error threshold in optimal coding numerical criteria and classes of universalities for complexity the free energy of the random energy model at the transition point between the ferromagnetic and spin glass phases is calculated at this point equivalent to the decoding error threshold in optimal codes the free energy has finite size corrections proportional to the square root of the number of degrees the response of the magnetization to an external ferromagnetic phase is maximal at values of magnetization equal to one half we give several criteria of complexity and define different universality classes according to our classification at the lowest class of complexity are random graphs markov models and hidden markov models at the next level is the sherrington kirkpatrick spin glass connected to neuron network models on a higher level are critical theories the spin glass phase of the random energy model percolation and self organized criticality the top level class involves highly optimized tolerance design error thresholds in optimal coding language and maybe financial markets living systems are also related to the last class the concept of antiresonance is suggested for complex systems

reductions of hidden information sources in all but special circumstances measurements of time dependent processes reflect internal structures and correlations only indirectly building predictive models of such hidden information sources requires discovering in some way the internal states and mechanisms unfortunately there are often many possible models that are observationally equivalent here we show that the situation is not as arbitrary as one would think we show that generators of hidden stochastic processes can be reduced to a minimal form and compare this reduced representation to that provided by computational mechanics the e machine on the way to developing deeper measure theoretic foundations for the latter we introduce a new two step reduction process the first step internal event reduction produces the smallest observationally equivalent algebra and the second internal state reduction removes algebra components that are redundant for optimal prediction for several classes of stochastic dynamical systems these reductions produce representations that are equivalent to e machines

automated pattern detection an algorithm for constructing optimally synchronizing multi regular language filters in the computational mechanics structural analysis of one dimensional cellular automata the following automata theoretic analogue of the change point problem from time series analysis arises given a string and a collection di offinite automata identify the regions of that belong to each di and in particular the boundaries separating them we present two methods for solving this multi regular language filtering problem the first although providing the ideal solution requires a stack has a worst case compute time that grows quadratically in s length and conditions its output at any point on arbitrarily long windows of future input the second method is to algorithmically construct a finite transducer that approximates the first algorithm in contrast to the stack based algorithm however the transducer requires only a finite amount of memory runs in linear time and gives immediate output for each letter read it is moreover the best possible finite state approximation with these three features

nonregular languages in the kicked rotor a symbolic dynamics of the markov chain model for trajectories of the kicked rotor standard map in sticky regions near resonant orbits is developed two sets of trajectories are shown to correspond to a context free and a context sensitive language respectively

on maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean abstract there are two approaches to maximum likelihood ml estimation of the parameter of fractionally integrated noise approximate frequency domain ml fox and taqqu 1986 and exact time domain ml sowell 1992b if the mean of the process is known then a clear finite sample mean squared error ranking of the estimators emerges the exact time domain estimator is superior we show in this paper however that the finite sample efficiency of approximate frequency domain ml relative to exact time domain ml rises dramatically when the mean is unknown and so must be estimated the intuition for our result is straightforward the frequency domain ml estimator is invariant to the true but unknown mean of the process while the time domain ml estimator is not feasible time domain estimation must therefore be based upon de meaned data but the long memory associated with fractional integration makes precise estimation of the mean difficult we conclude that the frequency domain estimator is an attractive and efficient alternative for situations in which large sample sizes render time domain estimation impractical

rigorous properties and uncertainty like relationships on product complexity measures application to atomic systems most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework valid for any arbitrary n dimensional distribution in the present work different product complexities constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated different results on product complexities such as rigorous bounds uncertainty like inequalities relationships among themselves and also statistical interpretations are given for the sake of completeness some of these analytical results on product complexities are numerically checked for the one particle densities of atomic systems in both conjugated spaces

coupled replicator equations for the dynamics of learning in multiagent systems starting with a group of reinforcement learning agents we derive coupled replicator equations that describe the dynamics of collective learning in multiagent systems we show that although agents model their environment in a self interested way without sharing knowledge a game dynamics emerges naturally through environment mediated interactions an application to rock scissors paper game interactions shows that the collective learning dynamics exhibits a diversity of competitive and cooperative behaviors these include quasiperiodicity stable limit cycles intermittency and deterministic chaos behaviors that should be expected in heterogeneous multiagent systems described by the general replicator equations we derive

evolving cellular automata to perform computations mechanisms and impediments abstract we present results from experiments in which a genetic algorithm ga was used to evolve cellular automata cas to perform a particular computational task one dimensional density classification we look in detail at the evolutionary mechanisms producing the ga s behavior on this task and the impediments faced by the ga in particular we identify four epochs of innovation in which new ca strategies for solving the problem are discovered by the ga describe how these strategies are implemented in ca rule tables and identify the ga mechanisms underlying their discovery the epochs are characterized by a breaking of the task s symmetries on the part of the ga the symmetry breaking results in a short term fitness gain but ultimately prevents the discovery of the most highly fit strategies we discuss the extent to which symmetry breaking and other impediments are general phenomena in any ga search

learning hidden markov models using nonnegative matrix factorization the baum welch algorithm together with its derivatives and variations has been the main technique for learning hidden markov models hmms from observational data we present an hmm learning algorithm based on the nonnegative matrix factorization nmf of higher order markovian statistics that is structurally different from the baum welch and its associated approaches the described algorithm supports estimation of the number of recurrent states of an hmm and iterates the nmf algorithm to improve the learned hmm parameters numerical examples are provided as well

role of finite populations in determining evolutionary dynamics the connection between the finite size of an evolving population and its dynamical behavior is examined through analytical and computational studies of a simple model of evolution the infinite population limit of the model is shown to be governed by a special case of the quasispecies equations a flat fitness landscape yields identical results for the dynamics of infinite and finite populations on the other hand a monotonically increasing fitness landscape shows epochs in the dynamics of finite populations that become more pronounced as the rate of mutation decreases the details of the dynamics are profoundly different for any two simulation runs in that events arising from the stochastic noise in the pseudorandom number sequence are amplified as the population size is increased or equivalently the mutation rate is increased these epochs become smaller but do not entirely disappear

on estimation of the integrals of the fourth order cumulant spectral density summary it is known that the asymptotic variance of a quasimaximum likelihood estimate for a non gaussian process contains certain integrals of the fourth order cumulant spectral density if we apply the asymptotic theory we are required to estimate these integrals here we shall propose some operational consistent estimates for them

generalized inverses of morphisms with kernels abstract let x y be a morphism with kernel k x in an additive category with an involution then has a moore penrose inverse with respect to iff is invertible in this case 1 if x y then has a group inverse iff has a cokernel x k and 2 is invertible in this case 2 1

the moore penrose inverse of a morphism with factorization abstract given an m by n matrix a of rank r over a field with an involutory automorphism it is well known that a has a moore penrose inverse if and only if rank a a r rank aa by use of the full rank factorization theorem this result may be restated in the category of finite matrices as follows if a 1 r a 2 is an epic monic factorization of a m n through r then a has a moore penrose inverse if and only if a a 1 r a 2 and a 1 r a 2 a are respectively epic monic factorizations of a a n n and aa m m through r this characterization of the existence of moore penrose inverses is extended to arbitrary morphisms with epic monic factorizations

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

whittle estimation of egarch and other exponential volatility models abstract the strong consistency and asymptotic normality of the whittle estimate of the parameters in a class of exponential volatility processes are established our main focus here are the egarch model of nelson d 1991 conditional heteroscedasticity in asset pricing a new approach econometrica 59 347 370 and other one shock models such as the gjr model of glosten l jaganathan r runkle d 1993 on the relation between the expected value and the volatility of the nominal excess returns on stocks journal of finance 48 1779 1801 but two shock models such as the sv model of taylor s 1986 modelling financial time series wiley chichester uk are also comprised by our assumptions the variable of interest might not have finite fractional moment of any order and so in particular finite variance is not imposed we allow for a wide range of degrees of persistence of shocks to conditional variance allowing for both short and long memory

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

quasi maximum likelihood estimation in conditionally heteroscedastic time series a stochastic recurrence equations approach this paper studies the quasi maximum likelihood estimator qmle in a general conditionally heteroscedastic time series model of multiplicative form xt t zt where the unobservable volatility t is a parametric function of xt 1 x t p t 1 t q for some p q 0 and zt is standardized i i d noise we assume that these models are solutions to stochastic recurrence equations which satisfy a contraction random lipschitz coefficient property these assumptions are satisfied for the popular garch asymmetric garch and exponential garch processes exploiting the contraction property we give conditions for the existence and uniqueness of a strictly stationary solution xt to the stochastic recurrence equation and establish consistency and asymptotic normality of the qmle we also discuss the problem of invertibility of such time series models 1 introduction gaussian quasi maximum likelihood estimation that is likelihood estimation under the hypothesis of gaussian innovations is a popular method which is widely used for inference in time series models however it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi maximum likelihood estimator qmle applied to specific models and therefore an in depth analysis of the probabilistic structure generated by the model is called for a classical example of this kind is the seminal paper by hannan 18 on estimation in linear arma time series in this paper we study the qmle for a general class of conditionally heteroscedastic time series models which includes garch asymmetric garch and exponential garch recall that a garch p q generalized autoregressive conditionally heteroscedastic of order p q process 4 i s def ined by

integration of large deviation kernels and applications to large deviations for evolutionary games we investigate large deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large deviation principle the results of our analysis are then used to demonstrate the large deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes

statistical dynamics of the royal road genetic algorithm abstract metastability is a common phenomenon many evolutionary processes both natural and artificial alternate between periods of stasis and brief periods of rapid change in their behavior in this paper an analytical model for the dynamics of a mutation only genetic algorithm ga is introduced that identifies a new and general mechanism causing metastability in evolutionary dynamics the ga s population dynamics is described in terms of flows in the space of fitness distributions the trajectories through fitness distribution space are derived in closed form in the limit of infinite populations we then show how finite populations induce metastability even in regions where fitness does not exhibit a local optimum in particular the model predicts the occurrence of fitness epochs periods of stasis in population fitness distributions at finite population size and identifies the locations of these fitness epochs with the flow s hyperbolic fixed points this enables exact predictions of the metastable fitness distributions during the fitness epochs as well as giving insight into the nature of the periods of stasis and the innovations between them all these results are obtained as closed form expressions in terms of the ga s parameters an analysis of the jacobian matrices in the neighborhood of an epoch s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space s topological structure more general quantitative features of the dynamics fitness fluctuation amplitudes epoch stability and speed of the innovations are also determined from the jacobian eigenvalues the analysis shows how quantitative predictions for a range of dynamical behaviors that are specific to the finite population dynamics can be derived from the solution of the infinite population dynamics the theoretical predictions are shown to agree very well with statistics from ga simulations we also discuss the connections of our results with those from population genetics and molecular evolution theory

sharp asymptotics for fixation times in stochastic population genetics models at low mutation probabilities this article describes new models in population genetics that extend the neutral wright fisher model by including strong selection and mutation fixation times are studied in the limit of small mutation probabilities within the framework of markov chains with rare transitions the main result outlines the role of the discrete geometry of the fitness landscape and provides a mean for estimating the expected number of generations for an individual with better fitness value to appear some connections to evolutionary algorithms are discussed as well

dynamics of boltzmann q learning in two player two action games we consider the dynamics of q learning in two player two action games with a boltzmann exploration mechanism for any nonzero exploration rate the dynamics is dissipative which guarantees that agent strategies converge to rest points that are generally different from the game s nash equlibria nes we provide a comprehensive characterization of the rest point structure for different games and examine the sensitivity of this structure with respect to the noise due to exploration our results indicate that for a class of games with multiple nes the asymptotic behavior of learning dynamics can undergo drastic changes at critical exploration rates furthermore we demonstrate that for certain games with a single ne it is possible to have additional rest points not corresponding to any ne that persist for a finite range of the exploration rates and disappear when the exploration rates of both players tend to zero

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

stability and diversity in collective adaptation we derive a class of macroscopic differential equations that describe collective adaptation starting from a discrete time stochastic microscopic model the behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior we show that although individual agents interact with their environment and other agents in a purely self interested way macroscopic behavior can be interpreted as game dynamics application to several familiar explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors the simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation we also analyze the adaptation dynamics from an information theoretic viewpoint and discuss self organization induced by the dynamics of uncertainty giving a novel view of collective adaptation

the detection and estimation of long memory in stochastic volatility abstract we propose a new time series representation of persistence in conditional variance called a long memory stochastic volatility lmsv model the lmsv model is constructed by incorporating an arfima process in a standard stochastic volatility scheme strongly consistent estimators of the parameters of the model are obtained by maximizing the spectral approximation to the gaussian likelihood the finite sample properties of the spectral likelihood estimator are analyzed by means of a monte carlo study an empirical example with a long time series of stock prices demonstrates the superiority of the lmsv model over existing short memory volatility models

the dynamics of a genetic algorithm for a simple learning problem a formalism for describing the dynamics of genetic algorithms gas using method s from statistical mechanics is applied to the problem of generalization in a perceptron with binary weights the dynamics are solved for the case where a new batch of training patterns is presented to each population member each generation which considerably simplifies the calculation the theory is shown to agree closely to simulations of a real ga averaged over many runs accurately predicting the mean best solution found for weak selection and large problem size the difference equations describing the dynamics can be expressed analytically and we find that the effects of noise due to the finite size of each training batch can be removed by increasing the population size appropriately if this population resizing is used one can deduce the most computationally efficient size of training batch each generation for independent patterns this choice also gives the minimum total number of training patterns used although using independent patterns is a very inefficient use of training patterns in general this work may also prove useful for determining the optimum batch size in the case where patterns are recycled

the quasispecies regime for the simple genetic algorithm with ranking selection we study the simple genetic algorithm with a ranking selection mechanism linear ranking or tournament we denote by the length of the chromosomes by m the population size by pc the crossover probability and by pm the mutation probability we introduce a parameter called the selection drift which measures the selection intensity of the fittest chromosome we show that the dynamics of the genetic algorithm depend in a critical way on the parameter 1 pc 1 pm if 1 then the genetic algorithm operates in a quasispecies regime an advantageous mutant invades a positive fraction of the population with probability larger than a constant p which does not depend on m we estimate next the probability of the occurrence of a catastrophe the whole population falls below a fitness level which was previously reached by a positive fraction of the population the asymptotic results suggest the following rules 1 pc 1 pm should be slightly larger than 1 pm should be of order 1 m should be larger than ln the running time should be of exponential order in m the first condition requires that pm pc ln these conclusions must be taken with great care they come from an asymptotic regime and it is a formidable task to understand the relevance of this regime for a real world problem at least we hope that these conclusions provide interesting guidelines for the practical implementation of the simple genetic algorithm

turbulence in diffusion replicator equation dynamical behaviors in the diffusion replicator equation of three species are numerically studied we point out the significant role of the heteroclinic cycle in the equation and analyze the details of the turbulent solution that appeared in this system firstly the bifurcation diagram for a certain parameter setting is drawn then it is shown that the turbulence appears with the supercritical hopf bifurcation of a stationary uniform solution and it disappears under a subcritical type bifurcation secondly the statistical property of the turbulence near the supercritical hopf onset point is analyzed precisely further the correlation lengths and correlation times obey some characteristic scaling laws

symmetric morphisms and the existence of moore penrose inverses abstract let x y be a morphism of an additive category with an involution then has a moore penrose inverse with respect to iff is invertible for some annihilator n x of iff there exists an object z and morphisms x z z z z y such that 2 and 1 z and 1 z are invertible in this case 1 1 z 1 1 z 1

drift analysis and average time complexity of evolutionary algorithms the computational time complexity is an important topic in the theory of evolutionary algorithms eas this paper reports some new results on the average time complexity of eas based on drift analysis some useful drift conditions for deriving the time complexity of eas are studied including conditions under which an ea will take no more than polynomial time in problem size to solve a problem and conditions under which an ea will take at least exponential time in problem size to solve a problem the paper first presents the general results and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of eas while previous work only considered 1 1 eas without any crossover the eas considered in this paper are fairly general which use a finite population crossover mutation and selection 2001 elsevier science b v all rights reserved

complexity of dynamics as variability of predictability we construct a complexity measure from first principles as an average over the obstruction against prediction of some observable that can be chosen by the observer our measure evaluates the variability of the predictability for characteristic system behaviors which we extract by means of the thermodynamic formalism using theoretical and experimental applications we show that complex and chaotic are different notions of perception in comparison to other proposed measures of complexity our measure is easily computable non divergent for the classical 1 d dynamical systems and has properties of non overuniversality the measure can also be computed for higher dimensional and experimental systems including systems composed of different attractors moreover the results of the computations made for classical 1 d dynamical systems imply that it is not the nonhyperbolicity but the existence of a continuum of characteristic system length scales that is at the heart of complexity

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

complexity and fisher information the connection between an observer based theory of measurement and a hierarchical classification of complex systems in terms of topological exponents is discussed this leads to generalized definitions of complexity that capture different aspects of the structure of the trajectory space of complex systems

reexamination of an information geometric construction of entropic indicators of complexity abstract information geometry and inductive inference methods can be used to model dynamical systems in terms of their probabilistic description on curved statistical manifolds in this article we present a formal conceptual reexamination of the information geometric construction of entropic indicators of complexity for statistical models specifically we present conceptual advances in the interpretation of the information geometric entropy ige a statistical indicator of temporal complexity chaoticity defined on curved statistical manifolds underlying the probabilistic dynamics of physical systems

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

rates of convergence and optimal spectral bandwidth for long range dependence for a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like 1 2h as 0 for 1 2 h 1 apart from a slowly varying factor which may be of unknown form we consider a discrete average of the periodogram across the frequencies 2 j n j 1 m wherem andm n 0 asn we study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates and deduce an optimal choice ofm

information accessibility and cryptic processes we give a systematic expansion of the crypticity a recently introduced measure of the inaccessibility of a stationary process s internal state information this leads to a hierarchy of k cryptic processes and allows us to identify finite state processes that have infinite cryptic order the internal state information is present across arbitrarily long observed sequences the crypticity expansion is exact in both the finite and infinite order cases it turns out that k crypticity is complementary to the markovian finite order property that describes state information in processes one application of these results is an efficient expansion of the excess entropy the mutual information between a process s infinite past and infinite future that is finite and exact for finite order cryptic processes

context tree modeling of observed symbolic dynamics modern techniques invented for data compression provide efficient automated algorithms for the modeling of the observed symbolic dynamics we demonstrate the relationship between coding and modeling motivating the well known minimum description length mdl principle and give concrete demonstrations of the context tree weighting and context tree maximizing algorithms the predictive modeling technique obviates many of the technical difficulties traditionally associated with the correct mdl analyses these symbolic models representing the symbol generating process as a finite state automaton with probabilistic emission probabilities provide excellent and reliable entropy estimations the resimulations of estimated tree models satisfying the mdl model selection criterion are faithful to the original in a number of measures the modeling suggests that the automated context tree model construction could replace fixed order word lengths in many traditional forms of empirical symbolic analysis of the data we provide an explicit pseudocode for implementation of the context tree weighting and maximizing algorithms as well as for the conversion to an equivalent markov chain

chaotic switching in a two person game abstract we study robust long term complex behaviour in the rock scissors paper game with two players played using reinforcement learning the complex behaviour is connected to the existence of a heteroclinic network for the dynamics this network is made of three heteroclinic cycles consisting of nine equilibria and the trajectories connecting them we provide analytical proof both for the existence of chaotic switching near the heteroclinic network and for the relative asymptotic stability of at least one cycle in the network leading to some trajectories paying alternate visits to only two nodes while others follow a complicated sequence of visits to all nodes in the network our results are obtained by making use of the symmetry of the original problem a new approach in the context of learning and they provide an explanation of numerical results previously obtained by other authors

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

comment i on simple measure for complexity we critique the measure of complexity introduced by shiner davison and landsberg phys rev e 59 1459 1999 in particular we point out that it is over universal in the sense that it has the same dependence on disorder for structurally distinct systems we then give counterexamples to the claim that complexity is synonymous with being out of equilibrium equilibrium systems can be structurally complex and nonequilibrium systems can be structurally simple we also correct a misinterpretation of a result given by two of the present authors j p crutchfield and d p feldman phys rev e 55 r1239 1997

optimal adaptive performance and delocalization in nk fitness landscapes abstract we investigate the evolutionary dynamics of a finite population of sequences adapting to nk fitness landscapes we find that unlike in the case of an infinite population the average fitness in a finite population is maximized at a small but finite rather than vanishing mutation rate the highest local maxima in the landscape are visited for even larger mutation rates close to a transition point at which the population delocalizes i e leaves the fitness peak at which it was localized and starts traversing the sequence space if the mutation rate is increased even further the population undergoes a second transition and loses all sensitivity to fitness peaks this second transition corresponds to the standard error threshold transition first described by eigen we discuss the implications of our results for biological evolution and for evolutionary optimization techniques

limit cycles and the benefits of a short memory in rock paper scissors games when playing games in groups it is an advantage for individuals to have accurate statistical information on the strategies of their opponents such information may be obtained by remembering previous interactions we consider a rock paper scissors game in which agents are able to recall their last m interactions used to estimate the behavior of their opponents at critical memory length a hopf bifurcation leads to the formation of stable limit cycles in a mixed population agents with longer memories have an advantage provided the system has a stable fixed point and there is some asymmetry in the payoffs of the pure strategies however at a critical concentration of long memory agents the appearance of limit cycles destroys their advantage by introducing population dynamics that favors successful agents we show that the system evolves toward the bifurcation point

is there switching for replicator dynamics and bimatrix games abstract we consider heteroclinic networks for replicator dynamics and bimatrix games that is in a simplex or product of simplices with equilibria at the vertices and connections at the edges edge networks switching dynamics near a heteroclinic network occurs whenever every infinite sequence of connections in the network is shadowed by at least one trajectory in its neighborhood aguiar and castro m a d aguiar s b s d castro chaotic switching in a two person game physica d 239 16 1598 1609 prove switching near an edge network for the dynamics of the rock scissors paper game here we give conditions for switching dynamics in general bimatrix games and show that switching near an edge network can never occur for replicator dynamics

estimating semiparametric arch models by kernel smoothing methods we investigate a class of semiparametric arch models that includes as a special case the partially nonparametric pnp model introduced by engle and ng 1993 and which allows for both flexible dynamics and flexible function form with regard to the news impact function we show that the functional part of the model satisfies a type ii linear integral equation and give simple conditions under which there is a unique solution we propose an estimation method that is based on kernel smoothing and profiled likelihood we establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model we also discuss efficiency of both the parametric part and the nonparametric part we investigate the performance of our procedures on simulated data and on a sample of s p500 index returns we find evidence of asymmetric news impact functions consistent with the parametric analysis

escort evolutionary game theory abstract a family of replicator like dynamics called the escort replicator dynamic is constructed using information geometric concepts and generalized information divergences from information geometry and statistical thermodynamics a single formula lyapunov function is given that covers the entire class of dynamics which includes the replicator dynamic and the projection dynamic as well as several new dynamics a further class is discussed that allows for more variation such as variable intensities of selection

group and moore penrose inverses of regular morphisms with kernel and cokernel abstract let y be a morphism with kernel k x and cokernel y l in an additive category c 1 if x y then has a group inverse iff is invertible and is regular 2 if c has an involution then has a moore penrose inverse with respect to iff and are invertible and is regular

pseudo maximum likelihood estimation of arch models strong consistency and asymptotic normality of the gaussian pseudo maximum likelihood estimate of the parameters in a wide class of arch 1 processes are established we require the arch weights to decay at least hyperbolically with a faster rate needed for the central limit theorem than for the law of large numbers various rates are illustrated in examples of particular parameteriza tions in which our conditions are shown to be satis ed

when a genetic algorithm outperforms hill climbing a toy optimisation problem is introduced which consists of a fitness gradient broken up by a series of hurdles the performance of a hill climber and a stochastic hill climber are computed these are compared with the empirically observed performance of a genetic algorithm ga with and without the hill climber with a sufficiently large neighbourhood outperforms the stochastic hill climber but is outperformed by a ga both with and without crossover the ga with crossover substantially outperforms all the other heuristics considered here the relevance of this result to real world problems is discussed

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

a note on matrix inversion abstract we generalize a recent result of thompson on inverses of block matrices over principal ideal domains to isomorphisms of direct sums of modules over an arbitrary ring

a new approach to estimating the expected first hitting time of evolutionary algorithms evolutionary algorithms ea have been shown to be very effective in solving practical problems yet many important theoretical issues of them are not clear the expected first hitting time is one of the most important theoretical issues of evolutionary algorithms since it implies the average computational time complexity in this paper we establish a bridge between the expected first hitting time and another important theoretical issue i e convergence rate through this bridge we propose a new general approach to estimating the expected first hitting time using this approach we analyze eas with different configurations including three mutation operators with without population a recombination operator and a time variant mutation operator on a hard problem the results show that the proposed approach is helpful for analyzing a broad range of evolutionary algorithms moreover we give an explanation of what makes a problem hard to eas and based on the recognition we prove the hardness of a general problem

dynamics near a periodically perturbed robust heteroclinic cycle abstract robust heteroclinic cycles rhcs arise naturally in collections of symmetric differential equations derived as dynamical models in many fields including fluid mechanics game theory and population dynamics in this paper we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle rhc we give a detailed derivation of the poincare map for trajectories near the rhc asymptotically correct in the limit of small amplitude perturbations this reduces the nonautonomous system in r 3 to a 2d map we identify three distinct dynamical regimes the distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem the first regime corresponds to the rhc being only weakly attracting here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque in the second regime the periodically perturbed rhc is more strongly attracting and the system dynamics corresponds to that of a non invertible or invertible circle map in the third regime of yet stronger attraction the dynamics of the return map is chaotic and no longer reducible to a one dimensional map this third regime has been noted previously our analysis in this paper focusses on providing quantitative results in the first two regimes

temporal aggregation of volatility models in this paper we consider temporal aggregation of volatility models we introduce semiparametric volatility models termed square root stochastic autoregressive volatility sr sarv which are characterized by autoregressive dynamics of the stochastic variance our class encompasses the usual garch models and various asymmetric garch models moreover our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables the sr sarv class is a natural extension of the class of weak garch models this extension has four advantages i we do not assume that fourth moments are finite ii we allow for asymmetries skewness leverage effect that are excluded from weak garch models iii we derive conditional moment restrictions and iv our framework allows us to study temporal aggregation of igarch models

finite statistical complexity for sofic systems we propose a measure of complexity for symbolic sequences which is based on conditional probabilities and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata dfa moreover if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures causal states

asymptotically stable equilibrium and limit cycles in the rock paper scissors game in a population of players with complex personalities abstract we investigate a population of individuals who play the rock paper scissors rps game the players choose strategies not only by optimizing their payoffs but also taking into account the popularity of the strategies for the standard rps game we find an asymptotically stable polymorphism with coexistence of all strategies for the general rps game we find the limit cycles their stability depends exclusively on two model parameters the sum of the entries of the rps payoff matrix and a sensitivity parameter which characterizes the personality of the players apart from the supercritical hopf bifurcation we found the subcritical bifurcation numerically for some intervals of the parameters of the model