splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

continuous selections of trajectories of hybrid systems we present results on existence of continuous selections of trajectories of hybrid systems evolving according to lipschitz nonlinear inclusions first we utilize the skorohod metric to define both a family of pseudo metrics and a metric on the set of trajectories accepted by the hybrid automaton we show that under a non zeno condition the latter metric is complete second the existence of continuous selections with respect to the family of pseudo metrics is proved under a relatively mild assumption of transversality

on prediction of future observation in growth curve model rao proposed and compared several approaches for predicting future observations in a growth curve model the assessment of associated prediction efficiency for different prediction methods were evaluated by cross validation assessment error cvae he used three data sets each with a limited number of subjects 13 27 and also with a limited number of repeated measurements 4 7 per subject to illustrate the prediction methods in the present paper we applied four of the prediction methods discussed by rao on a data set with a relatively large number of subjects 174 and also with a larger number of measurements 21 per subject using the polynomial function and log linear function we propose to use the restricted cubic spline function as an alternative growth curve model and compare its performance with the polynomial function and log linear function it turns out that at least for larger data sets such as that used in this paper the prediction methods perform somewhat better when the growth is described by restricted cubic spline function than when the growth is described by polynomial function and log linear function

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

tests and model selection for the general growth curve model the model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitution it is defined as yp x n xp x m tau m x r ar x n epsilon p x n where tau is unknown and x and a are known design matrices of ranks m less than p and r less than n respectively furthermore the columns of epsilon are independent p variate normal with mean vector 0 and common covariance matrix sigma in general p is the number of time or spatial points observed on each of the n cases m 1 is the degree of polynomial in time and r is the number of groups the main focus of this paper is the selection of models for the general growth curve model with regard to the covariance matrix sigma likelihood ratio tests and selection procedures based on sample reuse and predictions are proposed special emphasis is on the serial covariance structure for sigma which has been shown to be quite important in the prediction of biological data and technology substitution data one population and k population problems are considered some of the results are illustrated with two sets of biological data

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

flexible regression models with cubic splines we describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates this simple method can help prevent the problems that result from inappropriate linearity assumptions we compare restricted cubic spline regression to non parametric procedures for characterizing the relationship between age and survival in the stanford heart transplant data we also provide an illustrative example in cancer therapeutics

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

mean squared error properties of empirical bayes estimators in a multivariate random effects general linear model abstract estimation of an individual s regression coefficients is considered in a multivariate general linear model where it is assumed that the individual s coefficients k are subject to both fixed effects and random effects over different individuals the mean squared error matrix of a natural estimator of k is derived for any individual k in the general situation where all parameters of the model must be estimated and is shown to be smaller than the mean squared error matrix of the individual least squares estimator for every individual extension of this result to more general multiparameter estimation situations is also considered

martin s maximum saturated ideals and non regular ultrafilters part ii we prove assuming the existence of a huge cardinal the consistency of fully non regular ultrafilters on the successor of any regular cardinal we also construct ultrafilters with ultraproducts of small cardinality part ii is logically independent of part i

differential automata and their discrete simulators on presente la notion d automate differentiel pour la formulation d une classe de problemes aux valeurs initiales sur des champs vectoriels hysteretiques

a general stability criterion for time varying systems using a modified detectability condition uniform asymptotic stability for general nonlinear time varying systems is investigated from the output to state viewpoint a unified criterion is proposed using a modified detectability condition and certain integral inequalities the criterion extends the krasovskii lasalle theorem to nonperiodic time varying systems an example taken from the tracking control of mobile robots is given to illustrate that much as the invariance principle of lasalle is used in studying the stability of time invariant systems our results can be used to study the stability of nonperiodic time varying systems by checking the modified detectability condition

nonpathological lyapunov functions and discontinuous caratheodory systems differential equations with discontinuous right hand side and solutions intended in caratheodory sense are considered for these equations sufficient conditions which guarantee both lyapunov stability and asymptotic stability in terms of nonsmooth lyapunov functions are given an invariance principle is also proven

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

efficient inference for random coefficient growth curve models with unbalanced data growth and dose response curve studies often result in incomplete or unbalanced data random effects models together with a variety of computer intensive iterative techniques have been suggested for the analysis of such data this paper is concerned with a noniterative method for estimating and comparing location parameters in random coefficient growth curve models consistent and asymptotically efficient estimators of the location parameters are obtained using estimated generalized least squares two criteria for testing multivariate general linear hypotheses are introduced and their asymptotic properties are investigated the results are applied to clinical data obtained on the blood ultrafiltration performance of hemodialyzers used in the treatment of patients with end stage renal disease

an integral invariance principle for differential inclusions with applications in adaptive control the byrnes martin integral invariance principle for ordinary differential equations is extended to differential inclusions on bbb r n the extended result is applied in demonstrating the existence of adaptive stabilizers and servomechanisms for a variety of nonlinear system classes

gauss markov and weighted least squares estimation under a general growth curve model abstract gauss markov estimator of x 1 bx 2 under a general growth curve model y x 1 bx 2 v 2 v 1 is given necessary and sufficient conditions for equality between a 1 ya 2 and b 1 yb 2 where a i x i x i w i x i x i w i b i x i x i s i x i x i s i with w i s i being any matrix i 1 2 are derived finally a number of criteria for a 1 ya 2 to coincide with the gauss markov estimator of x 1 bx 2 are established

an invariance principle for nonlinear hybrid and impulsive dynamical systems in this paper we develop an invariance principle for dynamical systems possessing left continuous flows specifically we show that left continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems as a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems namely state dependent impulsive dynamical systems these results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems

towards a stability theory of general hybrid dynamical systems in recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time for such systems we introduced the usual concepts of lyapunov and lagrange stability we showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties but defined on real time r 0 the motions of this class of systems are in general discontinuous this class of systems may be finite or infinite dimensional for the above discontinuous dynamical systems and hence for the above hybrid dynamical systems we established the principal lyapunov stability theorems as well as lagrange stability theorems for some of these we also established converse theorems we demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems in the present paper we continue the work described above in doing so we first develop a general comparison theory for the class of hybrid dynamical systems resp discontinuous dynamical systems considered herein making use of stability preserving mappings we then show how these results can be applied to establish some of the principal lyaponov stability theorems for the latter we also state and prove a converse theorem not considered previously finally to demonstrate the applicability of our results we consider specific examples throughout the paper

lyapunov based hybrid loops for stability and performance of continuous time control systems we construct hybrid loops that augment continuous time control systems we consider a continuous time nonlinear plant in feedback with a possibly non stabilizing given nonlinear dynamic continuous time state feedback controller the arising hybrid closed loops are guaranteed to follow the underlying continuous time closed loop dynamics when flowing and to jump in suitable regions of the closed loop state space to guarantee that a positive definite function v of the closed loop state and or a positive definite function v p of the plant only state is non increasing along the hybrid trajectories sufficient conditions for the construction of these hybrid loops are given for the nonlinear case and then specialized for the linear case with the use of quadratic functions for the linear case we illustrate specific choices of the functions v and v p which allow for the reduction of the overshoot of a scalar output the proposed approaches are illustrated on linear and nonlinear examples

prediction and estimation of growth curves with special covariance structures abstract the model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitutions it is defined as yp n xp m m rar n p n where is unknown and x and a are known design matrices of ranks m p and r n respectively further the columns of are independent p variate normal with mean vector 0 and common covariance matrix in general p is the number of time or spatial points observed on each of the n cases m 1 is the degree of polynomial and r is the number of groups this article focuses mainly on predicting future observations including partially observed vectors and future values on a given number of cases using their past observations these prediction problems as well as the estimation of parameters are considered for two parsimonious covariance structures that are very important in practice the results are illustrated with three real data sets som

smooth lyapunov functions for hybrid systems part i existence is equivalent to robustness hybrid systems are dynamical systems where the state is allowed to either evolve continuously flow on certain subsets of the state space or evolve discontinuously jump from other subsets of the state space for a broad class of such systems characterized by mild regularity conditions on the data we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth lyapunov function this result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right hand sides furthermore we give a description of forward completeness of a hybrid system in terms of a smooth lyapunov like function

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

hierarchical nonlinear control for multi rotor asymptotic stabilization based on zero moment direction we consider the hovering control problem for a class of multi rotor aerial platforms with generically oriented propellers characterized by intrinsically coupled translational and rotational dynamics in doing this we first discuss some assumptions guaranteeing the rejection of generic disturbance torques while compensating for the gravity force these assumptions are translated into a geometric condition usually satisfied by both standard models and more general configurations then we propose a control strategy based on the identification of a zero moment direction for the exerted force and a dynamic state feedback linearization based on this preferential direction which locally asymptotically stabilizes the platform to a static hovering condition stability properties of the control law are rigorously proved through lyapunov based methods and reduction theorems for the stability of nested sets asymptotic zeroing of the error dynamics and convergence to the static hovering condition are then confirmed by simulation results on a star shaped hexarotor model with tilted propellers

stability and performance of siso control systems with first order reset elements we consider set point regulation and l2 robust stability properties of a class of reset control systems consisting of a minimum phase relative degree one linear siso plant controlled by a novel first order reset element fore these results rely on necessary and sufficient conditions for exponential and l2 finite gain stability of a class of planar reset systems consisting of a scalar linear plant controlled by the novel fore we show that the l2 gain of the planar reset system decreases to zero as the pole and or the gain of the fore are increased to infinity a number of stability results including lyapunov conditions for lp and exponential stability for a larger class of reset and hybrid systems are presented and used to prove our main results

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

hybrid feedback control and robust stabilization of nonlinear systems in this paper we show for any nonlinear system that is asymptotically controllable to a compact set that a logic based hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise actuator error and external disturbance the construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed loop hybrid system the robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties auxiliary results give uniformity of convergence and of overshoots for hybrid systems and give a characterization of asymptotic stability of compact sets

an invariance principle for nonlinear switched systems abstract in this paper we address the problem of extending lasalle invariance principle to switched system we prove an extension of the invariance principle relative to dwell time switched solutions and a second one relative to constrained switched systems

left separated spaces with point countable bases this is the published version also available here http www dx doi org 10 2307 2000207 first published in trans amer math soc in 1986 published by the american mathematical society

linear sufficiency in a general growth curve model abstract the notion of linear sufficiency for the whole set of estimable functions in the general gauss markov model is extended to the estimation of any special set of estimable functions in a general growth curve model some general results with respect to the concept of linear sufficiency are obtained from which a necessary and sufficient condition is established for a linear transformation f 1 f 2 of the observation matrix y to have the property that there exists a linear function of f 1 yf 2 which is the blue of the estimable functions k 1 bk 2

a nonsmooth hybrid invariance principle applied to robust event triggered design we first propose a nonsmooth hybrid invariance principle with relaxed conditions stemming from the fact that flowing solutions evolve only in the tangent cone and complete jumping solutions cannot jump outside the jump and flow sets we then show an application consisting in the design of event triggered rules to stabilize a class of uncertain linear control systems the event triggering rule depends only on local information that is it uses only the output signals available to the controller the approach proposed combines hybrid and sampled data tools the proposed design conditions are formulated in terms of linear matrix inequalities lmis ensuring global robust asymptotic stability of the closed loop system a tunable parameter is also available to guarantee an adjustable dwell time property of the solutions the effectiveness of the approach is evaluated through an example borrowed from the literature