testing for nonlinearity in unevenly sampled time series we generalize the method of surrogate data of testing for nonlinearity in time series to the case that the data are sampled with uneven time intervals the null hypothesis will be that the data have been generated by a linear stochastic process possibly rescaled and sampled at times chosen independently from the generating process the surrogate data are generated with their linear properties specified by the lomb periodogram the inversion problem is solved by combinatorial optimization

significance testing of information theoretic functionals abstract a transformation is derived such that under the transformation the mutual information of two unrelated variables is a normally distributed random variable with zero mean and unit variance it is demonstrated that the transformed mutual information can be used as a test statistic to test the statistical significance of the relationship between variables and provides an alternative to the method of surrogate data if the mutual information is regarded as a generalization of the linear correlation coefficient then this test statistic can be regarded as analogous to the fisher z test statistic

convergence speeding convergence and summability abstract four general principles are proposed for use in constructing convergence speeding methods five old methods and about 15 more recent non linear methods are discussed with reference to their range of applicability and rate of convergence speeding when applied to alternating monotonic slightly erratic fourier and alternating divergent series several of these methods are proved to give analytic continuations while even asymptotic series with zero radius of convergence can be summed accurately divergent series of positive terms may also in some cases be interpreted with moderate accuracy using four different methods convergence tests are reviewed in the light of these convergence speeding methods and analytic continuation

backlund calogero group for the general differential spectral problem of an arbitrary order the infinite dimensional abelian group of general backlund calogero transformations is constructed for the evolution equations which can be integrated by a gelfand dikij zakharov shabat spectral problem of an arbitrary order the structure of the recursion operator and transformation properties of the backlund calogero transformations under the gauge group are considered

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

quantifying strength of chaos in the population firing rate of neurons in this study i considered quantifying the strength of chaos in the population firing rate of a pulse coupled neural network in particular i considered the dynamics where the population firing rate is chaotic and the firing of each neuron is stochastic i calculated a time histogram of firings to show the variation in the population firing rate over time to smooth this histogram i used bayesian adaptive regression splines and a gaussian filter the nonlinear prediction method based on reconstruction was applied to a sequence of interpeak intervals in the smoothed time histogram of firings i propose the use of the sum of nonlinearity as a quantifier of the strength of chaos when applying this method to the firings of a pulse coupled neural network the sum of nonlinearity was seen to satisfy three properties for quantifying the strength of chaos first it can be calculated from spiking data alone second it takes large values when applied to firings that are confirmed theoretically or numerically to be chaotic third it reflects the strength of chaos of the original dynamics

test your surrogate data before you test for nonlinearity the schemes for the generation of surrogate data in order to test the null hypothesis of linear stochastic process undergoing nonlinear static transform are investigated as to their consistency in representing the null hypothesis in particular we pinpoint some important caveats of the prominent algorithm of amplitude adjusted fourier transform surrogates aaft and compare it to the iterated aaft which is more consistent in representing the null hypothesis it turns out that in many applications with real data the inferences of nonlinearity after marginal rejection of the null hypothesis were premature and have to be reinvestigated taking into account the inaccuracies in the aaft algorithm mainly concerning the mismatching of the linear correlations in order to deal with such inaccuracies we propose the use of linear together with nonlinear polynomials as discriminating statistics the application of this setup to some well known real data sets cautions against the use of the aaft algorithm

rewiring induced chaos in pulse coupled neural networks the dependence of the dynamics of pulse coupled neural networks on random rewiring of excitatory and inhibitory connections is examined when both excitatory and inhibitory connections are rewired periodic synchronization emerges with a hopf like bifurcation and a subsequent period doubling bifurcation chaotic synchronization is also observed when only excitatory connections are rewired periodic synchronization emerges with a saddle node like bifurcation and chaotic synchronization is also observed this result suggests that randomness in the system does not necessarily contaminate the system and sometimes it even introduces rich dynamics to the system such as chaos

completeness of the eigenfunctions for the caudrey beals coifman system the direct and the inverse scattering problem for the first order linear systems of the type l x i d dx q x j x 0 j h q x gj which generalizes the zakharov shabat system and the systems studied by caudrey beals and coifman cbc is analyzed herein here j is a regular complex constant element of the cartan subalgebra h g of the simple lie algebra g and the potential q x vanishes fast enough for x taking values in the image gj of adj the cbc results are generalized and the fundamental analytic solution m x for any choice of the irreducible finite dimensional representation v of g is constructed four pairwise equivalent minimal sets of scattering data for l invariant with respect to the choice of the representation of g are extracted from the asymptotics of m x for x from m x the resolvent of l is constructed in the adjoint representation vad and the completeness relation is proven for the eigenfunctions of l in vad it is also proven that the discrete spectrum

constrained realization monte carlo method for hypothesis testing abstract we compare two theoretically distinct approaches to generate artificial or surrogate data for testing hypotheses about a given data set the first and more straightforward approach is to fit a single best model to the original data and then to generate surrogate data sets that are typical realizations of that model the second approach concentrates not on the model but directly on the original data it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics examples of these two approaches are provided for two simple cases a test for deviations from a gaussian distribution and a test for serial dependence in a time series additionally we consider test for nonlinearity in time series based on a fourier transform ft method and on more conventional autoregressive moving average arma fits to the data the comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal a statistic is pivotal if its distribution is the same for all processes consistent with the null hypothesis the typical realization method requires that the discriminating statistic satisfy this property the constrained realization approach on the other hand does not share this requirement and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic

discrimination power of measures for nonlinearity in a time series the performance of a number of different measures of nonlinearity in a time series is compared numerically their power to distinguish noisy chaotic data from linear stochastic surrogates is determined by monte carlo simulation for a number of typical data problems the main result is that the ratings of the different measures vary from example to example it therefore seems preferable to use an algorithm with good overall performance that is higher order autocorrelations or nonlinear prediction errors

gauge covariant theory of the generating operator i a gauge covariant formulation of the generating operator operator theory for the zakharov shabat system is proposed the operator tilde lambda corresponding to the gauge equivalent system in the pole gauge is explicitly calculated thus the unified approach to the nonlinear schrodinger type equations based on is automatically reformulated with the help of tilde lambda for the heisenberg ferromagnet type equations consequently it is established that the conserved densities for the heisenberg ferromagnet type equations are polynomial ins x and itsx derivatives special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge equivalent equations it is shown that the geometrical properties of the conjugated operator are gaugeindependent

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

generalized redundancies for time series analysis abstract extensions to various information theoretic quantities such as entropy redundancy and mutual information are discussed in the context of their role in nonlinear time series analysis we also discuss linearized versions of these quantities and their use as benchmarks in tests for nonlinearity many of these quantities can be expressed in terms of the generalized correlation integral and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics such as the bds and green savit statistics further numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation finally we consider several local versions of these quantities including a local kolmogorov sinai entropy which gives an estimate of variability of the short term predictability

integrable equations and recursion operators related to the affine lie algebras ar 1 we have derived a family of equations related to the untwisted affine lie algebras ar 1 using a coxeter r 1 reduction they represent the third member of the hierarchy of soliton equations related to the algebra applying additional 2 involutions we also obtain the real hamiltonian forms of these equations we also give some particular examples and impose additional reductions

surrogates for finding unstable periodic orbits in noisy data sets recently searches for unstable periodic orbits in biological and medical applications have become of interest the motivations for this research range in order of ascending complexity from efforts to understand the dynamics of simple sensory neurons through speculations regarding neural coding to the hopeful development of new diagnostic and or control techniques for cardiac and epileptic pathologies biological and medical data are however noisy and nonstationary findings of unstable periodic orbits in such data thus require convincing assessments of their statistical significance such tests are accomplished by comparison with surrogate data files designed to test an appropriate null hypothesis in this paper we test surrogates generated by three different algorithms against correlated noise as well as stable periodic orbits one of the surrogates is new and has been specifically designed to preserve the shape of the attractor we discuss the suitability of these surrogates and argue that the simple shuffled one correctly tests the appropriate null hypothesis

measuring statistical dependences in a time series we propose two methods to measure all linear and nonlinear statistical dependences in a stationary time series presuming ergodicity the measures can be obtained from efficient numerical algorithms

bayesian curve fitting with free knot splines we describe a bayesian method for fitting curves to data drawn from an exponential family that uses splines for which the number and locations of knots are free parameters the method uses reversible jump markov chain monte carlo to change the knot configurations and a locality heuristic to speed up mixing for nonnormal models we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the bayesian information criterion bic under priors that make this approximation accurate our technique is based on a marginalised chain on the knot number and locations but we provide methods for inference about the regression coefficients and functions of them in both normal and nonnormal models simulation results suggest that the method performs well and we illustrate the method in two neuroscience applications

fourier series and the δ 2 process we discuss the effect of a particular sequence acceleration method the d 2 process on the partial sums of fourier series we show that for a very general class of functions this method fails on a dense set of points not only does it not speed up convergence it turns the sequence of partial sums into a sequence with multiple limit points

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

squared eigenfunctions and the perturbation theory for the nondegenerate n n operator a general outline we provide an overview of the soliton perturbation theory for the n n eigenvalue operator key to perturbation studies of integrable systems are the squared eigenfunctions and their adjoints which serve as mappings between variations in the potentials and variations in the scattering data we also address the problem of the normalization of the jost functions how this affects the structure and solvability of the inverse scattering equations and the definition of the scattering data we also discuss the inner products and closure relations for these squared eigenfunctions and their adjoints

testing for nonlinearity using redundancies quantitative and qualitative aspects abstract a method for testing nonlinearity in time series is described based on information theoretic functionals redundancies linear and nonlinear forms of which allow either qualitative or after incorporating the surrogate data technique quantitative evaluation of dynamical properties of scrutinized data an interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated evaluation of redundancies and redundancy based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study

general structure of nonlinear evolution equations in 1 2 dimensions integrable by the two dimensional gelfand dickey zakharov shabat spectral problem and their transformation properties the general form of nonlinear evolution equations connected with the matrix two dimensional gelfand dickey zakharov shabat spectral problem is found the infinite dimensional abelian group of general backlund transformations and infinite dimensional abelian symmetry group for these equations are constructed

on classification of soliton solutions of multicomponent nonlinear evolution equations we consider several ways of how one could classify the various types of soliton solutions related to multicomponent nonlinear evolution equations which are solvable by the inverse scattering method for the generalized zakharov shabat system related to a simple lie algebra g in doing so we make use of the fundamental analytic solutions the zakharov shabat dressing procedure the reduction technique and other tools characteristic for that method the multicomponent solitons are characterized by several important factors the subalgebras of g and the way these subalgebras are embedded in g the dimension of the corresponding eigensubspaces of the lax operator l as well as by additional constraints imposed by reductions

the generating operator for the n n linear system abstract the class of exactly integrable non linear evolution equations nlee related to the general first order n n linear problem is studied the set of independent scattering data j is determined and trace identities are obtained next we use a special integro differential operator and obtain the expansions of the potential and its variation in the eigenfunctions of these enable us to consider the inverse scattering method as a generalized fourier transform and to write down in a compact form the integrals of motion we prove that there exist a hierarchy of hamiltonian structures generated by at the end we calculate the symplectic form in terms of the scattering data variations

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

a method for selecting the bin size of a time histogram the time histogram method is the most basic tool for capturing a time dependent rate of neuronal spikes generally in the neurophysiological literature the bin size that critically determines the goodness of the fit of the time histogram to the underlying spike rate has been subjectively selected by individual researchers here we propose a method for objectively selecting the bin size from the spike count statistics alone so that the resulting bar or line graph time histogram best represents the unknown underlying spike rate for a small number of spike sequences generated from a modestly fluctuating rate the optimal bin size may diverge indicating that any time histogram is likely to capture a spurious rate given a paucity of data the method presented here can nevertheless suggest how many experimental trials should be added in order to obtain a meaningful time dependent histogram with the required accuracy

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package