local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

additive nonparametric regression with autocorrelated errors a bayesian approach is presented for nonparametric estimation of an additive regression model with autocorrelated errors each of the potentially non linear components is modelled as a regression spline using many knots while the errors are modelled by a high order stationary autoregressive process parameterized in terms of its autocorrelations the distribution of significant knots and partial autocorrelations is accounted for using subset selection our approach also allows the selection of a suitable transformation of the dependent variable all aspects of the model are estimated simultaneously by using the markov chain monte carlo method it is shown empirically that the approach proposed works well on several simulated and real examples

on the state minimization of fuzzy automata this paper investigates the minimization problem of fuzzy automata aiming to obtain a procedure for finding a minimal state fuzzy automaton equivalent to a given one the decision version of the minimization problem is as follows given a fuzzy automaton cal a and a natural number k i e a pair langle cal a k rangle is there a k state fuzzy automaton equivalent to cal a we prove that the above problem is decidable for fuzzy automata over totally ordered lattices and then obtain a procedure for minimizing a given fuzzy automaton to this end we introduce the concept of systems of fuzzy polynomial equations present a procedure for finding solutions of these systems and finally reduce the above decision problem to finding a solution of a system of fuzzy polynomial equations it is worth pointing out that although some algorithms in the literature were claimed to be minimization algorithms the term minimization there did not mean state minimization in our sense since these algorithms did not aim at a minimal fuzzy automaton but found reasonably small fuzzy automata

derivative estimation with local polynomial fitting we present a fully automated framework to estimate derivatives nonparametrically without estimating the regression function derivative estimation plays an important role in the exploration of structures in curves jump detection and discontinuities comparison of regression curves analysis of human growth data etc hence the study of estimating derivatives is equally important as regression estimation itself via empirical derivatives we approximate the qth order derivative and create a new data set which can be smoothed by any nonparametric regression estimator we derive l1 and l2 rates and establish consistency of the estimator the new data sets created by this technique are no longer independent and identically distributed i i d random variables anymore as a consequence automated model selection criteria data driven procedures break down therefore we propose a simple factor method based on bimodal kernels to effectively deal with correlated data in the local polynomial regression framework

fixed design regression for linear time series this investigation is concerned with recovering a regression function g x i on the basis of noisy observations taken at uniformly spaced design points x i it is presumed that the corresponding observations are corrupted by additive dependent noise and that the noise is in fact induced by a general linear process in which the summand law can be discrete as well as continuously distributed discreteness induces a complication because such noise is not known to be strong mixing the postulate by which regression estimates are often shown to be asymptotically normal in fact as cited there are processes of this character which have been proven not to be strong mixing the main analytic result of this study is that in general circumstances which include the non strong mixing example the smoothers we propose are asymptotically normal some motivation is offered and a simple illustrative example calculation concludes this investigation the innovative elements of this work mainly consist of compassing models with discrete noise important in practical applications and in dispensing with mixing assumptions the ensuing mathematical difficulties are overcome by sharpening standard arguments

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

alternative goodness of fit tests for linear models fan and huang 2001 presented a goodness of fit test for linear models based on fourier transformations of the residuals of the fitted model we present two more theoretically appealing tests in which the fourier transforms are incorporated into a fitted model we show that when suitably normalized the new test statistics have the same asymptotic distribution as fan and huang s test we propose modifications to the asymptotic normalization constants to improve the small sample sizes of our tests while retaining their asymptotic distributions small sample sizes and powers are examined via simulations an illustration is given

nonparametric detection of correlated errors summary in regression problems it is hard to detect correlated errors since the errors are not observed in this paper a nonparametric method is proposed for the detection of correlated errors when the design points are equally spaced it turns out that the first order sample autocovariance of the residuals from the kernel regression estimates provides essential information about correlated errors and its bootstrap is quite effective in implementing such information

nonparametric comparison of several regression functions exact and asymptotic theory a new test is proposed for the comparison of two regression curves f and g we prove an asymptotic normal law under fixed alternatives which can be applied for power calculations for constructing confidence regions and for testing precise hypotheses of a weighted l 2 distance between f and g in particular the problem of nonequal sample sizes is treated which is related to a peculiar formula of the area between two step functions these results are extended in various directions such as the comparison of k regression functions or the optimal allocation of the sample sizes when the total sample size is fixed the proposed pivot statistic is not based on a nonparametric estimator of the regression curves and therefore does not require the specification of any smoothing parameter

bias robust estimation in finite populations using nonparametric calibration a standard problem in sample survey inference is that of predicting the finite population total h of a function h y of a random variable y the model based approach to this problem first defines a working model for y and then predicts h by estimating its expectation under conditional on the sample values of y this approach leads to biased predictions if is incorrect we explore an automatic solution to this misspecification bias that uses nonparametric regression to define a robust but inefficient predictor of h and then calibrates this predictor for its bias under an application to prediction of the finite population distribution function of a population of australian beef farms is presented

autoregressive wild bootstrap inference for nonparametric trends in this paper we propose an autoregressive wild bootstrap method to construct confidence bands around a smooth deterministic trend the bootstrap method is easy to implement and does not require any adjustments in the presence of missing data which makes it particularly suitable for climatological applications we establish the asymptotic validity of the bootstrap method for both pointwise and simultaneous confidence bands under general conditions allowing for general patterns of missing data serial dependence and heteroskedasticity the finite sample properties of the method are studied in a simulation study we use the method to study the evolution of trends in daily measurements of atmospheric ethane obtained from a weather station in the swiss alps where the method can easily deal with the many missing observations due to adverse weather conditions

further improvements of determinization methods for fuzzy finite automata in this paper we provide further improvements of determinization methods for fuzzy finite automata these methods perform better than all previous determinization methods for fuzzy finite automata developed by b lohlavek 3 li and pedrycz 38 ignjatovic et al 25 and jancic et al 33 in the sense that they produce smaller automata while require the same computation time the only exception is the brzozowski type determinization algorithm developed recently by jancic and iric 34 which produces a minimal crisp deterministic fuzzy automaton but the algorithms created here can also be used within the brzozowski type algorithm and improve its performance

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

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

markov cross validation for time series model evaluations cross validation cv is a simple and universal tool to estimate generalization ability however existing cvs do not work well for periodicity overlapping or correlation of series the corresponding three criteria aimed at describing these properties are presented based on them we put forward a novel markov cross validation m cv whose data partition can be seen as a markov process the partition ensures that samples in each subset are neither too close nor too far in doing so overfitting model or information loss of series which may result in underestimation or overestimation of the error can be avoided furthermore subsets from m cv partition could well represent the original series and it may be extended to time series or stream data sampling theoretical analysis shows that m cv is the unique one which meets all of above criteria among current cvs in addition the error estimation on subsets is proved to have less variance than that on original series therefore it ensures the stability of m cv experimental results demonstrate that the proposed m cv has lower bias variance and time consumption than other cvs

behaviour reduction and minimization of finite g6 automata abstract finite fuzzy automata over a bounded chain l l automata are studied in a natural manner including max min fuzzy automata as a special case the solvability of systems of linear equations over l is investigated an algorithm for computing the behaviour matrix for each finite l automaton is proposed using the behaviour matrices we can decide whether two l automata have approximately the same behaviour i e if an l automaton makes an approximate model of the other for this purpose approximate equivalence reduction and minimization for finite l automata are defined and completely studied the paper arises as a refinement and natural extension of 2 4 7 the results are valid for all finite l automata and for a narrow class of stochastic automata

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

weak bisimulations for fuzzy automata abstract forward and backward bisimulations have been introduced recently by ciric et al 2012 9 as a means for modeling the equivalence between states of fuzzy automata and approximating the language equivalence as well as for use in the state reduction of fuzzy automata the main aim of the present paper is to introduce two new kinds of bisimulations weak forward and backward bisimulations which provide better approximations of the language equivalence than forward and backward bisimulations and when employed in the state reduction they provide better reductions we give procedures for deciding whether there exist weak forward and backward simulations and bisimulations and for computing the greatest ones whenever they exist using weak bisimulations in conjunction with the concept of a uniform fuzzy relation we determine necessary and sufficient conditions under which two fuzzy automata are weak bisimulation equivalent we also characterize uniform weak backward and forward bisimulations between two fuzzy automata in terms of isomorphisms between their nerode and reverse nerode automata

computing behavior of finite fuzzy machines algorithm and its application to reduction and minimization we define finite fuzzy machines and investigate their behavior algorithm and software are proposed for computing behavior for establishing equivalence and redundancy of states and for solving reduction and minimization problems computational complexity of the algorithm is discussed testing examples are supplied the results are valid for finite max min min max and max product fuzzy machines

minimization of fuzzy finite automata abstract in this paper we show that a fuzzy finite automation m 1 has an equivalent minimal fuzzy finite automation m we also show that m can be chsen so that if m 2 is equivalent to m 1 then m is a homomorphic image of m 2

the l 1 method for robust nonparametric regression abstract consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy tailed error distribution the two standard approaches to such a problem are each flawed robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods which are not resistant against outliers existing algorithms for robustifying nonparametric regression procedures use either nonlinear optimization of an influence function or iterative solution of local polynomial fitting using reweighted least squares neither of these two approaches combines computational ease with asymptotic theoretical results furthermore application of the robust procedure has been limited almost exclusively to the case of a single explanatory variable with the response variable in this article a new hybrid method is propose

nonuniform sequential sampling for signal analysis new electronic technologies for signal analysis raise the possibility of sampling very rapidly with a time varying density and determining empirically both the sampling rate and the window width as the signal evolves in time these opportunities also point to the possibility of sequentially sampling in a time varying way in more traditional problems motivated by these ideas we establish a sampling formula valid in cases where both sampling rate and window width may be varied the formula states that in terms of the ways in which these quantities should alter with time optimal performance is achieved when the window width is inversely proportional to squared sampling rate and sampling rate is directly proportional to squared bias

equivalence reduction and minimization of finite fuzzy automata abstract a new algebraic approach to the problem of equivalence reduction and minimization of some kinds of fuzzy automata is given a system of necessary and sufficient conditions for equivalence of weakly initial fuzzy automata is formulated some algorithms considering the equivalence for two fuzzy automata are constructed

finite l fuzzy machines abstract for finite l fuzzy machines we develop behaviour reduction and minimization theory here l stands for l 0 1 where l is a totally ordered set with universal bounds 0 and 1 and the operations are join and meet the most essential results include investigating behaviour equivalence reduction and minimization problems and their algorithmical decidability

equivalence reduction and minimization of finite automata over semirings abstract a new unified approach to equivalence reduction and minimization of finite automata over semirings based on free semimodules is presented it includes as special cases deterministic nondeterministic stochastic and fuzzy automata a necessary and sufficient condition for the existence of a finite behaviour matrix for a given automaton is proved different algorithmical decidability properties for equivalence reduction and minimization are connected to a noetherian property in the category s mod c of the semimodules over the semiring c

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

automata theory based on complete residuated lattice valued logic reduction and minimization automata theory based on complete residuated lattice valued logic called l valued finite automata l vfas has been established by qiu recently in this paper we define a kind of mealy type of l vfas mlfas a generalization of l vfas two kinds of statewise equivalence relations are introduced and a minimal form is defined we study the existence of the minimal form of an mlfa then we show that any two states can be distinguished by some word with finite length also a minimization algorithm of the mlfas is presented in addition we obtain a minimization algorithm for l vfas as well finally we define l valued languages l vls and l valued regular languages l vrls recognized by l vfas and provide some properties of l vrls

a measure of robust stability for an identified set of parametrized transfer functions defines a measure of robustness for a set of parameterized transfer functions as delivered by classical prediction error identification and that contains the true system at a prescribed probability level this measure of robustness is the worst case vinnicombe distance between the model and the plants in the uncertainty region we show how it can be computed exactly using lmi based optimization in addition we show that this measure is directly connected to the size of the set of controllers that are guaranteed to stabilize all plants in the uncertainty region i e the smaller the worst case vinnicombe distance for an uncertainty region the larger the set of model based controllers that are guaranteed to stabilize all systems in this uncertainty region

the robustness of feedback systems with bounded complexity controllers h sub spl infin methods for the analysis and design of robust feedback control systems have sometimes been criticized for an apparent conservatism however recent results have shown that they can provide the least conservative results possible under a particular set of assumptions for example a ball in the v gap metric is the largest set of plants that can be guaranteed to be stabilized a priori by a controller known only to satisfy a bound on the induced norm of a particular closed loop operator nevertheless there are examples of uncertainty which whilst large when measured by the v gap metric would be regarded as relatively benign by an experienced designer of control systems the present paper examines the possibility that in arriving at this judgment such a designer is implicitly using the knowledge that he will always choose the least complex controller necessary to do the job it is shown that given an appropriate bound on the complexity of the controller significantly stronger a priori robustness results can be obtained

understanding exponential smoothing via kernel regression exponential smoothing is the most common model free means of forecasting a future realization of a time series it requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one step ahead forecasts in this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation we also use theoretical developments from the kernel regression field to derive for the first time asymptotic properties of exponential smoothing forecasters

equivalence in automata theory based on complete residuated lattice valued logic automata theory based on complete residuated lattice valued logic called l valued finite automata abbr l vfas was introduced by the second author in 2001 in this paper we deal with the problems of equivalence between l valued sequential machines abbr l vsms and l vfas we define l vsms and particularly present a method for deciding the equivalence between l vsms as well an algorithm procedure for deciding the equivalence between l vsms is constructed we analyze the complexity and efficiency of the algorithm procedure and obtain the relative results to l vfas moreover the definitions of l valued languages abbr l vls and l valued regular languages abbr l vrls recognized by l vfas are given and some related properties are also discussed we show an equivalent relation between l vrls and conventional regular languages by using l valued pumping lemma we get a necessary and sufficient condition for an l vl to be nonconstant

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

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

infinite fuzzy computations we consider buchi automata weighted over the fuzzy semiring f 0 1 max min 0 1 a buchi theorem for the class of w recognizable series with coefficients in f i e for behaviors of buchi automata over f is established furthermore we show that an infinitary language is buchi recognizable iff its characteristic series is w recognizable

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

construction of fuzzy automata from fuzzy regular expressions li and pedrycz have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice ordered monoid and generalize the well known results of the classical theory of formal languages in particular they have shown that a fuzzy language over an integral lattice ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a fuzzy finite automaton however they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression in this paper we provide such an efficient method transforming scalars appearing in a fuzzy regular expression ainto letters of the new extended alphabet we convert the fuzzy regular expression ato an ordinary regular expression a r then starting from an arbitrary nondeterministic finite automaton a that recognizes the language a r represented by the regular expression a r we construct fuzzy finite automata a a and a a r with the same or even less number of states than the automaton a which recognize the fuzzy language a represented by the fuzzy regular expression a the starting nondeterministic finite automaton a can be obtained from a r using any of the well known constructions for converting regular expressions to nondeterministic finite automata such as glushkov mcnaughton yamada s position automaton brzozowski s derivative automaton antimirov s partial derivative automaton or ilie yu s follow automaton

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

from stein s unbiased risk estimates to the method of generalized cross validation this paper concerns the method of generalized cross validation gcv a promising way of choosing between linear estimates based on stein estimates and the associated unbiased risk estimates stein 1981 a new approach to gcv is developed many consistency results are obtained for the cross validated steinized estimates in the contexts of nearest neighbor nonparametric regression model selection ridge regression and smoothing splines moreover the associated stein s unbiased risk estimate is shown to be uniformly consistent in assessing the true loss not the risk consistency properties are examined as well when the sampling error is unknown finally we propose a variant of gcv to handle the case that the dimension of the raw data is known to be greater than that of their expected values

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

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

local polynomial regression with correlated errors in random design and unknown correlation structure summaryautomated or data driven bandwidth selection methods tend to break down in the presence of correlated errors while this problem has previously been studied in the fixed design setting for kernel regression the results were applicable only when there is knowledge about the correlation structure this article generalizes these results to the random design setting and addresses the problem in situations where no prior knowledge about the correlation structure is available we establish the asymptotic optimality of our proposed bandwidth selection criterion based on kernels k satisfying k 0 0

bootstrap test for difference between means in nonparametric regression abstract a bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression error distributions in the regression model are permitted to be arbitrary and unequal the test enjoys power properties akin to those in a parametric setting in the sense that it can distinguish between regression functions distant only n 1 2 apart where n is the sample size it has exceptional level accuracy with level error of only n 2 and uses a very accurate estimate of the critical point of an exact test being in error by only n 3 2 under the null hypothesis the test admits several generalizations for example to the case of testing for differences between several regression means this is a nonparametric regression analog of analysis of variance a simulation study using n as small as 15 corroborates the asymptotic result on level accuracy of the bootstrap test applications are illustrated with an example involving acid rain data

analysis of oldest old mortality lifetables revisited ing the hazard function we provide rigorous asymptotic results for the smoothing of lifetables and show that the transformation approach is supported by both asymptotic and simulation results we also demonstrate how the information contained in many samples of lifetables as typically obtained in aging experiments can be summarized in a twodimensional hazard surface

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

testing for trends in high dimensional time series the article considers statistical inference for trends of high dimensional time series based on a modified l2 distance between parametric and nonparametric trend estimators we propose a de diagon

non parametric estimation of residual moments and covariance the aim of non parametric regression is to model the behaviour of a response vector y in terms of an explanatory vector x based only on a finite set of empirical observations this is usually perf

why bandwidth selectors tend to choose smaller bandwidths and a remedy summary this note is concerned with the problem of automatic data driven bandwidth selectors for nonparametric regression though some selectors were shown to be consistent and asymptotically unbiased by rice 1984 and hardle hall marron 1988 it is often observed in simulation studies that most selectors are biased toward undersmoothing an explanation for this is given the source of the large sample variation in the bandwidth estimates is also pointed out this leads to the consideration of a new procedure which is a simple modification of a classical selector a simulation study suggests that the proposed selector is much more consistent than the classical one

comparison of two bandwidth selectors with dependent errors for nonparametric regression in the case of dependent observations cross validation is known to be severely affected by dependence this effect is precisely quantified through a limiting distribution for the cross validated bandwidth the performance of two methods the leave 2e 1 out version of cross validation and partitioned cross validation which adjust for the dependence effect on bandwidth selection is investigated the bandwidths produced by these two methods are analyzed by further limiting distributions which reveal significantly different characteristics simulations demonstrate that the asymptotic effects hold for reasonable sample sizes

bootstrapping in nonparametric regression local adaptive smoothing and confidence bands abstract the operation of the bootstrap in the context of nonparametric regression is considered bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator the application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances the technique of the bootstrap is to replace any occurrence of the unknown distribution in the definition of the statistical function of interest by the empirical distribution function of the observed errors in a regression context these errors are not directly observed although their role can be played by the residuals from the fitted model in this article the fitted model is a kernel nonparametric regression estimator since nonparametric smoothing is involved an additional difficulty is created by the bias incurred in smoothing this bias however can be estimated

reduction of fuzzy automata by means of fuzzy quasi orders abstract in our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations in that paper we have also studied reductions by means of those solutions which are fuzzy equivalences in this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi orders generally fuzzy quasi orders and fuzzy equivalences are equally good in the state reduction but we show that right and left invariant fuzzy quasi orders give better reductions than right and left invariant fuzzy equivalences we also show that alternate reductions by means of fuzzy quasi orders give better results than alternate reductions by means of fuzzy equivalences furthermore we study a more general type of fuzzy quasi orders weakly right and left invariant ones and we show that they are closely related to determinization of fuzzy automatons we also demonstrate some applications of weakly left invariant fuzzy quasi orders in conflict analysis of fuzzy discrete event systems

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

a bound on closed loop performance based on finite frequency response samples abstract a new computationally tractable bound is derived for the level of closed loop performance achieved by a given finite dimensional feedback compensator with a plant for which a finite number of frequency response samples are computable the bound involves quantities reflecting the performance of the controller with a finite dimensional nominal model of the plant quantities that can be determined from the finite number of frequency response samples of the true plant and quantities related to the complexity in the sense of vinnicombe of all systems involved this bound can be used to validate closed loop performance in the case that the true plant frequency response samples are of a plant which is not completely known or to measure the performance of a finite dimensional controller with a computationally intractable e g infinite dimensional model of the true plant

on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations in this paper we study systems of fuzzy relation inequalities and equations of the form u v i v i u i i where u is an unknown and v i i i are given fuzzy relations the dual systems v i u u v i i i their conjunctions the systems of the form u v i v i u i i and certain special types of these systems we call them weakly linear systems for each weakly linear system with a complete residuated lattice as the underlying structure of truth values we prove the existence of the greatest solution and we provide an algorithm for computing the greatest solution which works whenever the underlying complete residuated lattice is locally finite otherwise we determine some sufficient conditions under which the algorithm works the algorithm is iterative and each its single step can be viewed as solving of a particular linear system weakly linear systems emerged from the fuzzy automata theory but we show that they also have important applications in other fields e g in the concurrency theory and social network analysis

using difference based methods for inference in nonparametric regression with time series errors we show that difference based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors when the error process is gaussian our estimators are efficient but they are available well beyond the gaussian case as an illustration of their usefulness we show that difference based estimators can be used to produce a simplified version of time series cross validation this new approach produces a bandwidth selector that is equivalent to both first and second orders to that given by the full time series cross validation algorithm other applications of difference based methods are to variance estimation and construction of confidence bands in nonparametric regression

on the relation between equations with max product composition and the covering problem systems of equations with max product composition are considered it is shown that solving these equations is closely related with the covering problem which belongs to the category of np hard problems it is proved that minimal solutions of equations correspond to irredundant coverings in terms of the covering problem the conditions of compatibility of equations of redundancy of equations of uniqueness of solution of uniqueness of minimal solution are determined concepts of essential non essential semi essential and super essential variables are suggested ways of simplification of a covering problem and methods of its solving are considered

structural tests in additive regression we consider the component analysis problem for a regression model with an additive structure the problem is to test whether some of the additive components are of polynomial structure e g linear without specifying the structure of the remaining components a particular case is the problem of selecting the significant covariates the method that we present is based on the wavelet transform using the haar basis which allows for applications under mild conditions on the design and smoothness of the regression function the results demonstrate that each component of the model can be tested with the rate corresponding to the case if all of the remaining components were known the proposed procedure is also computationally straightforward simulation results and a real data example about female labor supply demonstrate the test s good performance

minimization of states in automata theory based on finite lattice ordered monoids in this study we consider a concept of complete l fuzzy matrix define complete lattice valued finite automata clfas and study their properties the definitions of statewise equivalence relations and automata equivalence relations of a clfa are given two algorithms are aimed at the minimization of states of a clfa

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

computation of the greatest simulations and bisimulations between fuzzy automata recently two types of simulations forward and backward simulations and four types of bisimulations forward backward forward backward and backward forward bisimulations between fuzzy automata have been introduced if there is at least one simulation bisimulation of some of these types between the given fuzzy automata it has been proved that there is the greatest simulation bisimulation of this kind in the present paper for any of the above mentioned types of simulations bisimulations we provide an efficient algorithm for deciding whether there is a simulation bisimulation of this type between the given fuzzy automata and for computing the greatest one whenever it exists the algorithms are based on the method developed in ignjatovic et al on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations fuzzy sets syst 161 2010 3081 3113 which comes down to the computing of the greatest post fixed point contained in a given fuzzy relation of an isotone function on the lattice of fuzzy relations

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

semiparametric regression in likelihood based models abstract a weighted likelihood is used to estimate the parameters in a semiparametric model involving two covariates and allowing an association between the covariates the development is for arbitrary but specified densities of the observations the estimators are consistent and asymptotically normal hypothesis testing of the parametric component can be performed using a wald test simulations and analysis of data with bernoulli observations demonstrate the estimators application speckman developed kernel estimators where the conditional density of the observations is normal with p parametric covariates speckman s estimators and the new estimators are asymptotically equivalent with the bias of speckman s estimators being smaller as an example we study the relationship between a binary response indicating the occurrence of an intraoperative cardiac complication icc in vascular surgery patients and two risk factors duration of the operation or and asa score which is an evaluation of the patien

curve fitting by polynomial trigonometric regression summary we consider a method of estimating an unknown regression curve by regression on a combination of low order polynomial terms and trigonometric terms estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions we show that these boundary problems are alleviated by adding low order polynomial terms the utility of the method is illustrated with examples and asymptotic results show that the estimators are competitive with other nonparametric procedures applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho