are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

cross validation for correlated data abstract k fold cross validation cv with squared error loss is widely used for evaluating predictive models especially when strong distributional assumptions cannot be taken however cv with sq

two step estimation of functional linear models with applications to longitudinal data functional linear models are useful in longitudinal data analysis they include many classical and recently proposed statistical models for longitudinal data and other functional data recently smoothing spline and kernel methods have been proposed for estimating their coefficient functions nonparametrically but these methods are either intensive in computation or inefficient in performance to overcome these drawbacks in this paper a simple and powerful two step alternative is proposed in particular the implementation of the proposed approach via local polynomial smoothing is discussed methods for estimating standard deviations of estimated coefficient functions are also proposed some asymptotic results for the local polynomial estimators are established two longitudinal data sets one of which involves time dependent covariates are used to demonstrate the approach proposed simulation studies show that our two step approach improves the kernel method proposed by hoover and co workers in several aspects such as accuracy computational time and visual appeal of the estimators

on the notion of concept ii it is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory recent work on compositionality and concept determination showed that the quantitative solution initially offered by classical fuzzy logic also led to important drawbacks several qualitative approaches were thereafter tempted that aimed at modeling membership through ordinal scales or lattice fuzzy sets most of the solutions obtained by these theoretical constructions however are of difficult use in categorization theory we propose a simple qualitative model in which membership relative to a given concept f is represented by a function that takes its value in a finite abstract set a f equipped with a total order this function is recursively built through a stratification of the set of concepts at hand based on a notion of complexity similarly the typicality associated with a concept f will be described using an ordering that takes into account the characteristic features of f once the basic notions of membership and typicality are set the study of compound concepts is possible and leads to interesting results in particular we investigate the internal structure of concepts and obtain the characterization of all smooth subconcepts of a given concept

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

belief functions on distributive lattices the dempster shafer theory of belief functions is an important approach to deal with uncertainty in ai in the theory belief functions are defined on boolean algebras of events in many applications of belief functions in real world problems however the objects that we manipulate is no more a boolean algebra but a distributive lattice in this paper we employ birkhoff s representation theorem for finite distributive lattices to extend the dempster shafer theory to the setting of distributive lattices which has a mathematical theory as attractive as in that of boolean algebras moreover we use this more general theory to provide a framework for reasoning about belief functions in a deductive approach on non classical formalisms which assume a setting of distributive lattices as an illustration of this approach we investigate the theory of belief functions for a simple epistemic logic the first degree entailment fragment of relevance logic r by providing an axiomatization for reasoning about belief functions for this logic and by showing that the complexity of the satisfiability problem of a belief formula with respect to the class of the corresponding dempster shafer structures is np complete

analysis of repeated measurements using nonparametric smoothers and randomization tests the mixed model analysis of variance anova which is commonly applied to repeated measurements taken over time depends on specialized assumptions about the error distribution and fails to exploit information contained in the ordering of the data points over time this paper describes a procedure that overcomes these disadvantages while preserving familiar features of the mixed model anova group profiles are estimated by nonparametric smoothing of observed mean profiles group and time main effects and the group by time interaction effect are tested using randomization tests results of zerbe 1979 journal of the american statistical association 74 215 221 are used to construct f test approximations for the randomization tests of the group and group by time effects a new approximate f test for time effect is proposed a simulation study demonstrates that the approximations perform well and that smoothing increases the power of the tests for time main effect and group by time interaction the procedure is applied to data on hormone levels in cows

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

nonparametric trending regression with cross sectional dependence panel data whose series length t is large but whose cross section size n need not be are assumed to have common time trend of unknown form the model includes additive unknown individual specific components and allows for spatial or other cross sectional dependence and or heteroscedasticity a simple smoothed nonparametric trend estimate is shown to be dominated by an estimate which exploits availability of cross sectional data asymptotically optimal bandwidth choices are justified for both estimates feasible optimal bandwidths and feasible optimal trend estimates are asymptotically justified finite sample performance of the latter being examined in a monte carlo study potential extensions are discussed

evaluating decision rules for nitrogen fertilization it is important both for farmer profit and for the environment to correctly dose nitrogen fertilizer for crop growth fertilizer recommendations are embodied in decision rules which give a recommended dose of nitrogen n as a function of information available at the time the decision is made in this paper we first propose a criterion for evaluating decision rules the proposed criterion is the expectation of the objective function when the decision rule is implemented the major problem here is the estimation of this criterion two estimators are considered a model based and a nonparametric estimator a simulation study shows that in essentially all cases the nonparametric estimator is better or no worse than the model based estimator the bias in the nonparametric estimator is always very small

smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables longitudinal samples i e datasets with repeated measurements over time are common in biomedical and epidemiological studies such as clinical trials and cohort observational studies an exploratory tool for the analyses of such data is the varying coefficient model y t xt t t t where y t and x t x 0 t x k t t are the response and covariates at time t t 0 t k t t are smooth coefficient curves of t and t is a mean zero stochastic process a special case that is of particular interest in many situations is data with time dependent response and time independent covariates we propose in this article a componentwise smoothing spline method for estimating 0 t k t nonparametrically based on the previous varying coefficient model and a longitudinal sample of t y t x with time independent covariates x x 0 x k t from n independent subjects a leave one subject out cross validation is suggested to choose the smoothing parameters asymptotic properties of our

nonparametric empirical bayes growth curve analysis abstract nonparametric and semiparametric regression have been suggested as alternatives to parametric models for growth curve analysis in this article we demonstrate that empirical bayes estimation can be used to improve linear smoothing estimates when multiple curves are available

a geometric approach to confidence regions and bands for functional parameters summary functional data analysis is now a well established discipline of statistics with its core concepts and perspectives in place despite this there are still fundamental statistical questions which have received relatively little attention one of these is the systematic construction of confidence regions for functional parameters this work is concerned with developing understanding and visualizing such regions we provide a general strategy for constructing confidence regions in a real separable hilbert space by using hyperellipsoids and hyper rectangles we then propose specific implementations which work especially well in practice they provide powerful hypothesis tests and useful visualization tools without relying on simulation we also demonstrate the negative result that nearly all regions including our own have zero coverage when working with empirical covariances to overcome this challenge we propose a new paradigm for evaluating confidence regions by showing that the distance between an estimated region and the desired region with proper coverage tends to 0 faster than the regions shrink to a point we call this phenomena ghosting and refer to the empirical regions as ghost regions we illustrate the proposed methods in a simulation study and an application to fractional anisotropy tract profile data

a framework for linguistic modelling a new framework for linguistic reasoning is proposed based on a random set model of the degree of appropriateness of a label labels are assumed to be chosen from a finite predefined set of labels and the set of appropriate labels for a value is defined as a random set valued function from a population of individuals into the set of subsets of labels appropriateness degrees are then evaluated relative to the distribution on this random set where the appropriateness degree of a label corresponds to the probability that it is contained in the set of appropriate labels this interpretation is referred to as label semantics a natural calculus for appropriateness degrees is described which is weakly functional while taking into account the logical structure of expressions given this framework it is shown that a bayesian approach can be adopted in order to infer probability distributions on the underlying variable given constraints both in the form of linguistic expressions and mass assignments in addition two conditional measures are introduced for evaluating the appropriateness of a linguistic expression given other linguistic information

selecting the smoothing parameter for estimation of slowly changing evoked potential signals summary brain evoked potential ep data consist of a true response signal and random background activity noise which are observed over repeated stimulus presentations trials a signal that changes slowly from trial to trial can be estimated by smoothing across trials and over time within trials we present a method for selecting the smoothing parameter by minimizing an estimate of the mean average squared error mase we evaluate the performance of this method using simulated ep data and apply the method to an example set of real flash evoked potentials

semiparametric regression analysis for recurrent event interval counts this paper deals with analysis of data from longitudinal studies where the rate of a recurrent event characterizing morbidity is the primary criterion for treatment evaluation we consider clinical trials which require patients to visit their clinical center at successive scheduled times as part of follow up at each visit the patient reports the number of events that occurred since the previous visit or an examination reveals the number of accumulated events such as skin cancers the exact occurrence times of the events are unavailable and the actual patient visit times typically vary randomly about the scheduled follow up times each patient s record thus consists of a sequence of clinic visit dates event counts corresponding to the successive time intervals between clinic visits and baseline covariates we propose a semiparametric regression model extending the fully parametric model of thall 1988 biometrics 44 197 209 to estimate and test for covariate effects on the rate of events over time while also accounting for the possibly time varying nature of the underlying event rate covariate effects enter the model parametrically while the underlying time varying event rate is modelled nonparametrically the method of severini and wong 1992 annals of statistics 20 1768 1802 is used to construct asymptotically efficient estimators of the parametric component and to specify their asymptotic distribution a simulation study and application to a data set are provided

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

set based representations of conjunctive and disjunctive knowledge abstract we look at the use of sets as a method of representing information we study the use of different set operators to implement different semantic operations careful distinction is made between conjunctive and disjunctive type information

covariance matrix estimation using repeated measurements when data are incomplete modeling repeated measurements data has been studied extensively lately in the parametric situation however it is significant to study the effect of various treatments over a period of time where the repeated measurements on the same subject are expected to be correlated the correlation among the repeated measurements for all the subjects will be studied via the covariance matrix the positive definite constraint is one of obstacles that encounters modeling the covariance structure however the cholesky decomposition removes this constraint and allows modeling the components of the covariance matrix m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 in this paper we adopt the estimation procedures introduced by diggle and verbyla nonparametric estimation of covariance structure in longitudinal data biometrics 54 1998 401 415 where the variogram cloud as well as the squared residuals are used to estimate the variogram and the variances via the kernel smoothing selecting the appropriate bandwidth value is one of the important steps in the estimation process thus in our data analysis we choose the bandwidth using one of the most simple straight forward methods which is the cross validation method developed by rice and silverman estimating the mean and covariance structure nonparametrically when the data are curves j roy statist soc b 53 1991 233 243 and adapted by others finally we apply these nonparametric techniques as well as a graphical method m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 to a real life data and use the penalized likelihood criterion like aic and bic to compare models of our interest our interest

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

semiparametric models for longitudinal data with application to cd4 cell numbers in hiv seroconverters summary the paper describes a semiparametric model for longitudinal data which is illustrated by its application to data on the time evolution of cd4 cell numbers in hiv seroconverters the essential ingredients of the model are a parametric linear model for covariate adjustment a nonparametric estimation of a smooth time trend serial correlation between measurements on an individual subject and random measurement error a back fitting algorithm is used in conjunction with a cross validation prescription to fit the model a notable feature in the application is that the onset of hiv infection is associated with a sudden drop in cd4 cells followed by a longer term slower decay the model is also used to estimate an individual s curve by combining his data with the population curve shrinkage toward the population mean trajectory is controlled in a natural way by the estimated covariance structure of the data

wavelet based functional mixed models increasingly scientific studies yield functional data in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid we present new methodology that generalizes the linear mixed model to the functional mixed model framework with model fitting done by using a bayesian wavelet based approach this method is flexible allowing functions of arbitrary form and the full range of fixed effects structures and between curve covariance structures that are available in the mixed model framework it yields nonparametric estimates of the fixed and random effects functions as well as the various between curve and within curve covariance matrices the functional fixed effects are adaptively regularized as a result of the non linear shrinkage prior that is imposed on the fixed effects wavelet coefficients and the random effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient because we have posterior samples for all model quantities we can perform pointwise or joint bayesian inference or prediction on the quantities of the model the adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks

semiparametric and nonparametric regression analysis of longitudinal data this article deals with the regression analysis of repeated measurements taken at irregular and possibly subject specific time points the proposed semiparametric and nonparametric models postulate that the marginal distribution for the repeatedly measured response variable y at time t is related to the vector of possibly time varying covariates x through the equations e y t x t 0 t 0x t and e y t x t 0 t 0 t x t where 0 t is an arbitrary function of t 0 is a vector of constant regression coefficients and 0 t is a vector of time varying regression coefficients the stochastic structure of the process y is completely unspecified we develop a class of least squares type estimators for 0 which is proven to be n consistent and asymptotically normal with simple variance estimators furthermore we develop a closed form estimator for a cumulative function of 0 t which is shown to be n consistent and on proper normalization converges weakly to a zero mean gaussian pr

conditional probability fuzzy sets and possibility a unifying view we propose an interpretation of fuzzy set theory both from a semantic and a syntactic point of view in terms of conditional events and coherent conditional probabilities during past years a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory in this paper we emphasize the role of conditioning in a proper framework i e de finetti s coherence to get rid of many controversial aspects moreover we introduce suitable operations between fuzzy subsets looked on as corresponding operations between conditional events endowed with the relevant conditional probability finally we show how the concept of possibility function naturally arises as a coherent conditional probability

a frequency domain selection criterion for regression with autocorrelated errors abstract we consider the regression model yt t e t t 0 1 n 1 where y t are scalar observations t is the unknown regression function and e t are unobservable errors generated by a zero mean weakly stationary process independent of t and with completely unknown autocorrelation structure we propose a data driven method for selecting a parametric or nonparametric estimator of t the method is based on cross validation in the frequency domain and requires no assumptions about the form of the estimator or the error correlations it does however require the discrete fourier transform dft of the signal t to be a smooth complex function of frequency as is the case for example with transient signals or growth and decay curves after giving some general motivations for the method we focus on the special case of linear estimators of a nonparametric regression function including both kernel and spline estimators for these estimators we develop efficient methods of evaluating t

mixed effects smoothing spline analysis of variance we propose a general family of nonparametric mixed effects models smoothing splines are used to model the fixed effects and are estimated by maximizing the penalized likelihood function the random effects are generic and are modelled parametrically by assuming that the covariance function depends on a parsimonious set of parameters these parameters and the smoothing parameter are estimated simultaneously by the generalized maximum likelihood method we derive a connection between a nonparametric mixed effects model and a linear mixed effects model this connection suggests a way of fitting a nonparametric mixed effects model by using existing programs the classical two way mixed models and growth curve models are used as examples to demonstrate how to use smoothing spline analysis of variance decompositions to build nonparametric mixed effects models similarly to the classical analysis of variance components of these nonparametric mixed effects models can be interpreted as main effects and interactions the penalized likelihood estimates of the fixed effects in a two way mixed model are extensions of james stein shrinkage estimates to correlated observations in an example three nested nonparametric mixed effects models are fitted to a longitudinal data set

asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data abstract we consider the estimation of the k 1 dimensional nonparametric component t of the varying coefficient model y t x t t t e t based on longitudinal observations yij x i tij tij i 1 n j 1 ni where tij is the jth observed design time point t of the ith subject and yij and x i tij are the real valued outcome and r k 1 valued covariate vectors of the ith subject at tij the subjects are independently selected but the repeated measurements within subject are possibly correlated asymptotic distributions are established for a kernel estimate of t that minimizes a local least squares criterion these asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for t applying these methods to an epidemiological study we show that our procedures are useful for predicting cd4 t helper lymphocytes cell changes among hiv human immunodeficiency virus infected persons the finite sample properties of o

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

the finite mixed sturm liouville problems of the dirichlet newton type abstract the mixed sturm liouville problem for the laplace operator in the unit circle is considered the boundary conditions consist in the dirichlet condition prescribed partially at the circumference while the complementary part is subjected to the newton condition the problem is reduced to a homogeneous wing singular integro differential equation and this in turn to an infinite system of algebraic equations whose solution can conveniently be obtained through truncation the procedures of the reduction suggest that almost any problem of the type considered in this paper can be reduced to wing equations of the same class the calculations are carried out right to the numerical steps the first few eigenvalues and their corresponding eigenfunctions are obtained

a distribution free test of constant mean in linear mixed effects models we propose a distribution free procedure an analogy of the dip test in non parametric regression to test whether the means of responses are constant over time in repeated measures data unlike the existing tests the proposed procedure requires very minimal assumptions to the distributions of both random effects and errors we study the asymptotic reference distribution of the test statistic analytically and propose a permutation procedure to approximate the finite sample reference distribution the size and power of the proposed test are illustrated and compared with competitors through several simulation studies we find that it performs well for data of small sizes regardless of model specification finally we apply our test to a data example to compare the effect of fatigue in two different methods used for cardiopulmonary resuscitation

on truth gaps bipolar belief and the assertability of vague propositions this paper proposes an integrated approach to indeterminacy and epistemic uncertainty in order to model an intelligent agent s decision making about the assertability of vague statements initially valuation pairs are introduced as a model of truth gaps for propositional logic sentences these take the form of lower and upper truth valuations representing absolutely true and not absolutely false respectively in particular we consider valuation pairs based on supervaluationist principles and also on kleene s three valued logic the relationship between kleene valuation pairs and supervaluation pairs is then explored in some detail with particular reference to a natural ordering on semantic precision in the second part of the paper we extend this approach by proposing bipolar belief pairs as an integrated model combining epistemic uncertainty and indeterminacy these comprise of lower and upper belief measures on propositional sentences defined by a probability distribution on a finite set of possible valuation pairs the properties of these measures are investigated together with their relationship to different types of uncertainty measure finally we apply bipolar belief measures in a preliminary decision theoretic study so as to begin to understand how the use of vague expressions can help to mitigate the risk associated with making forecasts or promises this then has potential applications to natural language generation systems

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

nonparametric regression analysis of longitudinal data abstract nonparametric methods are developed for estimating the dose effect when a response consists of correlated observations over time measured in a dose response experiment the methods can also be applied to data collected from a completely randomized design experiment methods are developed for the detection and description of the effects of dose time and their interaction the methods allow for individual variation in the timing and number of observations a generalization allowing baseline covariates to be incorporated is addressed these results may be used in an exploratory fashion in the process of building a random effects model for longitudinal data

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

measuring properties of fuzzy sets a general technique and its use in fuzzy query evaluation abstract a fuzzy set can be viewed as a weighted max combination or as a convex combination of a family of nested ordinary sets the latter point of view allows the interpretation of a fuzzy set as a consonant random set then any numerical set function applied to a fuzzy set leads to a result which is a random number the expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set function this approach is very general and can be used for various kinds of measurements cardinality confidence measures attached to the occurrence of events diameter or perimeter of a fuzzy region distance between two sets etc also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed the weighted max combination view which leads to another type of scalar evaluation is also briefly discussed in this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property this kind of query can be managed by languages like sql in case of a crisp set of items when this set is fuzzily specified a scalar answer can still be provided as well as a fuzzy valued or interval valued one if we want to keep track of the effect of the fuzziness of the query the case where the data base contains precise information but also the more general case of fuzzy information data bases are considered

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

kernel regression in the presence of correlated errors it is a well known problem that obtaining a correct bandwidth and or smoothing parameter in nonparametric regression is difficult in the presence of correlated errors there exist a wide variety of methods coping with this problem but they all critically depend on a tuning procedure which requires accurate information about the correlation structure we propose a bandwidth selection procedure based on bimodal kernels which successfully removes the correlation without requiring any prior knowledge about its structure and its parameters further we show that the form of the kernel is very important when errors are correlated which is in contrast to the independent and identically distributed i i d case finally some extensions are proposed to use the proposed criterion in support vector machines and least squares support vector machines for regression

additive extensions to generalized estimating equation methods the methods underlying vector generalized additive models are extended to provide additive extensions to the generalized estimating equations approaches to multivariate regression problems of liang and zeger and the subsequent literature the methods are illustrated by two examples using correlated binary data concerning the presence or absence of several plant species at the same geographical site and chest pain in male workers from a workforce study with minor modification the extensions are shown to apply to longitudinal data

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

test of significance when data are curves abstract with modern technology massive data can easily be collected in a form of multiple sets of curves new statistical challenge includes testing whether there is any statistically significant difference among these sets of curves in this article we propose some new tests for comparing two groups of curves based on the adaptive neyman test and the wavelet thresholding techniques introduced earlier by fan we demonstrate that these tests inherit the properties outlined by fan and that they are simple and powerful for detecting differences between two sets of curves we then further generalize the idea to compare multiple sets of curves resulting in an adaptive high dimensional analysis of variance called hanova these newly developed techniques are illustrated by using a dataset on pizza commercials where observations are curves and an analysis of cornea topography in ophthalmology where images of individuals are observed a simulation example is also presented to illustrate the power of the adapti

inference for smooth curves in longitudinal data with application to an aids clinical trial we discuss a longitudinal study where data for many subjects are collected at irregular intervals the study is a randomized trial of hiv infected subjects and the response variable of interest is serum neopterin the mean of the outcome variable taken over patients in each treatment group is assumed to follow a smooth curve piecewise cubic polynomials with a moderate number of knots are used to model the curves a general parametric form is assumed for the covariance structure maximum penalized likelihood estimation is used to smooth the over parameterized curves statistical inference for the mean curves including confidence bands and hypothesis tests is discussed two approaches one using a bayesian interpretation of the penalized likelihood and the other based on the asymptotic distribution of the maximum penalized likelihood estimates are discussed and contrasted the properties of the confidence bands obtained from these two approaches are evaluated by examining their coverage rates in a simulation study

kernel regression estimation using repeated measurements data abstract the estimation of growth curves has been studied extensively in parametric situations here we consider the nonparametric estimation of an average growth curve suppose that there are observations from several experimental units each following the regression model y xi f xj e j 1 n where e1 e n are correlated zero mean errors and 0 x1 xn 1 are fixed constants we study some of the properties of a kernel estimator of f x asymptotic and finite sample results concerning the mean squared error of the estimator are obtained in particular the influence of correlation on the bandwidth minimizing mean squared error is discussed a data based method for selecting the bandwidth is illustrated in a data analysis most previous research on kernel regression estimators has involved uncorrelated errors we investigate how dependence of the errors changes the behavior of a kernel estimator our theorems concerning the asymptotic mean squared error show that the estimator cannot be consistent un

estimation and hypothesis testing for collections of autoregressive time series summary the analysis of a number of independent first order autoregressive time series is considered in a normal theory context a model is studied which allows for nonstationary and nonidentical distribution of the series caused by both fixed effect and random effect components

multivariate repeated measurement or growth curve models with multivariate random effects covariance structure abstract the analysis of multiple response repeated measurement or growth curve models with a multivariate random effects covariance structure for individuals is considered results on estimation hypothesis testing and simultaneous confidence bounds for the parameters of the generalized linear model of potthoff and roy 1964 are derived for the multiple response repeated measurements model under the multivariate random effects covariance structure two numerical examples are presented to illustrate these techniques

modeling longitudinal data with ordinal response by varying coefficients the paper presents a smooth regression model for ordinal data with longitudinal dependence structure a marginal model with cumulative logit link mccullagh 1980 is applied to cope for the ordinal scale and the main and covariate effects in the model are allowed to vary with time local fitting is pursued and asymptotic properties of the estimates are discussed a data example demonstrates the exploratory flavor of the smooth model in a second step the longitudinal dependence of the observations is considered cumulative log odds ratios are fitted locally which provides insight how the dependence of the ordinal observations changes with time

the justification of the truncation applied to homogeneous integral equations with cauchy s kernel abstract this paper is devoted to studying the influence of the truncation applied to a homogeneous cauchy type singular integral equation on its eigenvalues and eigenfunctions this equation represents the class to which mixed sturm liouville problems of the dirichlet neuman type are reducible the study is illustrated through considering a concrete problem it has been shown that an eigenvalue of the truncated equation tends to the exact corresponding one on increasing the order of the truncation set on that equation as for a truncated eigenfunction it leads to the designation of the exact one in the limit where the corresponding eigenvalue is sufficiently precise an illustrative example is given

nonparametric estimation of covariance structure in longitudinal data summary in longitudinal studies the effect of various treatments over time is usually of prime interest however observations on the same subject are usually correlated and any analysis should account for the underlying covariance structure a nonparametric estimate of the covariance structure is useful either as a guide to the formulation of a parametric model or as the basis for formal inference without imposing parametric assumptions the sample covariance matrix provides such an estimate when the data consist of a short sequence of measurements at a common set of time points on each of many subjects but is impractical when the data are severely unbalanced or when the sequences of measurements on individual subjects are long relative to the number of subjects the variogram of residuals from a saturated model for the mean response has previously been suggested as a nonparametric estimator for covariance structure assuming stationarity in this paper we consider kernel weighted local linear regression smoothing of sample variogram ordinates and of squared residuals to provide a nonparametric estimator for the covariance structure without assuming stationarity the value of the estimator as a diagnostic tool is demonstrated in two applications one to a set of data concerning the blood pressure of newborn babies in an intensive care unit and the other to data on the time evolution of cd4 cell numbers in hiv seroconverters

varying coefficient models and basis function approximations for the analysis of repeated measurements summary a global smoothing procedure is developed using basis function approximations for estimating the parameters of a varying coefficient model with repeated measurements inference procedures based on a resampling subject bootstrap are proposed to construct confidence regions and to perform hypothesis testing conditional biases and variances of our estimators and their asymptotic consistency are developed explicitly finite sample properties of our procedures are investigated through a simulation study application of the proposed approach is demonstrated through an example in epidemiology in contrast to the existing methods this approach applies whether or not the covariates are timeinvariant and does not require binning of the data when observations are sparse at distinct observation times

a fourier method for initial value problems with mixed boundary conditions abstract works concerning solutions of initial problems with mixed boundary conditions are very rare in the literature and essentially depend on the geometry of some individual problems k b ranger for example considered the cooling of a finite needle and strip with prescribed cooling temperature in a medium bounded by two parallel planes perpendicular to the axis and maintained at zero temperature 1 because of the axial symmetries of these problems ranger could employ integral operators which map 2 d harmonics into the solution of the reduced wave equation and into that of the heat equation thereby in this work a systematic method for solving initial plane problems with mixed boundary conditions is proposed it can be considered as a natural generalization of the fourier method that deals with the uniform initial problems on the one hand and of that started by ju i cersi 2 dealing with steady mixed boundary problems on the other by gradual modifications the latter method has become of wide application to solutions of problems in several branches of mathematical physics this has recently been expounded 3 we shall start with presenting the method by means of a concrete example that geometrically as well as physically is rather simplified the heat conductivity in the unit circle with dirichlet neumann boundary conditions the method is applicable to problems with different configurations further it can be extended to deal with more complicated physical situations of mixed character such as dynamical problems of the theory of elasticity with the typical stress strain boundary conditions

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

on modeling of linguistic information using random sets this paper discusses the formal connection between possibility distributions zadeh 21 and the theory of random sets via choquet s theorem based upon these relationships it is suggested that plausible inferences and modeling of common sense can be derived from the statistics of random sets the analysis of subjectivity in meaning representation of natural languages can be carried out by taking account of covariates of individuals as in the statistical analysis of survival data

statistical inferences for functional data with modem technology development functional data are being observed frequently in many scientific fields a popular method for analyzing such functional data is smoothing first then estimation that is statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another however little is known about this substitution effect on functional data analysis in this paper this problem is investigated when the local polynomial kernel lpk smoothing technique is used for individual function reconstructions we find that under some mild conditions the substitution effect can be ignored asymptotically based on this we construct lpk reconstruction based estimators for the mean covariance and noise variance functions of a functional data set and derive their asymptotics we also propose a gcv rule for selecting good bandwidths for the lpk reconstructions when the mean function also depends on some time independent covariates we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time the lpk reconstruction based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived moreover we propose a l 2 norm based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression the effect of the bandwidths selected by the proposed gcv rule on the accuracy of the lpk reconstructions and the mean function estimator is investigated via a simulation study the proposed methodologies are illustrated via an application to a real functional data set collected in climatology

choosing a range for the amount of smoothing in nonparametric regression abstract most nonparametric regression estimates smooth the observed data in the process of generating a suitable estimate the resulting curve estimate is sensitive to the amount of smoothing introduced and it is important to examine the estimate over a range of smoothing choices an objective way to specify such a range is to estimate the optimal value of the smoothing parameter bandwidth with respect to some loss function and then report a confidence interval for this parameter estimate this interval suggests examining three regression estimates an undersmoothed curve evaluated at the lower endpoint of the confidence interval an oversmoothed curve evaluated at the upper endpoint and also an optimal curve chosen by the data based method this article describes two strategies for constructing such confidence intervals using asymptotic approximations and simulation techniques suppose that data are observed from the model yk f tk ek 1 k n where f is a function that is twice differentia

on bandwidth choice in nonparametric regression with both short and long range dependent errors we analyse methods based on the block bootstrap and leave out cross validation for choosing the bandwidth in nonparametric regression when errors have an almost arbitrarily long range of dependence a novel analytical device for modelling the dependence structure of errors is introduced this allows a concise theoretical description of the way in which the range of dependence affects optimal bandwidth choice it is shown that provided block length or leave out number respectively are chosen appropriately both techniques produce first order optimal bandwidths nevertheless the block bootstrap has far better empirical properties particularly under long range dependence

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

inferences for a semiparametric model with panel data summary in a longitudinal study suppose that for each subject repeated measurements of the response variable and covariates are collected at a set of distinct irregularly spaced time points we consider a semiparametric model which relates the mean of the response variable at each time point proportionally to a function of a time dependent covariate vector to analyse such panel data inference procedures for regression parameters are proposed without involving any nonparametric function estimation for the nuisance mean function a dataset from a recent aids clinical trial is used to illustate the new proposal

validity in a logic that combines supervaluation and fuzzy logic based theories of vagueness abstract supervaluationism and fuzzy logic are two complementary formalisms for reasoning with vague information we study a framework for combining both approaches supervaluationism is modeled by a space of precisifications essentially a kripke structure we equip this space with a probability measure to extract the truth value of each propositional variable by measuring the set of precisifications in which it is true complex formulas are evaluated by the truth functions given by a continuous t norm and its residuum we also add a universal modality to this logic besides unrestricted probability measures we motivate two other natural classes strictly positive and uniform probability measures the goal of this paper is to analyze how the choice of a probability measure and a t norm affects the set of valid formulas in our hybrid logic

function on function linear regression by signal compression abstractwe consider functional linear regression models with a functional response and multiple functional predictors with the goal of finding the best finite dimensional approximation to the signal part of the response function defining the integrated squared correlation coefficient between a random variable and a random function we propose to solve a penalized generalized functional eigenvalue problem whose solutions satisfy that projections on the original predictors generate new scalar uncorrelated variables and these variables have the largest integrated squared correlation coefficient with the signal function with these new variables we transform the original function on function regression model to a function on scalar regression model whose predictors are uncorrelated and estimate the model by penalized least square method this method is also extended to models with both multiple functional and scalar predictors we provide the asymptotic consistency and the corresponding convergence rates

local polynomial mixed effects models for longitudinal data we consider a nonparametric mixed effects model yi tij tij vi tij i tij j 1 2 ni i 1 2 n for longitudinal data we propose combining local polynomial kernel regression and linear mixed effects lme model techniques to estimate both fixedeffects population curve t and random effects curves vi t the resulting estimator called the local polynomial lme llme estimator takes the local correlation structure of the longitudinal data into account naturally we also propose new bandwidth selection strategies for estimating t and vi t simulation studies show that our estimator for t is superior to the existing estimators in the sense of mean squared errors the asymptotic bias variance mean squared errors and asymptotic normality are established for the llme estimators of t when ni is bounded and n tends to infinity our llme estimator converges in a standard nonparametric rate and the asymptotic bias and variance are essentially the same as those of the kernel generalized esti

a logic for reasoning about the probability of fuzzy events in this paper we present the logic fp l n l which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a mv algebra this logic is defined starting from a basic idea exposed by hajek metamathematics of fuzzy logic kluwer dordrecht 1998 two kinds of semantics have been introduced namely the class of weak and strong probabilistic models the main result of this paper is a completeness theorem for the logic fp l n l w r t both weak and strong models we also present two extensions of fp l n l the first one is the logic fp l n rpl obtained by expanding the fp l n l language with truth constants for the rationals in 0 1 while the second extension is the logic fcp l n email protected allowing to reason about conditional states

smoothing methods for epidemiologic analysis the development of techniques for fitting non parametric smooth curves has resulted in less restrictive regression models we discuss the ideas underlying such smoothing algorithms develop their application to epidemiologic studies and address specific issues such as coping with correlated errors an example illustrates a particular smoothing approach as applied to pulmonary function data the method provides new insight into the effect of smoking on pulmonary function the discussion offers some qualitative comparisons between smoothing methods and conventional linear models