an affine invariant bivariate version of the sign test abstract the generalized median of h oja yields a notion of bivariate quantile and in turn an affine invariant bivariate analogue of the sign test its properties include a simple null covariance formula facilitating a permutation or sign change test in the case of bivariate symmetry normal efficiency coinciding with that of the oja median and bounded influence hence strong robustness author

global fuzzy adaptive hierarchical path tracking control of a mobile robot with experimental validation due to the nature of the complete model with motor dynamics virtual control input i e desired motor current is designed by the first sliding surface so that the indirect output i e 3 d pose of mobile robot is controlled by the direct output i e motor current subsequently the linear dynamic tracking error of the virtual control input is employed to establish the second sliding surface then the hierarchical path tracking control hptc is constructed so that the direct output asymptotically and robustly tracks the virtual control input in the meanwhile the asymptotic tracking of the indirect output is achieved to improve only asymptotical tracking to convex set due to the existence of huge uncertainties e g different ground conditions time varying system parameters external disturbances two online fuzzy models of uncertainties are plunged into the hptc with a switching mechanism to design the so called global fuzzy adaptive hptc for a mobile robot moreover the global adaptive path tracking control for different initial system states outside of approximated set is achieved the simulations confirm the effectiveness efficiency and practicality of the proposed technique two compared experiments also validate the consistence with the simulation result

conjugacy classes in the symplectic renner monoid abstract in this paper we study conjugacy classes of the symplectic renner monoid r and find the following results each element in r is a join of strictly disjoint cycles and strings there is an explicit one to one correspondence between conjugacy classes and symplectic partitions a formula for calculating the number of conjugacy classes is obtained an explicit formula for computing the number of elements in each conjugacy class is given a relationship between conjugacy classes and munn conjugacy classes is described

monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

on an adaptive transformation retransformation estimate of multivariate location an affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied the work is primarily motivated by earlier work on different versions of the multivariate median and their properties we explore an issue related to efficiency and equivariance that was originally raised by bickel and subsequently investigated by brown and hettmansperger our estimate has better asymptotic performance than the vector of co ordinatewise medians when the variables are substantially correlated the finite sample performance of the estimate is investigated by using monte carlo simulations some examples are presented to demonstrate the effect of the adaptive transformation retransformation strategy in the construction of multivariate location estimates for real data

a frequency domain median time series abstract this article discusses the median of a batch of time series we focus on the frequency domain median fdm series zeger 1985 which is calculated from the mediancenters gower 1974 of the discrete fourier transforms like the univariate median it has a breakdown value of 50 like the mean the fdm time series of independent realizations from a stationary gaussian process has a spectrum proportional to that of the component series analogs of the univariate median have been described for multivariate e g brown 1983 survival brookmeyer and crowley 1982 and spherical fisher 1985 data as in the multivariate case there is not a unique definition of a median time series one possibility is the median value at each time or time domain median tdm a drawback of the tdm is that it can be qualitatively dissimilar in appearance to its component series because of discontinuities when the median value switches from one realization to another this is particularly disturbing for smooth tim

the smoothed median and the bootstrap summary even in one dimension the sample median exhibits very poor performance when used in conjunction with the bootstrap for example both the percentile t bootstrap and the calibrated percentile method fail to give second order accuracy when applied to the median the situation is generally similar for other rank based methods particularly in more than one dimension some of these problems can be overcome by smoothing but that usually requires explicit choice of the smoothing parameter in the present paper we suggest a new implicitly smoothed version of the k variate sample median based on a particularly smooth objective function our procedure preserves many features of the conventional median such as robustness and high efficiency in fact higher than for the conventional median in the case of normal data it is however substantially more amenable to application of the bootstrap focusing on the univariate case we demonstrate these properties both theoretically and numerically

conjugacy decomposition of reductive monoids we study the decomposition of a reductive monoid related to conjugacy classes and obtain a description of the associated finite poset open image in new window this poset is constructed via the gauss jordan elements of the renner monoid r for mn k the poset can be identified with the lattice of all partitions of m 0 m n where the order is an extension of the dominance order on partitions of n

affine invariant rank methods in the bivariate location model abstract the generalized multivariate median of h oja is used to define a multivariate notion of quantile or rank and to define a measure of scatter of multivariate linear models the latter when applied to the one and two sample bivariate location models yields affine invariant analogs of the wilcoxon rank sum and signed rank tests and of the corresponding estimates additional keywords dispersion measures rank statics author

adaptive fuzzy hierarchical sliding mode control for a class of mimo nonlinear time delay systems with input saturation in this paper an adaptive fuzzy hierarchical sliding mode control method for a class of multiinput multioutput unknown nonlinear time delay systems with input saturation is proposed the studied system is first transformed into an equivalent system subsequently based on sliding mode control technology and the concept of hierarchical design a set of adaptive fuzzy hierarchical sliding mode controllers are designed for the new defined system by using fuzzy systems to approximate uncertain functions and compensate input saturation choosing an appropriate lyapunov krasovskii function it is theoretically proved that all the signals in the closed loop system together with the proposed sliding surfaces are uniformly ultimately bounded under our designed adaptive fuzzy controllers simulation results demonstrate the effectiveness of the proposed design techniques for the systems under consideration

a nonparametric test for equality of survival medians in clinical trials researchers often encounter testing for equality of survival medians across study arms based on censored data even though brookmeyer and crowley introduced a method for comparing medians of several survival distributions still some researchers misuse procedures that are designed for testing the homogeneity of survival curves these procedures include the log rank wilcoxon and cox models this practice leads to inflation of the probability of a type i error particularly when the underlying assumptions of these procedures are not met we propose a new nonparametric method for testing the equality of several survival medians based on the kaplan meier estimation from randomly right censored data we derive asymptotic properties of this test statistic through simulations we compute and compare the empirical probabilities of type i errors and the power of this new procedure with those of the brookmeyer crowley log rank and wilcoxon methods our simulation results indicate that the performance of these test procedures depends on the level of censoring and appropriateness of the underlying assumptions when the objective is to test homogeneity of survival medians rather than survival curves and the assumptions of these tests are not met some of these procedures severely inflate the probability of a type i error in these situations our test statistic provides an alternative to the brookmeyer crowley test

conjugacy classes of renner monoids abstract in this paper we describe conjugacy classes of a renner monoid r with unit group w the weyl group we show that every element in r is conjugate to an element ue where u w and e is an idempotent in a cross section lattice denote by w e and w e the centralizer and stabilizer of e in w respectively let w e act by conjugation on the set of left cosets of w e in w we find that ue and ve u v w are conjugate if and only if u w e and v w e are in the same orbit as consequences there is a one to one correspondence between the conjugacy classes of r and the orbits of this action we then obtain a formula for calculating the number of conjugacy classes of r and describe in detail the conjugacy classes of the renner monoid of some j irreducible monoids we then generalize munn conjugacy on a rook monoid to any renner monoid and show that munn conjugacy coincides with semigroup conjugacy action conjugacy and character conjugacy we also show that the number of inequivalent irreducible representations of r over an algebraically closed field of characteristic zero equals the number of munn conjugacy classes in r

m estimates of rigid body motion on the sphere and in euclidean space this paper calculates the influence functions and asymptotic distributions of m estimators of the rotation a in a spherical regression model on the unit sphere in p dimensions with isotropic errors the problem arises in the reconstruction of the motion of a rigid body on the surface of the sphere the comparable model for p dimensional euclidean space data is that 1 n are independent with i symmetrically distributed around a u i b ui known where the real constant 0 p x p rotation matrix a and p vector b are the parameters to be estimated this paper also calculates the influence functions and asymptotic distributions of m estimators for a and b besides rigid body motion this problem arises in image registration from landmark data particular attention is paid to how the geometry of the rigid body or landmarks affects the statistical properties of the estimators

extremal depth for functional data and applications abstractwe propose a new notion called extremal depth ed for functional data discuss its properties and compare its performance with existing concepts the proposed notion is based on a measure of extreme outlyingness ed has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces in particular a the central region achieves the nominal desired simultaneous coverage probability b there is a correspondence between ed based simultaneous central regions and appropriate pointwise central regions and c the method is resistant to certain classes of functional outliers the article examines the performance of ed and compares it with other depth notions its usefulness is demonstrated through applications to constructing central regions functional boxplots outlier detection and simultaneous confidence bands in regression problems supplementary materials for this article are availab

spherical median axes in an earlier paper fisher defined the notion of a median direction for data from a unimodal distribution of three dimensional unit vectors and studied its statistical properties corresponding notions of a median axis for bipolar axial or great circle data were also suggested but not analysed this paper gives a statistical treatment of spherical median axes including their asymptotic relative efficiencies and a comparison with the customary principal or polar axis estimators using asymptotic relative efficiencies

orthogonalization of multivariate location estimators the orthomedian the coordinatewise median of a multivariate data set is a highly robust location estimator but it depends on the choice of coordinates a popular alternative which avoids this drawback is the spatial median defined as the value that minimizes the sum of distances to the individual data points in this paper we introduce and discuss another orthogonal equivariant version of the multivariate median obtained by averaging the coordinatewise median over all orthogonal transformations we investigate the asymptotic behavior of this estimator and compare it to the spatial median

on the influence functions of certain bivariate medians the asymptotic efficiency influence function and breakdown properties of the multivariate median proposed by oja are investigated and comparisons are made with two other generalizations of the median the spatial median and the vector of marginal medians it is shown that for distributions with finite expectations the oja median has a bounded influence function although this median has 0 breakdown for circular distributions the influence functions of the oja median and the spatial median coincide the oja median is then as efficient as the spatial median in the circular case but strictly better for any other elliptical case it is noteworthy that the spatial median however has the best possible 50 breakdown this phenomenon is analysed in the paper it is shown that the finite sample breakdown point of the oja median depends on the dispersion of the corrupted observations

a k sample median test for censored data abstract a k sample median test for censored data is proposed the test is more sensitive to differences in median survival times than to differences in shape of the survival curves the asymptotic distribution of the test statistic is derived under both the null and the alternative hypotheses and its variance is shown to take a simple form the median test is compared to several other tests for censored data by computing asymptotic relative efficiencies and performing a monte carlo simulation under various conditions the procedure described is applied to data from a colorectal cancer clinical trial

variance of the kaplan meier estimator and its quantiles under certain fixed censoring models for fixed censoring models that contain at most one intermediate censoring point we obtain exact algebraic expressions for the asymptotic variances of i the quantiles of the kaplan meier km 1958 survival estimator and ii the km estimator itself at fixed time points the relationship between i and ii is found to be the same as the one derived by sander 1975 and reid 1981b for the random censorship model confidence intervals for the quantiles based on i are briefly discussed and compared to previously known procedures although greenwood s formula is recommended over ii in practice because of its desirable conditioning on the observed censoring pattern ii is of theoretical interest as an asymptotic limit for greenwood s formula in closed form

median tests for censored survival data a contingency table approach the median failure time is often utilized to summarize survival data because it has a more straightforward interpretation for investigators in practice than the popular hazard function however existing methods for comparing median failure times for censored survival data either require estimation of the probability density function or involve complicated formulas to calculate the variance of the estimates in this article we modify a k sample median test for censored survival data brookmeyer and crowley 1982 journal of the american statistical association 77 433 440 through a simple contingency table approach where each cell counts the number of observations in each sample that are greater than the pooled median or vice versa under censoring this approach would generate noninteger entries for the cells in the contingency table we propose to construct a weighted asymptotic test statistic that aggregates dependent 2 statistics formed at the nearest integer points to the original noninteger entries we show that this statistic follows approximately a 2 distribution with k 1 degrees of freedom for a small sample case we propose a test statistic based on combined p values from fisher s exact tests which follows a 2 distribution with 2 degrees of freedom simulation studies are performed to show that the proposed method provides reasonable type i error probabilities and powers the proposed method is illustrated with two real datasets from phase iii breast cancer clinical trials

robust fuzzy design for nonlinear uncertain stochastic systems via sliding mode control this paper deals with the sliding mode control smc problem for nonlinear stochastic time delay systems by means of fuzzy approach the takagi sugeno t s fuzzy stochastic time delay model with parametric uncertainties and unknown nonlinearities is presented a sufficient condition for the exponential stability in mean square of the sliding motion is also derived moreover it is shown that when the linear matrix inequalities lmis with equality constraint are feasible the designs of both sliding surface and sliding mode controller can be easily obtained via convex optimization a simulation example illustrating the proposed method is given

analyzing survival curves at a fixed point in time a common problem encountered in many medical applications is the comparison of survival curves often rather than comparison of the entire survival curves interest is focused on the comparison at a fixed point in time in most cases the naive test based on a difference in the estimates of survival is used for this comparison in this note we examine the performance of alternatives to the naive test these include tests based on a number of transformations of the survival function and a test based on a generalized linear model for pseudo observations the type i errors and power of these tests for a variety of sample sizes are compared by a monte carlo study we also discuss how these tests may be extended to situations where the data are stratified the pseudo value approach is also applicable in more detailed regression analysis of the survival probability at a fixed point in time the methods are illustrated on a study comparing survival for autologous and allogeneic bone marrow transplants

estimating the median survival time summary two methods are proposed for setting confidence intervals for the median survival time it is shown how these are related to bootstrap methods for censored data and some numerical comparisons are presented in many medical follow up studies the survival experience of a group of patients is summarized by the product limit or kaplan meier estimate of the survival distribution this may follow an analysis of the relation between survival time and some explanatory variables by for example the proportional hazards model the median of the estimated survival distribution is sometimes a convenient summary statistic and may be preferred to the mean because it is less sensitive to long tails in the estimated survival distribution this paper is concerned with estimating the distribution of the estimated median and providing confidence intervals for the true median two methods are considered in 2 they are both based in different ways on generalizations of the solution for the uncensored case these methods turn out to be connected to two ways of extending bootstrap methods for use with censored data this connexion is discussed in 3 thinking about extensions of the bootstrap to censored data highlights the need for careful consideration of the role of the censoring mechanism and how it should be incorporated into the data analysis section 4 provides some examples and monte carlo simulations the work is closely related to that of efron 1981 and emerson 1981

multiobjective programming for type 2 hierarchical fuzzy inference trees this paper proposes a design of hierarchical fuzzy inference tree hfit an hfit produces an optimum tree like structure i e a natural hierarchical structure that accommodates simplicity by combining several low dimensional fuzzy inference systems fiss such a natural hierarchical structure provides a high degree of approximation accuracy the construction of the hfit takes place in two phases first a nondominated sorting based multiobjective genetic programming mogp is applied to obtain a simple tree structure a low complexity model with a high accuracy second the differential evolution algorithm is applied to optimize the obtained tree s parameters in the derived tree each node acquires a different input s combination where the evolutionary process governs the input s combination hence hfit nodes are heterogeneous in nature which leads to a high diversity among the rules generated by the hfit additionally the hfit provides an automatic feature selection because it uses mogp for the tree s structural optimization that accepts inputs only relevant to the knowledge contained in data the hfit was studied in the context of both type 1 and type 2 fiss and its performance was evaluated through six application problems moreover the proposed multiobjective hfit was compared both theoretically and empirically with recently proposed fiss methods from the literature such as mcit2fis tscit2fnn sit2fnn rit2fns wb et2fis mrit2nfs it2fnn svr etc from the obtained results it was found that the hfit provided less complex and highly accurate models compared to the models produced by the most of other methods hence the proposed hfit is an efficient and competitive alternative to the other fiss for function approximation and feature selection

on the concept of depth for functional data the statistical analysis of functional data is a growing need in many research areas in particular a robust methodology is important to study curves which are the output of many experiments in applied statistics as a starting point for this robust analysis we propose analyze and apply a new definition of depth for functional observations based on the graphic representation of the curves given a collection of functions it establishes the centrality of an observation and provides a natural center outward ordering of the sample curves robust statistics such as the median function or a trimmed mean function can be defined from this depth definition its finite dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high dimensional observations thus this new depth is also suitable for complex observations such as microarray data images and those arising in some recent marketing and financial studies natural properties of these

optimal control of underactuated nonholonomic mechanical systems in this paper we use an affine connection formulation to study an optimal control problem for a class of nonholonomic underactuated mechanical systems in particular we aim to minimize the norm squared of the control input to move the system from an initial to a terminal state we consider systems evolving on general manifolds the class of nonholonomic systems we study in this paper includes in particular wheeled type vehicles which are important for many robotic locomotion systems the two special aspects of this optimal control problem are the nonholonomic constraints and underactuation nonholonomic constraints restrict the evolution of the system to a distribution on the manifold the nonholonomic connection is used to express the constrained equations of motion many robotic systems are underactuated since control inputs are usually applied through the robot s internal configuration space only while we do not consider symmetries with respect to group actions in this paper the fact that the system is underactuated is taken into account in our problem formulation this allows one to compute reaction forces due to any inputs applied in directions orthogonal to the constraint distribution we illustrate our ideas by considering a simple example on a three dimensional manifold including obstacle avoidance using the method of navigation functions

stable adaptive fuzzy control of nonlinear systems a direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control is capable of incorporating fuzzy if then control rules directly into the controllers and guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded is developed the specific formula for the bounds is provided so that controller designers can determine the bounds based on their requirements the direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the duffing chaotic system to track a trajectory the simulation results show that the controller worked without using any fuzzy control rules and that after fuzzy control rules were incorporated the adaptation speed became much faster it is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control

monitoring a general class of two sample survival statistics with applications summary this paper considers a general class of statistics for testing the equality of two survival distributions in clinical trials with sequential monitoring the tests can be expressed as lebesgue stieltjes integrals of a weight function with respect to the difference between two survival distributions prominent members of this class include the two sample difference in kaplan meier estimates the test of medians brookmeyer crowley 1982 a truncated version of efron s 1967 test and the pepe fleming statistic pepe fleming 1989 1991 statistics in this class are shown to converge to a gaussian process indexed by information time under both null and local alternatives even if different statistics are used at different information times properly standardised statistics in a subclass converge to gaussian processes with independent increments so that the usual group sequential techniques for monitoring a clinical trial can be applied the design of a trial comparing two treatments with respect to mother to newborn transmission of hiv is used to illustrate practical aspects of monitoring

a high dimensional nonparametric multivariate test for mean vector this work is concerned with testing the population mean vector of nonnormal high dimensional multivariate data several tests for high dimensional mean vector based on modifying the classical hotelling t2 test have been proposed in the literature despite their usefulness they tend to have unsatisfactory power performance for heavy tailed multivariate data which frequently arise in genomics and quantitative finance this article proposes a novel high dimensional nonparametric test for the population mean vector for a general class of multivariate distributions with the aid of new tools in modern probability theory we proved that the limiting null distribution of the proposed test is normal under mild conditions when p is substantially larger than n we further study the local power of the proposed test and compare its relative efficiency with a modified hotelling t2 test for high dimensional data an interesting finding is that the newly proposed test can have even more substantial power gain with l

distance based tests for homogeneity of multivariate dispersions the traditional likelihood based test for differences in multivariate dispersions is known to be sensitive to nonnormality it is also impossible to use when the number of variables exceeds the number of observations many biological and ecological data sets have many variables are highly skewed and are zero inflated the traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non euclidean dissimilarity would be more appropriate distance based tests of homogeneity of multivariate dispersions which can be based on any dissimilarity measure of choice are proposed here they rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes the tests are straightforward multivariate extensions of levene s test with p values obtained either using the traditional f distribution or using permutation of either least squares or lad residuals examples illustrate the utility of the approach including the analysis of stabilizing selection in sparrows biodiversity of new zealand fish assemblages and the response of indonesian reef corals to an el nino monte carlo simulations from the real data sets show that the distance based tests are robust and powerful for relevant alternative hypotheses of real differences in spread

conjugacy classes in algebraic monoids let m be a connected linear algebraic monoid with zero and a reductive group of units g the following theorem is established theorem there exist affne subsets ml mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms 0t mi gi such that 1 every element of m is conjugate to an element of some ms and 2 two elements a b in mi are conjugate in m if and only if there exists x e gi such that x0 a x si b as a consequence it is shown that m is a union of its inverse submonoids introduction the objects of study in this paper are connected linear algebraic monoids m with zero this means by definition that the underlying set of m is an irreducible affine variety and that the product map is a morphism i e a polynomial map we will further assume that the group of units g of m is reductive this means 1 3 that the unipotent radical of g is trivial then by 6 10 m is unit regular i e m e m g where e e m e e mle2 e in this paper we study the conjugacy classes of m an initial study was made by the author 8 where the general problem was reduced to nilpotent elements the approach here is quite different yielding a more complete answer to be precise we show that there exist affine subsets m1 mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms si mi gi i 1 k such that 1 every element of m is conjugate to an element of some mi and 2 if a b e mi then a is conjugate to b in m if and only if there exists x e gi such that xsi a x si b as an application of this result we show that m is a union of its inverse submonoids an inverse semigroup is a semigroup s with the property that for each a e s there exists a unique a e s such that aaa a and aaa a see 2 finally in 3 we use our main results to briefly analyze the conjugacy classes of nilpotent elements 1 preliminaries throughout this paper z will denote the set of all positive integers and k an algebraically closed field let g be a connected linear algebraic group defined over k the radical r g is the maximal closed connected normal solvable subgroup of g and the unipotent radical ru g is the group of unipotent elements of r g we will assume throughout that g is a reductive group i e ru g 1 then r g c c g the center of g moreover g r g go where go g g is a semisimple group i e r go 1 also 3 theorem 27 5 go is a product of the simple closed normal subgroups of g in particular we have the following received by the editors november 3 1986 1980 mathematics subject classification 1985 revision primary 20g99 20m10 c 1987 american mathematical society 0002 9947 87 1 00 25 per page

classification of semisimple algebraic monoids let x be a semisimple algebraic monoid with unit group g associated with e is its polyhedral root system x c where x x t is the character group of the maximal torus t c g c x t is the set of roots and c x t is the character monoid of t c e zariski closure the correspondence e x c is a complete and discriminating invariant of the semisimple monoid e extending the well known classification of semisimple groups in establishing this result monoids with preassigned root data are first constructed from linear representations of g that done we then show that any other semisimple monoid must be isomorphic to one of those constructed to do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory thls ultimately yields the desired conclusions consider the classification problem for semisimple algebraic monoids over the algebraically closed field k what sort of problem is this first the definition a semisimple algebraic monoid is an irreducible affine algebraic variety e defined over k together with an associative morphlsm m e x e e and a two sided unit 1 e e for m we assume further that e has a 0 the unit group g which is always linear algebraic and dense in e is reductive e g gl k dim zg 1and e is a normal variety the problem then presents us with two familiar objects let t c g be a maximal torus then z t c e zariski closure is an affine torus embedding and g c e is a reductive algebraic group torus embeddings have been introduced by demazure in 6 in his study of cremona groups and are classified numerically using rational polyhedral cones 14 on the other hand reductive groups have been studied at least in principle since the nineteenth century their classification in the modern sense being achieved largely by chevalley 4 that numerical classification uses the now familiar root systems l l chapter 31 of killing that were introduced by him 15 in his penetrating formulation of the classification e cartan s of semisimple lie algebras thus in the classification of semisimple monoids we are compelled to consider the root system x x t t of g and the polyhedral cone c x z c x t of t c z the two objects are canonically related by the weyl group action on x which leaves c stable received by the editors november 16 1984 1980 mathematics subject classification primary 14m99 secondary 20m99

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

adaptive fuzzy control with guaranteed convergence of optimal approximation error with no a priori knowledge of plant boundary functions a novel direct adaptive fuzzy controller afc for a class of single input single output siso uncertain affine nonlinear systems is developed in this paper based on the theory of fuzzy logic systems flss with variable universes of discourse uds sufficient conditions that guarantee that the optimal fuzzy approximation error fae is locally convergent are given by the use of the output tracking error and its derivatives as input variables and by the selection of suitable adjusting parameters a variable ud fls with an optimal fae local convergence is constructed and its parameter adaptive law is derived by virtue of the lyapunov stability theorem under the assumption that the optimal fae is bounded it is proved that the closed loop system is asymptotically stable in the sense that all variables are uniformly ultimately bounded and that the tracking errors converge to zero the proposed approach eliminates the influence of the fae on the tracking errors by means of the inherent mechanism of the variable ud fls thus it has the potential to achieve high control performance without additional compensation under only a few fuzzy rules simulation studies demonstrate the superiority of the proposed afc in terms of the settling time tracking accuracy smoothness of the control input and robustness against external disturbances and parameter variations

adaptive control of a class of nonlinear systems with fuzzy logic an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture employs fuzzy systems which are expressed as a series expansion of basis functions to adaptively compensate for the plant nonlinearities global asymptotic stability of the algorithm is established in the lyapunov sense with tracking errors converging to a neighborhood of zero simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance

adaptive fuzzy terminal sliding mode control for a class of mimo uncertain nonlinear systems this paper presents a new adaptive terminal sliding mode tracking control design for a class of nonlinear systems using fuzzy logic system the terminal sliding mode control tsm was developed to provide faster convergence and higher precision control than the linear hyperplane based sliding control however the original tsm encountered singularity problem with discontinuous control action moreover a prior knowledge about the plant to be controlled is required the proposed controller combines a continuous non singular tsm with an adaptive learning algorithm and fuzzy logic system to estimate the dynamics of the controlled plant so that closed loop stability and finite time convergence of tracking errors can be guaranteed the performance of the proposed control strategy is evaluated through the control of a two link rigid robotic manipulator finally the effectiveness of the proposed scheme is demonstrated through the control of the ankle and knee movements in paraplegic subjects using functional electrical stimulation simulation and experimental results verify that the proposed control strategy can achieve favorable control performance with regard to system parameter variations and external disturbances 2011 elsevier b v all rights reserved

nonparametric quantile inference with competing risks data a conceptually simple quantile inference procedure is proposed for cause specific failure probabilities with competing risks data the quantiles are defined using the cumulative incidence function which is intuitively meaningful in the competing risks set up we establish the uniform consistency and weak convergence of a nonparametric estimator of this quantile function these results form the theoretical basis for extensions of standard one sample and two sample quantile inference for independently censored data this includes the construction of confidence intervals and bands for the quantile function and two sample tests simulation studies and a real data example illustrate the practical utility of the methodology

stable adaptive control using fuzzy systems and neural networks stable direct and indirect adaptive controllers are presented which use takagi sugeno fuzzy systems conventional fuzzy systems or a class of neural networks to provide asymptotic tracking of a reference signal for a class of continuous time nonlinear plants with poorly understood dynamics the indirect adaptive scheme allows for the inclusion of a priori knowledge about the plant dynamics in terms of exact mathematical equations or linguistics while the direct adaptive scheme allows for the incorporation of such a priori knowledge in specifying the controller we prove that with or without such knowledge both adaptive schemes can learn how to control the plant provide for bounded internal signals and achieve asymptotically stable tracking of a reference input in addition for the direct adaptive scheme a technique is presented in which linguistic knowledge of the inverse dynamics of the plant may be used to accelerate adaptation the performance of the indirect and direct adaptive schemes is demonstrated through the longitudinal control of an automobile within an automated lane

stabilization of uncertain fuzzy time delay systems via variable structure control approach in view of a recent new application of variable structure control vsc to the stabilization problem for takagi sugeno t s fuzzy models this paper aims to study the stabilization of uncertain fuzzy time delay systems in t s fuzzy model via vsc approach there are mainly two features in this paper one lies in the incorporation of time delays both smooth and nonsmooth delays in which case lyapunov functionals and razumikhin theorem are required to solve the stabilization problem the other feature is that not only matched uncertainties but also mismatched uncertainties in the state variables are considered as a sequence the contribution of this paper consists of various control schemes proposed for the vsc design and the present results are in terms of linear matrix inequalities lmis an illustrative example is given to show the effectiveness of our various results

on empirical likelihood statistical functions abstract we consider the empirical likelihood method for estimation of distribution and quantile functions where side information is incorporated through moment conditions we systematically study the asymptotic properties of the estimators such as the uniform strong laws of large numbers and weak convergence over classes of functions two monte carlo examples are also given to illustrate the practical utility of the method

the spatial distribution in infinite dimensional spaces and related quantiles and depths the spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data in this paper we investigate the concept of spatial distribution for data in infinite dimensional banach spaces many technical difficulties are encountered in such spaces that are primarily due to the non compactness of the closed unit ball in this work we prove some glivenko cantelli and donsker type results for the empirical spatial distribution process in infinite dimensional spaces the spatial quantiles in such spaces can be obtained by inverting the spatial distribution function a bahadur type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived a study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces the spatial distribution can be used to define the spatial depth in infinite dimensional banach spaces and we study the asymptotic properties of the empirical spatial depth in such spaces we also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data

a practical affine equivariant multivariate median summary a robust affine equivariant estimator of location for multivariate data is proposed which becomes the univariate median for data of dimension one the estimator is robust in the sense that it has a bounded influence function a positive breakdown value and has high efficiency compared to the sample mean for heavy tailed distributions perhaps its greatest strength is that unlike other affine equivariant multivariate medians it is easily computed for data in any practical dimension

a confidence interval for the median survival time a nonparametric asymptotic confidence interval for the median survival time is developed for the case where data are subject to arbitrary right censoring this is accomplished by inverting a generalization of the sign test for censored data a simulation study shows that this nonparametric confidence interval performs well for a variety of underlying survival functions the procedure is applied to data from a clinical trial that compared f our dosage regimens of 5 uorouracil

inference on multivariate m estimators based on bivariate censored data consider two random variables subject to random right censoring for example the survival times of twins or two recurrence times of a certain disease recorded on the same individual of interest is the estimation of a multivariate m functional of these two random variables asymptotic results for the proposed estimator are established under general conditions on the criterion function of the m estimator the obtained results include the asymptotic normality of the estimator its local power and construction of a confidence region our primary example of interest is the bivariate l1 median we compare the small sample behavior of this estimator with that of the vector of marginal medians by means of a simulation study finally we consider a dataset on kidney dialysis patients and estimate the median time to two different infections for these individuals

affine invariant multivariate multisample sign tests summary invariant sign type tests for the multivariate multisample location models are developed the definition of sign is based on the oja criterion function emphasis is on the twosample location model and the associated equivariant estimate of shift asymptotic theory is developed to provide large sample approximations for the null distribution of the test and to provide a framework for a discussion of efficiency tests are then developed for the one way and two way lay outs

a novel takagi sugeno based robust adaptive fuzzy sliding mode controller in this paper a nonlinear dynamic system is first approximated by n fuzzy based linear state space subsystems to track a trajectory dominant by a specific frequency the reference models with desired amplitude and phase features are established by the same fuzzy sets of the system rule similarly the same fuzzy sets of the system rule are employed to design robust fuzzy sliding mode control rfsmc and adaptive fuzzy sliding mode control afsmc the difference between rfsmc and afsmc is that afsmc contains an updating law to learn system uncertainties and then an extra compensation is designed it is different from the most previous papers to learn the whole nonlinear functions as the norm of the sliding surface is inside of a defined set the updating law starts simultaneously as it is outside of the other set the updating law stops for the purpose of smoothing the possibility of discontinuous control input a transition between rfsmc and afsmc is also assigned under the circumstances the proposed control robust adaptive fuzzy sliding mode control rafsmc can automatically tune as a rfsmc or an afsmc then the advantages coming from the rfsmc and afsmc are obtained finally the stabilities of the overall system of rfsmc afsmc and rafsmc are verified by lyapunov stability theory the compared simulations among rfsmc afsmc and rafsmc are also carried out to confirm the usefulness of the proposed control scheme

adaptive fuzzy asymptotic tracking control of uncertain nonaffine nonlinear systems with non symmetric dead zone nonlinearities this paper is concerned with the problem of robust adaptive fuzzy tracking controller design for a class of uncertain nonaffine nonlinear systems with external disturbances and non symmetric dead zone nonlinearities with the help of fuzzy logic systems flss to approximate the desired nonlinear functions a novel direct adaptive tracking control scheme is established first the implicit function theorem and the mean value theorem are used to deal with the nonaffine term with unknown dead zone input then based on the estimation information for the unknown upper bound of the compounded disturbance an adaptive fuzzy tracking controller with the corresponding parameter updating laws is designed to eliminate the effect of the compounded disturbance furthermore it is proved that all the resulting closed loop system signals are uniformly bounded and the tracking error can asymptotically converge to zero via the lyapunov stability analysis finally a numerical example is provided to verify the effectiveness of the proposed adaptive fuzzy asymptotic tracking design method

a comparison of reflected versus test based confidence intervals for the median survival time based on censored data the small sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time based on randomly right censored data is compared with that of two new methods most of these methods are equivalent for large samples all proposed intervals are either test based or reflected intervals in the sense defined in the paper coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data and by stimulation for three life distributions and four censoring patterns in the range of situations studied test based methods often have less than nominal coverage while the coverage of the new reflected confidence intervals is closer to nominal although somewhat conservative and these intervals are easy to compute

a multivariate sign ewma control chart nonparametric control charts are useful in statistical process control spc when there is a lack of or limited knowledge about the underlying process distribution especially when the process measurement is multivariate this article develops a new multivariate spc methodology for monitoring location parameters it is based on adapting a powerful multivariate sign test to online sequential monitoring the weighted version of the sign test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control ewma scheme which results in a nonparametric counterpart of the classical multivariate ewma mewma it is affine invariant and has a strictly distribution free property over a broad class of population models that is the in control ic run length distribution can attain or is always very close to the nominal one when using the same control limit designed for a multivariate normal distribution moreover when the process distribution comes from the elli

adaptive output feedback fuzzy tracking control for a class of nonlinear systems this paper is concerned with the problem of adaptive fuzzy tracking control via output feedback for a class of uncertain single input single output siso strict feedback nonlinear systems the dynamic feedback strategy begins with an input driven filter by utilizing fuzzy logic systems to approximate unknown and desired control input signals directly instead of the unknown nonlinear functions an output feedback fuzzy tracking controller is designed via a backstepping approach it is shown that the proposed fuzzy adaptive output controller can guarantee that all the signals remain bounded and that the tracking error converges to a small neighborhood of the origin simulations results are presented to demonstrate the effectiveness of the proposed methods

directional processing of color images theory and experimental results the processing of color image data using directional information is studied the class of vector directional filters vdf which was introduced by the authors in a previous work is further considered the analogy of vdf to the spherical median is shown and their relation to the spatial median is examined moreover their statistical and deterministic properties are studied which demonstrate their appropriateness in image processing vdf result in optimal estimates of the image vectors in the directional sense this is very important in the case of color images where the vectors direction signifies the chromaticity of a given color issues regarding the practical implementation of vdf are also considered in addition efficient filtering schemes based on vdf are proposed which include adaptive and or double window structures experimental and comparative results in image filtering show very good performance measures when the error is measured in the l a b space l a b is known as a space where equal color differences result in equal distances and therefore it is very close to the human perception of colors moreover an indication of the chromaticity error is obtained by measuring the error on the maxwell triangle the results demonstrate that vdf are very accurate chromaticity estimators

chaos synchronization of uncertain fractional order chaotic systems with time delay based on adaptive fuzzy sliding mode control this paper proposes an adaptive fuzzy sliding mode control afsmc to synchronize two different uncertain fractional order time delay chaotic systems which are infinite dimensional in nature and time delay is a source of instability because modeling the behavior of dynamical systems by fractional order differential equations has more advantages than integer order modeling the adaptive time delay fuzzy logic system is constructed to approximate the unknown fractional order time delay system functions by using lyapunov stability criterion the free parameters of the adaptive fuzzy controller can be tuned online by output feedback control law and adaptive law the sliding mode design procedure not only guarantees the stability and robustness of the proposed afsmc but it also guarantees that the external disturbance on the synchronization error can be attenuated the simulation example is included to confirm validity and synchronization performance of the advocated design methodology

a stable adaptive fuzzy sliding mode control for affine nonlinear systems with application to four bar linkage systems in this paper a stable adaptive fuzzy sliding mode control for affine highly nonlinear systems is developed first the external part of a transformed system via a feedback linearizing control evolves a linear dynamic system with uncertainties a reference model with the desired amplitude and phase properties is given to obtain an error model since the uncertainties are assumed to be large a fuzzy model is employed to model these uncertainties a learning law with e modification for the weight of a fuzzy model is considered to ensure the boundedness of learning weight without the requirement of persistent excitation condition then an equivalent control using the known part of system dynamics and the learning fuzzy model is designed to achieve the desired control behavior furthermore the uncertainties caused by the approximation of fuzzy model and the error of learning weight are tackled by a switching control finally the stability of the overall system is verified by the lyapunov theory simulations and experiments of the velocity control of a four bar linkage system are presented to verify the usefulness of the proposed control

on the efficiency of multivariate spatial sign and rank tests conditionally and asymptotically distribution free rotation but not scale invariant competitors of the well known hotelling s t 2 tests in this paper efficiency results are found in the one sample location case but they natu rally hold in the multivariate two sample multisample and general linear model case the lack of means of fully ordering or ranking observations in the multivariate setting seems at first sight to be an obstacle to generalizing the concepts of sign and rank and signed rank however these concepts can be defined also by using l criterion functions as follows consider k 1 variate observations y y utilizing the euclidean distance y s 1 n 2 2 1r2 y q qy the three objective functions 1 k t y s s y y i i i i i

optimal shrinkage covariance matrix estimation under random sampling from elliptical distributions this paper considers the problem of estimating a high dimensional covariance matrix in a low sample support situation where the sample size is smaller or not much larger than the dimensionality of the data which could potentially be very large we develop a regularized sample covariance matrix rscm estimator which can be applied in commonly occurring high dimensional data problems the proposed rscm estimator is based on estimators of the unknown optimal oracle shrinkage parameters that yield the minimum mean squared error between the rscm and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution we propose two variants of the rscm estimator which differ in the approach in which they estimate the underlying sphericity parameter involved in the theoretical optimal shrinkage parameter the performance of the proposed rscm estimators are evaluated with numerical simulation studies in particular when the sample sizes are low the proposed rscm estimators often show a significant improvement over the conventional rscm estimator by ledoit and wolf 2004 we further evaluate the performance of the proposed estimators in a portfolio optimization problem with real data wherein the proposed methods are able to outperform the benchmark methods

tests for location with k samples and censored data summary tests which are specific for location are developed in the k sample situation with right censored data the tests are based on general estimating functions and do not require full parametric assumptions for large samples but in general they are not based only on ranks one class of tests allows shapes to differ in the k populations and includes as a special case the median test of brookmeyer crowley 1982 if the distributions are assumed to differ at most in location modified tests are suggested in which survival function estimates are obtained by pooling the data from all samples prentice s 1978 median test with minor modifications is included in this class

two sample inference for median survival times based on one sample procedures for censored survival data abstract confidence intervals for median survival times are derived for censored survival data the intervals are obtained by using the quantiles of the kaplan meier product limit estimator and have the same pitman efficiency as the intervals found by inverting the sign test two sample tests and confidence intervals for the difference in median survival times are then developed based on the comparison of the one sample confidence intervals several methods for choosing the confidence coefficients of the corresponding one sample confidence intervals are developed the pitman efficiencies of these two sample tests are the same as that of the median test proposed by brookmeyer and crowley 1982a the procedures can also be used for the accelerated failure time model proportional hazard model and the behrens fisher problem nonparametric two sample inference procedures are useful in comparing the responses of treatment and control groups in medical follow up studies the data are usually subjected to cens

reductive monoids are von neumann regular let k be an algebraically closed field an a e algebraic monoid e is an affme algebraic variety defined over k together with an associative morphism m e x e e and a two sided unit 1 e e for m in categorical terminology an afftne algebraic monoid is a representable functor from the category of afftne varieties to the category of monoids algebraic group theory is concerned mainly with reductive groups and their representations whereas the theory of monoids is concerned largely with von neumann regular monoids the main result of this paper provides a fundamental link between these two objects

convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces the computation of robust regression estimates often relies on minimization of a convex functional on a convex set in this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization minimization algorithms our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case we prove convergence on convex function spaces for general coercive and convex functionals f and derive geometric convergence in certain unconstrained settings the algorithm is applied to total variation tv penalized quantile regression and is compared with a step size corrected newton raphson algorithm it is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better whereas the newton type method outpaces the former only after many iterations finally in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows one to utilize highly efficient algorithms for special quadratic programs in more complex settings

the metrically trimmed mean as a robust estimator of location the metrically trimmed mean is defined as the average of observations remaining after a fixed number of outlying observations have been removed a metric the distance from the median is used to determine which points are outlying the influence curve and the asymptotic normality of the metrically trimmed mean are derived using von mises expansions the relative merits of the median the trimmed mean and the metrically trimmed mean are discussed in neighborhoods of nonparametric models with natural parameters it is observed that the metrically trimmed mean works well for the center of symmetry of a symmetric distribution function with asymmetric contamination a multivariate extension of the metrically trimmed mean is discussed

adaptive fuzzy hierarchical sliding mode control for the trajectory tracking of uncertain underactuated nonlinear dynamic systems the trajectory tracking of uncertain underactuated nonlinear dynamic systems is tackled by an adaptive fuzzy hierarchical sliding mode control afhsmc first one of the subsystems is assigned as the first layer sliding surface next a second layer sliding surface from the first layer sliding surface and the sliding surface of another subsystem is constructed in this paper the nth layer is supposed to be the top layer or hierarchical layer for including the sliding surfaces of all subsystems because two nonlinear system functions and the time varying external disturbance of each subsystem are supposed to be unknown different online fuzzy models are employed to approximate these nonlinear system functions and the upper bounded functions of external disturbances moreover the upper bound of uncertainties caused by these fuzzy modeling errors is estimated online based on these learning fuzzy models and the estimated upper bound of these modeling errors an afhsmc is developed the stability analysis and tracking performance of the closed loop system are verified by lyapunov stability theory finally two simulation examples including different amplitudes of external disturbance and comparison with hierarchical sliding mode control confirm the effectiveness and robustness of the proposed control

adaptive fuzzy robust tracking controller design via small gain approach and its application an adaptive fuzzy robust tracking control afrtc algorithm is proposed for a class of nonlinear systems with the uncertain system function and uncertain gain function which are all the unstructured or nonrepeatable state dependent unknown nonlinear functions arising from modeling errors and external disturbances the takagi sugeno type fuzzy logic systems are used to approximate unknown uncertain functions and the afrtc algorithm is designed by use of the input to state stability approach and small gain theorem the algorithm is highlighted by three advantages 1 the uniform ultimate boundedness of the closed loop adaptive systems in the presence of nonrepeatable uncertainties can be guaranteed 2 the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be removed and 3 the adaptive mechanism with minimal learning parameterizations can be obtained the performance and limitations of the proposed method are discussed the uses of the afrtc for the tracking control design of a pole balancing robot system and a ship autopilot system to maintain the ship on a predetermined heading are demonstrated through two numerical examples simulation results show the effectiveness of the control scheme