splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

a single blind controlled competition among tests for nonlinearity and chaos interest has been growing in testing for nonlinearity or chaos in economic data but much controversy has arisen about the available results this paper explores the reasons for these empirical difficulties we designed and ran a single blind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series the data generating mechanisms include linear processes chaotic recursions and nonchaotic stochastic processes and both large and small samples were included in the experiment the data series were produced in a single blind manner by the competition manager and sent by e mail without identifying information to the experiment participants each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test the results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos

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

the backward euler method for numerical solution of the hodgkin huxley equations of nerve conduction a widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations pdes derived for the giant axon of the squid loligo by hodgkin and huxley when consider

nonlinear modeling of time series using multivariate adaptive regression splines mars abstract multivariate adaptive regression splines mars is a new methodology due to friedman for nonlinear regression modeling mars can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions given a set of predictor variables mars fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy mars produces continuous models for high dimensional data that can have multiple partitions and predictor variable interactions predictor variable contributions and interactions in a mars model may be analyzed using an anova style decomposition by letting the predictor variables in mars be lagged values of a time series one obtains a new method for nonlinear autoregressive threshold modeling of time series a significant feature of this extension of mars is its ability to produce models with limit cycles when modeling time series data that

on estimation of a regression model with long memory stationary errors we consider estimation of a regression model with long memory stationary errors first we estimate the regression parameters by the least squares estimator lse and next those describing the correlation structure of the error terms by using the residuals obtained from the lse certain regularity conditions introduced to develop the asymptotic theory no longer hold in this model in this situation we derive asymptotic properties of the preceding estimation procedure

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

prime fuzzy ideals in rings abstract in this paper give a characterization of prime fuzzy ideals we introduce weakly completely prime fuzzy ideals and establish a relation between completely prime fuzzy ideals and weakly completely prime fuzzy ideals

burmann expansion and test for additivity summary we propose a lagrange multiplier test for additivity based on the btirmann expansion of a conditional mean function the asymptotic null distribution of the test is shown to be x2 under some regularity conditions in contrast the lagrange multiplier test proposed by chen et al 1995 is based on the volterra expansion of the conditional mean function we discuss some desirable advantages of the bt rmann expansion over the volterra expansion for nonlinear time series modelling we also reported an empirical study which shows that in terms of empirical power the lagrange multiplier test motivated by the biirmann expansion outperforms the test of chen et al 1995 for the cases for which the lagrange multiplier test is designed for other cases for which none of the tests is specifically designed the empirical powers of the two tests are comparable finally we illustrated the use of the lagrange multiplier test with a blowfly experimental system

analytic numerical solutions with priori error bounds for a class of strongly coupled mixed partial differential systems abstract this paper deals with the construction of analytic numerical solutions with a priori error bounds for systems of the type u t au xx u 0 t u x 0 t 0 bu 1 t cu x 1 t 0 0 x t 0 u x 0 f x here a b c are matrices for which no diagonalizable hypothesis is assumed first an exact series solution is obtained after solving appropriate vector sturm liouville type problems given an admissible error e and a bounded subdomain d after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy e uniformly in d

on the lattice of fuzzy ideals of a ring abstract first we prove that in the lattice of all fuzzy left right ideals with the same tip t using the technique of strong cut by this we obtain the following results 1 we point out that the lattice of all fuzzy ideals of a ring is not distributive thus the main theorem l f1 r is distributive in majumdar and sultana 1996 is erroneous 2 the lattice of all fuzzy left right ideals with same tip t of a ring r is modular this generalizes the main theorem in ajmal and thomas 1995

smoothing data with local instabilities for the identification of chaotic systems chaotic systems are characterized by the local divergence of nearby orbits in state space this provokes a sensitive dependence on initial conditions and in turn drastically limits the accuracy of long term predictions this has important implications in the filtering of data generated by such systems in the present paper the use of global smoothers for chaotic data is investigated the ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed the objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system such as the geometry of the attractors in state space the largest lyapunov exponent fractal dimensions and poincare sections numerical examples are included which illustrate the main points of the paper

fuzzy nil radicals and fuzzy primary ideals abstract the purpose of this paper is to extend the concept of a fuzzy set by introducing the concept of a fuzzy nil radical of a fuzzy ideal in a ring using this concept a fuzzy primary ideal is defined some fundamental results concerning these notions are proved

identification of non linear additive autoregressive models summary we propose a lag selection method for non linear additive autoregressive models that is based on spline estimation and the bayes information criterion the additive structure of the autoregression function is used to overcome the curse of dimensionality whereas the spline estimators effectively take into account such a structure in estimation a stepwise procedure is suggested to implement the method proposed a comprehensive monte carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local polynomial based methods consistency of the lag selection method based on the bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing which provides the first theoretical result of this kind for spline smoothing of weakly dependent data

time series and dependent variables we present a new method for analyzing time series which is designed to extract inherent deterministic dependencies in the series the method is particularly suited to series with broad band spectra such as chaotic series with or without noise we derive quantities j e based on conditional probabilities whose magnitude roughly speaking is an indicator of the extent to which the kth element in the series is a deterministic function of the k j th element to within a measurement uncertainty e we apply our method to a number of deterministic time series generated by chaotic processes such as the tent logistic and h non maps as well as to sequences of quasi random numbers in all cases the 6j correctly indicate the expected dependencies we also show that the j are robust to the addition of substantial noise in a deterministic process in addition we derive a predictability index which is a measure of the extent to which a time series is predictable given some tolerance e finally we discuss the behavior of the 6i as e approaches zero

nonparametric identification of nonlinear time series selecting significant lags abstract in this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series the procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series the procedure is illustrated by simulations and sunspot data lynx data and blowfly data are analyzed it is indicated that projectors can be used in conjunction with the procedure for selecting significant lags to check the adequacy of an additive time series model

nonparametric transfer function models in this paper a class of nonparametric transfer function models is proposed to model nonlinear relationships between input and output time series the transfer function is smooth with unknown functional forms and the noise is assumed to be a stationary autoregressive moving average arma process the nonparametric transfer function is estimated jointly with the arma parameters by modelling the correlation in the noise the transfer function can be estimated more efficiently the parsimonious arma structure improves the estimation efficiency in finite samples the asymptotic properties of the estimators are investigated the finite sample properties are illustrated through simulations and one empirical example

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

dynamic adaptive partitioning for nonlinear time series summary we propose a dynamic adaptive partitioning scheme for nonparametric analysis of stationary nonlinear time series it yields estimates of the whole probability distribution of the underlying process we use information from past values to construct adaptive partitioning in a dynamic fashion which is then different from the more common static schemes in the regression set up the idea of dynamic partitioning is new we make it constructive by an approach based on quantisation of the data and adaptively modelling partition cells with a parsimonious markov chain the methodology is formulated in terms of a new model class the so called quantised variable length markov chains it is a new extension of finite valued variable length markov chains to processes with values in rd we discuss estimation explore asymptotic properties of the new method and give some numerical results which reflect the finite sample behaviour

on the general linear coupled system for diffusion in media with two diffusivities abstract for the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity simple uniqueness criteria and a method of solution of boundary value problems are established the equations studied retain the so called cross terms which have been neglected in all previous investigations moreover these equations arise as generalizations of a number of existing theories for example heat flow in heterogeneous multicomponent systems flow of water in fissured rocks and a model of an arms race the simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants the solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems the paper emphasizes the general equations without special reference to particular physical applications or boundary value problems

wavelet based parameter estimation for polynomial contaminated fractionally differenced processes we consider the problem of estimating the parameters for a stochastic process using a time series containing a trend component trend i e large scale variations in the series that are best modeled outside of a stochastic framework is often confounded with low frequency stochastic fluctuations this problem is particularly evident in models such as fractionally differenced fd processes which exhibit slowly decaying autocorrelations and can be extended to encompass nonstationary processes with substantial low frequency components we use the discrete wavelet transform dwt to estimate parameters for stationary and nonstationary fd processes in a model of polynomial trend plus fd noise using daubechies wavelet filters allows for automatic elimination of polynomial trends due to embedded differencing operations parameter estimation is based on an approximate maximum likelihood approach made possible by the fact that the dwt decorrelates fd processes approximately we consider this decorrelation in detail examining the between and within scale wavelet correlations separately better between scale decorrelation can be achieved by increasing the length of the wavelet filter whereas the within scale correlations can be handled via explicit modeling by a low order autoregressive process we demonstrate our methodology by applying it to a popular climate dataset

a matrix approach to the analytic numerical solution of mixed partial differential systems e navarro e ponsoda and l jodar departamento de matem tica aplicada universidad polit cnica de valencia p o box 22012 valencia spain received and accepted march 1994 abstract in this paper analytic numerical solutions for initial boundary value problems related to systems of partial differential equations are proposed given an admissible error e 0 and a finite domain g a numerical approximation is constructed in terms of the data so that the error is uniformly upper bounded by e in g keywords partial differential system eigenvalues bound error bound schur decomposition matrix norms matrix functions

the lattice of fuzzy ideals of a ring the set of all fuzzy ideals of a ring r is partially ordered with respect to the natural ordering of fuzzy sets in r it has been shown that this is a lattice but not a sublattice of the lattice of all fuzzy sets in r this lattice of all fuzzy ideals of r is complete distributive and bounded and has neither atoms nor dual atoms

continuous numerical solutions of coupled mixed partial differential systems using fer s factorization in this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time dependent systems of the type ut a t uxx b t u 0 0 0 u 0 t u p t 0 u x 0 f x 06 x6p after truncation of an exact series solution the numerical solution is constructed using fer s factorization given 0 and t0 t1 with 0 t0 t1 and d t0 t1 f x t 06x6p t06t6t1g the error of the approximated solution with respect to the exact series solution is less than uniformly in d t0 t1 an algorithm is also included c 1999 elsevier science b v all rights reserved ams classication 65m15 34a50 35c10 35a50

the stable discrete numerical solution of strongly coupled mixed partial differential systems abstract this paper is concerned with the discrete numerical solution of coupled partial differential mixed problems with non dirichlet coupled boundary value conditions by application of a discrete separation of variables method the proposed numerical solution of the problem turns out to be the exact solution of certain coupled partial difference system appearing from the discretization of the continuous partial differential system our approach avoids the iterative solution of algebraic systems which appears when one uses simple discretization methods existence stability and the construction of solutions are considered

fuzzy ideals and fuzzy prime ideals of a ring abstract besides obtaining certain ring theoretic analogues some peculiarities of the fuzzy setting are also discussed in particular a study of i the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals ii the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms and iii the ring of all the fuzzy cosets of a fuzzy prime fuzzy maximal ideal is carried out

a goodness of fit test for time series with long range dependence we propose a test statistic for goodness of fit in time series with slowly decaying serial correlations the asymptotic distribution of the test statistic originally proposed by milhoj for time series with smooth spectra turns out to be the same under the null hypothesis even if the spectrum has a pole at 0 in particular the test is suitable to detect lack of independence in the observations or estimated residuals if the first few correlations are small but the decay of the correlations is slow

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

embedding lattices of fuzzy subalgebras into lattices of crisp subalgebras abstract given a universal algebra a we define an algebra a where a and a are of the same type and generate the same variety of algebra next we define a map e lf a lc a where lf a is the lattice of fuzzy subalgebras of a and lc a is the lattice of crisp subalgebras of a for many algebras a the map e is shown to be a lattice embedding as a corollary of this fact we derive the results that the lattice of fuzzy normal subgroups of a group and the lattice of fuzzy ideals of a ring are both modular lattices

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

testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression abstract broad classes of diagnostics for serial correlation and or dynamic conditional heteroskedasticity of regression disturbances are considered the classes include statistics with good power against strongly dependent alternatives along with the usual ones that test against weak dependence and many others limiting null distributions are obtained under mild conditions on the dependence structure of the alternative against which the test is derived on moments of the disturbances and on the regressors the various test statistics have a similar overall structure and while tests against strongly dependent alternatives entail more computation than ones against weakly dependent alternatives the difference can be slight if the fast fourier transform is used

fuzzy semiprimary ideals of rings abstract the concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings this class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals it is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary a study of i the level ideals of a fuzzy semiprimary ideal ii the characteristic function of a semiprimary ideal iii the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and iv the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms is carried out

modeling long range dependence nonlinearity and periodic phenomena in sea surface temperatures using tsmars abstract we analyze a time series of 20 years of daily sea surface temperatures measured off the california coast the temperatures exhibit quite complicated features such as effects on many different time scales nonlinear effects and long range dependence we show how a time series version of the multivariate adaptive regression splines mars algorithm tsmars can be used to obtain univariate adaptive spline threshold autoregressive models that capture many of the physical characteristics of the temperatures and are useful for short and long term prediction we also discuss practical modeling issues such as handling cycles long range dependence and concurrent predictor time series using tsmars models for the temperatures are evaluated using out of sample forecast comparisons residual diagnostics model skeletons and sample functions of simulated series we show that a categorical seasonal indicator variable can be used to model nonlinear structure in the data that is changing with time of yea

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

modeling persistence in hydrological time series using fractional differencing the class of autoregressive integrated moving average arima time series models may be generalized by permitting the degree of differencing d to take fractional values models including fractional differencing are capable of representing persistent series d 0 or short memory series d 0 the class of fractionally differenced arima processes provides a more flexible way than has hitherto been available of simultaneously modeling the long term and short term behavior of a time series in this paper some fundamental properties of fractionally differenced arima processes are presented methods of simulating these processes are described estimation of the parameters of fractionally differenced arima models is discussed and an approximate maximum likelihood method is proposed the methodology is illustrated by fitting fractionally differenced models to time series of streamflows and annual temperatures

fuzzy irreducible ideals in rings abstract in this paper the concept of a fuzzy set is extended by introducing the concept of a fuzzy irreducible ideal in a ring the characteristic function of an irreducible ideal and the level ideals of a fuzzy irreducible ideal are studied the algebraic nature of homomorphic images and preimages of fuzzy irreducible ideals is also examined some other fundamental results pertaining to this concept are discussed

the mechanism of the increase of the generalized dimension of a filtered chaotic time series the determination of the attractor dimension from an experimental time series may be affected by the influence of filters which are incorporated into many measuring processes while this is expected from the kaplan yorke conjecture we show that for one dimensional maps a weak filter can induce a self similarity which is responsible for the increase of the hausdorff dimension we are able to calculate the increase of the generalized dimensiond q for the filtered time series of the logistic mapx i 1 rx i 1 x i atr 4 analytically

fuzzy prime ideals of a ring abstract this paper characterizes all fuzzy prime ideals p of an arbitrary ring r we show that a nonconstant fuzzy ideal p of r is prime if and only if p0 x r p x p 0 is a prime ideal of r p is two valued and p 0 1 examples are given showing that p0 is a prime ideal is not sufficient for p to be a fuzzy prime ideal and that p0 may be a prime ideal of r with the image of p being of arbitrary finite cardinality or being countably infinite

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

a flexible coefficient smooth transition time series model we consider a flexible smooth transition autoregressive star model with multiple regimes and multiple transition variables this formulation can be interpreted as a time varying linear model where the coefficients are the outputs of a single hidden layer feedforward neural network this proposal has the major advantage of nesting several nonlinear models such as the self exciting threshold autoregressive setar the autoregressive neural network ar nn and the logistic star models furthermore if the neural network is interpreted as a nonparametric universal approximation to any borel measurable function our formulation is directly comparable to the functional coefficient autoregressive far and the single index coefficient regression models a model building procedure is developed based on statistical inference arguments a monte carlo experiment showed that the procedure works in small samples and its performance improves as it should in medium size samples several real examples are also addressed

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

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

weighted multiscale cumulative residual renyi permutation entropy of financial time series abstract in this paper based on cumulative residual entropy cre and renyi permutation entropy rpe multiscale cumulative residual renyi permutation entropy mcrrrpe and weighted multiscale cumulative residual renyi permutation entropy wmcrrpe are proposed as novelty measures to quantify the uncertainty of the nonlinear time series first the mcrrpe and wmcrrpe methods are performed on the synthetic data and the impact of the changes to parameters is discussed then the mcrrpe and wmcrrpe methods are applied to the closing prices of the us european and chinese stock markets we analyze the statistics and draw some conclusions from the comparison the stock markets can be divided into four categories 1 nasdaq and ftse100 2 s p500 dji and hsi 3 dax30 and cac40 and 4 sse and szse the standard deviation has a certain relationship with wmcrrpe compared with the mcrrpe method the wmcrrpe method can distinguish these financial stock markets effectively and better estimate the complexity of the time series containing amplitude information

testing for a change of the long memory parameter summary long range dependence is often observed in long time series correlations decay approximately like ik i2 2 with h e 0 5 1 as the lag k tends to infinity the long term features of the data are essentially characterised by the parameter h small changes of h have strong implications for the long term behaviour of the process in particular rates of convergence of estimators for the mean and for many other parameters of interest differ for different values of h for some data sets h appears to change with time in this paper we consider a simple test of the null hypothesis that h is constant the test is based on a functional central limit theorem for quadratic forms critical values for the test statistic are given simulations confirm the validity of the test a data example illustrates its practical application

linear filters and non linear systems it has been asserted in the literature that the low pass filtering of time series data may lead to erroneous results when calculating attractor dimensions here we prove that finite order non recursive filters do not have this effect in fact a generic finite order non recursive filter leaves invariant all the quantities that can be estimated by using embedding techniques such as the method of delays

strong consistency of least squares estimators in regression with correlated disturbances this note considers under minimal assumptions the strong consistency of least squares estimates in regression with correlated errors

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

regularized local linear prediction of chaotic time series abstract local linear prediction based on the ordinary least squares ols approach is one of several methods that have been applied to prediction of chaotic time series apart from potential linearization errors a drawback of this approach is the high variance of the predictions under certain conditions here a different set of so called linear regularization techniques originally derived to solve ill posed regression problems are compared to ols for chaotic time series corrupted by additive measurement noise these methods reduce the variance compared to ols but introduce more bias a main tool of analysis is the singular value decomposition svd and a key to successful regularization is to damp the higher order svd components several of the methods achieve improved prediction compared to ols for synthetic noise corrupted data from well known chaotic systems similar results were found for real world data from the r r intervals of ecg signals good results are also obtained for real sunspot data compared to published predictions using nonlinear techniques

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 characterization of l fuzzy prime ideals abstract in this paper the concepts of a l fuzzy prime and a l fuzzy completely prime ideal are given and some fundamental lemmas are proved in particular bijections among some of the l fuzzy prime ideals of two homomorphic rings are constructed also a characterization of a l fuzzy prime ideal of a ring is given

certain fuzzy ideals of rings redefined abstract an attempt is made to redefine fuzzy semiprimary fuzzy primary fuzzy prime and fuzzy maximal ideals of rings these fuzzy ideals are characterized by their level ideals fuzzy analogues of various results concerning semiprimary primary prime and maximal ideals are obtained the concept of a fuzzy nil radical of a fuzzy ideal is also formulated and discussed

prime l fuzzy ideals and primary l fuzzy ideals abstract in this paper the concepts of a primary l fuzzy ideal and a prime l fuzzy ideal and p primary are introduced and some fundamental propositions proved in particular a structural theorem for a primary l fuzzy ideal belonging to a prime l fuzzy ideal is given

the lattice of l ideals of a ring is modular in this paper we extend the notion of a tip extended pair of fuzzy subgroups to l ideals of a ring we prove that the sum of two tip extended l ideals of an arbitrary pair of l ideals of a ring is the least l ideal containing the union of the given l ideals using this construction of join of l ideals we prove that the lattice of all l ideals of a given ring is modular

nonparametric identification of nonlinear time series projections abstract we study the possibility of identifying general linear and nonlinear time series models using nonparametric methods the kernel estimators of the conditional mean and variance are used as a basis and the properties of these quantities as model indicators are briefly discussed some drawbacks are pointed out and motivated by these we introduce projections as tools of identification the projections are especially useful for additive modeling expressions for the asymptotic bias and variance are obtained the projection of the conditional variance is suggested as a tool for identifying heteroscedastic time series the results are illustrated by simulations for both the estimators of the projections and the estimators of the conditional mean and variance

tarso modeling of water table depths threshold autoregressive self exciting open loop tarso models are fitted to six time series of water table depths with precipitation excess as input variable basically these models are nonlinear in structure because they incorporate several regimes which are separated by so called thresholds for each well a subset tarso ss tarso model is selected using a bayes information criterion bic ss tarso models are used to simulate realizations of water table depths with lengths of 30 years from which characteristics such as durations of exceedance are computed the simulation performance of the fitted ss tarso models is compared with results obtained from transfer function noise tfn models dynamic regression dr models and with a physical descriptive model called swatre extended with additional autoregressive moving average arma processes for the noise swatre arma as compared to the linear tfn and dr models the ss tarso models perform better because they incorporate several regimes these regimes are the result of different soil layers or drainage levels furthermore it is interesting that ss tarso models show a good relative performance as compared to the swatre arma models a possible reason may be that inputs of swatre are uncertain

efficiency improvements in inference on stationary and nonstationary fractional time series we consider a time series model involving a fractional stochastic component whose integration order can lie in the stationary invertible or nonstationary regions and be unknown and an additive deterministic component consisting of a generalized polynomial the model can thus incorporate competing descriptions of trending behavior the stationary input to the stochastic component has parametric autocorrelation but innovation with distribution of unknown form the model is thus semiparametric and we develop estimates of the parametric component which are asymptotically normal and achieve an m estimation efficiency bound equal to that found in work using an adaptive lam lan approach a major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies this is relevant also when the innovation density is parameterized and we provide a result for that case also our semiparametric estimates employ nonparametric series estimation which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models our work thus also contributes to methods and theory for nonfractional time series models such as autoregressive moving averages a monte carlo study of finite sample performance of the semiparametric estimates is included

miscellanea a note on tests for nonlinearity in a vector time series a multivariate extension of the univariate nonlinearity test of tsay 1986 is presented simulation results show that the multivariate test is more powerful than its univariate counterpart especially for series having nonlinear structure involving several components of the vector process and weakly or moderately cross correlated process error terms for illustration the test is applied to a set of seasonally adjusted quarterly capital expenditures and appropriations in u s manufacturing and a vector nonlinear model for the data is constructed

on a class of m estimators for gaussian long memory models summary we consider estimation for parametric stationary gaussian models with long memory the spectral density f x 0 is assumed to be characterised by a vector 0 01 022 03 om such that 02 he 1 1 and f x 0 is proportional to xl 2h as x tends to zero an approximate maximum likelihood estimator based on the autoregressive representation of the process is proposed its asymptotic distribution is derived more generally the approach leads to a class of m estimators for which a central limit theorem holds by choosing an appropriate t function robustness against additive outliers can be achieved while keeping high efficiency under the ideal model this is illustrated by a small simulation study a simple algorithm and an explicit formula for the efficiency and thus for choosing an appropriate tuning parameter are given for hampel s redescending function

time series regression with long range dependence a central limit theorem is established for time series regression estimates which include generalized least squares in the presence of long range dependence in both errors and stochastic regressors the setting and results differ significantly from earlier work on regression withlong range dependent errors spectral singularities are permitted at any frequency when sufficiently strong spectral singularities in the error and a regressor coincide at the same frequency least squares need no longer be n 1 2 consistent where n is the sample size however we show that our class of estimates is n 1 2 consistent and asymptotically normal in the generalized least squares case we show that efficient estimation is still possible when the error autocorrelation is known only up to finitely many parameters we include a monte carlo study of finite sample performance and provide an extension to nonlinear least squares

reconstructing attractors from filtered time series abstract in this paper we investigate the criteria necessary to reconstruct an attractor from a time series that has been passed through an iir filter to guarantee that the whole of the attractor is reconstructed requires uniform bounds on the contraction rate of the dynamics however we show that if we use a condition that is only based on the average contraction rates the lyapunov exponents then a set of high probability can still be reconstructed we argue that in practice this is all that is necessary