corrected confidence bands for functional data using principal components functional principal components fpc analysis is widely used to decompose and express functional observations curve estimates implicitly condition on basis functions and other quantities derived from fpc decompositions however these objects are unknown in practice in this article we propose a method for obtaining correct curve estimates by accounting for uncertainty in fpc decompositions additionally pointwise and simultaneous confidence intervals that account for both model and decomposition based variability are constructed standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition iterated expectation and variance formulas combine model based conditional estimates across the distribution of decompositions a bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions we apply our method to sparse observations of cd4 cell counts and to dense white matter tract profiles code for the analyses and simulations is publicly available and our method is implemented in the r package refund on cran

marginal longitudinal nonparametric regression locality and efficiency of spline and kernel methods we consider nonparametric regression in a longitudinal marginal model of generalized estimating equation gee type with a time varying covariate in the situation where the number of observations per subject is finite and the number of subjects is large in such models the basic shape of the regression function is affected only by the covariate values and not otherwise by the ordering of the observations two methods of estimating the nonparametric function can be considered kernel methods and spline methods recently surprising evidence has emerged suggesting that for kernel methods previously proposed in the literature it is generally asymptotically preferable to ignore the correlation structure in our marginal model and instead assume that the data are independent that is working independence in the gee jargon as seen through equivalent kernel results in univariate independent data problems splines and kernels have similar behavior smoothing splines are equivalent to kernel regression with a s

a natural approach to the numerical integration of riccati differential equations this paper introduces a new class of methods which we call mobius schemes for the numerical solution of matrix riccati differential equations the approach is based on viewing the riccati equation in its natural geometric setting as a flow on the grassmannian of m dimensional subspaces of an n m dimensional vector space since the grassmannians are compact differentiable manifolds and the coefficients of the equation are assumed continuous there are no singularities or intrinsic instabilities in the associated flow the presence of singularities and numerical instabilities is an artifact of the coordinate system but since mobius schemes are based on the natural geometry they are able to deal with numerical instability and pass accurately through the singularities a number of examples are given to demonstrate these properties

on testing an unspecified function through a linear mixed effects model with multiple variance components summary we examine a generalized f test of a nonparametric function through penalized splines and a linear mixed effects model representation with a mixed effects model representation of penalized splines we imbed the test of an unspecified function into a test of some fixed effects and a variance component in a linear mixed effects model with nuisance variance components under the null the procedure can be used to test a nonparametric function or varying coefficient with clustered data compare two spline functions test the significance of an unspecified function in an additive model with multiple components and test a row or a column effect in a two way analysis of variance model through a spectral decomposition of the residual sum of squares we provide a fast algorithm for computing the null distribution of the test which significantly improves the computational efficiency over bootstrap the spectral representation reveals a connection between the likelihood ratio test lrt in a multiple variance components model and a single component model we examine our methods through simulations where we show that the power of the generalized f test may be higher than the lrt depending on the hypothesis of interest and the true model under the alternative we apply these methods to compute the genome wide critical value and p value of a genetic association test in a genome wide association study gwas where the usual bootstrap is computationally intensive up to 108 simulations and asymptotic approximation may be unreliable and conservative

penalized estimating equations les modeles avec penalisation comme l estimateur ridge l estimateur de stein l estimateur en pont et le lasso ont ete proposes pour prendre en compte la collineariteen regression le lasso a ete applique au modele lineaire a la regression logistique au modele des hasards proportionnels de cox et aux reseaux de neurones cet article etudie le modele avec penalisation en pont pour une penalite j j applique aux equations d estimation et applique ce modele avec penalisation aux equations d estimation generalisees gee dans des etudes longitudinales le defaut de vraisemblance jointe dans les gee est surmonte par les equations d estimation penalisees pour lesquelles aucune vraisemblance jointe n est requise on donne les resultats asymptotiques pour l estimateur sons penalite on montre par des simulations et une application que les cee avec penalisation ameliorent potentiellement les resultats de l estimateur gee et presentent les memes proprietes que les modeles lineaires penalises

boundary optimal lq control of coupled hyperbolic pdes and odes this contribution addresses the development of a linear quadratic lq regulator for a set of hyperbolic pdes coupled with a set of odes through the boundary the approach is based on an infinite dimensional hilbert state space description of the system and the well known operator riccati equation ore in order to solve the optimal control problem the ore is converted to a set of matrix riccati equations the feedback operator is found by solving the resulting matrix riccati equations the performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor cstr and a plug flow reactor pfr through a numerical simulation

solving the matrix differential riccati equation a lyapunov equation approach in this technical note we investigate a solution of the matrix differential riccati equation that plays an important role in the linear quadratic optimal control problem unlike many methods in the literature the approach that we propose employs the negative definite anti stabilizing solution of the matrix algebraic riccati equation and the solution of the matrix differential lyapunov equation an illustrative numerical example is provided to show the efficiency of our approach

magnus integrators for solving linear quadratic differential games we consider magnus integrators to solve linear quadratic n player differential games these problems require to solve backward in time non autonomous matrix riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game to be integrated forward in time we analyze different magnus integrators which can provide either analytical or numerical approximations to the equations they can be considered as time averaging methods and frequently are used as exponential integrators we show that they preserve some of the most relevant qualitative properties of the solution for the matrix riccati differential equations as well as for the remaining equations the analytical approximations allow us to study the problem in terms of the parameters involved some numerical examples are also considered which show that exponential methods are in general superior to standard methods

joint modelling of paired sparse functional data using principal components we propose a modelling framework to study the relationship between two paired longitudinally observed variables the data for each variable are viewed as smooth curves measured at discrete time points plus random errors while the curves for each variable are summarized using a few important principal components the association of the two longitudinal variables is modelled through the association of the principal component scores we use penalized splines to model the mean curves and the principal component curves and cast the proposed model into a mixed effects model framework for model fitting prediction and inference the proposed method can be applied in the difficult case in which the measurement times are irregular and sparse and may differ widely across individuals use of functional principal components enhances model interpretation and improves statistical and numerical stability of the parameter estimates

the recursive reduced order numerical solution of the singularly perturbed matrix differential riccati equation under stability observability conditions imposed on a singularly perturbed system an efficient numerical method for solving the corresponding matrix differential riccati equation is obtained in terms of the reduced order problems the order reduction is achieved via the use of the chang transformation applied to the hamiltonian matrix of a singularly perturbed linear quadratic control problem an efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the chang transformation

static duality and a stationary action application abstract conservative dynamical systems propagate as stationary points of the action functional using this representation it has previously been demonstrated that one may obtain fundamental solutions for two point boundary value problems for some classes of conservative systems via solution of an associated dynamic program further such a fundamental solution may be represented as a set of solutions of differential riccati equations dres where the solutions may need to be propagated past escape times notions of static duality and stat quad duality are developed where the relationship between the two is loosely analogous to that between convex and semiconvex duality static duality is useful for smooth functionals where one may not be guaranteed of convexity or concavity some simple properties of this duality are examined particularly commutativity application to stationary action is considered which leads to propagation of dres past escape times via propagation of stat quad dual dres

reduced order solution to the finite time optimal control problems of linear weakly coupled systems the optimal solution to the finite time optimal control problem of weakly coupled linear systems is found in terms of completely decoupled reduced order differential equations for both closed loop and open loop control this is achieved through the use of the decoupling transformation that block diagonalizes the hamiltonian matrix of the weakly coupled linear quadratic control problem the convergence to the optimal solution is pretty rapid the proposed technique is very well suited for parallel computations

efficient matrix valued algorithms for solving stiff riccati differential equations in the time varying case a classical approach which has been widely used to compute the solution of the riccati matrix equation of for example size n n is to unroll the matrices into vectors and integrate the resulting system of n sup 2 vector differential equations directly if the system of vectorized differential equations is stiff the cost computation time and storage requirements of applying the popular backward differentiation formulas bdfs to the stiff equations will be very high for large n because a linear system of algebraic equations of size n sup 2 n sup 2 must be solved at each time step new matrix valued algorithms based on a matrix generalization of the bdfs are proposed for solving stiff riccatti differential equations the amount of work required to compute the solution per time step is only o n sup 3 flops by using the matrix valued algorithms whereas the classical approach requires o n sup 6 flops per time step

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

on the solution of the riccati differential equation arising from the lq optimal control problem in this paper we consider the matrix riccati differential equation rde that arises from linear quadratic lq optimal control problems in particular we establish explicit closed formulae for the solution of the rde with a terminal condition using particular solutions of the associated algebraic riccati equation we discuss how these formulae change as assumptions are progressively weakened an application to lq optimal control is briefly analysed

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis in this article we present new methods to analyze data from an experiment using rodent models to investigate the role of p27 an important cell cycle mediator in early colon carcinogenesis the responses modeled here are essentially functions nested within a two stage hierarchy standard functional data analysis literature focuses on a single stage of hierarchy and conditionally independent functions with near white noise however in our experiment there is substantial biological motivation for the existence of spatial correlation among the functions which arise from the locations of biological structures called colonic crypts this possible functional correlation is a phenomenon we term crypt signaling thus as a point of general methodology we require an analysis that allows for functions to be correlated at the deepest level of the hierarchy our approach is fully bayesian and uses markov chain monte carlo methods for inference and estimation analysis of this data set gives new insights into the structure of p27 expression in early colon carcinogenesis and suggests the existence of significant crypt signaling our methodology uses regression splines and because of the hierarchical nature of the data dimension reduction of the covariance matrix of the spline coefficients is important we suggest simple methods for overcoming this problem

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

relaxation to equilibrium for two dimensional disordered ising systems in the griffiths phase we consider glauber type dynamics for two dimensional disordered magnets of ising type we prove that if the disorder averaged influence of the boundary condition is sufficiently small in the equilibrium system then the corresponding glauber dynamics is ergodic with probability one and the disorder average c t of time autocorrelation function satisfies for large t for the standard two dimensional dilute ising ferromagnet with i i d random nearest neighbor couplings taking the values 0 or j 0 0 our results apply even if the active bonds percolate and j 0 is larger than the critical value j c of the corresponding pure ising model for the same model we also prove that in the whole griffiths phase the previous upper bound is optimal this implies the existence of a dynamical phase transition which occurs when j crosses j c

matching conditional and marginal shapes in binary random intercept models using a bridge distribution function summary random effects logistic regression models are often used to model clustered binary response data regression parameters in these models have a conditional subject specific interpretation in that they quantify regression effects for each cluster very often the logistic functional shape conditional on the random effects does not carry over to the marginal scale thus parameters in these models usually do not have an explicit marginal population averaged interpretation we study a bridge distribution function for the random effect in the random intercept logistic regression model under this distributional assumption the marginal functional shape is still of logistic form and thus regression parameters have an explicit marginal interpretation the main advantage of this approach is that likelihood inference can be obtained for either marginal or conditional regression inference within a single model framework the generality of the results and some properties of the bridge distribution functions are discussed an example is used for illustration

a penalized spline approach to functional mixed effects model analysis summary in this work we propose penalized spline based methods for functional mixed effects models with varying coefficients we decompose longitudinal outcomes as a sum of several terms a population mean function covariates with time varying coefficients functional subject specific random effects and residual measurement error processes using penalized splines we propose nonparametric estimation of the population mean function varying coefficient random subject specific curves and the associated covariance function which represents between subject variation and the variance function of the residual measurement errors which represents within subject variation proposed methods offer flexible estimation of both the population level and subject level curves in addition decomposing variability of the outcomes as a between subject and a within subject source is useful in identifying the dominant variance component therefore optimally model a covariance function we use a likelihood based method to select multiple smoothing parameters furthermore we study the asymptotics of the baseline p spline estimator with longitudinal data we conduct simulation studies to investigate performance of the proposed methods the benefit of the between and within subject covariance decomposition is illustrated through an analysis of berkeley growth data where we identified clearly distinct patterns of the between and within subject covariance functions of children s heights we also apply the proposed methods to estimate the effect of anti hypertensive treatment from the framingham heart study data

brief paper computing the l2 gain for linear periodic continuous time systems a method to compute the l 2 gain is developed for the class of linear periodic continuous time systems that admit a finite dimensional state space realisation a bisection search for the smallest upper bound on the gain is employed where at each step an equivalent discrete time problem is considered via the well known technique of time domain lifting the equivalent problem involves testing a bound on the gain of a linear shift invariant discrete time system with the same state dimension as the periodic continuous time system it is shown that a state space realisation of the discrete time system can be constructed from point solutions to a linear differential equation and two differential riccati equations all subject to only single point boundary conditions these are well behaved over the corresponding one period intervals of integration and as such the required point solutions can be computed via standard methods for ordinary differential equations a numerical example is presented and comparisons made with alternative techniques

bootstrap based inference on the difference in the means of two correlated functional processes we propose nonparametric inference methods on the mean difference between two correlated functional processes we compare methods that 1 incorporate different levels of smoothing of the mean and covariance 2 preserve the sampling design and 3 use parametric and nonparametric estimation of the mean functions we apply our method to estimating the mean difference between average normalized power of sleep electroencephalograms for 51 subjects with severe sleep apnea and 51 matched controls in the first 4 h after sleep onset we obtain data from the sleep heart health study the largest community cohort study of sleep although methods are applied to a single case study they can be applied to a large number of studies that have correlated functional data copyright 2012 john wiley sons ltd

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

percolation techniques in disordered spin flip dynamics relaxation to the unique invariant measure we consider lattice spin systems with short range but random and unbounded interactions we give criteria for ergodicity of spin flip dynamics and estimate the speed of convergence to the unique invariant measure we find for this convergence a stretched exponential in time for a class of directed dynamics such as in the disordered toom or stavskaya model for the general case we show that the relaxation is faster than any power in time no assumptions of reversibility are made the methods are based on relating the problem to an oriented percolation problem contact process and for the general case using a slightly modified version of the multiscale analysis of e g klein 1993

a gmres based bdf method for solving differential riccati equations abstract differential riccati equations play a fundamental role in control theory for example optimal control filtering and estimation decoupling and order reduction etc the most popular codes to solve stiff differential riccati equations use backward differentiation formula bdf methods in this paper a new approach to solve differential riccati equations by means of a bdf method is described in each step of these methods an algebraic riccati equation is obtained which is solved by means of newton s method in the standard approach this system is transformed into a sylvester equation which could be solved by means of the well known bartels stewart method in our code we obtain a system of linear equations defined from a kronecker product of matrices related to coefficient matrices of the differential riccati equation that is solved by means of the iterative generalized minimum residual gmres method we have also implemented an efficient matrix vector product in order to reduce the computational and storage cost of the gmres method the above approach has been applied in the development of an algorithm to solve differential riccati equations the accuracy and efficiency of this algorithm has been compared with the bdf algorithm that uses the bartels stewart method experimental results show the advantages of the new algorithm

marginal nonparametric kernel regression accounting for within subject correlation there has been substantial recent interest in non and semiparametric methods for longitudinal or clustered data with dependence within clusters it has been shown rather inexplicably that when standard kernel smoothing methods are used in a natural way higher efficiency is obtained by assuming independence than by using the true correlation structure it is shown here that this result is a natural consequence of how standard kernel methods incorporate the within subject correlation in the asymptotic setting considered where the cluster sizes are fixed and the cluster number increases in this paper an alternative kernel smoothing method is proposed unlike the standard methods the smallest variance of the new estimator is achieved when the true correlation is assumed asymptotically the variance of the proposed method is uniformly smaller than that of the most efficient working independence approach a small simulation study shows that significant improvement is obtained for finite samples

a fixed point based bdf method for solving differential riccati equations abstract this paper describes an approach for solving differential riccati equations dre by means of the backward differentiation formula bdf and resolution of the corresponding implicit equation using newton s method with a fixed point approach the role and use of dre is especially important in several applications such as optimal control filtering and estimation the goodness of this new method is compared with respect to the so called dieci method l dieci numerical integration of the differential riccati equation and some related issues siam j numer anal 29 3 1992 781 815

rational approximate solutions and error bounds for the nonsymmetric riccati matrix differential equation in this paper we determine an interval where the exact solution of the nonsymmetric riccati matrix differential equation is defined rational approximate solutions are whose error is smaller than a prefixed number e 0 are constructed in terms of the data

on the asymptotics of marginal regression splines with longitudinal data summary there have been studies on how the asymptotic efficiency of a nonparametric function estimator depends on the handling of the within cluster correlation when nonparametric regression models are used on longitudinal or cluster data in particular methods based on smoothing splines and local polynomial kernels exhibit different behaviour we show that the generalized estimation equations based on weighted least squares regression splines for the nonparametric function have an interesting property the asymptotic bias of the estimator does not depend on the working correlation matrix but the asymptotic variance and therefore the mean squared error is minimized when the true correlation structure is specified this property of the asymptotic bias distinguishes regression splines from smoothing splines

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

aberrant crypt foci and semiparametric modeling of correlated binary data motivated by the spatial modeling of aberrant crypt foci acf in colon carcinogenesis we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent gaussian spatial process that accounts for short range dependencies the mean is modeled in a general way using regression splines the mean function can be viewed as a fixed effect and is estimated with a penalty for regularization with the latent process viewed as another random effect the model becomes a generalized linear mixed model in our motivating data set and other applications the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation reml so pairwise likelihood a special case of composite likelihood is used instead we develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications including but not limited to the problem that motivated this work the splines have penalty parameters that must converge to zero asymptotically we derive theory for this along with a data driven method for selecting the penalty parameter a method that is shown in simulations to improve greatly upon standard devices such as likelihood crossvalidation finally we apply the methods to the data from our experiment acf we discover an unexpected location for peak formation of acf

metastability in the dilute ising model consider glauber dynamics for the ising model on the hypercubic lattice with a positive magnetic field starting from the minus configuration the system initially settles into a metastable state with negative magnetization slowly the system relaxes to a stable state with positive magnetization schonmann and shlosman showed that in the two dimensional case the relaxation time is a simple function of the energy required to create a critical wulff droplet the dilute ising model is obtained from the regular ising model by deleting a fraction of the edges of the underlying graph in this paper we show that even an arbitrarily small dilution can dramatically reduce the relaxation time this is because of a catalytic effect rare regions of high dilution speed up the transition from minus phase to plus phase

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

a shishkin mesh for a singularly perturbed riccati equation in this paper a singularly perturbed riccati initial value problem is examined parameter explicit bounds on the solution and its derivatives are given a numerical method composed of an implicit difference operator and a piecewise uniform shishkin mesh is constructed a theoretical parameter independently bound on the errors in the numerical approximations is established numerical results are presented which are in agreement with the theoretical error bound

a survey of recursive algorithms for the solution of the discrete time riccati equation abstract the riccati equation plays a fundamental role in many fields of mathematics science and engineering its solution constitutes an integral prerequisite to the solution of important problems in the above fields due to the importance of the riccati equation there exists considerable literature on its algebraic as well as algorithmic solution a very large number of those studies are devoted to the continuous time riccati equation in this paper we present a survey of classical as well as more recent recursive algorithms that solve the discrete time riccati equatbn emanating from the kaiman filter as well as from the lainiotis filter equations either using per step calculations or the doubling principle it is established that these algorithms converge fast and are numerically stable

relaxation of disordered magnets in the griffiths regime we study the relaxation to equilibrium of discrete spin systems with random finite range not necessarily ferromagnetic interactions in the griffiths regime we prove that the speed of convergence to the unique reversible gibbs measure is almost surely faster than any stretched exponential at least if the probability distribution of the interaction decays faster than exponential e g gaussian furthermore if the interac tion is uniformly bounded the average over the disorder of the time autocorrelation function goes to equilibrium as exp k log t d d 1 in d 1 in agreement with previous results obtained for the dilute ising model

a family of anadromic numerical methods for matrix riccati differential equations matrix riccati differential equations mrdes x a21 xa11 a22x xa12x x 0 x0 where aij aij t appear frequently throughout applied mathematics science and engineering naturally the existing conventional runge kutta methods and linear multi step methods can be adapted to solve mrdes numerically indeed they have been adapted there are a few unconventional numerical methods too but they are suited more for time invariant mrdes than time varying ones for stiff mrdes existing implicit methods which are preferred to explicit ones require solving nonlinear systems of equations of possibly much higher dimensions than the original problem itself of for example implicit runge kutta methods and thus they can pose implementation difficulties and also be expensive in the past the property of an mrde which has been most preserved is the symmetry property for a symmetric mrde many other crucial properties have been discarded besides the symmetry property our proposed methods also preserve two other important properties bilinear rational dependence on the initial value and a generalized inverse relation between an mrde and its complementary mrde by preserving the generalized inverse relation our methods are accurately able to integrate an mrde whose solution has singularities by preserving the property of bilinear dependence on the initial value our methods also conserve the rank of change to the initial value and a solution s monotonicity property our methods are anadromic 1meaning if an mrde is integrated by one of our methods from t to and then integrated backward from t to using the same method the value at t is recovered in the absence of rounding errors this implies that our methods are necessarily of even order of convergence for time invariant mrdes methods of any even order of convergence are established while for time varying mrdes methods of order as high as 10 are established but only methods of order up to 6 are stated in detail our methods are semi implicit in the sense that there are no nonlinear systems of matrix equations to solve only linear ones unlike any pre existing implicit method given the availability of high quality codes for linear matrix equations our methods can easily be implemented and embedded into any application software package that needs a robust mrde solver numerical examples are presented to support our claims 1we coined anadromic from the greek roots back up and o o act of running of anadromous which describes fish that must return from the ocean to spawn in the same streams whence they hatched after we had tried to describe our numerical methods with words such as reflexive 27 28 symmetric which is still used today in the literature e g 22 and reversible which turned out to be overused received by the editor received august 25 2009 and in revised form october 30 2010 2010 mathematics subject classification primary 65l05

eigenfactor solution of the matrix riccati equation a continuous square root algorithm this paper introduces a new algorithm for solving the matrix riccati equation differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix the solution of these equations yields then the time behavior of the eigenvalues and eigenvectors of the solution matrix a reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation this algorithm serves two purposes first being a square root solution it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy secondly it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors the algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples through these examples it is seen that the algorithm is indeed more accurate than the ordinary one moreover it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed form solution cannot be computed as a result of numerical difficulties

on the benefits of the ldlt factorization for large scale differential matrix equation solvers abstract we propose efficient algorithms for solving large scale matrix differential equations in particular we deal with the differential riccati equations dre and state the applicability to the differential lyapunov equations dle we focus on methods based on standard versions of ordinary differential equations in the matrix setting the application of these methods yields algebraic lyapunov equations ales with a certain structure to be solved in every step the alternating direction implicit adi algorithm and krylov subspace based methods allow to exploit this special structure however a direct application of classic low rank formulations requires the use of complex arithmetic using an l d l t type decomposition of both the right hand side and the solution of the equation we avoid this problem thus the proposed methods are a more practical alternative for large scale problems arising in applications also they make the application of higher order methods feasible the numerical results show the better performance of the proposed methods compared to earlier formulations

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

asymptotic properties of penalized spline estimators we study the class of penalized spline estimators which enjoy similarities to both regression splines without penalty and with fewer knots than data points and smoothing splines with knots equal to the data points and a penalty controlling the roughness of the fit depending on the number of knots sample size and penalty we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines with a clear breakpoint distinguishing the cases we prove that using fewer knots results in better asymptotic rates than when using a large number of knots we obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter

simultaneous confidence bands for penalized spline estimators in this article we construct simultaneous confidence bands for a smooth curve using penalized spline estimators we consider three types of estimation methods a as a standard fixed effect nonparametric model b using the mixed model framework with the spline coefficients as random effects and c a full bayesian approach the volume of tube formula is applied for the first two methods and compared with bayesian simultaneous confidence bands from a frequentist perspective we show that the mixed model formulation of penalized splines can help obtain at least approximately confidence bands with either bayesian or frequentist properties simulations and data analysis support the proposed methods the r package confbands accompanies the article

rosenbrock methods for solving riccati differential equations the riccati differential equation rde arises in several fields like optimal control optimal filtering h control of linear time varying systems differential games etc in the literature there is a large variety of approaches to compute its solution particularly for stiff rdes matrix valued versions of the standard multi step methods for solving ordinary differential equations have given good results in this technical note we discuss a particular class of one step methods these are the linear implicit runge kutta methods or rosenbrock methods we show that they offer a practical alternative for solving stiff rdes they can be implemented with good stability properties and allow for a cheap step size control the matrix valued version of the rosenbrock methods for rdes requires the solution of one sylvester equation in each stage of the method for the case in which the coefficient matrices of the sylvester equation are dense the bartels stewart method can be efficiently applied for solving the equations the computational cost computing time and memory requirements is smaller than for multi step methods

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

almost analytic representation for the solution of the differential matrix riccati equation an almost analytic representation is presented for the solution of the nonhomogeneous and homogeneous time invariant and time variant differential matrix riccati equation this representation gives a new insight into the behavior of the optimal estimator on a finite time interval for the time invariant case this solution is a basis for an exact true iterative algorithm for the time invariant case an approximate iterative algorithm is derived

tight bounds for mixing of the swendsen wang algorithm at the potts transition point we study two widely used algorithms for the potts model on rectangular subsets of the hypercubic lattice mathbb z d heat bath dynamics and the swendsen wang algorithm and prove that under certain circumstances the mixing in these algorithms is torpid or slow in particular we show that for heat bath dynamics throughout the region of phase coexistence and for the swendsen wang algorithm at the transition point the mixing time in a box of side length l with periodic boundary conditions has upper and lower bounds which are exponential in ld 1 this work provides the first upper bound of this form for the swendsen wang algorithm and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in l log l 2

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

gip integrators for matrix riccati differential equations matrix riccati differential equations mrdes are initial value problems of the form x a 2 1 xa 1 1 a 2 2x xa 1 2x x 0 x 0 these equations arise frequently throughout applied mathematics science and engineering it can happen that even when the a i j are smooth functions of t or constant the solution x may have a singularity or even infinitely many singularities this paper shows several classes of numerical algorithms which we call gip integrators that can solve for x past its singularities furthermore none of the algorithms require knowledge of the placement or even existence of singularities in x also it is shown how embedded runge kutta methods can be used to construct gip integrators to not only approximate x past singularities but also provide for error estimation to allow efficient time stepping finally several examples are shown to validate the theory

stretched exponential relaxation in the biased random voter model we study the relaxation properties of the voter model with i i d random bias we prove under mild conditions that the disorder averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d d where 0 2 depends on the transition rates of the non biased voter model under an additional assumption we show that the above upper bound is optimal the main ingredient of our proof is a result of donsker and varadhan

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

upper and lower bounds on dynamic correlations in the griffiths phase dilute ferromagnetic spin systems are considered in the griffiths phase below the percolation threshold it is shown that for relaxational dynamics the spin autocorrelation functions are bounded both above and below by functions of the form exp a 1n t d d 1 ising systems and exp bt12 heisenberg systems establishing these as the correct asymptotic forms

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

coexistence of infinite clusters ii ising percolation in two dimensions we show a strong type of conditionally mixing property for the gibbs states ofd dimensional ising model when the temperature is above the critical one by using this property we show that there is always coexistence of infinite and clusters when is smaller than c andh 0 in two dimensions it is also possible to show that this coexistence region extends to some non zero external field case i e for every c there exists someh c 0 such that h h c implies the coexistence of infinite clusters with respect to the gibbs state for h

fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models summary recent work by reiss and ogden provides a theoretical basis for sometimes preferring restricted maximum likelihood reml to generalized cross validation gcv for smoothing parameter selection in semiparametric regression however existing reml or marginal likelihood ml based methods for semiparametric generalized linear models glms use iterative reml or ml estimation of the smoothing parameters of working linear approximations to the glm such indirect schemes need not converge and fail to do so in a non negligible proportion of practical analyses by contrast very reliable prediction error criteria smoothing parameter selection methods are available based on direct optimization of gcv or related criteria for the glm itself since such methods directly optimize properly defined functions of the smoothing parameters they have much more reliable convergence properties the paper develops the first such method for reml or ml estimation of smoothing parameters a laplace approximation is used to obtain an approximate reml or ml for any glm which is suitable for efficient direct optimization this reml or ml criterion requires that newton raphson iteration rather than fisher scoring be used for glm fitting and a computationally stable approach to this is proposed the reml or ml criterion itself is optimized by a newton method with the derivatives required obtained by a mixture of implicit differentiation and direct methods the method will cope with numerical rank deficiency in the fitted model and in fact provides a slight improvement in numerical robustness on the earlier method of wood for prediction error criteria based smoothness selection simulation results suggest that the new reml and ml methods offer some improvement in mean square error performance relative to gcv or akaike s information criterion in most cases without the small number of severe undersmoothing failures to which akaike s information criterion and gcv are prone this is achieved at the same computational cost as gcv or akaike s information criterion the new approach also eliminates the convergence failures of previous reml or ml based approaches for penalized glms and usually has lower computational cost than these alternatives example applications are presented in adaptive smoothing scalar on function regression and generalized additive model selection

equivalent kernels of smoothing splines in nonparametric regression for clustered longitudinal data correlation we show that a smoothing spline estimator is asymptotically equivalent to a recently proposed seemingly unrelated kernel estimator of wang 2003 for any working covariance matrix we show that both estimators can be obtained iteratively by applying conventional kernel or spline smoothing to pseudo observations this result allows us to study the asymptotic properties of the smoothing spline estimator by deriving its asymptotic bias and variance we show that smoothing splines are consistent for an arbitrary working covariance and have the smallest variance when assuming the true covariance we further show that both the seemingly unrelated kernel estimator and the smoothing spline estimator are nonlocal unless working independence is assumed but have asymptotically negligible bias their finite sample performance is compared through simulations our results justify the use of efficient non local estimators such as smoothing splines for clustered longitudinal data

a marginal approach to reduced rank penalized spline smoothing with application to multilevel functional data multilevel functional data are collected in many biomedical studies for example in a study of the effect of nimodipine on patients with subarachnoid hemorrhage sah patients underwent multiple 4 hr treatment cycles within each treatment cycle subjects vital signs were reported every 10 min these data have a natural multilevel structure with treatment cycles nested within subjects and measurements nested within cycles most literature on nonparametric analysis of such multilevel functional data focuses on conditional approaches using functional mixed effects models however parameters obtained from the conditional models do not have direct interpretations as population average effects when population effects are of interest we may employ marginal regression models in this work we propose marginal approaches to fit multilevel functional data through penalized spline generalized estimating equation penalized spline gee the procedure is effective for modeling multilevel correlated generalized

max plus representation for the fundamental solution of the time varying differential riccati equation using the tools of optimal control semiconvex duality and max plus algebra this work derives a unifying representation of the solution for the matrix differential riccati equation dre with time varying coefficients it is based upon a special case of the max plus fundamental solution first proposed in fleming and mceneaney 2000 such a fundamental solution can extend a particular solution of certain bivariate dres into the general solution and the dres can be analytically solved from any initial condition this paper also shows that under a fixed duality kernel the semiconvex dual of a dre solution satisfies another dual dre whose coefficients satisfy the matrix compatibility conditions involving hamiltonian and certain symplectic matrices for the time invariant dre this allows us to make dual dre linear and thereby solve the primal dre analytically this paper also derives various kernel duality relationships between the primal and time shifted dual dres which lead to an array of dre solution methods time invariant analogue of one of these methods was first proposed in mceneaney 2008

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

accurate numerical integration of stiff differential riccati equations abstract in this paper accurate numerical integration methods for four times continuously differentiable coefficient stiff riccati matrix differential equations are proposed first a discrete numerical solution is constructed at a net of points of the predetermined existence interval using implicit one step matrix methods finally using linear b spline matrix functions a continuous numerical solution whose error is uniformly upper bounded by a prefixed error in the existence domain is constructed

taming griffiths singularities infinite differentiability of quenched correlation functions we prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of griffiths singularities we also show uniqueness of the gibbs state and exponential decay of truncated correlation functions with probability one our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments our results require no assumptions on the probability distributions of the random parameters except for the obvious one of no percolation of infinite couplings and in the strong field situation for the also obvious requirement that zero magnetic fields do not percolate

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

a new fundamental solution for differential riccati equations arising in control the matrix differential riccati equation dre is ubiquitous in control and systems theory the presence of the quadratic term implies that a simple linear systems fundamental solution does not exist of course it is well known that the bernoulli substitution may be applied to obtain a linear system of doubled size here however tools from max plus analysis and semiconvex duality are brought to bear on the dre we consider the dre as a finite dimensional solution to a deterministic linear quadratic control problem taking the semiconvex dual of the associated semigroup one obtains the solution operator as a max plus integral operator with quadratic kernel the kernel is equivalently represented as a matrix using the semigroup property of the dual operator one obtains a matrix operation whereby the kernel matrix propagates as a semigroup the propagation forward is through some simple matrix operations this time indexed family of matrices forms a new fundamental solution for the dre solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition in analogy with standard algebra linear systems the fundamental solution can be viewed as an exponential form over a certain idempotent semiring this fundamental solution has a particularly nice control interpretation and might lead to improved dre solution speeds

a constructive mixing condition for 2 d gibbs measures with random interactions for spin systems with random finite range interactions we define an analog of the usual weak mixing property which we call weak mixing in expectation wme this property implies almost sure uniqueness of the gibbs measure we concentrate on the two dimensional case for which we present finite volume conditions which are sufficient for wme we also show the reverse if the system is wme then the condition is satisfied for some sufficiently large volume simultaneously we obtain an extension to random interactions of the result by martinelli olivieri and schonmann that weak mixing implies strong mixing our method is based on a rescaled version of the disagreement percolation approach of van den berg and maes combined with ideas and techniques of gielis and maes and martinelli olivieri and schonmann however apart from some general results on coupling stated in section 2 this paper is self contained

dynamics at a smeared phase transition we investigate the effects of rare regions on the dynamics of ising magnets with planar defects i e disorder perfectly correlated in two dimensions in these systems the magnetic phase transition is smeared because static long range order can develop on isolated rare regions we first study an infinite range model by numerically solving local dynamic mean field equations then we use extremal statistics and scaling arguments to discuss the dynamics beyond mean field theory in the tail region of the smeared transition the dynamics is even slower than in a conventional griffiths phase the spin autocorrelation function decays like a stretched exponential at intermediate times before approaching the exponentially small equilibrium value following a power law at late times

nonparametric function estimation for clustered data when the predictor is measured without with error abstract we consider local polynomial kernel regression with a single covariate for clustered data using estimating equations we assume that at most m observations are available on each cluster in the case of random regressors with no measurement error in the predictor we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent in the further special case of longitudinal data on individuals with fixed common observation times we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average we also consider what happens when the predictor is measured with error using the simex approach to correct for measurement error we construct an asymptotic theory for both the pooled and the weighted average estimators surprisingly for the same amount of smoothing the weighted a

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough