished on random partitions of an original data set was superior as the degree of overlap in population distributions increased neither method performed adequately when overlap obliterated visually discernible density nodes

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

statistical significance of clustering for high dimension low sample size data clustering methods provide a powerful tool for the exploratory analysis of high dimension low sample size hdlss data sets such as gene expression microarray data a fundamental statistical issue in clustering is which clusters are really there as opposed to being artifacts of the natural sampling variation we propose sigclust as a simple and natural approach to this fundamental statistical problem in particular we define a cluster as data coming from a single gaussian distribution and formulate the problem of assessing statistical significance of clustering as a testing procedure this gaussian null assumption allows direct formulation of p values that effectively quantify the significance of a given clustering hdlss covariance estimation for sigclust is achieved by a combination of invariance principles together with a factor analysis model the properties of sigclust are studied simulated examples as well as an application to a real cancer microarray data set show that the proposed method

a hybrid clustering method for identifying high density clusters abstract a hybrid method for clustering multivariate observations is proposed which combines elements of the k means and the single linkage clustering techniques one purpose of the proposed method is to discover the high density clusters given a random sample of size n from some underlying population a high density cluster at level c in a population with density f is defined as a maximal connected set of points x with f x c this clustering procedure is practicable for very large numbers of observations and is shown to be consistent under certain regularity conditions in one dimension

on comparing two sequences of numbers and its applications to clustering analysis a conceptual problem that appears in different contexts of clustering analysis is that of measuring the degree of compatibility between two sequences of numbers this problem is usually addressed by means of numerical indexes referred to as sequence correlation indexes this paper elaborates on why some specific sequence correlation indexes may not be good choices depending on the application scenario in hand a variant of the product moment correlation coefficient and a weighted formulation for the goodman kruskal and kendall s indexes are derived that may be more appropriate for some particular application scenarios the proposed and existing indexes are analyzed from different perspectives such as their sensitivity to the ranks and magnitudes of the sequences under evaluation among other relevant aspects of the problem the results help suggesting scenarios within the context of clustering analysis that are possibly more appropriate for the application of each index

proper and stable minimal macmillan degrees bases of rational vector spaces the structure of proper and stable bases of rational vector spaces is investigated we prove that if t s is a rational vector space then among the proper bases of 3 s there is a subfamily of proper bases which are 1 stable 2 have no zeros in c bigcup infty and therefore are column row reduced at infinity and 3 their macmillan degree is minimum among the macmillan degrees of all other proper bases of 3 s and it is given by the sum of the macmillan degrees of their columns taken separately it is shown that this notion is the counterpart of forney s concept of a minimal polynomial basis for the family of proper and stable bases of 3 s

performance evaluation of some clustering algorithms and validity indices in this article we evaluate the performance of three clustering algorithms hard k means single linkage and a simulated annealing sa based technique in conjunction with four cluster validity indices namely davies bouldin index dunn s index calinski harabasz index and a recently developed index i based on a relation between the index i and the dunn s index a lower bound of the value of the former is theoretically estimated in order to get unique hard k partition when the data set has distinct substructures the effectiveness of the different validity indices and clustering methods in automatically evolving the appropriate number of clusters is demonstrated experimentally for both artificial and real life data sets with the number of clusters varying from two to ten once the appropriate number of clusters is determined the sa based clustering technique is used for proper partitioning of the data into the said number of clusters

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

three mode component analysis with crisp or fuzzy partition of units a new methodology is proposed for the simultaneous reduction of units variables and occasions of a three mode data set units are partitioned into a reduced number of classes while simultaneously components for variables and occasions accounting for the largest common information for the classification are identified the model is a constrained three mode factor analysis and it can be seen as a generalization of the redkm model proposed by de soete and carroll for two mode data the least squares fitting problem is mathematically formalized as a constrained problem in continuous and discrete variables an iterative alternating least squares algorithm is proposed to give an efficient solution to this minimization problem in the crisp and fuzzy classification context the performances of the proposed methodology are investigated by a simulation study comparing our model with other competing methodologies different procedures for starting the proposed algorithm have also been tested a discussion of some interesting differences in the results follows finally an application to real data illustrates the ability of the proposed model to provide substantive insights into the data complexities

combining multiple clusterings using evidence accumulation we explore the idea of evidence accumulation eac for combining the results of multiple clusterings first a clustering ensemble a set of object partitions is produced given a data set n objects or patterns in d dimensions different ways of producing data partitions are 1 applying different clustering algorithms and 2 applying the same clustering algorithm with different values of parameters or initializations further combinations of different data representations feature spaces and clustering algorithms can also provide a multitude of significantly different data partitionings we propose a simple framework for extracting a consistent clustering given the various partitions in a clustering ensemble according to the eac concept each partition is viewed as an independent evidence of data organization individual data partitions being combined based on a voting mechanism to generate a new n times n similarity matrix between the n patterns the final data partition of the n patterns is obtained by applying a hierarchical agglomerative clustering algorithm on this matrix we have developed a theoretical framework for the analysis of the proposed clustering combination strategy and its evaluation based on the concept of mutual information between data partitions stability of the results is evaluated using bootstrapping techniques a detailed discussion of an evidence accumulation based clustering algorithm using a split and merge strategy based on the k means clustering algorithm is presented experimental results of the proposed method on several synthetic and real data sets are compared with other combination strategies and with individual clustering results produced by well known clustering algorithms

feedback block decoupling and infinite structure of linear systems abstract using factorizations at infinity of the transfer matrix two interesting necessary and sufficient conditions arc obtained for the block decoupling problem of linear systems by state feedback and a constant non singular input transformation these conditions are expressed in a very simple way in terms of integers which are feedback invariants for the system under consideration these feedback invariants are related either to the system infinite structure or to the reachability indices of the feedback reduced system

cluster validity profiles abstract the quantitative evaluation of clusters has lagged far behind the development of clustering algorithms this paper introduces a new procedure based on probability profiles for judging the validity of clusters established from rank order proximity data probability profiles furnish a comprehensive picture of the compactness and isolation of a cluster scaled in probability units given a rank order proximity matrix and a cluster to be examined profiles compare the cluster s upper bounds on the best compactness and isolation indices one would expect in a randomly chosen graph after reviewing the pertinent literature this paper explains the background from graph theory and cluster analysis needed to treat cluster validity the probabilities and bounds needed to form cluster profiles are derived and strategies for using profiles are suggested special attention is given to the underlying probability models profiles are demonstrated on four artificially generated data sets two of which have good hierarchical structure and on data from a speaker recognition project they reject spurious clusters and accept apparently valid clusters since profiles quantify the interaction between a cluster and its environment they provide a much richer source of information on cluster structure than single number indices proposed in the literature

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

parameterized graph separation problems we consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible the following five problems are studied delete k vertices such that a each of the given l terminals is separated from the others b each of the given l pairs of terminals is separated c exactly l vertices are cut away from the graph d exactly l connected vertices are cut away from the graph e the graph is separated into at least l components we show that if both k and l are parameters then a b and d are fixed parameter tractable while c and e are w 1 hard

segmentation of textured images based on fractals and image filtering this paper describes a new approach to the segmentation of textured gray scale images based on image pre ltering and fractal features traditionally lter bank decomposition methods consider the energy in each band as the textural feature a parameter that is highly dependent on image intensity in this paper we use fractal based features which depend more on textural characteristics and not intensity information to reduce the total number of features used in the segmentation the signi cance of each feature is examined using a test similar to the f test and less signi cant features are not used in the clustering process the commonly used k means algorithm is extended to an iterative k means by using a variable window size that preserves boundary details the number of clusters is estimated using an improved hierarchical approach that ignores information extracted around region boundaries 2001 pattern recognition society published by elsevier science ltd all rights reserved

automatic aspect discrimination in data clustering the attributes describing a data set may often be arranged in meaningful subsets each of which corresponds to a different aspect of the data an unsupervised algorithm scad that simultaneously performs fuzzy clustering and aspects weighting was proposed in the literature however scad may fail and halt given certain conditions to fix this problem its steps are modified and then reordered to reduce the number of parameters required to be set by the user in this paper we prove that each step of the resulting algorithm named ascad globally minimizes its cost function with respect to the argument being optimized the asymptotic analysis of ascad leads to a time complexity which is the same as that of fuzzy c means a hard version of the algorithm and a novel validity criterion that considers aspect weights in order to estimate the number of clusters are also described the proposed method is assessed over several artificial and real data sets

invariant spaces at infinity of linear systems application to block decoupling the dynamic state feedback block decoupling problem in the general case i e with singular or nonsingular input transformations is solved it is proven that the problem is solvable if the number of inputs is large enough the main tools for this study are some invariant spaces associated with the columns infinite behavior of a rational matrix it turns out that when the problem is solvable it is also solvable with stability

simultaneous classification and multidimensional scaling with external information for the exploratory analysis of a matrix of proximities or dis similarities between objects one often uses cluster analysis ca or multidimensional scaling mds solutions resulting from such analyses are sometimes interpreted using external information on the objects usually the procedures of ca mds and using external information are carried out independently and sequentially although combinations of two of the three procedures ca and mds or multidimensional scaling and using external information have been proposed in the literature the present paper offers a procedure that combines all three procedures in one analysis using a model that describes a partition of objects with cluster centroids represented in a low dimensional space which in turn is related to the information in the external variables a simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure also to illustrate the method it is applied to two empirical data sets

channel dependent constrained combinatorial clustering constrained combinatorial clustering ccc is a new approach for grouping multiple features where the clustering metric depends on an unknown communication channel assignment features assigned to the same channel cannot be from the same source and conversely channels assigned to the same source must be distinct while the number of sources and their states are unknown the channels are assumed to be known except for additive noise potential clustering assignments are checked for compatibility with the constraints in a structured way that results in significant computational savings with respect to exhaustive enumeration especially when combined with a k best channel assignment algorithm that has polynomial complexity by combining two channel assignment methods exhaustive and k best with two clustering techniques ccc and greedy four new algorithms are presented to solve this novel problem along with detailed computational complexity analyses the main algorithm ccc is a top down clustering strategy based on assignment aggregation in a channel constrained environment the approaches are compared on a one dimensional simulation significant performance differences in source estimation accuracy and estimated number of sources are observed at low signal to noise ratios snrs and for low values of k some novel permutation symmetry properties arising from the study which lead to a new type of self affine set or discrete fractal are also presented

fuzzy clustering based on k nearest neighbours rule abstract in cluster analysis several algorithms have been made for partitioning a set of objects into c natural clusters in general this problem is formulated as being an objective function optimization approach however it is known that the function being minimized is nonconvex and hence it may lead to convergence to many local minima i e to different partitions thus the clustering is repeated with different initializations hoping that some runs will lead to the global minimum therefore the performance of these algorithms depends largely on good choice of these initializations the most widely used algorithm using this function is called fuzzy c means algorithm fcma in this paper a new algorithm is proposed to carry out fuzzy clustering without a priori assumptions on initial guesses this algorithm is based on two layer clustering strategy during the first layer the k nearest neighbours decision rule is used then to achieve an optimal partition the second layer involves one iteration of fcma the performance of the proposed algorithm and that of fcma have been tested on six data sets the results obtained show that the new algorithm possesses a number of advantages over the fcma

an extensive comparative study of cluster validity indices the validation of the results obtained by clustering algorithms is a fundamental part of the clustering process the most used approaches for cluster validation are based on internal cluster validity indices although many indices have been proposed there is no recent extensive comparative study of their performance in this paper we show the results of an experimental work that compares 30 cluster validity indices in many different environments with different characteristics these results can serve as a guideline for selecting the most suitable index for each possible application and provide a deep insight into the performance differences between the currently available indices

a monte carlo study of thirty internal criterion measures for cluster analysis a monte carlo evaluation of thirty internal criterion measures for cluster analysis was conducted artificial data sets were constructed with clusters which exhibited the properties of internal cohesion and external isolation the data sets were analyzed by four hierarchical clustering methods the resulting values of the internal criteria were compared with two external criterion indices which determined the degree of recovery of correct cluster structure by the algorithms the results indicated that a subset of internal criterion measures could be identified which appear to be valid indices of correct cluster recovery indices from this subset could form the basis of a permutation test for the existence of cluster structure or a clustering algorithm

the effect of cluster size dimensionality and the number of clusters on recovery of true cluster structure an evaluation of four clustering methods and four external criterion measures was conducted with respect to the effect of the number of clusters dimensionality and relative cluster sizes on the recovery of true cluster structure the four methods were the single link complete link group average upgma and ward s minimum variance algorithms the results indicated that the four criterion measures were generally consistent with each other of which two highly similar pairs were identified the tirst pair consisted of the rand and corrected rand statistics and the second pair was the jaccard and the fowlkes and mallows indexes with respect to the methods recovery was found to improve as the number of clusters increased and as the number of dimensions increased the relative cluster size factor produced differential performance effects with ward s procedure providing the best recovery when the clusters were of equal size the group average method gave equivalent or better recovery when the clusters were of unequal size

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

measure projection analysis a probabilistic approach to eeg source comparison and multi subject inference a crucial question for the analysis of multi subject and or multi session electroencephalographic eeg data is how to combine information across multiple recordings from different subjects and or sessions each associated with its own set of source processes and scalp projections here we introduce a novel statistical method for characterizing the spatial consistency of eeg dynamics across a set of data records measure projection analysis mpa first finds voxels in a common template brain space at which a given dynamic measure is consistent across nearby source locations then computes local mean eeg measure values for this voxel subspace using a statistical model of source localization error and between subject anatomical variation finally clustering the mean measure voxel values in this locally consistent brain subspace finds brain spatial domains exhibiting distinguishable measure features and provides 3 d maps plus statistical significance estimates for each eeg measure of interest applied to sufficient high quality data the scalp projections of many maximally independent component ic processes contributing to recorded high density eeg data closely match the projection of a single equivalent dipole located in or near brain cortex we demonstrate the application of mpa to a multi subject eeg study decomposed using independent component analysis ica compare the results to k means ic clustering in eeglab sccn ucsd edu eeglab and use surrogate data to test mpa robustness a measure projection toolbox mpt plug in for eeglab is available for download sccn ucsd edu wiki mpt together mpa and ica allow use of eeg as a 3 d cortical imaging modality with near cm scale spatial resolution

smith mcmillan factorizations at infinity of rational matrix functions and their control interpretation in this paper the smith mcmillan factorizations at infinity of a transfer function are studied the non uniqueness of these factorizations is characterized via a multiplicative group of bicausal isomorphisms control interpretations are given in terms of system equivalence under some groups of transformations a characterization of the stabilizer of morse group at a b c is given for irreducible systems

clustered common factor exploration in factor analysis the factor analysis fa model does not permit unique estimation of the common and unique factor scores this weakness is notorious as the factor indeterminacy in fa luckily some part of the factor scores can be uniquely determined thus as a whole they can be viewed as a sum of determined and undetermined parts the paper proposes to select the undetermined part such that the resulting common factor scores have the following feature the rows i e individuals of the common factor score matrix are as well classified as possible into few clusters the clear benefit is that we can easily interpret the factor scores simply by focusing on the clusters the procedure is called clustered common factor exploration ccfe an alternating least squares algorithm is developed for ccfe it is illustrated with real data examples the proposed approach can be viewed as a parallel to the rotation techniques in fa they exploit another fa indeterminacy the rotation indeterminacy which is resolved by choosing the rotation that transforms the loading matrix into the most interpretable one according to a pre specified criterion in contrast to the rotational indeterminacy the factor indeterminacy is utilized to achieve well clustered factor scores by ccfe to the best of our knowledge such an approach to the fa interpretation has not been studied yet

a statistical model of cluster stability in the current paper we present a method for assessing cluster stability this method combined with a clustering algorithm yields an estimate of the data partition namely the number of clusters we adopt the cluster stability standpoint where clusters are imagined as islands of high density in a sea of low density explicitly a cluster is associated with its high density core our approach offers to evaluate the goodness of a cluster by the similarity amongst the entire cluster and its core we propose to measure this resemblance by two sample tests or by probability distances between appropriate probability distributions the distances are calculated on clustered samples drawn from the source population according to two different distributions the first law is the underlying set distribution the second law is constructed so that it represents the clusters cores here a variant of the k nearest neighbor density estimation is applied so that items belonging to cores have a much higher chance to be selected as the sample distribution is unknown a distribution free two sample test is required to examine the mentioned correspondence for constructing such a test we use distance functions built on negative definite kernels in practice outliers in the samples and limitations of the clustering algorithm heavily contribute to the noise level as a result of this shortcoming the distance values have to be determined for many pairs of samples and therefore an empirical distance s distribution is obtained the distribution is dependent on the examined number of clusters to prevent this property for biasing the results we normalize the distances it is conjectured that the true number of clusters yields the most concentrated normalized distribution to measure the concentration we use the sample mean and the sample 25th percentile the paper exhibits the good performance of the proposed method on synthetic and real world data

causal factorization and linear feedback an algebraic framework for the investigation of linear dynamic output feedback is introduced pivotal in the present theory is the problem of causal factorization i e the problem of factoring two systems over each other through a causal factor the basic issues are resolved with the aid of the new concept of latency kernels

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

resampling method for unsupervised estimation of cluster validity we introduce a method for validation of results obtained by clustering analysis of data the method is based on resampling the available data a figure of merit that measures the stability of clustering solutions against resampling is introduced clusters that are stable against resampling give rise to local maxima of this figure of merit this is presented first for a one dimensional data set for which an analytic approximation for the figure of merit is derived and compared with numerical measurements next the applicability of the method is demonstrated for higher dimensional data including gene microarray expression data

estimating the number of clusters in a numerical data set via quantization error modeling in this paper we consider the problem of unsupervised clustering vector quantization of multidimensional numerical data we propose a new method for determining an optimal number of clusters in the data set the method is based on parametric modeling of the quantization error the model parameter can be treated as the effective dimensionality of the data set the proposed method was tested with artificial and real numerical data sets and the results of the experiments demonstrate empirically not only the effectiveness of the method but its ability to cope with difficult cases where other known methods fail a parameterized model for the clustering error is introduced the model parameter is a measure of the data dimension and homogeneity a new cost criterion is derived from the properties of the model the method demonstrates good results for numerical data sets

order constrained solutions in k means clustering even better than being globally optimal this paper proposes an order constrained k means cluster analysis strategy and implements that strategy through an auxiliary quadratic assignment optimization heuristic that identifies an initial object order a subsequent dynamic programming recursion is applied to optimally subdivide the object set subject to the order constraint we show that although the usual k means sum of squared error criterion is not guaranteed to be minimal a true underlying cluster structure may be more accurately recovered also substantive interpretability seems generally improved when constrained solutions are considered we illustrate the procedure with several data sets from the literature

recovering the number of clusters in data sets with noise features using feature rescaling factors in this paper we introduce three methods for re scaling data sets aiming at improving the likelihood of clustering validity indexes to return the true number of spherical gaussian clusters with additional noise features our method obtains feature re scaling factors taking into account the structure of a given data set and the intuitive idea that different features may have different degrees of relevance at different clusters we experiment with the silhouette using squared euclidean manhattan and the pth power of the minkowski distance dunn s calinski harabasz and hartigan indexes on data sets with spherical gaussian clusters with and without noise features we conclude that our methods indeed increase the chances of estimating the true number of clusters in a data set

diagonal decoupling of linear systems by static state feedback a solution is presented to the long standing open problem of linear systems theory diagonal decoupling by static state feedback the earliest known investigation of system decoupling dates back to 1934 a state space formulation of the problem appeared in 1964 and a solution for square and invertible systems followed in 1967 the case of right invertible systems however has withstood all past efforts to obtain a solution save for several special cases the present formulation avoids restrictive hypotheses concerning system and decoupling feedback the existence of a solution is shown to depend on the existence of three lists of integers conditioned solely by system invariants with respect to the permissible transformations

new decoupling invariants the essential orders the aim of this paper is to introduce a new list of integers which is invariant under the action of biproper compensation and which plays a key role in the solution of the row by row decoupling problem several equivalent characterizations are provided within both the geometric and the transfer matrix approaches

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o

approximate spectral clustering with utilized similarity information using geodesic based hybrid distance measures spectral clustering has been popular thanks to its ability to extract clusters of varying characteristics without using a parametric model in expense of high computational cost required for eigendecomposition of pairwise similarities in order to utilize its advantages in large datasets where it is infeasible due to its computational burden approximate spectral clustering asc methods apply spectral clustering on a reduced set of points data representatives selected by sampling or quantization this two step approach i e finding the representatives and then clustering them brings new opportunities for precise similarity definition such as manifold based topological relations data distribution within the voronoi polyhedra of the representatives and their geodesic distance information which are often ignored in similarity definition for asc in this study we propose geodesic based hybrid similarity criteria which enable the use of different types of information for accurate similarity representation in asc despite the fact that geodesic concept has been widely used in clustering our contribution is the unique way of representing data topology to form geodesic relations and jointly harnessing various information types including topology distance and density the proposed criteria are tested using both sampling selective sampling and quantization neural gas and k means approaches experiments on artificial datasets well known small medium size real datasets and four large datasets four remote sensing images with different types of clusters show that the proposed geodesic based hybrid similarity criteria outperform traditional similarity criteria in terms of clustering accuracies and several cluster validity indices highlightsgeodesic based hybrid similarity measures are proposed for approximate spectral clustering neighborhood graph required for geodesic approach is effectively constructed by weighted delaunay triangulation conn the proposed geodesic based hybrid similarities outperform in terms of both accuracies and cluster validity indices the proposed geodesic based hybrid similarities can be powerful for clustering of large remote sensing images the proposed similarities are significant especially for quantization based approximate spectral clustering

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

fitting an unknown number of lines and planes to image data through compatible cluster merging abstract a compatible cluster merging algorithm is presented that is specially designed to find the optimum number of linear planar and hyperplanar clusters i e clusters that lie in a subspace of the original space when an upper bound on the number of clusters present is known this algorithm is shown to be superior to more traditional validity measure based approaches the effectiveness and advantages of the proposed technique in 2d and 3d applications is demonstrated with both synthetic and real data the proposed applications include character recognition obtaining straight line descriptions of intensity edge images and obtaining planar approximations of 3d range data