gridding with continuous curvature splines in tension a gridding method commonly called minimum curvature is widely used in the earth sciences the method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature the minimum curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points minimum curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used these extraneous inflection points can be eliminated by adding tension to the elastic plate flexure equation it is straightforward to generalize minimum curvature gridding algorithms to include a tension parameter the same system of equations must be solved in either case and only the relative weights of the coefficients change therefore solutions under tension require no more computational effort than minimum curvature solutions and any algorithm which can solve the minimum curvature equations can solve the more general system we give common geologic examples where minimum curvature gridding produces erroneous results but gridding with tension yields a good solution we also outline how to improve the convergence of an iterative method of solution for the gridding equations

small subset sums abstract let be a norm in r d whose unit ball is b assume that v b is a finite set of cardinality n with v v v 0 we show that for every integer k with 0 k n there exists a subset u of v consisting of k elements such that v u v d 2 we also prove that this bound is sharp in general we improve the estimate to o d for the euclidean and the max norms an application on vector sums in the plane is also given

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

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

multiparametric response surface construction by means of proper generalized decomposition an extension of the parafac procedure abstract multidimensional problems are found in almost every scientific field in particular this is standard in parametric design inverse analysis in optimization and in metamodeling analysis whereby statistical or deterministic approximations of multiparametric solutions are built from the results of experimental campaigns or computer simulations multidimensional fitting or approximation of response functions exponentially increase their complexity and computational cost with the number of dimensions responding to the well known curse of dimensionality to reduce the order of complexity and make the solution of many parameter problems affordable we propose to combine the model reduction technique known as proper generalized decomposition pgd and the response surface rsm methodology as a proof of concept we have used a simple fitting procedure as it is least squares although other more complex fitting procedures may be easily included the combined algorithm is presented and its capabilities discussed in a set of multidimensional examples the data samples to be fit in each of these examples are obtained by means of appropriate discretizing the interval of interest for each design factor and then generating the output values exact or stochastically modified by means of virtual experiments to have an idea of the number of discretization points needed along each direction the taguchi s design of experiments is used the obtained results show evident improvements in computer time and accuracy when compared to other traditional multiparametric approximation techniques based on polynomial functions and the standard levenberg marquardt algorithm especially in problems with non linear behavior and with high number of design parameters further comparison was done with the parafac als algorithm the combination of pgd and rsm seems to be an appealing tool for accurately modelling multiparametric problems in almost real time if a sufficient set of previous off line results is available despite the intrinsic complexity of the problem and of the number of parameters involved

the randomization technique as a modeling tool and solution procedure for transient markov processes we present a randomization procedure for computing transient solutions to discrete state space continuous time markov processes this procedure computes transient state probabilities it is based on a construction relating a continuous time markov process to a discrete time markov chain modifications and extensions of the randomization method allow for computation of distributions of first passage times and sojourn times in markov processes and also the computation of expected cumulative occupancy times and expected number of events occurring during a time interval several implementations of the randomization procedure are discussed in particular we present an implementation for a general class of markov processes that can be described in terms of state space s event set e rate vectors r and target vectors t abbreviated as sert this general approach can handle systems whose state spaces are quite large if they have sparse generators

brief optimal trajectory planning and smoothing splines in this paper some of the relationships between optimal control and trajectory planning are examined when planning trajectories for linear control systems a demand that arises naturally in air traffic control or noise contaminated data interpolation is that the curve passes close to given points or through intervals at given times in this paper we produce these curves by solving an optimal control problem for linear control systems while driving the output of the system close to the waypoints we furthermore show how this optimal control problem reduces to a finite quadratic programming problem and we thus provide a constructive yet theoretically sound framework for producing a rich set of curves called smoothing splines

nonlinear modeling of time series using multivariate adaptive regression splines mars abstract multivariate adaptive regression splines mars is a new methodology due to friedman for nonlinear regression modeling mars can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions given a set of predictor variables mars fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy mars produces continuous models for high dimensional data that can have multiple partitions and predictor variable interactions predictor variable contributions and interactions in a mars model may be analyzed using an anova style decomposition by letting the predictor variables in mars be lagged values of a time series one obtains a new method for nonlinear autoregressive threshold modeling of time series a significant feature of this extension of mars is its ability to produce models with limit cycles when modeling time series data that

assessing time by covariate interactions in proportional hazards regression models using cubic spline functions proportional hazards or cox regression is a popular method for modelling the effects of prognostic factors on survival use of cubic spline functions to model time by covariate interactions in cox regression allows investigation of the shape of a possible covariate time dependence without having to specify a specific functional form cubic spline functions allow one to graph such time by covariate interactions to test formally for the proportional hazards assumption and also to test for non linearity of the time by covariate interaction the functions can be fitted with existing software using relatively few parameters the regression coefficients are estimated using standard maximum likelihood methods

a parametric cubic element with tension properties in this paper we present the construction and study the properties of a new parametric cubic element having tension properties the element is based on the clough tocher split of a given triangle due to its tension properties the element can be used to obtain a local interpolation scheme for scattered data

c 1 comonotone hermite interpolation via parametric cubics abstract we present a local scheme for c 1 hermite interpolation the interpolant constructed is a parametric piecewise cubic and it is comonotone with the data the approximation order is also investigated under appropriate assumptions the method is fourth order accurate

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

computing cumulative measures of stiff markov chains using aggregation an aggregation method for computing transient cumulative measures of large stiff markov models is presented the method is based on classifying the states of the original problem into slow fast transient and fast current states the authors aggregate fast transient states and fast recurrent states so that an approximate value to the desired cumulative measure can be obtained by solving a nonstiff set of linear differential equations defined over a reduced subset of slow states only several examples are included to illustrate how stiffness arises naturally in actual queuing and reliability models and to show that cumulative measures provide a better characterization of the time dependent system behavior

a semiparametric approach to density estimation abstract one method of fitting a parametric density function f x is first to estimate by maximum likelihood say and then to estimate f x by on the other hand when the parametric model does not hold the true density f x may be estimated nonparametrically as in the case of a kernel estimate the key idea proposed is to fit a combination of the parametric and nonparametric estimates namely where 0 1 is unknown the parameter is estimated from the data and its estimate is then used in as the proposed density estimate the main point is that we expect to be close to unity when the parametric model prevails and close to zero when it does not hold we show that under certain conditions converges to the parametric density when the parametric model holds and approaches the true f x when the parametric model does not hold the procedure was applied to a number of actual data sets in each case the maximum likelihood estimate was readily obtained and the semiparametric density es

interpolation methods comparison abstract in many fields spatial interpolation is used to evaluate physical data in a continuous domain the many different techniques offer different performances according to the characteristics of initial data points the aim of this paper is to provide help in choosing and evaluating the technique that best suits the data set a few indices measure the quality of interpolation by different viewpoints an extensive test case is reported involving four different interpolation methods data sets are generally connected with environmental topics or taken from literature

fitting monotone surfaces to scattered data using c 1 piecewise cubics we derive sufficient conditions on the bezier net of a bernstein bezier polynomial defined on a triangle in the plane to insure that the corresponding surface is monotone we then apply these conditions to construct a new algorithm for fitting a monotone surface to gridded data the method uses c1 cubic splines defined on the triangulation obtained by drawing both diagonals of each subrectangle in addition we present an algorithm for the monotone scattered data interpolation problem which is based on a method for creating gridded data from the scattered data numerical results for several test examples are presented

non parametric survival analysis of infectious disease data this paper develops nonparametric methods based on contact intervals for the analysis of infectious disease data the contact interval from person i to person j is the time between the onset of infectiousness in i and infectious contact from i to j where we define infectious contact as a contact sufficient to infect a susceptible individual the hazard function of the contact interval distribution equals the hazard of infectious contact from i to j so it provides a summary of the evolution of infectiousness over time when who infects whom is observed the nelson aalen estimator produces an unbiased estimate of the cumulative hazard function of the contact interval distribution when who infects whom is not observed we use an em algorithm to average the nelson aalen estimates from all possible combinations of who infected whom consistent with the observed data this converges to a nonparametric maximum likelihood estimate of the cumulative hazard function that we call the marginal nelson aalen estimate we study the behavior of these methods in simulations and use them to analyze household surveillance data from the 2009 influenza a h1n1 pandemic

a cubic spline extension of the durbin watson test summary we propose a new test for model misspecification in linear regression based on a measure of smoothness of the regression residuals this curvature statistic is related to a property of cubic splines and leads to a convenient powerful test for patterns the distribution of the curvature statistic may be derived from assumptions of normality for equally spaced data the calculations are simplified and a table of significance points of the distribution is given smoothness considerations also lead to a natural generalization of the durbin watson or mean square successive differences statistic for unequally spaced observations this generalized mean squared successive differences statistic arises when linear splines are substituted for cubic splines in the development of the curvature statistic the power of the curvature statistic is investigated by simulation compared to several other commonly used tests for patterns the curvature test is shown to be more powerful where noise levels are small and the complexity of the underlying pattern is moderate the calculation and use of this statistic is illustrated on a problem involving nonlinear regression residuals

extended hazard regression for censored survival data with covariates a spline approximation for the baseline hazard function a regression model for censored survival data with covariates is introduced and termed extended hazard regression ehr ehr includes the proportional hazards ph and the accelerated failure time aft models as special cases by approximating the baseline hazard function with quadratic splines we develop a maximum likelihood estimation procedure to provide a simultaneous estimate of both the hazard function and the regression coefficients the aft and ph assumptions can then be compared by likelihood ratio tests within the more general ehr model several examples based on artificial and real data are discussed

multi level repeated measures growth modelling using extended spline functions this paper explores the fitting of multi level models to growth data over a wide age range using a new class of extended spline models these extend conventional splines based on a function representation by allowing variable order functions and by including fractional polynomial terms the work focuses on modelling human growth in height and head circumference with example data sets the procedures can be used with covariates and for comparing population parameters

inverse passive learning of an input output map through update spline smoothing this paper presents a robust method of learning passively a one dimensional input output map when receiving only indirect information about the correct input output map e g only the sign of the deviation between the actual estimated output value and the correct output value is obtained this information is obtained for only one input output combination per updating cycle the approach is to increment or decrement step by step the output values of the actually stored map and then to apply global or local cubic spline smoothing in order to avoid adaptation holes at points which are never updated or less frequently updated than other points this method works with noisy measurements as well as slowly time varying systems even discontinuous changes of the desired input output relation do not result in instability problems of convergence and stability are treated and design rules are given

surface smoothing with finite elements abstract spline smoothing is a very good technique to fit a surface to a noisy scattered data set surface splines are obtained while minimizing a rotation invariant inner product in an hilbert space when the sampling size n is very large the surface splines are very hard to determine numerically and don t provide any data compression an approximate solution of the spline smoothing problem given by finite elements can be very useful to reduce the dimension bicubic splines are for this problem very efficient piecewise bicubic rectangular finite elements

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

a tension approach to controlling the shape of cubic spline surfaces on fvs triangulations we propose a parametric tensioned version of the fvs macro element to control the shape of the composite surface and remove artificial oscillations bumps and other undesired behaviour in particular this approach is applied to c 1 cubic spline surfaces over a four directional mesh produced by two stage scattered data fitting methods

a two stage algorithm for discontinuity preserving surface reconstruction a two stage algorithm for visual surface reconstruction from scattered data while preserving discontinuities is presented the first stage consists of a robust local approximation algorithm the moving least median of squares mlms of error to clean the data and create a grid from the original scattered data points this process is discontinuity preserving the second stage introduces a weighted bicubic spline wbs as a surface descriptor the wbs has a factor in the regularizing term that adapts the behavior of the spline across discontinuities the weighted bicubic approximating spline can approximate data with step discontinuities with no discernible distortion in the approximating surface the combination of robust surface fitting and wbss removes outliers and reduces gaussian noise either stage by itself would not effectively remove both kinds of noise experimental results with the two stage algorithm are presented

an aggregation technique for the transient analysis of stiff markov chains an approximation algorithm for systematically converting a stiff markov chain into a nonstiff chain with a smaller state space is discussed in this paper after classifying the set of all states into fast and slow states the algorithm proceeds by further classifying fast states into fast recurrent subsets and a fast transient subset a separate analysis of each of these fast subsets is done and each fast recurrent subset is replaced by a single slow state while the fast transient subset is replaced by a probabilistic switch after this reduction the remaining small and nonstiff markov chain is analyzed by a conventional technique

influence diagnostics for the cross validated smoothing parameter in spline smoothing abstract this article addresses the problem of influence in estimating the smoothing parameter when fitting a univariate smoothing spline diagnostics are presented that can identify observations that locally influence the choice of smoothing parameter by generalized cross validation either through their case weights or observed responses generalized cross validation and the diagnostic methods are given a frequency interpretation and illustrated with examples

vector splines and the estimation of filter functions in this paper we consider the estimation of vector valued periodic functions by use of splines an error criterion is constructed and it is shown that the vector spline is simply the vector of univariate splines the vector valued spline is shown to be an approach to the estimation of the spectral density matrix in a multivariate setting two approaches are given for estimation of filter related functions including gain transfer function coherency and so on some results on consistency are given and an application is made to sonar systems

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

comonotone parametric c1 interpolation of nongridded data we present a local scheme for constructing a c 1 interpolating function comonotone with a set of data having a tensor product topology the constructed interpolant is a parametric surface with piecewise bicubic components which locally maintain the monotonicity of the data along curves parallel to the related coordinate lines error estimates and graphical examples are provided

international migration and air travel global smoothing and estimation the relative likelihood of a migration from any one point on the earth s land surface to any other point is viewed as a function of four variables the latitudes and longitudes of the origin and destination it is expressed as a sum of six bivariate functions one for each pairing of the four variables two of the six attractivity and emissivity terms are fitted by bicubic splines over a sphere the earth and the other four by bicubic splines over rectangular grids the model is estimated with un data on immigration streams from 169 nations to a subset of 93 a missing value procedure is implemented national differences in definitions of immigrants are exposed through the examination of residuals 1982 international air travel is modelled as well exceptionally large passenger volumes are associated with pilgrimages to mecca estimates of air traffic over potential or nonreported confidential routes can be obtained

helly type theorems for the sum of vectors in a normed plane abstract the main results here are two helly type theorems for the sum of at most unit vectors in a normed plane also we give a new characterization of centrally symmetric convex sets in the plane

flexible modeling of the effects of continuous prognostic factors in relative survival relative survival methods permit separating the effects of prognostic factors on disease related excess mortality from their effects on other causes natural mortality even when individual causes of death are unknown as in conventional crude survival accurate assessment of prognostic factors requires testing and possibly modeling of non proportional effects and for continuous covariates of non linear relationships with the hazard we propose a flexible extension of the additive hazards relative survival model in which the observed all causes mortality hazard is represented by a sum of disease related excess and natural mortality hazards in our flexible model the three functions representing i the baseline hazard for excess mortality ii the time dependent effects and iii for continuous covariates non linear effects on the logarithm of this hazard are all modeled by low dimension cubic regression splines non parametric likelihood ratio tests are proposed to test the time dependent and non linear effects the accuracy of the estimated functions is evaluated in multivariable simulations to illustrate the new insights offered by the proposed model we apply it to re assess the effects of patient age and of secular trends on disease related mortality in colon cancer

ray tracing in 3 d complex isotropic media an analysis of the problem procedures for accurate ray tracing in complex three dimensional media with interfaces are proposed the ray tracing equations and the associated paraxial linear equations are solved either by a numerical solver or by an analytical perturbation approach interfaces are described with an explicit representa tion or an implicit representation using b spline inter polation for the implicit representation we exploit two important properties of b splines the convex hull and subdivision properties in order to determine the intersection of the ray with the interface at the free surface where the recording system is located a sampling strategy is proposed limits of branches at caustics shadow zones and medium boundaries are detected for a fixed azimuth while the take off angle is automatically adjusted in order to have a roughly homogeneous spacing between end points of the rays the same strategy is also possible for a fixed take off angle the assumed continuity of the traveltime surface between two adjacent azimuths enables one to obtain the initial condition of a ray arriving at any station located on the portion of surface delimited by these two azimuths this procedure al lows for the classification of rays arriving at a given station as we show on different synthetic examples

minimizing the laplacian of a function squared with prescribed values on interior boundaries theory of polysplines in this paper we consider the minimization of the integral of the laplacian of a real valued function squared and more general functionals with prescribed values on some interior boundaries r with the integral taken over the domain d we prove that the solution is a biharmonic function in d except on the interior boundaries r and satisfies some matching conditions on r there is a close analogy with the one dimensional cubic splines which is the reason for calling the solution a polyspline of order 2 or biharmonic polyspline similarly when the quadratic functional is the integral of qf 2 q a positive integer then the solution is a polyharmonic function of order 2q 2qf x o for x e d r satisfying matching conditions on r and is called a polyspline of order 2q uniqueness and existence for polysplines of order 2q provided that the interior boundaries r are sufficiently smooth surfaces arld ad c r is proved three examples of data sets r possessing symmetry are considered in which the computation of polysplines is reduced to computation of onedimensional l splines

reproducing kernel hilbert space method for optimal interpolation of potential field data the rkhs based optimal image interpolation method presented by chen and de figueiredo 1993 is applied to scattered potential field measurements the rkhs which admits only interpolants consistent with laplace s equation is defined and its kernel derived the algorithm is compared to bicubic spline interpolation and is found to yield vastly superior results

fitting time dependent multicompartment models a case study a multicompartment model with time dependent transfer rates is fitted to data on ovarian follicle dynamics in mice the fitted model arises from an interplay between parametric and nonparametric approaches to fitting curves to these data nonparametric regression estimates in the form of spline smoothers are used in conjunction with the biologically meaningful general class of multicompartment models to suggest refinements to the parametric model one result of this interplay is the suggestion of using three stage step functions for the compartmental transition rates the resulting curves mimic closely the nonparametric regression estimates

a penalized likelihood approach for arbitrarily censored and truncated data application to age specific incidence of dementia the cos model is the model of choice when analyzing survival data presenting only right censoring and left truncation there is a need for methods that can accommodate more complex observation schemes involving general censoring and truncation in addition it is important in many epidemiological applications to have a smooth estimate of the hazard function we show that the penalized likelihood approach gives a solution to these problems the solution of the maximum of the penalized likelihood is approximated on a basis of splines the smoothing parameter is estimated using approximate cross validation confidence bands can be given a simulation study shows that this approach gives better results than the smoothed nelson aalen estimator we apply this method to the analysis of data from a large cohort study on cerebral aging the age specific incidence of dementia is estimated and risk factors of dementia studied

a general parametric framework for functional tension schemes the paper describes a general approach that can be used to obtain local tension schemes for functional approximation and interpolation examples of one and two dimensional schemes are presented and analysed

multiresolution parameterization for geophysical inverse problems a model parameterization based on a multiresolution wavelet representation is proposed for geophysical inverse problems in this study the matrix equation for the wavelet representation of the model is constructed by dual wavelet transforms of a readily built matrix based on the pixel parameterization we demonstrate the merits of the proposed parameterization using the simple examples of geophysical data gridding and 3d seismic tomography and comparing the results with those obtained by the conventional pixel parameterization it is shown that for the conventional damped least squares solutions of pixel parameterized models regularization by norm damping deteriorates the underlying model correlation due to sparse and heterogeneous sampling whereas roughness damping schemes impose a priori arbitrary correlation length irrespective of the heterogeneous sampling the scale hierarchy embedded in the proposed multiresolution parameterization facilitates the compromise between the dual spatial and scale resolution depending on the local richness of the data constraints across different scales at a site model variation of various scale spectra can be robustly resolved through the scale hierarchy there is thus no need to invoke additional smoothness regularization

application of the pocs inversion method to cross borehole imaging cross borehole tomography suffers from a well known problem of data incompleteness the limited ray coverage dictated by the poor experimental geometry implies that certain features of the velocity field are not determined by the data construction of a tomographic image of the velocity field therefore requires the addition of prior constraints to the inversion in the fourier wavenumber domain assuming straight line rays the process of adding prior constraints is equivalent to specifying unmeasured wavenumber coefficients the projection onto convex sets pocs algorithm can impose physically plausible constraints that allow high quality tomographic images to be produced each constraint is viewed as defining a set in function space of images that satisfy that particular constraint the pocs method finds one or more images in the intersection of the constraining sets which is equivalent to finding an image that simutaneously satisfies a number of constraints including the observed data the sets of

additive colourful caratheodory type results with an application to radii abstract in this paper we study the behaviour of the circumradius with respect to the minkowski addition in generalized minkowski spaces to do so we solve additive colourful caratheodory type results under certain equilibria conditions

a green s function based bi dimensional empirical mode decomposition a new git bemd algorithm is developed based on green s function with tension much faster than canonical bemd algorithms easy to implement and very robust superior performance tested on artificial and real images bidimensional empirical mode decomposition bemd interprets an image as a superposition of bidimensional intrinsic mode functions bimfs they are extracted by a process called sifting which encompasses two dimensional surface interpolations connecting a set of local maxima or minima to form corresponding envelope surfaces existing surface interpolation schemes are computationally very demanding and often induce artifacts in the extracted modes this paper suggests a novel method of envelope surface interpolation based on green s functions including surface tension greatly improves the stability of the new method which we call green s function in tension based bemd git bemd simulation results using toy images with various textures facial images and functional neuroimages demonstrate the superior performance of the new method when compared to its canonical bemd counterpart git bemd strongly speeds up computations and achieves a higher quality of the extracted bimfs furthermore git bemd can be extended simply to an ensemble based variant git beemd if needed in summary the study suggests the new variant git bemd as a highly competitive fast and stable alternative to existing bemd techniques for image analysis

diagnostics for smoothing splines summary diagnostic measures appropriate for use with smoothing splines are derived and their properties are investigated the proposed measures focus on detection of observations which substantially influence the fit and provide additional information over that obtained from examination of residuals alone a numerical example illustrates the technique

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

flexible estimation of survival curves conditional on non linear and time dependent predictor effects prognostic studies often estimate survival curves for patients with different covariate vectors but the validity of their results depends largely on the accuracy of the estimated covariate effects to avoid conventional proportional hazards and linearity assumptions flexible extensions of cox s proportional hazards model incorporate non linear nl and or time dependent td covariate effects however their impact on survival curves estimation is unclear our primary goal is to develop and validate a flexible method for estimating individual patients survival curves conditional on multiple predictors with possibly nl and or td effects we first obtain maximum partial likelihood estimates of nl and td effects and use backward elimination to select statistically significant effects into a final multivariable model we then plug the selected nl and td estimates in the full likelihood function and estimate the baseline hazard function and the resulting survival curves conditional on individual covariate vectors the td and nl functions and the log hazard are modeled with unpenalized regression b splines in simulations our flexible survival curve estimates were unbiased and had much lower mean square errors than the conventional estimates in real life analyses of mortality after a septic shock our model improved significantly the deviance likelihood ratio test 84 8 df 20 p 0 0001 and changed substantially the predicted survival for several subjects

simplification of markov chains with infinite state space and the mathematical theory of random gene expression bursts here we develop an effective approach to simplify two time scale markov chains with infinite state spaces by removal of states with fast leaving rates which improves the simplification method of finite markov chains we introduce the concept of fast transition paths and show that the effective transitions of the reduced chain can be represented as the superposition of the direct transitions and the indirect transitions via all the fast transition paths furthermore we apply our simplification approach to the standard markov model of single cell stochastic gene expression and provide a mathematical theory of random gene expression bursts we give the precise mathematical conditions for the bursting kinetics of both mrnas and proteins it turns out that random bursts exactly correspond to the fast transition paths of the markov model this helps us gain a better understanding of the physics behind the bursting kinetics as an emergent behavior from the fundamental multiscale biochemical reaction kinetics of stochastic gene expression

algebraic phase unwrapping based on two dimensional spline smoothing over triangles two dimensional 2d phase unwrapping is an estimation problem of a continuous phase function over a 2d domain from its wrapped samples in this paper we propose a novel approach for high resolution 2d phase unwrapping in the first step spline smoothing sps we construct a pair of the smoothest spline functions which minimize the energies of their local changes while interpolating respectively the cosine and the sine of given wrapped samples if these functions have no common zero over the domain the proposed estimate of the continuous phase function can be obtained by algebraic phase unwrapping in the second step algebraic phase unwrapping apu to avoid the occurrence of common zeros in sps due to phase noise in the observed wrapped samples we also propose a denoising step denoising by selective smoothing dss as preprocessing which selectively smooths unreliable wrapped samples by using convex optimization the smoothness of the proposed unwrapped phase function is guaranteed globally over the domain without losing any desired consistency with all reliable wrapped samples numerical experiments for terrain height estimation demonstrate the effectiveness of the proposed 2d phase unwrapping scheme

minimax estimates in a semiparametric model abstract the statistical analysis of multidimensional data has been greatly influenced by the widespread use of computer intensive smoothing techniques these techniques allow one to look at the dependency of a response on several covariates without actually specifying the exact form of that dependency nevertheless when the response s dependency on some subset of the covariates can be assumed to be say linear common sense dictates incorporating this assumption into the estimation technique in the semiparametric model with covariates x and t the response y is modeled as the sum of three components g t with g unknown lying in an infinite dimensional space a linear component x and random error if were the parameter of interest with the function g treated as a nuisance parameter the conservative statistician would be led to a minimax approach with the estimate of minimizing the worst that could happen over a specified class of functions of course the class of functions should be one

unit graph estimation and stabilization using quadratic programming and difference norms for a rainfall runoff event quadratic programming is particularly suited to the estimation of unit graph as linear and positivity constraints on unit graph ordinates can be naturally implemented in this paper the quadratic programming framework is invoked in a novel way to stabilize the unit graph estimation procedure via the use of difference norms the advantage of the latter over standard ridge regression is that penalties are placed on oscillations of the unit graph rather than on the size of its ordinates application of the methodology to real rainfall runoff data is provided and comparisons with existing approaches are made the latter indicate that our approach generally yields unit graph estimates of more realistic shape and smaller variance resulting in a better fit to the runoff data

principal components analysis of sampled functions this paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values this problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time in such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful as for example in the analysis of growth or learning curves it is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis the reproducing kernel for the hilbert space of functions plays a central role and defines the best interpolating functions which are generalized spline functions an example is offered of how sensitivity to derivative information can reveal interesting aspects of the data

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

a hierarchical model for aggregated functional data in many areas of science one aims to estimate latent subpopulation mean curves based only on observations of aggregated population curves by aggregated curves we mean linear combination of functional data that cannot be observed individually we assume that several aggregated curves with linearly independent coefficients are available and each aggregated curve is an independent partial realization of a gaussian process with mean modeled through a weighted linear combination of the disaggregated curves the mean of the gaussian process is modeled using b splines basis expansion methods we propose a semiparametric valid covariance function that is modeled as the product of a nonparametric variance function by a correlation function the variance function is described as the square of a function that is expanded using b splines basis functions this results in a nonstationary covariance function and includes constant variance models as special cases inference is performed following the bayesian paradig

six standard deviations suffice given n sets on n elements it is shown that there exists a two coloring such that all sets have discrepancy at most knl 2 k an absolute constant this improves the basic probabilistic method with which k c ln n l 2 the result is extended to n finite sets of arbitrary size probabilistic techniques are melded with the pigeonhole principle an alternate proof of the existence of rudin shapiro functions is given showing that they are exponential in number given n linear forms in n variables with all coefficients in 1 1 it is shown that initial values pl p e 0 1 may be approximated by el e 0 1 so that the forms have small error 1 we state our main result first in the language of linear forms theorem 1 let 1 1 li xl x l ailxl xe ai txn 1 i n be n linear forms in n variables with all laijl 1 then there exist e1 esl e 1 1 such that 1 2 li el e l ks for all i 1 i n here k is an absolute constant when all aij e 0 1 we may consider e1 aij as the incidence matrix for a family of n sets on n elements that is we may set ai j aij 1 given a two coloring say red and blue of 1 n the discrepancy disc x of a set x c 1 n is defined as the number of red points in x minus the number of blue points in x if we interpret si 1 as meaning i is to be colored red and ej 1 as blue then we obtain the following result corollary 2 let al an c 1 n then there exists a two coloring of 1 n so that 1 3 disc aj kw foralli 1 i n received by the editors july 1 1984 1980 muthemcltics subject clclssific ltion primary 05b20 secondary 41a28 42a61 this work was initiated while the author was an irex exchange fellow at the mathematical institute budapest and completed with support of the national science foundation for the many kindnesses and for the creative research environment provided by institute staff and colleagues koszonom szepen t1985 american mathematical society 0002 9947 85 1 00 25 per page

a least squares smooth fitting for irregularly spaced data finite element approach using the cubic b spline basis a new method of multivariate smooth fitting of scattered noisy data using cubic b splines was developed an optimum smoothing function was defined to minimize the x1d4c1 2 norm composed of the data residuals and the first and the second derivatives which represent the total misfit fluctuation and roughness of the function respectively the function is approximated by a cubic b spline expansion with equispaced knots the solution can be interpreted in three ways from the stochastic viewpoint it is the maximum likelihood estimate among the admissible functions under the a priori information that the first and second derivatives are zero everywhere due to random errors i e white noise from the physical viewpoint it is the finite element approximation for a lateral displacement of a bar or a plate under tension which is pulled to the data points by springs from a technical viewpoint it is an improved spline fitting algorithm the additional condition of minimizing the derivative norms stabilizes the l

on some combinatorial questions in finite dimensional spaces abstract bounds are given for supinf p i 1 i where sup is taken over all set systems v 1 v p of r n with 0 p i 1 conv v i and sup v v i 1 for i 1 p and inf is taken over all possible choices i v i for i 1 p another similar problem is considered the bounds are sharp

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

inequality constrained multivariate smoothing splines with application to the estimation of posterior probabilities abstract we consider the problem of estimating a smooth function of several variables given discrete scattered noisy observations of its values and given that it satisfies a family of linear inequality constraints constraints such as positivity over some region are included the model is z i f y 1 i yd e i 1 n where the e i s are independent zero mean random variables f is assumed to be smooth with smoothness defined in terms of square integrability of certain derivatives and f is known to satisfy a given set of linear inequality constraints it is proposed that f be estimated as the minimizer in an appropriate function space of subject to f satisfying the constraints if they are a finite family or satisfying a finite approximating set of constraints if they are not a finite family where j m f is the thin plate penalty functional defined by j m f 1 d m m 1 d m f y 11 y d d 2 dy 1 dy d more generally the results apply to the model wh

the reduction of perturbed markov generators an algorithm exposing the role of transient states a new algorithm for the hierarchical aggregation of singularly perturbed finite state markov processes is derived the approach taken bridges the gap between conceptually simple results for a relatively restricted class of processes and the significantly more complex results for the general case the critical role played by almost transient states is exposed resulting in a straightforward algorithm for the construction of a sequence of aggregate generators associated with various time scales these generators together provide a uniform asymptotic approximation of the original probability transition function

monotone nonparametric regression in monotone regression procedures one utilizes only the monotonicity of the regression function in nonparametric regression one utilizes only the assumed smoothness the analytic and asymptotic properties of the estimator are superior in the latter case however monotonicity is not guaranteed we study a hybrid procedure that produces monotone estimators with properties similar to those of nonparametric regression estimators

penalized partial likelihood for frailties and smoothing splines in time to first insemination models for dairy cows in many epidemiological studies time to event data are clustered and the physiological relationship between time dependent covariates and the log hazard is often not linear as assumed in the cox model introducing frailties in the cox model can account for the clustering of the data and smoothing splines can be used to describe nonlinear relations these two extensions of the cox model are introduced jointly and it is shown how penalized partial likelihood techniques can be used to fit the extended model we demonstrate the need for such a model to study the relation between the physiological covariates milk ureum and protein concentration and the log hazard of first insemination in dairy cows with the farms as clusters

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

capturing flexible heterogeneous utility curves a bayesian spline approach empirical evidence suggests that decision makers often weight successive additional units of a valued attribute or monetary endowment unequally so that their utility functions are intrinsically nonlinear or irregularly shaped although the analyst may impose various functional specifications exogenously this approach is ad hoc tedious and reliant on various metrics to decide which specification is best in this paper we develop a method that yields individual level flexibly shaped utility functions for use in choice models this flexibility at the individual level is accomplished through splines of the truncated power basis type in a general additive regression framework for latent utility because the number and location of spline knots are unknown we use the birth death process of denison et al 1998 and green s 1995 reversible jump method we further show how exogenous constraints suggested by theory such as monotonicity of price response can be accommodated our formulation is particularly suited to estimating reaction to pricing where individual level monotonicity is justified theoretically and empirically but linearity is typically not the method is illustrated in a conjoint application in which all covariates are splined simultaneously and in three panel data sets each of which has a single price spline empirical results indicate that piecewise linear splines with a modest number of knots fit these data well substantially better than heterogeneous linear and log linear a priori specifications in terms of price response specifically we find that although aggregate market level curves can be nearly linear or log linear individuals often deviate widely from either using splines hold out prediction improvement over the standard heterogeneous probit model ranges from 6 to 14 in the scanner applications and exceeds 20 in the conjoint study moreover optimal profiles in conjoint and aggregate price response curves in the scanner applications can differ markedly under the standard and the spline based models

model simplification and loss of irreversibility in this paper we reveal a general relationship between model simplification and irreversibility based on the model of continuous time markov chains with time scale separation according to the topological structure of the fast process we divide the states of the chain into the transient states and the recurrent states we show that a two time scale chain can be simplified to a reduced chain in two different ways removal of the transient states and aggregation of the recurrent states both the two operations will lead to a decrease in the entropy production rate and its adiabatic part and will keep its nonadiabatic part the same this suggests that although model simplification can retain almost all the dynamic information of the chain it will lose some thermodynamic information as a trade off

multicategorical spline model for item response theory additional information contained in incorrect responses calls for a multicategorical rather than a binary analysis of multiple choice data a nonparametric divided by total model for joint maximum likelihood estimation of probability of choice functions for particular responses and of latent ability is proposed the model approximates probability functions by rational splines some illustrative examples of real test data analysis and the results of a monte carlo study are presented

spline estimation of paths using bearings only tracking data abstract in many applications bearings are measured to a moving object with the goal of estimating the object s course of movement if movement is appropriately modeled as a smooth deterministic curve in the plane then a cubic spline is a reasonable representation of the curve maximum likelihood estimators are presented for parameters of regression splines assuming that observation errors follow a von mises distribution location estimates are obtainable even when data are sparse path estimation error number and placement of knots and outlier detection are discussed examples including both simulated paths and observations from a wildlife radio tracking study are presented

admissible slopes for monotone and convex interpolation in many applications interpolation of experimental data exhibiting some geometric property such as nonnegativity monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics this paper identifies admissible slopes at data points of variousc 1 interpolants which ensure a desirable shape we discuss this question in turn for the following function classes commonly used for shape preserving interpolations monotone polynomials c 1 monotone piecewise polynomials convex polynomials parametric cubic curves and rational functions

uniqueness and eventual uniqueness of optimal designs in some time series models abstract earlier results on the uniqueness and eventual uniqueness of optimal designs for certain time series models are extended to a wider class of processes which includes those with covariance structures such as that of multiple integrals of brownian motion and brownian bridge processes the relationship between the problems of regression design for time series and piecewise polynomial approximation with free breakpoints is discussed and consequently asymptotic results obtained by sacks and ylvisaker 1970 are seen to hold under weaker assumptions for these processes

multiple scale analysis of complex networks using the empirical mode decomposition method abstract empirical mode decomposition emd method can decompose any complicated data into finite intrinsic mode functions imfs in this paper we use emd method to analyze and discuss the structural properties of complex networks a random walk method is used to collect the data series of network systems utilizing the emd method we decompose the obtained data into finite imfs under different spatial scales the analysis results show that emd method is an effective tool for capturing the topological properties of network systems under different spatial scales such as the modular structures of network systems and their energy densities

nonparametric maximum likelihood estimation by the method of sieves maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space for example the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an operatorname iid sample the maximum of the likelihood is not attained by any density in this example as in many other examples the parameter space positive functions with area one is too big but the likelihood method can often be salvaged if we first maximize over a constrained subspace of the parameter space and then relax the constraint as the sample size grows this is grenander s method of sieves application of the method sometimes leads to new estimators for familiar problems or to a new motivation for an already well studied technique we will establish some general consistency results for the method and then we will focus on three applications

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

smoothness priors and nonlinear regression abstract smoothness priors represent prior information that an unknown function does not change slope quickly and hence that the function describes a simple curve e g wahba 1978 in this article such priors for the multiple nonlinear regression model are developed in such a way that estimates and standard errors can be obtained as a natural and conceptually straightforward extension of linear multiple regression estimation with the addition of dummy variables and dummy observations relations to spline and polynomial interpolation are described an illustrative example of cost function estimation is provided

the restricted cubic spline as baseline hazard in the proportional hazards model with step function time dependent covariables we incorporate a cubic spline function where the tails are linearly constrained as the baseline hazard into the proportional hazards model we show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the cox proportional hazards model when covariables are fixed examples show that the inclusion of time dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the cox model advantages of the spline model also include flexibility of the hazard smooth survival curves and confidence limits for the survival and hazard estimates when there are time dependent covariables present

segmented polynomials for incidence rate estimation from prevalence data the study considers the problem of estimating incidence of a non remissible infection or disease with possibly differential mortality using data from a several cross sectional prevalence survey s fitting segmented polynomial models is proposed to estimate the incidence as a function of age using the maximum likelihood method the approach allows automatic search for optimal position of knots and model selection is performed using the akaike information criterion the method is applied to simulated data and to estimate hiv incidence among men in zimbabwe using data from both the nimh project accept hptn 043 and zimbabwe demographic health surveys 2005 2006 copyright 2016 john wiley sons ltd

a method for fitting regression splines with varying polynomial order in the linear mixed model the linear mixed model has become a widely used tool for longitudinal analysis of continuous variables the use of regression splines in these models offers the analyst additional flexibility in the formulation of descriptive analyses exploratory analyses and hypothesis driven confirmatory analyses we propose a method for fitting piecewise polynomial regression splines with varying polynomial order in the fixed effects and or random effects of the linear mixed model the polynomial segments are explicity constrained by side conditions for continuity and some smoothness at the points where they join by using a reparameterization of this explicitly constrained linear mixed model an implicitly constrained linear mixed model is constructed that simplifies implementation of fixed knot regression splines the proposed approach is relatively simple handles splines in one variable or multiple variables and can be easily programmed using existing commercial software such as sas or s plus the method is illustrated using two examples an analysis of longitudinal viral load data from a study of subjects with acute hiv 1 infection and an analysis of 24 hour ambulatory blood pressure profiles copyright 2005 john wiley sons ltd

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