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

on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

nonparametric quantile estimations for dynamic smooth coefficient models in this article quantile regression methods are suggested for a class of smooth coefficient time series models we use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework we establish the asymptotic properties of both the local polynomial and local constant estimators for mixing time series also a bandwidth selector based on the nonparametric version of the akaike information criterion is suggested together with a consistent estimate of the asymptotic covariance matrix furthermore the asymptotic behaviors of the estimators at boundaries are examined a comparison of the local polynomial quantile estimator with the local constant estimator is presented a simulation study is carried out to illustrate the performance of estimates an empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures

random walks on networks cumulative distribution of cover time we derive an exact closed form analytical expression for the distribution of the cover time for a random walk over an arbitrary graph in special case we derive simplified exact expressions for the distributions of cover time for a complete graph a cycle graph and a path graph an accurate approximation for the cover time distribution with computational complexity of o2n is also presented the approximation is numerically tested only for graphs with n1000 nodes the random walk is a fundamental dynamic process which can be used to model random processes inherent to many important applications such as transport in disordered media 1 neuron firing dynamics 2 spreading of diseases 3 or transport and search processes 4 8 in this paper we investigate random walks on graphs 9 and derive exact expressions for the cumulative distribution functions for three quantities of a random walk that play the most important role in the theory of random walks 1 hit ting time hij or first passage time which is the number of steps before node j is visited starting from node i 2 com mute time ij hij hji and cover time that is the number of steps to reach every node average hitting time average commute time and average cover time have been recently studied in several papers in 10 the authors investigated random walks on complex net works and derived an exact expression for the mean first passage time between two nodes for each node the random walk centrality is introduced which determines the relative speed by which a node can receive and spread information over the network in a random process using both numerical simulations and scaling arguments the behavior of a random walker on a one dimensional small world network is studied in 11 the average number of distinct sites visited by the random walker the mean square displacement of the walker and the distribution of first return times obey a characteristic scaling form the expected time for a random walk to traverse between two arbitrary sites of the erdos renyi ran dom graph is studied in 12 the properties of random walks on complex trees are studied in 13 both the vertex discov ery rate and the mean topological displacement from the ori gin present a considerable slowing down in the tree case moreover the mean first passage time displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks 13 the random walk on net works has also much relevance to algorithmic applications the expected time taken to visit every vertex of connected graphs has recently been extensively studied in a series of papers cooper and frieze studied the average cover time of various models of a random graph see for example 14 this is an outline of the paper in sec ii we derive closed formulas of the cumulative distribution function for hitting time commute time and cover time we also present a simple example of a graph with four nodes and derive closed formulas of the cumulative distribution function for cover time of complete graphs cycle and path graphs an approximation of the cumulative distribution function for cover time is proposed in the sec iii we also present some numerical results of the cumulative distribution function for cover time of different graphs sec iv we finish the paper with conclusions

on hotelling s formula for the volume of tubes and naiman s inequality abstract motivation by the question of testing for a nonlinear parameter in a regression model with independent homoscedastic normal residuals hotelling 1939 was led to consider the geometric problem of computing the volume of a tube of given radius around a curve in s to the n 1 power the unit sphere in ir to the nth power the answer involves only the arc length of the curve and not its curvature providing the radius of the tube is sufficiently small that there is no self overlap in the tube starting from a somewhat different statistical setting naiman 1986 arrived at the same geometric problem and showed that hotelling s result properly interpreted is an upper bound for the volume of a tube of arbitrary radius the purpose of this note is to give two new derivations of the hotelling naiman results the first involves differential inequalities the second is probabilistic using the concept of upcrossing borrowed from the theory of gaussian processes in the context of gaussian processes knowles 1987 has observed that approximations obtained from hotelling s result and bounds derived via upcrossings are related

abstract tubes improved inclusion exclusion identities and inequalities and importance sampling numerous statistical applications require the evaluation of the probability content of a convex polyhedron we demonstrate for a given polyhedron in r d that there is a depth d inclusion exclusion identity for its indicator function which is a linear combination of indicator functions of intersections of at most d half spaces terms in the identity are determined by the incidence of the facets of the polyhedron which can be found using linear programming this identity can be truncated at any depth to give a lower or upper bound in addition the resulting inequalities lead to importance sampling schemes for evaluating the probability content and these methods tend to be more efficient than the naive hit or miss monte carlo method these results arise in a more general setting which we introduce an tube consists of a pair a s where a a1 a n is a collection of sets s is a simplicial complex and where each subcomplex s x f s x i f a i is contractible whenever x u i n 1 a i the notion presented here is stronger than the one introduced earlier by naiman and wynn several examples are given and key consequences are demonstrated in particular arrangements of points and half spaces in r d give rise to abstract tubes via voronoi decompositions and their associated delauney dual complexes every abstract tube is shown to give rise to an inclusion exclusion identity for i u n i 1 a i and upper and lower bounds are obtained by truncating the identity at an even or an odd depth this property is analogous to the truncation inequality property of the classical inclusion exclusion identity which may be viewed as a special case the notion of an abstract subtube is introduced and it is shown that if a s 1 is a subtube of a s 2 then the truncation inequality gotten from the depth m truncation for a s 1 is at least as sharp as the corresponding inequality from a s 2 as a consequence the generalized inclusion exclusion inequalities are always at least as sharp as their classical counterparts

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series we distinguish between two dimensions of a dynamical system given by experimental time series statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data taking into account sampling and noise problems the dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data singular spectrum analysis ssa provides estimates of the statistical dimension ssa also describes the main physical phenomena reflected by the data it gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data we apply ssa to four paleoclimatic records the principal climatic oscillations and the regime changes in their amplitude are detected about 10 degrees of freedom are statistically significant in the data large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension

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

detecting sparse signals in random fields with an application to brain mapping brain mapping data have been modeled as gaussian random fields and local increases in mean are detected by local maxima of a random field of test statistics derived from these data accurate p values for local maxima are readily available for isotropic data based on the expected euler characteristic of the excursion set of the test statistic random field in this article we give a simple method for dealing with nonisotropic data our approach has connections to the model of sampson and guttorp for nonisotropy in which there exists an unknown mapping of the support of the data to a space in which the random fields are isotropic heuristic justification for our approach comes from the nash embedding theorem formally we show that our method gives consistent unbiased estimators for the true p values based on new results of taylor and adler for random fields on manifolds which replace the euclidean metric by the variogram the results are used to detect gender differences in the cortical thickness of the b

discussion on the meeting on chaos robert m may university of oxford until relatively recently attempts to understand how the world works have been based often implicitly on the assumption that simple laws expressed as simple equations will result in simple predictable behaviour conversely complex dynamics were thought typically to result from complicated concatenations of causes roulette wheels may ultimately obey deterministic laws but their behaviour can only effectively be described in stochastic terms the recognition that many simple dynamical systems of which the quadratic map may be the simplest example can exhibit behaviour which not merely looks random but which is so sensitive to initial conditions as to be effectively unpredictable is revolutionizing this view the flip side of the chaos coin is the even more recent recognition that apparently random time series may be produced by very simple low dimensional deterministic systems operating in chaotic regimes thus researchers with backgrounds in theoretical physics or classical applied mathematics are converging on territory previously occupied solely by a subset of statisticians to ask whether we can distinguish the apparent randomness of deterministic chaos from the real randomness of the roulette wheel both kinds of noisy system are unpredictable in the long term but chaotic systems may differ from truly stochastic systems by admitting short term predictability this question has both fundamental intellectual importance and obvious potential applications in many fields not least economics the set of papers brought together for this meeting give in my opinion a superb overview of the current state of the art the authors have avoided becoming bogged down in semantics as casdagli observes it is philosophically possible to argue that ultimately everything is deterministic or oppositely that ultimately noise is all pervasive rather they have taken the commonsense view that whether we classify a system as exhibiting deterministic chaos described by some underlying non linear model or noise described in stochastic terms depends on the details broadly simple low dimensional chaotic dynamical systems may be recognized as deterministic and short term predictable whereas complicated high dimensional systems are best recognized in purely stochastic terms with only probabilistic prediction at any timescale in so far as short term prediction is central to distinctions between low and high dimensional chaotic dynamical systems an exercise that for physicists and applied mathematicians began with the desire to distinguish chaos from noise begins to take pragmatic shape as new ideas about forecasting on the basis of past time series data inter alia papers at this meeting offer important insights into how best to determine the dimension variously defined of a putative attractor given a relatively low dimensional chaotic system plus noise in particular richard smith has clarified the distinction between problems posed by observational noise and system noise broomhead has offered onion skin layers of complications introduced by filtering the data and cheng and tong have explored new approaches to non linear autoregressive methods rather than to attempt any overview and synthesis of the present papers i wish to highlight what i see as some of the likely next steps some touched on at the meeting some not

nonparametric tests of linearity for time series summary we introduce tests of linearity for time series based on nonparametric estimates of the conditional mean and the conditional variance the tests are compared to a number of parametric tests and to nonparametric tests based on the bispectrum asymptotic expressions give bad approximations and the null distribution under linearity is constructed using resampling of the best linear approximation the new tests perform well on the examples tested

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

nonlinear additive models for environmental time series with applications to ground level ozone data analysis abstract environmental time series usually vary systematically in response to meteorological conditions and thus often are not stationary in this article a class of additive models are introduced for environmental time series in which both mean levels and variances of the series are nonlinear functions of relevant meteorological variables backfitting algorithms in nonlinear regression are adopted to estimate the unknown functions in the model and the maximum likelihood method is used to estimate the parameters in the noise component asymptotic properties of the parameter estimates including consistency and limiting distribution are derived under mild conditions the model is applied to daily maxima of ground level ozone concentrations in the chicago area for possible long term trend assessment compared to alternative models the proposed models gave more accurate estimations for the 95th and 99th percentiles of the ozone distribution

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

statics of thin walled pretwisted beams the displacement and strain fields of thin walled pretwisted beams are prescribed in terms of generalized displacements for extension bending torsion and warping differential equations and boundary conditions are obtained from the elastic potential energy functional without assuming coincidence of the beam axis with any of the structural axes this procedure gives a unique consistent definition of sectional moments and generalized forces some simple explicit formulae are derived for homogeneous tension torsion for the general case a computer code is developed on the basis of discretized generalized displacements and a modified energy functional devised to obtain consistent lengthwise variation of the stresses and a unique decomposition of the torsional moment examples show agreement with analytical results for cylindrical beams and illustrate the various coupling effects for beams with pretwist they also demonstrate the usefulness of the explicit formulae for homogeneous tension torsion

a new efficient mixed formulation for thin shell finite element models abstract a nine node shell element is developed by a new and more efficient mixed formulation the new shell element formulation is based on the hellinger reissner principle with independent strain and the concept of degenerate solid shell the new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower part and a higher order part numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable in fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix

an analysis of arithmetic averaging in approximate riemann solvers with an application to steady supercritical flows an analysis of various arithmetic averaging procedures for approximate riemann solvers is made with a specific emphasis on efficiency and a jump capturing property the various alternatives discussed are intended for future work as well as the more immediate problem of steady supercritical free surface flows numerical results are shown for two test problems

the relationship between the direct and weak formulations of a linearised riemann solver for systems of conservation laws abstract the direct and weak formulations of a linearised riemann solver attributable to roe and toumi are shown to be equivalent for certain systems of conservation laws and particular parameter vectors it is demonstrated however that in other cases this is not necessarily true

hotelling s theorem on the volume of tubes some illustrations in simultaneous inference and data analysis we illustrate with contemporary examples hotelling s geometric approach to simultaneous probability calculations hotelling reduces the evaluation of certain normal theory significance probabilities to finding the volume of a tube about a curve in a hypersphere and shows that this volume is often exactly given by length times cross sectional area we review hotelling s result together with some recent complements and then use the approach to set simultaneous prediction regions for some data from gait analysis to study andrews plots in multivariate data analysis and to construct significance tests for projection pursuit regression a by product is a numerical criterion for tube self overlap relevant for example to uniqueness of certain nonlinear least squares estimates

a consistent characteristic function based test for conditional independence abstract y is conditionally independent of z given x if pr f y x z f y x 1 for all y on its support where f denotes the conditional density of y given x z or x this paper proposes a nonparametric test of conditional independence based on the notion that two conditional distributions are equal if and only if the corresponding conditional characteristic functions are equal we extend the test of su and white 2005 a hellinger metric nonparametric test for conditional independence discussion paper department of economics ucsd in two directions 1 our test is less sensitive to the choice of bandwidth sequences 2 our test has power against deviations on the full support of the density of x y z we establish asymptotic normality for our test statistic under weak data dependence conditions simulation results suggest that the test is well behaved in finite samples applications to stock market data indicate that our test can reveal some interesting nonlinear dependence that a traditional linear granger causality test fails to detect

application of a least absolute shrinkage and selection operator to aeroelastic flight test data identification of non linear systems involves estimating unknown parameters and model regressor selection selection of a subset of candidate terms that best describes the observed output model selection is an important step in black box modelling of any observed process this procedure is concerned with selecting a subset of parameters to give a parsimonious description of the system which may afford greater insight into the functionality of the system or a simpler controller design in this study a least absolute shrinkage and selection operator lasso technique is investigated for computing efficient model descriptions of non linear aeroelastic systems the lasso minimises the residual sum of squares by the addition of an l1 penalty term on the parameter vector of the traditional l2 minimisation problem its use for model selection is a natural extension of this constrained minimisation approach to pseudolinear regression problems which produces some model parameters that are exactly zero and the

conservative confidence bands in curvilinear regression this paper gives a method for constructing conservative scheffe type simultaneous confidence bands for curvilinear regression functions over finite intervals the method is based on the use of a geometric inequality giving an upper bound for the uniform measure of the set of points within a given distance from y an arbitrary piecewise differentiable path with finite length in s k 1 the unit sphere in r k the upper bound is obtained by straightening the path so that it lies in a great circle in s k 1

the detection of local shape changes via the geometry of hotelling s t 2 fields this paper is motivated by the problem of detecting local changes or differences in shape between two samples of objects via the nonlinear deformations required to map each object to an atlas standard local shape changes are then detected by high values of the random field of hotelling s t 2 statistics for detecting a change in mean of the vector deformations at each point in the object to control the null probability of detecting a local shape change we use the recent result of adler that the probability that a random field crosses a high threshold is very accurately approximated by the expected euler characteristic ec of the excursion set of the random field above the threshold we give an exact expression for the expected ec of a hotelling s t 2 field and we study the behavior of the field near local extrema this extends previous results for gaussian random fields by adler and x 2 t and f fields by worsley and cao for illustration these results are applied to the detection of differences in brain shape between a sample of 29 males and 23 females

shock capturing for supercritical free surface flows in curvilinear channel geometries abstract a shock capturing scheme is developed for the shallow water equations governing steady supercritical flow in two dimensions in channels with a curvilinear coordinate geometry the scheme incorporates a special averaging procedure of the flow variables as well as details of the transformation that maps a nonrectangular physical domain for a general channel transition

an efficient numerical scheme for the two dimensional shallow water equations using arithmetic averaging a finite difference scheme based on flux difference splitting is presented for the solution of the two dimensional shallow water equations of ideal fluid flow a linearised problem analogous to that of riemann for gas dynamics is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem and incorporates the technique of operator splitting an average of the flow variables across the interface between cells is required and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging this is in contrast to usual square root averages found in this type of riemann solver where the computational expense can be prohibitive the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids nonphysical spurious oscillations an extension to the two dimensional equations with source terms is included the scheme is applied to the one dimensional problems of a breaking dam and reflection of a bore and in each case the approximate solution is compared to the exact solution of ideal fluid flow the scheme is also applied to a problem of stationary bore generation in a channel of variable cross section finally the scheme is applied to two other dam break problems this time in two dimensions with one having cylindrical symmetry each approximate solution compares well with those given by other authors

memory universal prediction of stationary random processes we consider the problem of one step ahead prediction of a real valued stationary strongly mixing random process xi sub i spl infin sup spl infin the best mean square predictor of x sub 0 is its conditional mean given the entire infinite past x sub i sub i spl infin sup 1 given a sequence of observations x sub 1 x sub 2 x sub n we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension for example neural networks and legendre polynomials the proposed estimators select both the model memory and the model dimension in a data driven fashion by minimizing certain complexity regularized least squares criteria when the underlying predictor function has a finite memory we establish that the proposed estimators are memory universal the proposed estimators which do not know the true memory deliver the same statistical performance rates of integrated mean squared error as that delivered by estimators that know the true memory furthermore when the underlying predictor function does not have a finite memory we establish that the estimator based on legendre polynomials is consistent

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria

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

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

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

relationship between the fractal dimension and the power law index for a time series a numerical investigation abstract a time series showing self affine over all time scales follows a power law spectrum when such a time series x j j 1 2 n is transformed into the discrete fourier transformation i e x j n 2 k 0 c k cos 2 jkn k where c k and k are the amplitude and phase for wavenumber k respectively the power spectrum density for wavenumber k is given by p k c 2 k n k the relationship between the power law index and the fractal dimension d for a time series following a power law spectrum is investigated by using a numerical experiment it should be noted that the time series with the same power law index shows various behaviors in a time domain depending on distribution of k and gives different values of the fractal dimension accordingly in short the phase distribution strongly affects the irregularity represented in terms of the fractal dimension in addition the relationships between and d are also examined both for the differenced and for the integrated time series

a proof of the conjecture that the tukey kramer multiple comparisons procedure is conservative in this paper we present a first general proof of tukey s 1953 conjecture concerning the extension of the tukey multiple comparisons procedure to the case of unequal sample sizes thereby proving that the so called tukey kramer procedure is conservative in all cases also a brief history of the conjecture is given and some extensions and further problems concerning the procedure are discussed

an efficient numerical scheme for the euler equations a finite difference scheme based on flux difference splitting is presented for the solution of the euler equations for the compressible flow of an ideal gas a linearised riemann problem is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem an average of the flow variables across the interface between cells is required and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging this is in contrast to the usual square root averages found in this type of riemann solver where the computational expense can be prohibitive the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids nonphysical spurious oscillations the scheme is applied to a shock tube problem and a blast wave problem each approximate solution compares well with those given by other schemes and for the shock tube problem is in agreement with the exact solution

multiple testing of general contrasts using logical constraints and correlations abstract use of logical constraints among hypotheses and correlations among test statistics can greatly improve the power of step down tests an algorithm for uncovering these logically constrained subsets in a given dataset is described the multiple testing results are summarized using adjusted p values which incorporate the relevant dependence structures and logical constraints these adjusted p values are computed consistently and efficiently using a generalized least squares hybrid of simple and control variate monte carlo methods and the results are compared to alternative stepwise testing procedures

inclusion exclusion bonferroni identities and inequalities for discrete tube like problems via euler characteristics identity for the union of finitely many balls in d dimensional euclidean space with a single correction term this result also holds in the d dimensional sphere these results form the basis for a discrete theory of tubes which up to now has been continuous in nature the spherical result is used to give a simulation method for finding critical probabilities for multiple comparisons procedures and a computer program implementing the method is described numerical results are presented which demonstrate that in the tails of the distribution probability estimates based on the method tend to exhibit less variability than estimates based on naive simulation 1 introduction this paper touches on several areas of mathematics statistics and elementary probability theory the main results are somewhat technical in proof but easily stated consider a finite collection of measurable subsets ai n the classical inclusion exclusion identity is n

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

the kinematic formula in complex integral geometry given two nonsingular projective algebraic varieties x y c p meeting transversely it is classical that one may express the chern classes of their intersec tion x n y in terms of the chern classes of x and y and the kahler form hyperplane class of p this depends on global considerations however by putting a hermitian connection on the tangent bundle of x we may interpret the chern classes of x as invariant polynomials in the curvature form of the connec tion armed with this local formulation of chern classes we now consider two complex submanifolds not necessarily compact x y c p and investigate the geometry of their intersection the pointwise relation between the chern forms of x n y and those of x and y is rather complicated however when we average integrals of chern forms of x n g y over all elements g of the group of motions of p these can be expressed in a universal fashion in terms of integrals of chem forms of x and y this is then the kinematic formula for the unitary group

an analysis of arithmetic averaging in a riemann solver for the one dimensional euler equations abstract an analysis of various arithmetic averaging procedures is undertaken for an approximate riemann solver for the one dimensional euler equations governing the compressible flow of an ideal gas this study draws on earlier work for both the reduced set of equations governing shallow water flows and the full euler equations and is ultimately intended for use with the equations governing two dimensional steady and unsteady compressible flows

a path length inequality for the multivariate t distribution with applications to multiple comparisons abstract this article presents a new inequality for the multivariate t distribution which implies a new method for multiple comparisons whose foundation rests on a recent inequality due to naiman the new method is promising in view of the fact that it utilizes information estimator intercorrelations ignored by the most widely used multiple comparison methods yet is not computationally prohibitive requiring only the numerical evaluation of a single one dimensional integral in this article the validity of the new method in the normal theoretic general linear model is established and efficiency studies relative to the methods of scheffe bonferroni sidak and hunter worsley are presented the new method is shown to always improve on scheffe s method the new method is also shown to perform well that is to lead to a smaller critical point than its competitors with low degrees of freedom but the method is not as efficient as the hunter worsley method for high degrees of freedom in addition the me

improved multiple comparisons with a control in response surface analysis quadratic response surface methodology often focuses on finding the levels of some coded predictor variables x x 1 x 2 x k that optimize the expected value of a response variable y typically the experimenter starts from some best guess or control combination of the predictors usually coded to x 0 and performs an experiment varying them in a region around this center point the question of interest addressed here is whether any x in the experimental region provides a long run mean response e y preferable to that of the control and if so by what amount this article approaches this question via simultaneous confidence intervals for x e y x e y 0 for all x within a specified distance of 0 a new method for two or more predictors is introduced that gives sharper intervals than the scheffe method and also the sa and edwards adaptation of the casella and strawderman method the new method does not require a rotatable design and allows for one sided simultaneous bounds for x appr

a new approach to the finite element modelling of beams with warping effect as a first step toward developing a finite element formulation that can model coupling among extensional bending and torsional behaviour of beams a new method is proposed to properly represent the warping of arbitrary cross sections the basic approach is to introduce a small warping displacement superimposed over flat cross sections of a shear flexible beam in a deformed configuration numerical tests involving simple isotropic beams undergoing a small elastic displacement demonstrate the validity of the new approach the present approach can be extended to composite beams as well as isotropic beams experiencing a large deflection or finite rotation

the length heuristic for simultaneous hypothesis tests we observe a sequence of test statistics t t 1 t 2 t j each of which is approximately n 0 1 under the null hypothesis h 0 but which are correlated with each other not being certain which t j is best we use the test statistic t max max t 1 t j for a given observed value of t max say c what is the significance probability pr t max c define the length of t to be j l 2 arccos p j 1 j where p j 1 j is the null hypothesis correlation between t j 1 and t j hotelling s theorem on the volume of tubes leads to a length based bound on pr t max c that usually beats the bonferroni bound it is easy to improve hotelling s bound several examples show that in favourable circumstances the improved bounds can be good approximations to the actual value of pr t max c