generalization effects of k neighbor interpolation training this paper describes a new training method for a continuous mapping and or pattern classification neural network that performs local sample density smoothing a conventional training method uses point to point mapping from an input space to an output space even though the mapping may be precise at two given training sample points there are no guarantees of mapping accuracy at points on a line segment connecting the sample points this paper first discusses a theory for formulating line to line mapping the theory is called interpolation training this paper then expands the theory to k nearest neighbor interpolation the k neighbor interpolation training knit method connects an input sample training point to its k neighbor points via k line segments then the method maps these k line segments in the input space for each training sample to linear line segments in the output space that interpolate between training output values thus a web structure made by connecting input samples is mapped into the same structure in an output space the knit method reduces the over learning problem caused by point to point training by smoothing input output functions simulation tasks show that knit improves vowel recognition on a small speech database

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

testing for symmetry summary we introduce the characteristic symmetry function based on the characteristic function of the underlying distribution whose behaviour is indicative of symmetry or its absence a statistic is proposed for testing symmetry about an unspecified centre derived from the empirical characteristic symmetry function the statistic is readily computible it utilizes information in the empirical characteristic function over an interval and does not require the estimation of the centre of symmetry under general symmetry the asymptotic null distribution of the statistic is folded normal the empirical power for selected alternatives is studied by a small scale simulation and a numerical illustration is given

equilibrium states for factor maps between subshifts abstract let x y be a factor map where x x and y y are subshifts over finite alphabets assume that x satisfies weak specification let a a 1 a 2 r 2 with a 1 0 and a 2 0 let f be a continuous function on x with sufficient regularity holder continuity for instance we show that there is a unique shift invariant measure on x that maximizes f d a 1 h x a 2 h 1 y in particular taking f 0 we see that there is a unique invariant measure on x that maximizes the weighted entropy a 1 h x a 2 h 1 y which answers an open question raised by gatzouras and peres 1996 in 15 an extension is given to high dimensional cases as an application we show that for each compact invariant set k on the k torus under a diagonal endomorphism if the symbolic coding of k satisfies weak specification then there is a unique invariant measure supported on k so that dim h dim h k

dimension and product structure of hyperbolic measures we prove that every hyperbolic measure invariant under a c 1 fi difieomorphism of a smooth riemannian manifold possesses asymptotically almost local product structure i e its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials this has not been known even for measures supported on locally maximal hyperbolic sets using this property of hyperbolic measures we prove the long standing eckmann ruelle conjecture in dimension theory of smooth dynamical systems the pointwise dimension of every hyperbolic measure invariant under a c 1 fi difieomorphism exists almost everywhere this implies the crucial fact that virtually all the characteristics of dimension type of the measure including the hausdorfi dimension box dimension and information dimension coincide this provides the rigorous mathematical justiflcation of the concept of fractal dimension for hyperbolic measures

local dimensions of measures on infinitely generated self affine sets abstract we show the existence of the local dimension of an invariant probability measure on an infinitely generated self affine set for almost all translations this implies that an ergodic probability measure is exactly dimensional furthermore the local dimension equals the minimum of the local lyapunov dimension and the dimension of the space

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

a test for asymmetry associated with the hodges lehmann estimator abstract one version of the hodges lehmann estimator median xi xj 2 1 i j n is also the minimum distance estimator derived from a particular cramer von mises distance this distance evaluated at that is the minimized distance provides a natural statistic for testing asymmetry of the underlying distribution

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

fuzzy pair wise dominance and fuzzy indices an evaluation of productive performance abstract usually efficiency measurement and evaluation are based on the definition of a frontier that envelops the observed production plans measurement and evaluation of productive performance is also achieved with the concept of pair wise dominance that does not require the existence of a frontier along with the required technological assumptions needed for its definition in situations where measurement inaccuracies occur the traditional assumption of crisp production plans can be substituted with the alternative assumption of fuzzy production plans as proposed by fuzzy set theory this research presents indices that capture the degree to which pair wise dominance occurs between two fuzzy production plans the proposed approach is based on the various comparison indices known from the literature that are used to compare fuzzy intervals and is compared with an earlier fuzzy pair wise classification scheme finally the approach is used to evaluate the productive performance of suspect production plans from the preprint insertion manufacturing process

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

generating fuzzy membership functions a monotonic neural network model abstract a crucial issue in the practical applications of fuzzy sets is to find a fuzzy membership function this paper suggests that neural networks which use the back propagation learning algorithm under monotonic function constraints can be used in generating fuzzy membership functions

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

hausdorff dimension for randomly perturbed self affine attractors in this paper we shall consider a self affine iterated function system in mathbb r d d 2 where we allow a small random translation at each application of the contractions we compute the dimension of a typical attractor of the resulting random iterated function system complementing a famous deterministic result of falconer which necessarily requires restrictions on the norms of the contraction however our result has the advantage that we do not need to impose any additional assumptions on the norms this is of benefit in practical applications where such perturbations would correspond to the effect of random noise we also give analogous results for the dimension of ergodic measures in terms of their lyapunov dimension finally we apply our method to a problem originating in the theory of fractal image compression

uncertainty estimation using fuzzy measures for multiclass classification uncertainty arises in classification problems when the input pattern is not perfect or measurement error is unavoidable in many applications it would be beneficial to obtain an estimate of the uncertainty associated with a new observation and its membership within a particular class although statistical classification techniques base decision boundaries according to the probability distributions of the patterns belonging to each class they are poor at supplying uncertainty information for new observations previous research has documented a multiarchitecture monotonic function neural network model for the representation of uncertainty associated with a new observation for two class classification this paper proposes a modification to the monotonic function model to estimate the uncertainty associated with a new observation for multiclass classification the model therefore overcomes a limitation of traditional classifiers that base decisions on sharp classification boundaries as such it is believed that this method will have advantages for applications such as biometric recognition in which the estimation of classification uncertainty is an important issue this approach is based on the transformation of the input pattern vector relative to each classification class separate monotonic single output neural networks are then used to represent the degree of similarity between each input pattern vector and each class an algorithm for the implementation of this approach is proposed and tested with publicly available face recognition data sets the results indicate that the suggested approach provides similar classification performance to conventional principle component analysis pca and linear discriminant analysis lda techniques for multiclass pattern recognition problems as well as providing uncertainty information caused by misclassification

a data driven smooth test of symmetry in this paper we propose a data driven smooth test of symmetry we first transform the raw data via the probability integral transformation according to a symmetrized empirical distribution and show that under the null hypothesis of symmetry the transformed data has a limiting uniform distribution reducing testing for symmetry to testing for uniformity employing neyman s smooth test of uniformity we show that only odd ordered orthogonal moments of the transformed data are required in constructing the test statistic we present a standardized smooth test that is distribution free asymptotically and derive the asymptotic behavior of the test and establish its consistency extension to dependent data case is discussed we investigate the finite sample performance of the proposed tests on both homogeneous and mixed distributions with unobserved heterogeneity an empirical application on testing symmetry of wage adjustment process based on heterogeneous wage contracts with different durations is provided

strong limit theorems for oscillation moduli of the uniform empirical process let u n t n tfrac 1 2 gamma n t t 0 leqq t leqq 1 denote the uniform empirical process based on the first n of a sequence 1 2 of iid uniform 0 1 random variables where gamma n t n 1 sum limits i 1 n 1 0 t xi i is the empirical distribution function the oscillation modulus of u n is defined by omega n a sup u n t h u n t 0 leqq t leqq 1 h h leqq a and the lipschitz 1 2 modulus of u n is defined by tilde omega n a sup u n t h u n t h tfrac 1 2 0 leqq t leqq 1 h a leqq h leqq 1

gaussian characterization of uniform donsker classes of functions it is proved that for classes of functions y satisfying some measurability the empirical processes indexed by f and based on p e 9 s satisfy the central limit theorem uniformly in p e 9w s if and only if the p brownian bridges gop indexed by y are sample bounded and pp uniformly continuous uniformly in p e 9 s uniform exponential bounds for empirical processes indexed by universal bounded donsker and uniform donsker classes of functions are also obtained

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

bootstrap and randomization tests of some nonparametric hypotheses in this paper the asymptotic behavior of some nonparametric tests is studied in situations where both bootstrap tests and randomization tests are applicable under fairly general conditions the tests are asymptotically equivalent in the sense that the resulting critical values and power functions are appropriately close this implies among other things that the difference in the critical functions of the tests evaluated at the observed data tends to 0 in probability randomization tests may be preferable since an exact desired level of the test may be obtained for finite samples examples considered are testing independence testing for spherical symmetry testing for exchangeability testing for homogeneity and testing for a change point

hausdorff dimension of the limit sets of some planar geometric constructions abstract we determine the hausdorff and box dimension of the limit sets for some class of planar non moran like geometric constructions generalizing the bedford mcmullen general sierpinski carpets the class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines as well as some non affine examples e g the flexed sierpinski gasket

dimensions of invariant sets of expanding maps we consider a compact invariant set of an expanding map of a manifoldm and give upper and lowerbounds for the hausdorff dimension dimh and box dimensionsdimb and dimb these bounds are given in terms of the topological entropy topological pressure and uniform lyapunov exponents of the map

dimension theory of iterated function systems let si l i 1 be an iterated function system ifs on d with attractor k let denote the one sided full shift over the alphabet 1 l we define the projection entropy function h on the space of invariant measures on e associated with the coding map k and develop some basic ergodic properties about it this concept turns out to be crucial in the study of dimensional properties of invariant measures on k we show that for any conformal ifs respectively the direct product of finitely many conformal ifss without any separation condition the projection of an ergodic measure under is always exactly dimensional and its hausdorff dimension can be represented as the ratio of its projection entropy to its lyapunov exponent respectively the linear combination of projection entropies associated with several coding maps furthermore for any conformal ifs and certain affine ifss we prove a variational principle between the hausdorff dimension of the attractors and that of projections of ergodic measures 2008 wiley periodicals inc

geometrical representations for mcda abstract in this paper geometrical representations for multicriteria decision problems are proposed this new approach provides assistance to understand the conflictual aspects of the criteria and to tackle the problem of the weights associated to them a generalized criterion including a preference function is first generated for each criterion this allows to define unicriterion preference flows for which a geometrical representation can be obtained by using the principal components analysis the actions are represented by points and criteria by axes in the pca plane a decision axis taking into account the weights associated to the criteria can be defined this technique provides the decision maker with a considerable enrichment for the understanding of his problem clusters of actions can be considered the importance of the criteria can be evaluated conflictual criteria are immediately detected incomparability between actions is emphasized and explained best compromise actions are easily selected new decision axes representing possible clusters of criteria can be considered undesirable actions can be eliminated the technique consists in a powerful new qualitative decision tool it is illustrated in the paper on some examples treated by the promethee i and ii methods a didactic and user friendly microcomputer code is available