testing for arch in the presence of a possibly misspecified conditional mean abstract ever since the development of the autoregressive conditional heteroskedasticity arch model engle 1982 testing for the presence of arch has become a routine diagnostic one popular method of testing for arch is t times the r2 from a regression of the squared residuals on p of its lags this test has been shown to have a lagrange multiplier interpretation and is asymptotically distributed as a 2 p random variable underlying this test is the assumption of a correctly specified conditional mean in this paper we consider the properties of the arch test when there is a possibly misspecified conditional mean examples of misspecification include omitted variables structural change and parameter instability we show that in general misspecification will lead to overrejection of the null hypothesis of conditional homoskedasticity we demonstrate the use of recursive residuals to improve the fit of a first stage conditional mean regression we illustrate these results via monte carlo simulations and consider two empirical examples

interval valued membership function estimation for fuzzy modeling abstract fuzzy modeling is an important topic in fuzzy sets theory and applications a powerful method for constructing an interval valued takagi sugeno fuzzy model ivfm based on input output data of the identified system is presented in this investigation a takagi sugeno fuzzy model is automatically generated in three steps 1 structure identification 2 envelope detection and 3 parameters identification in the structure identification phase a clustering method based on gustafson kessel algorithm is used in order to detect the linear subsystems of the whole nonlinear system local linearization then an envelope detection algorithm eda based on derivative concept is proposed to estimate both the upper and lower functions of the interval valued membership function defined point wise in the parameter identification step the least squares algorithm is applied to compute the best parameter values of the premises gaussians and the kalman filter algorithm to compute the consequences straight lines parameters the effectiveness of this approach is demonstrated on approximating some nonlinear static functions real world data and dynamical systems

quantitative recurrence properties in conformal iterated function systems abstract we consider a quantitative version of poincare s recurrence theorem in a conformal iterated function system let i i be a conformal iterated function system on 0 1 d with a countable index set denote by j the attractor of let f 0 1 d r be a positive function s n f x be the sum f x f w 1 1 x f w 1 w n 1 1 x analogous to an ergodic sum and consider the set of points for which the inequality x w 1 w n 0 1 d x w 1 w n 1 x e s n f x is realized for infinitely many n n with w 1 w n n the set above contains the points x whose orbit returns very close to x infinitely many times with the quality of approximation depending on the time n and the point x this can be viewed as a quantitative version of poincare s recurrence theorem it is shown that its hausdorff dimension is the solution to some pressure function the setting considered in this paper includes the quantitative recurrence properties in the dynamical systems of b adic expansions continued fraction expansion as well as some dynamical systems defined on fractal sets applying the main result to diophantine approximation gives a partial answer to a question by k mahler

on dichotomy law for beta dynamical system in parameter space let varphi mathbb n rightarrow 0 1 be a positive function and t beta be the beta transformation for any beta 1 we prove that the set begin aligned e 0 varphi beta 1 t n beta 1 0 varphi n text for infinitely many n in mathbb n end aligned is of zero or full lebesgue measure in 1 infty according to sum varphi n infty or not as an application we determine the exact lebesgue measure of the set begin aligned mathfrak e 0 l n beta 1 t n beta 1 0 beta l n text for infinitely many n in mathbb n end aligned where l n n ge 1 is a sequence of non negative real numbers

a type 2 fuzzy c regression clustering algorithm for takagi sugeno system identification and its application in the steel industry this paper proposes a new type 2 fuzzy c regression clustering algorithm for the structure identification phase of takagi sugeno t s systems we present uncertainties with fuzzifier parameter m in order to identify the parameters of interval type 2 fuzzy sets two fuzzifiers m 1 and m 2 are used then by utilizing these two fuzzifiers in a fuzzy c regression clustering algorithm the interval type 2 fuzzy membership functions are generated the proposed model in this paper is an extended version of a type 1 fcrm algorithm 25 which is extended to an interval type 2 fuzzy model the gaussian mixture model is used to create the partition matrix of the fuzzy c regression clustering algorithm finally in order to validate the proposed model several numerical examples are presented the model is tested on a real data set from a steel company in canada our computational results show that our model is more effective for robustness and error reduction than type 1 nfcrm and the multiple regression

fuzzy system modeling using linear distance rules abstract an approach of determining significant variables for multi input systems is proposed by using a modified fuzzy rule form called linear distance rules ldr by defining an index we can know the influences of input signals on output and can determine significant variables from the index we also propose a simple three layered generalized neural network to realize the ldrs furthermore this approach can be extended to decompose a multi input multi output mimo system into several simpler multi input single output miso systems

kurzweil type metrical diophantine properties in the field of formal laurent series abstract in this article we consider diophantine properties of the orbit of irrational rotations over the field of formal laurent series f q x 1 more precisely for a given f f q x 1 and a sequence r n we investigate the size of the set g f q x 1 min deg q n q f g r n for infinitely many n n in the sense of the haar measure and the hausdorff dimension

non stationary almost sure invariance principle for hyperbolic systems with singularities we investigate a wide class of two dimensional hyperbolic systems with singularities and prove the almost sure invariance principle asip for the random process generated by sequences of dynamically holder observables the observables could be unbounded and the process may be non stationary and need not have linearly growing variances our results apply to anosov diffeomorphisms sinai dispersing billiards and their perturbations the random processes under consideration are related to the fluctuation of lyapunov exponents the shrinking target problem etc

a localized jarnik besicovitch theorem abstract fundamental questions in diophantine approximation are related to the hausdorff dimension of sets of the form x r x where 1 and x is the diophantine approximation exponent of an irrational number x we go beyond the classical results by computing the hausdorff dimension of the sets x r x f x where f is a continuous function our theorem applies to the study of the approximation exponents by various approximation families it also applies to functions f which are continuous outside a set of prescribed hausdorff dimension

a dynamical dimension transference principle for dynamical diophantine approximation diophantine approximation in dynamical systems concerns the diophantine properties of the orbits in classic diophantine approximation the powerful mass transference principle established by beresnevich and velani provides a general principle to the dimension for a limsup set in this paper we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system more precisely let x t be a topological dynamical system where x is a compact metric space and t x rightarrow x is an expanding continuous transformation given y o in x we consider the following limsup set mathcal w t f driven by the dynamical system x t begin aligned big x in x x in b z e s n f log t z text for some z in t n y o text with infinitely many n in mathbb n big end aligned where log t is a function reflecting the local conformality of the transformation t f is a non negative continuous function over x and s n f log t z denotes the ergodic sum f log t z cdots f log t t n 1 z by proposing a dynamical ubiquity property assumed on the system x t we obtain that the dimensions of x and mathcal w t f are both related to the bowen manning mccluskey formulae namely the solution to the pressure functions begin aligned texttt p t log t 0 text and texttt p t log t f 0 text respectively end aligned we call this phenomenon a dynamical dimension transference principle because of its partial analogy with the mass transference principle this general principle unifies and extends some known results which were considered only separatedly before these include the b adic expansions expanding rational maps over julia sets inhomogeneous diophantine approximation on the triadic cantor set and finite conformal iterated function systems

decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors we consider maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium iterates of lebesgue measure converge exponentially fast to physical measure we prove that these maps have exponential decay of correlations over a large class of observables we use this result to deduce exponential decay of correlations for suitable poincare maps of a large class of singular hyperbolic flows from this we deduce a logarithm law for these flows

diophantine approximation and badly approximable sets let x d be a metric space and d a compact subspace of x which supports a non atomic finite measure m we consider natural classes of badly approximable subsets of loosely speaking these consist of points in which stay clear of some given set of points in x the classical set bad of badly approximable numbers in the theory of diophantine approximation falls within our framework as do the sets bad i j of simultaneously badly approximable numbers under various natural conditions we prove that the badly approximable subsets of have full hausdorff dimension applications of our general framework include those from number theory classical complex p adic and formal power series and dynamical systems iterated function schemes rational maps and kleinian groups

diophantine approximation of the orbit of 1 in the dynamical system of beta expansions we consider the distribution of the orbits of the number 1 under the transformations as varies mainly the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its hausdorff dimension the dimension of the following set is determined where is a given point in and is a sequence of integers tending to infinity as for the proof of this result the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders the set of with a common prefix in the expansion of 1 in the parameter space

the ergodic theory of shrinking targets to any dynamical system equipped with a metric we associate a class of well approximable sets in the case of an expanding rational map of the riemann sphere acting on its julia set we estimate and in some cases compute the hausdorff dimension of the associated well approximable sets the methods used show a clear link between distortion properties and the type of results obtained in this paper via ergodic theory and ubiquity

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

a multifractal mass transference principle for gibbs measures with applications to dynamical diophantine approximation let mu be a gibbs measure of the doubling map t of the circle for a mu generic point x and a given sequence r n subset of r consider the intervals t n x r n mod 1 t n x r n mod 1 in analogy to the classical dvoretzky covering of the circle we study the covering properties of this sequence of intervals this study is closely related to the local entropy function of the gibbs measure and to hitting times for moving targets a mass transference principle is obtained for gibbs measures that are multifractal such a principle was proved by beresnevich and velani ann math 164 2006 971 992 for monofractal measures in the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence in particular we can describe the occurrence of atypical relatively long words our results have a direct and deep number theoretical interpretation via inhomogeneous dyadic diophantine approximation by numbers belonging to a given dyadic diophantine class less

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov

approximation orders of real numbers by beta expansions we prove that almost all real numbers with respect to lebesgue measure are approximated by the convergents of their beta expansions with the exponential order beta n moreover the hausdorff dimensions of sets of the real numbers which are approximated by all other orders are determined these results are also applied to investigate the orbits of real numbers under beta transformation the shrinking target type problem the diophantine approximation and the run length function of beta expansions

modified shrinking target problem in beta dynamical systems abstract in the paper we investigate a modified version of the shrinking target problem in beta dynamical systems which is induced by the transformation t x x mod 1 the hausdorff dimension of the lim sup set w f x 0 1 t n x f x n for infinitely many n n is determined where f 0 1 0 1 is a lipschitz function and is a positive function defined on n the set w f can be viewed as a dynamical analogue of the inhomogeneous diophantine approximation in which the inhomogeneous factor is allowed to vary