measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

a new method to detect nonlinearity in a time series synthesizing surrogate data using a kolmogorov smirnoff tested hidden markov model abstract a way of statistically testing for nonlinearity in a time series is to employ the method of surrogate data this method often makes use of the fourier transform ft in order to generate the surrogate as various authors have shown this can lead to artefacts in the surrogates and spurious detection of nonlinearity can result this paper documents a new method to synthesize surrogate data using a 1st order hidden markov model hmm combined with a kolmogorov smirnoff test ks test to determine the required resolution of the hmm significance test results for a sinewave henon map and gaussian noise time series are presented it is demonstrated that ks tested hmm surrogates can be successfully used to distinguish between a deterministic and stochastic time series then by applying a simple test for linearity using linear and nonlinear predictors it is possible to determine the nature of the deterministic class and hence conclude whether the system is linear deterministic or nonlinear deterministic furthermore it is demonstrated that the method works for periodic functions too where ft surrogates break down

a measure of multivariate phase synchrony using hyperdimensional geometry phase synchrony has been used to investigate the dynamics of subsystems that make up a complex system current measures of phase synchrony are mostly bivariate focusing on the synchrony between pairs of time series bivariate measures do not necessarily lead to a complete picture of the global interactions within a complex system current multivariate synchrony measures are based on either averaging all possible pairwise synchrony values or eigendecomposition of the pairwise bivariate synchrony matrix these approaches are sensitive to the accuracy of the bivariate synchrony indices computationally complex and indirect ways of quantifying the multivariate synchrony recently we had proposed a method to compute the multivariate phase synchrony using a hyperdimensional coordinate system this method referred to as hyperspherical phase synchrony hps has been found to be dependent on the ordering of the phase differences in this paper we propose a more general hyperspherical coordinate system along with a new higher dimensional manifold representation to eliminate the dependency on the ordering of the signals phases this new framework referred to as hyper torus synchrony hts is shown to be equivalent to the root mean square of a sufficient set of squared phase locking values whose phase differences contain information about all oscillators in the network the statistical properties of hts are given analytically and its performance is evaluated thoroughly for both synthetic and real signals

a practical shortest path algorithm with linear expected time we present an improvement of the multilevel bucket shortest path algorithm of denardo and fox oper res 27 1979 pp 161 186 and justify this improvement both theoretically and experimentally we prove that if the input arc lengths come from a natural probability distribution the new algorithm runs in linear average time while the original algorithm does not we also describe an implementation of the new algorithm our experimental data suggests that the new algorithm is preferable to the original one in practice furthermore for integral arc lengths that fit into a word of today s computers the performance is close to that of breadth first search suggesting limitations on further practical improvements

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble a method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail mutual nonlinear prediction also provides information on the directionality of the coupling between systems furthermore the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony numerical examples studied include three classes of unidirectionally coupled systems systems with identical parameters nonidentical parameters and stochastic driving of a nonlinear system this technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool the interrelationships examined include single neuron unit firing the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex and intracellular measurements of integrated excitatory postsynaptic potentials epsp s dynamical interdependence perhaps generalized synchrony was identified in this neuronal network between simultaneous single unit firings between units and the population and between units and intracellular epsp s

synchronization likelihood an unbiased measure of generalized synchronization in multivariate data sets the study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems typical examples are the electroencephalogram eeg and magnetoencephalogram meg which may involve the simultaneous recording of 150 or more time series coherency which is often used to study such data is only sensitive to linear and symmetric interdependencies and cannot deal with non stationarity recently several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates e g physica d 134 1999 419 brain res 792 1998 24 however these methods are biased by the degrees of freedom of the interacting subsystems physica d 134 1999 419 physica d 148 2001 147 we propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non stationary dynamics

a comparative study of synchrony measures for the early diagnosis of alzheimer s disease based on eeg it is well known that eeg signals of alzheimer s disease ad patients are generally less synchronous than in age matched control subjects however this effect is not always easily detectable this is especially the case for patients in the pre symptomatic phase commonly referred to as mild cognitive impairment mci during which neuronal degeneration is occurring prior to the clinical symptoms appearance in this paper various synchrony measures are studied in the context of ad diagnosis including the correlation coefficient mean square and phase coherence granger causality phase synchrony indices information theoretic divergence measures state space based measures and the recently proposed stochastic event synchrony measures experiments with eeg data show that many of those measures are strongly correlated or anti correlated with the correlation coefficient and hence provide little complementary information about eeg synchrony measures that are only weakly correlated with the correlation coefficient include the phase synchrony indices granger causality measures and stochastic event synchrony measures in addition those three families of synchrony measures are mutually uncorrelated and therefore they each seem to capture a specific kind of interdependence for the data set at hand only two synchrony measures are able to convincingly distinguish mci patients from age matched control patients i e granger causality in particular full frequency directed transfer function and stochastic event synchrony those two measures are used as features to distinguish mci patients from age matched control subjects yielding a leave one out classification rate of 83 the classification performance may be further improved by adding complementary features from eeg this approach may eventually lead to a reliable eeg based diagnostic tool for mci and ad

estimating coupling direction between neuronal populations with permutation conditional mutual information to further understand functional connectivity in the brain we need to identify the coupling direction between neuronal signals recorded from different brain areas in this paper we present a novel methodology based on permutation analysis and conditional mutual information for estimation of a directionality index between two neuronal populations first the reliability of this method is numerically assessed with a coupled mass neural model the simulations show that this method is superior to the conditional mutual information method and the granger causality method for identifying the coupling direction between unidirectional or bidirectional neuronal populations that are generated by the mass neuronal model the method is also applied to investigate the coupling direction between neuronal populations in ca1 and ca3 in the rat hippocampal tetanus toxin model of focal epilepsy the propagation direction of the seizure events could be elucidated through this coupling direction estimation method all together these results suggest that the permutation conditional mutual information method is a promising technique for estimating directional coupling between mutually interconnected neuronal populations

symmetrizing the kullback leibler distance we define a new distance measure the resistor average distance between two probability distributions that is closely related to the kullback leibler distance while the kullbackleibler distance is asymmetric in the two distributions the resistor average distance is not it arises from geometric considerations similar to those used to derive the chernoff distance determining its relation to well known distance measures reveals a new way to depict how commonly used distance measures relate to each other

computing the zeros of periodic descriptor systems in this paper we give a numerically reliable algorithm to compute the zeros of a periodic descriptor system the algorithm is a variant of the staircase algorithm applied to the system pencil of an equivalent lifted time invariant state space system and extracts a low order pencil which contains the zeros both finite and infinite as well as the kronecker structure of the periodic descriptor system the proposed algorithm is efficient in terms of complexity by exploiting the structure of the pencil and is exclusively based on orthogonal transformations which ensures some form of numerical stability

convex p partitions of bipartite graphs a set of vertices x of a graph g is convex if no shortest path between two vertices in x contains a vertex outside x we prove that for fixed p 1 all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time

on the hausdorff dimension of fractal attractors we consider such mappingsx n 1 f xn of an interval into itself for which the attractor is a cantor set for the same class of mappings for which the feigenbaum scaling laws hold we show that the hausdorff dimension of the attractor is universally equal tod 0 538

stable inversion of lptv systems with application in position dependent and non equidistantly sampled systems abstractmany control applications including feedforward and learning control involve the inverse of a dynamical system for nonminimum phase systems the response of the inverse system is unbounded for linear time invariant lti nonminimum phase systems a bounded noncausal inverse response can be obtained through an exponential dichotomy for general linear time varying ltv systems such a dichotomy does not exist in general the aim of this paper is to develop an inversion approach for an important class of ltv systems namely linear periodically time varying lptv systems which occur in e g position dependent systems with periodic tasks and non equidistantly sampled systems the proposed methodology exploits the periodicity to determine a bounded inverse for general lptv systems conditions for existence are provided the method is successfully demonstrated in several application cases including position dependent and non equidistantly sampled systems

decremental single source shortest paths on undirected graphs in near linear total update time in the decremental single source shortest paths sssp problem we want to maintain the distances between a given source node s and every other node in an n node m edge graph g undergoing edge deletions while its static counterpart can be solved in near linear time this decremental problem is much more challenging even in the undirected unweighted case in this case the classic o mn total update time of even and shiloach 16 has been the fastest known algorithm for three decades at the cost of a 1 e approximation factor the running time was recently improved to n2 o 1 by bernstein and roditty 9 in this article we bring the running time down to near linear we give a 1 e approximation algorithm with m1 o 1 expected total update time thus obtaining near linear time moreover we obtain m1 o 1 log w time for the weighted case where the edge weights are integers from 1 to w the only prior work on weighted graphs in o mn time is the mn0 9 o 1 time algorithm by henzinger et al 18 19 which works for directed graphs with quasi polynomial edge weights the expected running time bound of our algorithm holds against an oblivious adversary in contrast to the previous results which rely on maintaining a sparse emulator our algorithm relies on maintaining a so called sparse h e hop set introduced by cohen 12 in the pram literature an h e hop set of a graph ge v e is a set f of weighted edges such that the distance between any pair of nodes in g can be 1 e approximated by their h hop distance given by a path containing at most h edges on g e v e f our algorithm can maintain an no 1 e hop set of near linear size in near linear time under edge deletions it is the first of its kind to the best of our knowledge to maintain approximate distances using this hop set we extend the monotone even shiloach tree of henzinger et al 20 and combine it with the bounded hop sssp technique of bernstein 4 5 and m dry 27 these two new tools might be of independent interest

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

improving space efficiency with path length prediction for finding k shortest simple paths finding mbi k shortest simple paths in a directed graph is a fundamental problem in many engineering applications most existing algorithms such as yen s algorithm and its variants have polynomial worst case time complexity but their average case running time is very high the heuristic algorithm mps can run significantly faster in practice however it requires an excessive amount of memory space in this paper we provide a new heuristic algorithm that achieves high space efficiency while maintaining similar average case running time we first propose a sidetrack representation of path with which a path can be stored in mbi o bf 1 space we then show how to categorize a candidate path as either partial or complete and restrict the number of paths added to the queue in addition we provide an empirical equation that can very accurately predict the mbi k th shortest path length provided that a much smaller number of shortest paths have been found extensive experiments prove that our algorithm can achieve an mbi o n speedup in practice over yen s algorithm in comparison with mps it runs up to three times faster and uses less space by an order of magnitude

on the detection of determinism in a time series the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with that derived from its surrogate counterpart to assess its chaoticity or the stochastic nature chaoticity manifests in the relatively decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process the distributions of the strengths of the modes for the data and the surrogate are compared statistically singular value decomposition is used for the implementation the ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated series including linear and nonlinear stochastic noise and processes described by lorenz rossler and mackey glass equations henon map and ecological map and some real life examples

fast and compact regular expression matching we study 4 problems in string matching namely regular expression matching approximate regular expression matching string edit distance and subsequence indexing on a standard word ram model of computation that allows logarithmic sized words to be manipulated in constant time we show how to improve the space and or remove a dependency on the alphabet size for each problem using either an improved tabulation technique of an existing algorithm or by combining known algorithms in a new way

quantifying statistical interdependence part iii n 2 point processes stochastic event synchrony ses is a recently proposed family of similarity measures first events are extracted from the given signals next one tries to align events across the different time series the better the alignment the more similar the n time series are considered to be the similarity measures quantify the reliability of the events the fraction of nonaligned events and the timing precision so far ses has been developed for pairs of one dimensional part i and multidimensional part ii point processes in this letter part iii ses is extended from pairs of signals to n 2 signals the alignment and ses parameters are again determined through statistical inference more specifically by alternating two steps 1 estimating the ses parameters from a given alignment and 2 with the resulting estimates refining the alignment the ses parameters are computed by maximum a posteriori map estimation step 1 in analogy to the pairwise case the alignment stepi 2 is solved by linear integer programming in order to test the robustness and reliability of the proposed n variate ses method it is first applied to synthetic data we show that n variate ses results in more reliable estimates than bivariate ses next n variate ses is applied to two problems in neuroscience to quantify the firing reliability of morris lecar neurons and to detect anomalies in eeg synchrony of patients with mild cognitive impairment those problems were also considered in parts i and ii respectively in both cases the n variate ses approach yields a more detailed analysis

assessment of eeg synchronization based on state space analysis cortical computation involves the formation of cooperative neuronal assemblies characterized by synchronous oscillatory activity a traditional method for the identification of synchronous neuronal assemblies has been the coherence analysis of eeg signals here we suggest a new method called s estimator whereby cortical synchrony is defined from the embedding dimension in a state space we first validated the method on clusters of chaotic coupled oscillators and compared its performance to that of other methods for assessing synchronization then nine adult subjects were studied with high density eeg recordings while they viewed in the two hemifields hence with separate hemispheres identical sinusoidal gratings either arranged collinearly and moving together or orthogonally oriented and moving at 90 degrees the estimated synchronization increased with the collinear gratings over a cluster of occipital electrodes spanning both hemispheres whereas over temporo parietal regions of both hemispheres it decreased with the same stimulus and it increased with the orthogonal gratings separate calculations for different eeg frequencies showed that the occipital clusters involved synchronization in the beta band and the temporal clusters in the alpha band the gamma band appeared to be insensitive to stimulus diversity different stimulus configurations therefore appear to cause a complex rearrangement of synchronous neuronal assemblies distributed over the cortex in particular over the visual cortex

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

linear time approximation for maximum weight matching the maximum cardinality and maximum weight matching problems can be solved in o m n time a bound that has resisted improvement despite decades of research here m and n are the number of edges and vertices in this article we demonstrate that this m n barrier can be bypassed by approximation for any e 0 we give an algorithm that computes a 1 e approximate maximum weight matching in o me 1 log e 1 time that is optimal linear time for any fixed e our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error

a shortest path algorithm for real weighted undirected graphs we present a new scheme for computing shortest paths on real weighted undirected graphs in the fundamental comparison addition model in an efficient preprocessing phase our algorithm creates a linear size structure that facilitates single source shortest path computations in o m log alpha time where alpha alpha m n is the very slowly growing inverse ackermann function m the number of edges and n the number of vertices as special cases our algorithm implies new bounds on both the all pairs and single source shortest paths problems we solve the all pairs problem in o mn log alpha m n time and if the ratio between the maximum and minimum edge lengths is bounded by n log n o 1 we can solve the single source problem in o m n log log n time both these results are theoretical improvements over dijkstra s algorithm which was the previous best for real weighted undirected graphs our algorithm takes the hierarchy based approach invented by thorup

all pairs shortest paths using bridging sets and rectangular matrix multiplication we present two new algorithms for solving the all pairs shortest paths apsp problem for weighted directed graphs both algorithms use fast matrix multiplication algorithms the first algorithm solves the apsp problem for weighted directed graphs in which the edge weights are integers of small absolute value in o n2 time where satisfies the equation 1 1 1 2 and 1 1 is the exponent of the multiplication of an n n matrix by an n n matrix currently the best available bounds on 1 1 obtained by coppersmith imply that 0 is an error parameter and w is the largest edge weight in the graph after the edge weights are scaled so that the smallest non zero edge weight in the graph is 1 it returns estimates of all the distances in the graph with a stretch of at most 1 e corresponding paths can also be found efficiently

fusion trees can be implemented with ac 0 instructions only addressing a problem of fredman and willard we implement fusion trees in deterministic linear space using ac o instructions only more precisely we show that a subset of 0 2 w 1 of size n

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

danco an intrinsic dimensionality estimator exploiting angle and norm concentration abstract in the past decade the development of automatic intrinsic dimensionality estimators has gained considerable attention due to its relevance in several application fields however most of the proposed solutions prove to be not robust on noisy datasets and provide unreliable results when the intrinsic dimensionality of the input dataset is high and the manifold where the points are assumed to lie is nonlinearly embedded in a higher dimensional space in this paper we propose a novel intrinsic dimensionality estimator danco and its faster variant fastdanco which exploit the information conveyed both by the normalized nearest neighbor distances and by the angles computed on couples of neighboring points the effectiveness and robustness of the proposed algorithms are assessed by experiments on synthetic and real datasets by the comparative evaluation with state of the art methodologies and by significance tests

on the zero freeness of tall multirate linear systems in this technical note tall discrete time linear systems with multirate outputs are studied in particular we focus on their zeros in systems and control literature zeros of multirate systems are defined as those of their corresponding time invariant systems obtained through blocking of the original multirate systems we assume that blocked systems are tall i e have more outputs than inputs it is demonstrated that for generic choice of the parameter matrices linear systems with multirate outputs generically have no finite nonzero zeros however they may have zeros at the origin or at infinity depending on the choice of blocking delay and the input state and output dimensions

intrinsic dimension estimation of manifolds by incising balls dimensionality reduction is a very important tool in data mining intrinsic dimension of data sets is a key parameter for dimensionality reduction however finding the correct intrinsic dimension is a challenging task in this paper a new intrinsic dimension estimation method is presented the estimator is derived by finding the exponential relationship between the radius of an incising ball and the number of samples included in the ball the method is compared with the previous dimension estimation methods experiments have been conducted on synthetic and high dimensional image data sets and on data sets of the santa fe time series competition and the results show that the new method is accurate and robust

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

compact oracles for reachability and approximate distances in planar digraphs it is shown that a planar digraph can be preprocessed in near linear time producing a near linear space oracle that can answer reachability queries in constant time the oracle can be distributed as an o log n space label for each vertex and then we can determine if one vertex can reach another considering their two labels only the approach generalizes to give a near linear space approximate distances oracle for a weighted planar digraph with weights drawn from 0 n it approximates distances within a factor 1 e in o log log nn 1 e time our scheme can be extended to find and route along correspondingly short dipaths

improved parallel integer sorting without concurrent writing we show thatnintegers in the range 1 ncan be sorted stably on an erew pram usingo t time ando n lognloglogn logn 2 t operations for arbitrary givent lognloglogn and on a crew pram usingo t time ando n logn logn 2t logn operations for arbitrary givent logn in addition we are able to sortnarbitrary integers on a randomized crew pram within the same resource bounds with high probability in each case our algorithm is a factor of almost logn closer to optimality than all previous algorithms for the stated problem in the stated model and our third result matches the operation count of the best previous sequential algorithm we also show thatnintegers in the range 1 mcan be sorted ino logn 2 time witho n operations on an erew pram using a nonstandard word length ofo lognloglognlogm bits thereby greatly improving the upper bound on the word length necessary to sort integers with a linear time processor product even sequentially our algorithms were inspired by and in one case directly use the fusion trees of fredman and willard

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

kulback leibler and renormalized entropies applications to electroencephalograms of epilepsy patients recently renormalized entropy was proposed as a novel measure of relative entropy p saparin et al chaos solitons and fractals 4 1907 1994 and applied to several physiological time sequences including electroencephalograms eegs of patients with epilepsy we show here that this measure is just a modified kullback leibler kl relative entropy and it gives similar numerical results to the standard kl entropy the latter better distinguishes frequency contents of e g seizure and background eegs than renormalized entropy we thus propose that renormalized entropy might not be as useful as claimed by its proponents in passing we also make some critical remarks about the implementation of these methods

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

detection of synchronization between chaotic signals an adaptive similarity based approach we present an adaptive similarity based approach to detect generalized synchronization gs with n m phase synchronization ps where n and m are integers and one of them is 1 this approach is based on the similarity index si and gaussian mixture model with the minimum description length criterion the clustering method which is shown to be superior to the closeness and connectivity of a continuous function is employed in this study to detect the existence of gs with n m ps we conducted a computer simulation and a finger lifting experiment to illustrate the effectiveness of the proposed method in the simulation of a rossler lorenz system our method outperformed the conventional si and gs with 2 1 ps within the coupled system was found in the experiment of self paced finger lifting movement cortico muscular gs with 1 2 and 1 3 ps was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area the gs with n m ps n or m 1 has been simultaneously resolved from both simulation and experiment the proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science

functional brain networks in alzheimer s disease eeg analysis based on limited penetrable visibility graph and phase space method abstract in this paper eeg series are applied to construct functional connections with the correlation between different regions in order to investigate the nonlinear characteristic and the cognitive function of the brain with alzheimer s disease ad first limited penetrable visibility graph lpvg and phase space method map single eeg series into networks and investigate the underlying chaotic system dynamics of ad brain topological properties of the networks are extracted such as average path length and clustering coefficient it is found that the network topology of ad in several local brain regions are different from that of the control group with no statistically significant difference existing all over the brain furthermore in order to detect the abnormality of ad brain as a whole functional connections among different brain regions are reconstructed based on similarity of clustering coefficient sequence ccss of eeg series in the four frequency bands delta theta alpha and beta which exhibit obvious small world properties graph analysis demonstrates that for both methodologies the functional connections between regions of ad brain decrease particularly in the alpha frequency band ad causes the graph index complexity of the functional network decreased the small world properties weakened and the vulnerability increased the obtained results show that the brain functional network constructed by lpvg and phase space method might be more effective to distinguish ad from the normal control than the analysis of single series which is helpful for revealing the underlying pathological mechanism of the disease

asymptotic wavelet and gabor analysis extraction of instantaneous frequencies the behavior of the continuous wavelet and gabor coefficients in the asymptotic limit using stationary phase approximations are investigated in particular it is shown how under some additional assumptions these coefficients allow the extraction of some characteristics of the analyzed signal such as frequency and amplitude modulation laws applications to spectral line estimations and matched filtering are briefly discussed

zeros of discrete time linear periodic systems in this note the concept of zero for linear discrete time siso periodic systems is discussed the definition is based on a time invariant mimo equivalent representation an input output characterization of the zeros is provided which extends a well known property for time invariant systems

sampled data and discrete time h_2 optimal control this paper deals with the sampled data h 2 optimal control problem given a linear time invariant continuous time system the problem of minimizing the h 2 performance over all sampled data controllers with a fixed sampling period can be reduced to a pure discrete time h 2 optimal control problem this discrete time h 2 problem is always singular motivated by this in this paper we give a treatment of the discrete time h 2 optimal control problem in its full generality the results we obtain are then applied to the singular discrete time h 2 problem arising from the sampled data h 2 problem in particular we give conditions for the existence of optimal sampled data controllers we also show that the h 2 performance of a continuous time controller can always be recovered asymptotically by choosing the sampling period sufficiently small finally we show that the optimal sampled data h 2 performance converges to the continuous time optimal h 2 performance as the sampling period converges to zero

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

absence of long range order in one dimensional spin systems for a one dimensional ising model with interaction energy e left mu right sum limits 1 leqslant i j leqslant n j j i mu iota mu j left j k geqslant 0 mu iota pm 1 right it is proved that there is no long range order at any temperature when s n sum limits k 1 n kj left k right o left log n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right the same result is shown to hold for the corresponding plane rotator model when s n o left left log n mathord left vphantom log n log log n right kern nulldelimiterspace log log n right right

granger causality and information flow in multivariate processes the multivariate versus bivariate measures of granger causality were considered granger causality in the application to multivariate physiological time series has the meaning of the information flow between channels it was shown by means of simulations and by the example of experimental electroencephalogram signals that bivariate estimates of directionality in case of mutually interdependent channels give erroneous results therefore multivariate measures such as directed transfer function should be used for determination of the information flow

fast greedy algorithms for constructing sparse geometric spanners given a set v of n points in ir d and a real constant t 1 we present the first o nlog n time algorithm to compute a geometric t spanner on v a geometric t spanner on v is a connected graph g v e with edge weights equal to the euclidean distances between the endpoints and with the property that for all u v in v the distance between u and v in g is at most t times the euclidean distance between u and v the spanner output by the algorithm has o n edges and weight o 1 cdot wt mst and its degree is bounded by a constant

towards a comprehensive theory of brain activity coupled oscillator systems under external forces abstract recently jirsa et al and haken discussed a theory comprising the brain wave equation proposed by nunez the wilson cowan ermentrout cowan model and hopfield networks this theory was applied to model findings obtained in an experiment that relates brain activity and behavior the so called julliard experiment in previous works of jirsa et al and frank et al the focus was on the brain wave aspect recently we conducted similar experiments the results obtained are modeled in this paper in terms of coupled oscillator systems coupled oscillator systems are considered to represent the wilson cowan ermentrout cowan model aspect of the unifying theory building on a model proposed by haken kelso and bunz and the theory of weakly coupled oscillators established by winfree and by kuramoto we derive a nonlinear fokker planck equation whose stationary solutions mimic the neocortical brain activity observed in our experiments

partial synchronization of chaotic systems with uncertainty we suggest an approach to partial synchronization of chaotic systems with uncertainty this method contains two steps i transforming the synchronization system into the canonical form by the well known feedback linearization theory and ii finding a control signal to ensure the asymptotic stability of the canonical system this partial synchronization approach requires very little system information by applying a finite time convergence technique to estimate uncertainties caused by unknown states parameters or structure we also argue in detail that this partial synchronization method can be extended to parameter identification sub structure estimation and even phase detection several examples are presented to illustrate the partial synchronization approach suggested

quantifying time varying coordination of multimodal speech signals using correlation map analysis this paper demonstrates an algorithm for computing the instantaneous correlation coefficient between two signals the algorithm is the computational engine for analyzing the time varying coordination between signals which is called correlation map analysis cma correlation is computed around any pair of points in the two input signals thus coordination can be assessed across a continuous range of temporal offsets and be detected even when changing over time due to temporal fluctuations the correlation algorithm has two major features i it is structurally similar to a tunable filter requiring only one parameter to set its cutoff frequency and sensitivity ii it can be applied either uni directionally computing correlation based only on previous samples or bi directionally computing correlation based on both previous and future samples computing instantaneous correlation for a range of time offsets between two signals produces a 2d correlation map in which correlation is characterized as

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

optimal detection of functional connectivity from high dimensional eeg synchrony data computing phase locking values between eeg signals is a popular method for quantifying functional connectivity however this method involves large scale high resolution datasets which impose a serious multiple testing problem standard multiple testing methods fail to exploit the information from the complex dependence structure that varies across hypotheses in spectral temporal and spatial dimensions and result in a severe loss of power they tend to control the false positives at the cost of hiding true positives we introduce a new approach called optimal discovery procedure odp for identifying synchrony that is statistically significant odp maximizes the number of true positives for a given number of false positives and thus offers a theoretical optimum for detecting significant synchrony in a multiple testing situation we demonstrate the utility of this method with plv data obtained from a visual search study we also present simulation analysis to confirm the validity and relevance of using odp in comparison with the standard fdr method for given configurations of true synchrony we also compare the effectiveness of odp with our previously published investigation of hierarchical fdr method singh and phillips 2010

all pairs shortest paths in o n 2 time with high probability we present an all pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from 0 1 is o n2 in expectation and with high probability this resolves a long standing open problem the algorithm is a variant of the dynamic all pairs shortest paths algorithm of demetrescu and italiano 2006 the analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is o n2 in expectation and with high probability we also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in o log2n expected time

time delay embedding increases estimation precision of models of intraindividual variability this paper investigates the precision of parameters estimated from local samples of time dependent functions we find that time delay embedding i e structuring data prior to analysis by constructing a data matrix of overlapping samples increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data we show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function hence we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions we support these conclusions both by mathematical analysis and two simulations

a permutation entropy based test for causality the volume stock price relation abstract the purpose of this paper is to propose a newly developed non parametric test for linear and nonlinear causality based on permutation entropy and to show its usefulness in analyzing the potential causal relationship between trading volume and security prices most of the empirical applications and tests for causality rely on using granger causality based test for linear models although these tests have high power in uncovering linear causal relations their power against nonlinear causal relations can be low our test is designed to deal with the detection of linear and non linear causality we also compare our permutation entropy based test with other granger causality tests monte carlo simulations show excellent performance in terms of size and power of the new test for detecting linear and non linear causality under different scenarios our conclusions point that there is a bidirectional causal relation from volume to price returns not only in the mean but also in the variance

radial basis function approach to nonlinear granger causality of time series we consider an extension of granger causality to nonlinear bivariate time series in this frame if the prediction error of the first time series is reduced by including measurements from the second time series then the second time series is said to have a causal influence on the first one not all the nonlinear prediction schemes are suitable to evaluate causality indeed not all of them allow one to quantify how much knowledge of the other time series counts to improve prediction error we present an approach with bivariate time series modeled by a generalization of radial basis functions and show its application to a pair of unidirectionally coupled chaotic maps and to physiological examples

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

the necessity for a time local dimension in systems with time varying attractors abstract we show that a simple non linear system for ordinary differential equations may possess a time varying attractor dimension this indicates that it is infeasible to characterize eeg and meg time series with a single time global dimension we suggest another measure for the description of non stationary attractors

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

undirected single source shortest paths with positive integer weights in linear time the single source shortest paths problem sssp is one of the classic problems in algorithmic graph theory given a positively weighted graph g with a source vertex s find the shortest path from s to all other vertices in the graph since 1959 all theoretical developments in sssp for general directed and undirected graphs have been based on dijkstra s algorithm visiting the vertices in order of increasing distance from s thus any implementation of dijkstra s algorithm sorts the vertices according to their distances from s however we do not know how to sort in linear time here a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights the algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure identifying vertex pairs that may be visited in any order

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems