nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

a locally linear rbf network based state dependent ar model for nonlinear time series modeling this paper presents a modeling approach to nonlinear time series that uses a set of locally linear radial basis function networks llrbfns to approximate the functional coefficients of the state dependent autoregressive sd ar model the resulting model called the locally linear radial basis function network based autoregressive llrbf ar model combines the advantages of the llrbfn in function approximation and of the sd ar model in nonlinear dynamics description the llrbfn weights that connect the hidden units with the output are linear functions of the input variables this differs from the conventional rbf network weight structure a structured nonlinear parameter optimization method snpom is applied to estimate the llrbf ar model parameters case studies on various time series and chaotic systems show that the llrbf ar modeling approach exhibits much better prediction accuracy compared to some other existing methods

weighted projective embeddings stability of orbifolds and constant scalar curvature kahler metrics we embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of kodaira embedding dividing by the non reductive automorphisms of weighted projective space then formally gives a moduli space of orbifolds we show how to express this as a reductive quotient and so a git problem thus defining a notion of stability for orbifolds we then prove an orbifold version of donaldson s theorem the existence of an orbifold kahler metric of constant scalar curvature implies k semistability by extending the notion of slope stability to orbifolds we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold kahler metrics we describe the manifold applications of this orbifold result and show how many previously known results troyanov ghigi kollar rollin singer the adscft sasaki einstein obstructions of gauntlett martelli sparks yau fit into this framework

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

a stabilization theorem for dynamics of continuous opinions abstract a stabilization theorem for processes of opinion dynamics is presented the theorem is applicable to a wide class of models of continuous opinion dynamics based on averaging like the models of hegselmann krause and weisbuch deffuant the analysis detects self confidence as a driving force of stabilization

a portmanteau test for self exciting threshold autoregressive type nonlinearity in time series summary a portmanteau test to detect self exciting threshold autoregressive type nonlinearity in time series data is proposed the test is based on cumulative sums of standardized residuals from autoregressive fits to the data significance levels for the test under the hypothesis of linearity are obtain from the asymptotic distribution of the cumulative sums as brownian motion the performance of the test is evaluated for simulated data from linear bilinear and self exciting threshold autoregressive models it is also compared with another test which has been suggested for detecting general nonlinearity features of the proposed test which make it useful in identifying the autoregressive order and the lag in threshold models are discussed

current developments in time series modelling abstract in this paper a survey is given of some current developments in time series analysis after a brief discussion of linear models arma models and state space reprentations the discussion concentrates on a fairly general class of non linear models called state dependent models and the applications of these to the problem of identifying general forms of non linear structure the effectiveness of the estimation technique is studied by means of numerical examples on simulated data towards the end of the paper it is indicated how these ideas may be applied to the study of systems governed by non linear differential equations

a global local optimization approach to parameter estimation of rbf type models we present a hybrid global local optimization algorithm for parameter estimation of radial basis function rbf networks and the rbf type autoregressive models without exogenous inputs rbf ar or with exogenous inputs rbf arx the rbf ar x models are quasi linear time varying ar x models with gaussian rbf network style coefficients which have been used to effectively model the nonlinear behavior of various complex systems however the identification of these models is a difficult optimization problem because of the large number of local minima a hybrid approach is proposed in this paper to achieve better optimization results for these rbf type models the applied hybrid search strategy ea snpom is developed by combining an evolutionary algorithm ea with a gradient based algorithm known as the structured nonlinear parameter optimization method snpom this strategy makes use of the robustness of the ea to cover an entire global search space and the efficiency of the gradient search to converge to a local optimum several examples of time series modeling and system identification are presented the simulation results indicate that the performance of the proposed hybrid approach is better than the performance obtained from using each method ea or snpom alone furthermore the rbf ar x models estimated by the ea snpom achieve much better modeling accuracy relative to other neural networks or fuzzy models in the simulations

statistical analysis for multiplicatively modulated nonlinear autoregressive model and its applications to electrophysiological signal analysis in humans modulating the dynamics of a nonlinear autoregressive model with a radial basis function rbf of exogenous variables is known to reduce the prediction error here rbf is a function that decays to zero exponentially if the deviation between the exogenous variables and a center location becomes large this paper introduces a class of rbf based multiplicatively modulated nonlinear autoregressive mmnar models first we establish the local asymptotic normality lan for vector conditional heteroscedastic autoregressive nonlinear charn models which include the mmnar and many other well known time series models as special cases asymptotic optimality for estimation and testing is described in terms of lan properties the mmnar model indicates goodness of fit for surface electromyograms emg using electrocorticograms ecog as the exogenous variables concretely it is found that the negative potential of the motor cortex forces change in the frequency of emg which is reasonable from a physiological point of view the proposed mmnar model fitting is both useful and efficient as a signal processing technique for extracting information on the action potential which is associated with the postsynaptic potential

modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model summary the behaviour of nonlinear deterministic vibrations has been studied by many authors and may typically include such features as jump phenomena and limit cycles nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude dependent frequency a discrete time series model is introduced which may be demonstrated to have properties similar to those of nonlinear random vibrations this model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models the model is fitted to the canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self exciting mechanism

additivity tests for nonlinear autoregression summary additivity is commonly used in the statistical literature to simplify data analysis especially in analysis of variance and in multivariate smoothing in this paper we propose three procedures for testing additivity in nonlinear time series analysis the first procedure combines some smoothing techniques with analysis of variance the second is a lagrange multiplier test using nonparametric estimation and the third is a permutation test which uses smoothing techniques to obtain the test statistic and its reference distribution we investigate properties of the proposed tests and use simulation to check their performance in finite samples applications of the tests to nonlinear time series analysis are discussed and illustrated by real examples

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

on single index coefficient regression models abstract in this article we investigate a class of single index coefficient regression models under dependence this includes many existing models such as the smooth transition threshold autoregressive star model of chan and tong the functional coefficient autoregressive far model of chen and tsay and the single index model of ichimura compared to the varying coefficient model of hastie and tibshirani our model can avoid the curse of dimensionality in multivariate nonparametric estimations another advantage of this model is that a threshold variable is chosen automatically an estimation method is proposed and the corresponding estimators are shown to be consistent and asymptotically normal some simulations and applications are also reported

opinion fluctuations and disagreement in social networks we study a tractable opinion dynamics model that generates long run disagreements and persistent opinion fluctuations our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders political parties or media sources attempting to influence the beliefs in the rest of the society when the society contains stubborn agents with different opinions the belief dynamics never lead to a consensus among the regular agents instead beliefs in the society fail to converge almost surely the belief profile keeps on fluctuating in an ergodic fashion and it converges in law to a nondegenerate random vector the structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics the expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the kolmogorov backward equation of a continuous time markov chain on the graph with absorbing states corresponding to the stubborn agents and hence to converge to a harmonic vector with every regular agent s value being the weighted average of its neighbors values and boundary conditions corresponding to the stubborn agents beliefs expected cross products of the agents beliefs allow for a similar characterization in terms of coupled markov chains on the graph describing the social network we prove that in large scale societies which are highly fluid meaning that the product of the mixing time of the markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large a condition of homogeneous influence emerges whereby the stationary beliefs marginal distributions of most of the regular agents have approximately equal first and second moments

shape description by time series time series modeling techniques are adapted to represent or describe two dimensional closed contours both linear and nonlinear models are fitted it is found that to detect small changes in shape nonlinear modeling is necessary even though linear models may be sufficient to differentiate between shapes which differ widely a nonlinear model called the noncausal quadratic volterra model is developed for the purpose implementation is illustrated with shapes of aircraft

nonparametric multistep ahead prediction in time series analysis we consider the problem of multistep ahead prediction in time series analysis by using nonparametric smoothing techniques forecasting is always one of the main objectives in time series analysis research has shown that non linear time series models have certain advantages in multistep ahead forecasting traditionally nonparametric k step ahead least squares prediction for non linear autoregressive ar d models is done by estimating e x t k mid x t cdots x t d 1 via nonparametric smoothing of x t k on x t cdots x t d 1 directly we propose a multistage nonparametric predictor we show that the new predictor has smaller asymptotic mean squared error than the direct smoother though the convergence rate is the same hence the predictor proposed is more efficient some simulation results advice for practical bandwidth selection and a real data example are provided copyright 2004 royal statistical society

on the existence of nontrivial extremal metrics on complete noncompact surfaces in this paper we consider the 2 dimensional local calabi flow on a complete noncompact surface sigma g 0 then based on the harnack type estimate we show the long time existence and asymptotic convergence of a subsequence of solutions of such a flow on sigma g 0 with r 0 leq 0 and r 0 bounded from above by a negative constant on a ball for its applications this will lead to the existence of extremal metrics on a complete noncompact surface of finite topological type in particular there exists an extremal metric of nonconstant gaussian curvature on sigma mathbf h 2 or mathbf r 2

testing for neglected nonlinearity in regression models based on the theory of random fields abstract within a flexible regression model j d hamilton econometrica 69 3 2001 537 we offer a battery of new lagrange multiplier statistics that circumvent the problem of unidentified nuisance parameters under the null hypothesis of linearity and that are robust to the specification of the covariance function that defines the random field these advantages are the result of i switching from the l 2 to the l 1 norm and ii assuming that the random field is sufficiently smooth for its covariance function to be locally approximated by a high order taylor expansion a monte carlo simulation suggests that our statistics have superior power performance on detecting bilinear neural network and smooth transition autoregressive specifications we also provide an application to the industrial production index of sixteen oecd countries

strategic influence in social networks we consider a model of influence with a set of non strategic agents and two strategic agents the non strategic agents have initial opinions and are linked through a simply connected network they update their opinions as in the degroot model the two strategic agents have fixed opinions 1 and 0 respectively and are characterized by the magnitude of the impact they can exert on non strategic agents each strategic agent forms a link with one non strategic agent in order to alter the average opinion that eventually emerges in the network this procedure defines a zero sum game whose players are the two strategic agents and whose strategy set is the set of non strategic agents we focus on the existence and the characterization of equilibria in pure strategy in this setting simple examples show that the existence of a pure strategy equilibrium does depend on the structure of the network the characterization of equilibrium we obtain emphasizes on the one hand the influenceability of target agents and on the other hand their centrality whose natural measure in our context defines a new concept related to betweenness centrality that we call intermediacy we also show that in the case where the two strategic agents have the same impact symmetric equilibria emerge as natural solutions whereas in the case where the impacts are uneven the strategic players generally have differentiated equilibrium targets the high impacts agent focusing on centrality and the low impact agent on influenceability

specification estimation and evaluation of smooth transition autoregressive models abstract this article considers the application of two families of nonlinear autoregressive models the logistic lstar and exponential estar autoregressive models this includes the specification of the model based on simple statistical tests linearity testing against smooth transition autoregression determining the delay parameter and choosing between lstar and estar models are discussed estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model the proposed techniques are illustrated by examples using both simulated and real time series

nonlinear additive arx models abstract we consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models the procedures proposed use two backfitting techniques the ace and bruto algorithms to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model simulated and real examples are used to illustrate the analysis

a parameter optimization method for radial basis function type models this paper considers the nonlinear systems modeling problem for control a structured nonlinear parameter optimization method snpom adapted to radial basis function rbf networks and an rbf network style coefficients autoregressive model with exogenous variable model parameter estimation is presented this is an off line nonlinear model parameter optimization method depending partly on the levenberg marquardt method for nonlinear parameter optimization and partly on the least squares method using singular value decomposition for linear parameter estimation when compared with some other algorithms the snpom accelerates the computational convergence of the parameter optimization search process of rbf type models the usefulness of this approach is illustrated by means of several examples

combining nonparametric and optimal linear time series predictions we introduce a semiparametric procedure for more efficient prediction of a strictly stationary process admitting an arma representation the procedure is based on the estimation of the arma representation followed by a nonparametric regression where the arma residuals are used as explanatory variables compared to standard nonparametric regression methods the number of explanatory variables can be reduced because our approach exploits the linear dependence of the process we establish consistency and asymptotic normality results for our estimator numerical experiments show that significant gains can be achieved with our approach all the supplemental materials used by this article are available online

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

functional coefficient regression models for nonlinear time series abstract the local linear regression technique is applied to estimation of functional coefficient regression models for time series data the models include threshold autoregressive models and functional coefficient autoregressive models as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance also proposed are a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross validatory estimation for the expected forecasting errors the proposed methodology is data analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality the asymptotic properties of the proposed estimators are investigated under the mixing condition both simulated and real data examples are used for illustration

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

blow up analysis for su 3 toda system abstract using the method of symmetrization and the rescaling we study non compact solution sequence to the su 3 toda system in non abelian relativistic self dual gauge theory i e the quantization of the total mass and classification of the singular limit

density estimation for nonlinear parametric models with conditional heteroscedasticity abstract this article studies density and parameter estimation problems for nonlinear parametric models with conditional heteroscedasticity we propose a simple density estimate that is particularly useful for studying the stationary density of nonlinear time series models under a general dependence structure we establish the root n consistency of the proposed density estimate for parameter estimation a bahadur type representation is obtained for the conditional maximum likelihood estimate the parameter estimate is shown to be asymptotically efficient in the sense that its limiting variance attains the cramer rao lower bound the performance of our density estimate is studied by simulations